WO2002068937A1 - Device and method for measuring the concentration of at least one substance in a sample - Google Patents

Abstract

The invention relates to a device and method for measuring the concentration of at least one substance in a sample, particularly in human skin. The inventive device comprises a means for exciting at least one substance by means of electromagnetic radiation and comprises a measuring cell for carrying out a detection based on the excitation of generated acoustic waves out of and/or inside the substrate. A calculating means serves to automatically convert the fluctuations in pressure that are induced by temperature changes on a surface of the sample into at least one measure that is dependent on the concentration of at least one of the substances. This enables a concentration measurement to be carried out in a rapid and precise manner by using the photoacoustic effect.

Description

 Device and ner driving for measuring the concentration of at least one

Substance in a sample

description

The invention relates to a device for measuring the concentration of at least one substance in a sample according to the preamble of claim 1 and a ner driving with the features of claim 11.

The inventive direction and the driving according to the invention are based on photothermal, in particular photoacoustic, effects.

A photothermal signal is generated when a photostimulated material (e.g. human skin) changes from an energy-rich to a low-energy state. Some of the energy released in the process is dissipated to the environment in the form of heat.

Various principles can be used to detect the heat released, and the measuring methods can be divided into direct and indirect ner driving. In the case of direct samples, the temperature or the resulting pressure wave in the sample are measured directly. In indirect ner driving, the reaction of a coupling medium to the emitted heat is used.

Photoacoustics is the oldest form of photothermal measurement and was developed by Alexander G. Bell in 1880. It is an indirect measurement method, since it uses the heating and subsequent expansion of a thin gas layer adjacent to the sample. The expansion of the gas layer leads to an increase in pressure in a closed measuring cell, which can be detected with a sensitive pressure sensor (eg a microphone). The photoacoustic effect itself is, for example, in G. Rosencwaig "Theory of the photoacustic effect with solids" Journal of Applied Physics Vol. 52, pp. 64-69 (1976).

A photoacoustic measuring cell is e.g. known from German utility model 296 17 790.

A disadvantage of these known ner drives is that a functional relationship between the measured values and the spatial and temporal changes in the concentration profile cannot be determined reliably and efficiently.

The present invention has for its object to provide a direction and a ner driving with which the determination of a concentration profile of a substance by means of the photoacoustic effect is possible in a fast and accurate manner.

This object is achieved according to the invention by a standard with the features of claim 1.

According to the invention, a computing means (e.g. software on a computer) is used for the automatic processing of the pressure fluctuations caused by temperature changes on a surface of the sample. According to the invention, the measurement signals are automatically converted into at least one measure which is dependent on the concentration of at least one of the substances. The concentration of a substance in the sample can thus advantageously be determined simply and precisely in spatial and temporal resolution.

It is particularly advantageous if the computing means has a means for automatically inverting an integral equation relating to heat conduction by means of a regulation approach. The use of a regulatory approach can be efficiently implemented on a computer and leads to stable solutions of the integral equation.

An advantageous embodiment of the device according to the invention has a pulsed laser device, in particular a Q-switched ΝdY AG laser, as a means for exciting the substance. Due to the coherent laser light with pulses in the range of eg Νa- noseconds and the good control possibilities of the laser light enable a targeted excitation of the substance.

The measuring chamber is advantageously designed as an acoustic resonator, the frequency tuning being adjustable as a function of a depth extension of the sample to be examined.

A means is advantageously provided for averaging at least two signals of the pressure fluctuation measurement over a predeterminable period of time. Here, e.g. averaging takes place over two light pulses of a laser. A particularly simple way of averaging is to use a digital storage oscilloscope as a means of averaging.

It is also advantageous if a means for calibrating the measuring cell, in particular a reference measuring cell, is provided. The reference measuring cell measures independently of the surface structure of the sample and therefore provides a real reference value.

It is also advantageous if a correction means is used for frequency-dependent transmission functions of the measuring chamber and / or a measuring head. The dynamic properties of the measuring chamber and / or of the measuring head can thus be calculated out of the measured values, so that only the measured values that originate from the substance remain in the result.

To evaluate the measurement, it is advantageous if the measuring cell is arranged in a measuring head which is coupled to a data processing device via a data line.

The object is also achieved by a method having the features of claim 11. The concentration of a substance in the sample can be determined quickly if the measured values of the pressure fluctuations caused by temperature changes on a surface of the sample are automatically converted into at least one measure which is dependent on the concentration of at least one of the substances. loading It is particularly advantageous if a time-dependent and / or location-dependent concentration profile of at least one substance is automatically determined.

The automatic conversion is particularly efficient if the computing means automatically solves an integral equation relating to heat conduction using a regulation approach.

A particularly advantageous embodiment of the method according to the invention is present when a pulsed laser device is used for excitation. It is also advantageous to design the measuring chamber as an acoustic resonator.

A further advantageous embodiment of the method according to the invention is when at least two signals of the pressure fluctuation measurement are averaged over a predeterminable period of time. This improves the signal / noise ratio and minimizes the influence of low-frequency interference from the measurement environment.

To improve the evaluation, it is advantageous if a measurement is carried out on a reference measuring cell in order to calibrate the measuring cell.

The influence of the measuring device itself on the measured signals can advantageously be minimized in that the frequency-dependent transfer functions of the measuring chamber and / or a measuring head are corrected by a correction function.

The invention is explained in more detail below with reference to the figures of the drawings using several exemplary embodiments. Show it:

1 shows a representation of the photoacoustic measuring method;

Fig. 2 is a schematic circuit diagram of an embodiment of the device according to the invention.

In Fig. 1 the basic sequence of the measurement process is shown. For a photoacoustic measurement, electromagnetic radiation 101, in particular in the range of visible light, is radiated onto a sample (not shown here). Sample is to be understood here to mean any material in which there is at least one substance whose concentration is to be measured

The sample has an optical absorption coefficient 102 so that the radiation is absorbed 103 in the sample.

The sample also has a certain conversion efficiency 104, which indicates what proportion of the incident light is converted into tub 105.

The photoacoustic effect can be used to measure properties of a sample, e.g. of the human skin, since the optical material parameters, absorption and scattering coefficients determine the spatial distribution of the optical energy in the sample. The local temperature changes (and therefore also pressure changes) depend on these parameters.

The heat 105 generated in the absorption area spreads according to the laws of heat conduction and can be detected on the surface of the sample with a suitable measuring instrument. This creates thermal waves 106 that depend on the temperature conductivity 106a and acoustic waves 107 that depend on the sound conductivity 107a.

If a photoacoustic measuring device is to be used to determine the concentrations of at least one substance in a sample, it is necessary to convert the pressure values of the acoustic waves 107 measured with the measuring cell into a concentration profile that is determined over time and in space.

This requires a mathematical model that creates a connection between the pressure fluctuations and the concentration profile of the substance to be determined. Such a model involves solving the heat transport equation in the sample and in the adjacent gas. The theoretical foundations for solving the heat transfer equation are summarized in Appendix 1. It shows that the problem underlying the present invention is the determination of a time and location resolved concentration profile. The resulting integral equation cannot be solved closed, so that a numerical calculation is necessary.

The various methods of solving and the use of a regularization method according to the invention are shown in Appendix 2.

The numerical treatment of the integral equation is broken down into discrete time and location steps. The resulting system of equations is then solved using a regularization process. The Tikhonov method has proven itself for practical evaluation. If a higher accuracy is to be achieved, the EM-NNLS method can be used.

For the first investigations, the integral equation was solved on a data processing device, in which software was used as the computing means. Alternatively, a specially manufactured chip can of course also be used for this.

2 shows the basic structure of a device by means of which the concentration of at least one substance in a sample 1 can be determined by means of the photoacoustic effect.

The measurement of the estradiol concentration in skin 1 is shown below as an example. Such a measurement is particularly important for examining the absorption of this substance into the skin.

The device according to the invention can in principle be used for all substances, which can be excited by electromagnetic radiation so that they emit acoustic waves. A typical area of application is, for example, the investigation of the penetration behavior of sun protection creams into the skin. In principle, materials other than skin 1 can also be used.

To generate an acoustic wave in the skin 1, a laser 10 is used as a means for excitation. The laser 1 is designed as an Nd-Yag laser. Laser 1 emits light with a wavelength of 355nm in pulses of 9ns. The light output on the skin is between 150 and 300 microjoules.

The laser beam is guided via a variable attenuator 11 to a first mirror 12, which is semi-transparent. The first mirror 12 conducts the light via an optical device 15 (symbolized here by a lens), via a fiber bundle 16, into a fundamentally known photoacoustic measuring head 18.

The laser beam then hits sample 1, here the human skin, and excites the substance in the sample.

The acoustic vibrations picked up by the measuring head 18 are transmitted to an evaluation computer 20 via a preamplifier 17 and an ozilloscope 21.

A partial beam of the laser beam passes through the first mirror 12 and is directed onto the second mirror 13, which brings the beam via a reference measuring cell 14, the oscilloscope 21, to the evaluation computer 20.

On the evaluation computer 20, the measured values are used as input values for the mathematical model, with which statements about the spatial and possibly temporal course of the concentration of the substance in the skin can be obtained.

In Appendix 3, the implementation of experiments with a device according to the invention is described in detail. The evaluation of the experiments is shown in Appendix 4. Based on photoacoustic spectroscopy, a method for the non-destructive determination of spatial concentration profiles of chromophores in biological tissues was developed. The necessary experimental arrangement consists of a periodically modulated or pulsed light source, the absorption of which in the tissue to be examined leads to a heat density that changes with time and place. The resulting temperature field on the sample surface can be determined indirectly with a special photoacoustic measuring cell in which a microphone registers the pressure fluctuations. The expansion of the medium adjacent to the sample due to the heat transfer from the sample material into the medium is used. The temperature field on the sample surface is a solution to the heat conduction problem of a thermally homogeneous body with a periodically changing or pulsed, location-dependent heat density. The dependence of the heat density on the location in the sample is determined by the local value of the optical absorption coefficient and thus the concentration of the chromophores.

The heat conduction problem leads to a connection between the time or frequency-dependent temperature field on the sample surface and the heat density inside the sample in the form of a Fredholm integral equation of the first kind. The integral equation can only be solved with the aid of numerical methods. The continuous problem is adapted to numerical processing by discretizing the variables in the local and time domain. Four numerical solution methods are selected and the accuracy of their solutions is checked using selected representative absorption profiles. An experimental situation is simulated by changing the signal / noise ratio. There are differences in the suitability of the solution methods. The EM NNLS method is well suited for absorption profiles with a discontinuous course. The Tikhonov method, on the other hand, is suitable for slowly and continuously changing profiles of the optical absorption coefficient.

The experimental investigations started with layers of optical colored glass filters and transparent cover glasses. A step profile of the optical absorption coefficient could thus be generated in a defined manner. The calculation of the absorption profile from the photoacoustic measurement values tended to lead to a correct result. Due to the The thermal inhomogeneities present in the layer structure were falsified, in particular in the deeper layers of the sample.

The examination of artificially produced gelatin layers with and without dye allowed the observation of the diffusion process of the dye in a colorless top layer. The limiting influence of the frequency-dependent transmission behavior of the measuring system became particularly clear in this experiment. The different weighting of frequencies in the measurement signal leads to errors in the calculation of the absorption profile. The correction of the transmission behavior by mathematical means was only of limited success.

The studies on isolated pig skin in the visible spectral range and the near UV confirmed the theoretically described possibility of determining concentration profiles of externally applied substances in the skin.

The final examinations on living people were carried out with sunscreen. The distribution of the sunscreen substance in the horny layer of the skin could be calculated. The concentration distribution in the skin is in accordance with the legal requirements for a sun protection product.

The measuring arrangement according to the invention with an open photoacoustic measuring cell is suitable for the in vivo investigation of concentration profiles of absorbing substances in human skin. Because of the microscopic and macroscopic movements of the skin, devices for reducing the interference in the measurement signal are very important. Therefore, it is only possible to achieve an acceptable signal-to-noise ratio with the pulsed excitation. With the help of the presented regulatory risk procedures, the absorption coefficient depending on the location in the sample can be calculated from the measured pressure curve in the photoacoustic cell.

The present work developed a non-destructive optical detection method for determining the concentration distribution of chromophores in biological matrices with sufficient accuracy for pharmacophysiological requirements. The literature cited here and in Appendices 1 to 4 is summarized in Appendix 5. In addition, an implementation for solving a partial differential equation is given in Appendix 5.

The embodiment of the invention is not limited to the preferred exemplary embodiments given above. Rather, a number of variants are conceivable that make use of the device according to the invention and the method according to the invention even in the case of fundamentally different types.

Appendix 1 ⁿ

Theoretical basics

A photothermal signal is generated when a photostimulated system changes from an energy-rich to a low-energy state. Some of the energy released in the process is dissipated to the environment in the form of heat. The relaxation of the excited electrons is called non-radiative. Competing ways of delivering energy are the emission of photons (fluorescence) or the use of energy to initiate a chemical reaction (e.g. photosynthesis).

Various principles can be used to detect the heat released. They have given the photothermal methods their respective names. The possible measuring methods can be divided into direct and indirect, depending on whether the temperature or the pressure wave in the material is measured directly, or the reaction of a coupling medium to the emitted heat is used IVargasf. The main procedures are briefly described and explained below.

• Photoacoustics as the oldest form of photothermal measurement was discovered in 1880 by A. G. Bell. It is an indirect method, since it uses the heating and subsequent expansion of a thin gas layer adjacent to the sample. This expansion leads to an increase in pressure in a closed measuring cell, which can be detected with a sensitive pressure sensor (e.g. microphone) / Rosencwaigl.

• The photothermal beam deflection or Mirage effect IBocarral uses the change of the refractive index with the temperature. A laser beam guided parallel to the sample surface is deflected as a result of the heating of the medium passing through. The deflection is proportional to the heat emitted. This method is an indirect method because the reaction of a coupling medium to the heat generated is evaluated. • Piezoelectric detection is one of the direct methods. With the help of a pressure-sensitive piezo element, the generated pressure or heat wave is measured directly in the absorbing medium or on the container thereof IFarrow, Jackson, Tarn I.

• Another direct measurement method consists in the detection of the emitted heat radiation in the half space above the sample with the help of an infrared detector ISantosl.

The intensity of the stimulating light is periodically modulated in all processes. The resulting heat emission is modulated to the same extent and can be evaluated with frequency-selective measurement technology.

There are two main reasons that make photothermal measurement methods interesting for many applications.

• These methods are "zero background" methods which have a sensitivity that is ten to one hundred times higher than that of transmission or reflection methods IZharov /.

Photothermal measuring methods occupy an intermediate position between optical spectroscopic and calorimetric methods. The choice of experimental conditions is decisive. This is the basis for interesting spectroscopic applications that would not be possible with purely optical methods, such as the non-destructive spectral depth analysis, which is dealt with in the course of the work.

2.1 Generation of the photoacoustic measurement signal

As part of the work, the photoacoustic measurement in the gas phase is to be applied. The principle diagram in Figure 2.1 illustrates the physical steps from the impact of light on the sample to the generation of thermal or acoustic waves.

The absorption of the light striking the sample causes an increase in temperature in the material. The optical material parameters, absorption and scattering coefficient, as well as the shape of the incident light spot on the sample surface determine the spatial distribution of the optical energy in the sample. The temperature change can vary locally depending on the degree of absorption and conversion efficiency for non-radiative processes in the material. The increase in temperature causes the illuminated volume in the sample to expand. The associated pressure wave is measurable. It is a direct proof in the sample material.

The heat generated in the absorption area spreads according to the laws of heat conduction and can be detected on the surface of the sample with a suitable measuring instrument. The photoacoustic measurement in the gas phase detects in a closed volume the pressure fluctuations resulting from the transfer of the thermal energy into the surrounding gas. It is an indirect proof.

Two cases are considered in the work, which differ in the type of light modulation. The excitation radiation, which is periodically modulated by a mechanical chopper, and the pulsed excitation lead to two different systems of equations for the heat transport in the sample and in the gas. The spatial and temporal temperature distribution at the sample - gas interface is derived from the solution of the respective system of equations. The temperature of this interface is directly proportional to the measurable photoacoustic signal / McDonald !.

2.2 Properties of human skin

Biological tissues, especially skin, are to be examined within the scope of the thesis. Human skin is a complex organ, which essentially consists of the layers subcutis (subcutis), leather skin (corium) and epidermis. The subcutaneous tissue consists of adipose and connective tissue as well as larger blood vessels. The subsequent dermis is formed from connective tissue with numerous small blood vessels. It contains the sweat and sebum glands as well as the hair roots. This is followed by the epidermis, which itself no longer has any blood vessels. Nutrients are supplied exclusively by diffusion from the deeper IRemanel tissues. Figure 2.2 shows a cross section through the human skin with the dimensions of the individual sub-layers.

The epidermis, which is only considered in this work, can be subdivided into several areas. The upper parts of the epidermis, the horny layer and the differentiation layer have a common extension of approx. 30 μm. They hardly differ in their optical properties IBrulsl. In the following, therefore, an optically homogeneous sample for the theoretical consideration of the photothermal measurement process is assumed as a first approximation. The approximation errors due to scattering processes in the skin are discussed at the end of the derivation. I (z, t, Ä) = (1 -R (Ä)) I ₀ (Ä) e-fr (l + e ^{to <} ) ^{2 2} >

R (λ) reflection coefficient ß optical absorption coefficient in m ^"1 z location coordinate in m

The explicit wavelength dependency of the light intensity and reflection on the sample surface is suppressed in the following, since the investigations are carried out as part of the work with monochromatic light. The partial reflection of the excitation light on the sample surface is minimized by vertical radiation. The remaining reflection loss takes into account the effective intensity In on the sample surface. The intensity of the electromagnetic wave at depth Z thus results in (2.3).

_ff effective intensity of light on the sample surface in Wm *

The amount of the derivation of the spatial intensity distribution of the light in the sample (2.3) according to the coordinate z is a measure of the radiation absorbed in the layer <z, z + dz>.

It is believed that all of the optical energy absorbed is immediately converted to heat. This approximation is justified since typical lifetimes of excited states are in the nanosecond range and the pressure oscillations are measured in the upper microsecond or millisecond range. The efficiency of the transformation of light into heat is taken into account by the factor η. With these assumptions, the heat arising from the absorption of light in the sample material can be described by the source term Q (z, t).

Q heat density in Wm ^"3 η conversion efficiency of optical energy into heat

Q (z, t) describes a location-dependent heat density in the sample that changes periodically with the angular frequency of the modulation ω. The amplitude of the vibrations is determined by the optical absorption coefficient, the incident light intensity and the conversion efficiency for non-radiative relaxation processes.

The diffusion of chromophores in biological matrices leads to an absorption coefficient ß (z) that changes with depth depending on the location-dependent concentration of the substance. This results in a modified expression for the heat density / Afromowitz /:

Equation (2.6) describes the contribution of all layers <z, z + dz> with the local absorption coefficient ß (z) to the resulting spatial distribution of the heat in the sample. Through the integration in the exponent, the energy components absorbed in different layers are taken into account. This expression is used to solve the heat conduction equations (WLG) for the sample and the gas-filled measuring volume. The structure of the two equations differs only by the source term Q (z, t) for the area of the sample.

Pro ^b e. z ≥ O

^(Z7 >

T _{G / P} temperature in K.

D _{G /} P thermal conductivity in mV ¹ p thermal conductivity of the sample material in W m ^"1 K ^{" 1}

The index G denotes the sizes in the gas, the index P the sizes in the area of the sample. The solution of the system of differential equations describes this spatial and temporal behavior of the temperature of the sample surface depending on the optical properties of the sample material. As a boundary condition, it must be demanded that the surface temperature of the sample corresponds to that of the surrounding gas at all times (2.9). Furthermore, the interface between sample and gas must allow a continuous flow of heat (2.10).

T _P (0, t) = T _G (0, t) (2.9 ⁾

, (2.10)

k _G thermal conductivity of the gas in W rrr -1K | -'- 1

This is a boundary value problem of an inhomogeneous parabolic differential equation IBronsteinl.

It is assumed that all periodically changing variables follow the time dependence e ^{(, ωt)} . This assumption is justified since only steady-state solutions from the WLG are interesting / McDonald /. The time dependency can be separated and the following product approach can be selected.

T (z, t) = Ψ (z, ω} e " ^{(2 11)}

The insertion of the solution (2.11) into the WLG for the sample space and the measurement volume provides equations (2.12) and (2.14).

Sample: z ≥ O ₍₂ .ι ₂ )

With the product approach (2.11), the problem has been reduced to an ordinary second order differential equation system in z. The separated equations for the location-dependent quantities of the system are (2.15) and (2.16).

The general solution of (2.15) results as a superposition of the solution of the homogeneous equation with a special solution of the inhomogeneous equation. The solutions of the homogeneous equation for the sample space and thus at the same time the solutions for the measurement volume can be written down directly.

ψ ₁ p _σ = ^ - ^GZ

The solution of the inhomogeneous equation in the area of the sample (2.15) consists of a linear combination of the solutions of the homogeneous equation Ψ ₁₂ and a special solution Ψ _{εp of} the inhomogeneous equation.

Ψ (z, ώ) = c _l Ψ _l (z, ώ) + c ₂ Ψ ₂ (z, ώ) + Ψsp (z, ώ) & ^{Λ 8} >

The special solution Ψ ^ can be obtained by the method of varying the constants. The invoice is shown in the appendix.

For physical reasons, the solution for z = oo must not grow indefinitely. For this reason, for the constant c ₁ in the area of the sample,

Cχ_ _P ~ ΔS _P fi - ^Spz QV - (2.20)

For the same reason, the constant c ₂ results in the measurement volume

c _2jG = 0 (2-2D

The boundary conditions (2.9) and (2.10) are used to determine the remaining constants. They lead to a linear system of equations with two unknowns for determining the parameters in the solutions in the measurement volume and in the sample. The results are

Inserting the constants obtained and the special solution in (2.18) leads to the sought solutions (2.24) and (2.25) of the system of differential equations.

Equations (2.24) and (2.25) describe the spatial and temporal temperature distribution in the sample and in the measurement volume as a result of the absorption of light harmonically modulated with the angular frequency ω. They have the structure of plane waves. For this reason, Kirkbright coined the term thermal wave. The wavelength of the thermal waves (2.26a) results from the imaginary part of s (2.26). It varies with the angular frequency of the modulation.

The damping of the wave in the material is described by the thermal diffusion length μ IRosencwaigl. It depends on the temperature conductivity of the sample material and the modulation frequency of the excitation radiation.

The thermal diffusion length μ describes the distance after which the amplitude of the wave has decayed to the 1 / e th part. It can thus be equated to a spatial information horizon. Only thermal waves that arise within a diffusion length μ make a significant contribution to the temperature oscillations on the sample surface. The analogue to the optical absorption coefficient ß is the reciprocal thermal diffusion length μ ^"1 for the thermal waves.

2.3.1 Properties of the solution

The properties of the solution found are discussed below. The measurable photoacoustic pressure is directly proportional to the temperature of the sample surface (z = 0). The considerations therefore only take place on the sample surface. The expression for the time-dependent temperature at the sample-gas interface is

The integrations via z 'and z "both run in the area of the sample. The integration via z" takes into account the energy components already absorbed up to the current integration variable z'. The temperature oscillations are difficult to evaluate in the present form. To demonstrate the basic properties, a constant optical absorption coefficient ß (z) = ß in the sample is assumed. The integrations can then be carried out and equation (2.28) goes into a simpler form.

After inserting s _P and taking into account the physically relevant real part of the surface temperature, one obtains (2.30).

μ _p = 1 / ß optical absorption length in m

The solution is decisively determined by the relationship between the optical absorption coefficient β or the optical absorption length μ _p and the thermal diffusion length μp. Two cases can be distinguished: a) μ _P «1 / ß or μ _P « μ _ß

Equation (2.30) is simplified

ηpκl _i eff

(l + g) k _P

The amplitude and phase of the surface temperature of the sample is determined by the optical properties. This case can be used for spectroscopic examinations. b) μ _P »1 / ß or μ _P » μ _ß

ηpκl C = eff

(l + g) k _p

The surface temperature oscillations are independent of the material's absorption properties as long as the condition μ »1 / ß is met. The amplitude of the temperature oscillations is dominated by the thermal material parameters. The phase of the temperature fluctuations is constant. In this case, one also speaks of photoacoustic saturation IRosencwaigl.

The special position of photothermal measurement methods between optics and calorimetry described in Chapter 1 is confirmed by the properties of the solution. They are remarkable in that it is possible to obtain information about the optical absorption coefficient β of an opaque material using a technique based on light absorption, because only the energy absorbed within a thermal diffusion length contributes to the measurement signal.

Cases a and b show the limitations of the solution with regard to the ratio of the absorption coefficient β to the thermal diffusion length μ _P. The thermal diffusion length is dependent on the modulation frequency of the exciting light, so increasing the modulation frequency can almost always avoid photoacoustic saturation.

2.3.1.1 Frequency dependence of the amplitude

The discussion of the frequency dependence of the amplitude of the temperature oscillations starts from the solution of the WLG on the sample surface (2.24). Inserting s gives the following form:

Equation (2.31) describes the periodic changes in the temperature of the sample surface with the angular frequency ω. The amplitude of the oscillations is dependent on the thermal diffusion length μ from the angular frequency of the modulation of the exciting light.

Only the amount of (2.31) is relevant for the physical measurement of the amplitude of the temperature fluctuations. By transforming the complex parts in (2.31) we get (2.32).

Equation (2.32) describes the measurable temperature oscillations on the sample surface and their relationship with the location-dependent heat density in the sample. The surface temperature results from the superposition of the heat waves from different depths in the sample, the proportion of the heat from deeper layers falling exponentially with the depth z. In contrast, the phase of the oscillations increases linearly with the depth in the sample.

To remove the time dependency of the derived temperature expression, it is necessary to consider the measuring process. A selective phase-sensitive amplifier, a lock-in amplifier, is used to measure the photoacoustic signals generated by means of periodic modulation. It precisely determines the amplitude and phase of the temperature oscillations at the specified reference frequency. If a fixed frequency ω is specified and the real part relevant for the physical measurement is taken into account, (2.33) results.

The explicit frequency dependency of the amplitude of the temperature oscillations contains the information about the location dependency of the heat density Q (z) and thus also the information about the location dependence of the optical absorption coefficient β (z).

The exponential function in the integrand ensures effective damping of the contributions of the heat density at positions z> μ to the temperature oscillations of the surface. In the summation of the integral, the proportions of heat in the areas of the sample for which z> μ applies are strongly suppressed. Only heat that is localized within the limits 0 <z / μ <1 contributes to the measurable surface temperature. The layer thickness for which this condition is met can be controlled on the basis of the frequency dependence of the thermal diffusion length μ. Increasing the modulation frequency from fi to f ₂ reduces the diffusion length from μi to \ ι ₂ . The difference in surface temperatures measured at fi and i is proportional to the energy absorbed at a depth between μi and μ ₂ . By gradually increasing the thermal diffusion length, i.e. reducing the modulation frequency, more layers of the sample contribute to the surface temperature. In this way, a depth profile of the absorption coefficient ß can be obtained. Rosencwaig and Gersho, the founders of photoacoustic theory, recognized these special properties and proposed a number of applications in the fields of medicine and non-destructive material analysis.

The dependence of the amplitude of the temperature oscillations on the sample surface on the modulation frequency of the exciting light is given by equation (2.33). It thus provides the mathematical basis for the vividly described determination of spatial depth profiles of the optical absorption coefficient from a photoacoustic measurement.

The solution of the heat conduction problem is represented in its frequency dependence as an integral equation. The equation belongs to the class of Fredholm integral equations of the first kind. The solution of Heat conduction equation for the pulsed excitation will lead to a similarly structured expression. Possible solutions are discussed in Sections 3.2 ff.

2.3.1.2 Frequency dependence of the phase

In addition to the frequency dependency of the solution, the phase must be considered separately. As already indicated, it offers a second option for obtaining depth information with periodically modulated lighting. In principle, this type of measurement can be compared directly with the pulse-shaped measurement, since the transit time of the thermal wave is evaluated.

The process can be clearly described as follows. It is believed that the process of light absorption and the conversion into temperature vibrations is instantaneous on the modulation time scale. Then all layers of the sample vibrate in phase with the modulation frequency of the exciting light ω. Since the temperature is only measured on the sample surface, layers at different depths contribute to the measurable temperature oscillations after different times. Layers near the surface contribute to the resulting surface temperature with a small phase shift and layers from deeper areas of the sample with a larger phase shift.

Equation (2.28) is also used for the mathematical derivation of the temperature variations on the surface depending on the phase of the thermal wave at the modulation frequency ω and s _{p is used} explicitly.

When considering the measuring process as described in section 2.2.1.1 and taking the real part into account, the equation (2.35) results for the phase Φ _{G of} the temperature oscillations.

ω, _ef reference frequency in s ^"1

The frequency dependence of the phase of the sinusoidal temperature oscillations on the sample surface, like the amplitude, is determined by the ratio z / μ in the integrand. High values of z are synonymous with large depths in the sample and correspondingly large phase shifts. As in the case of the amplitude, the exponential term in equation (2.35) ensures effective damping of the thermal waves, for which z / μ> 1 applies. Only the thermal waves generated within the thermal diffusion length make a measurable contribution to the surface temperature. Within this layer 0 <z / μ <1, the heat density at location z leads to a thermal wave with the phase φ = z / μ, which contributes to the temperature oscillations on the sample surface. A phase-sensitive measurement of the surface temperature vibrations according to equation (2.36) enables the location-dependent heat density to be separated at different depths in the sample.

<P = p

This property will be demonstrated using a simple example. Assuming the absorption in the sample is such that a heat density distribution arises which is localized at the dimensionless depth z / μ = 1/50 at the sample surface and at z / μ = 1. The temperature oscillation that can be measured on the sample surface has the course shown in Figure 2.4 as the resultant (1). For comparison, the intensity curve of the excitation radiation is shown as curve 4. The two curves 2 and 3 28

illustrate the vibration of the surface temperature in the event that the sample contains only one of the two absorbent layers. The resulting surface temperature is the superposition of the contributions of the individual absorbent layers, which are illustrated by curves 2 and 3.

The heat originating from the dimensionless depth z / μ = 1 contributes with a clear phase shift compared to the heat that was generated in the near-surface layer at z / μ = 1/50. The amplitude difference is explained by the damping of the thermal wave in the material, which is described by the exponential term in equation (2.35).

As this simple example shows, the temperature of the sample surface can be understood as the sum of different vibrations of the same frequency with different amplitude and phase position / Du /. The location-dependence of the heat density Q (z) and the optical absorption coefficient β (z) can therefore be determined by a phase-sensitive measurement of the temperature on the sample surface. The evaluation of the Frequency dependence of the phase from equation (2.35) is only possible approximately due to the complex structure of the expression.

2.4 Solution of the heat conduction equations for pulsed lighting

The one-dimensional arrangement from Figure 2.3 is used to solve the heat conduction problem for pulsed excitation. The sample is illuminated with short light pulses, such as those emitted by Q-switched lasers. The duration of the light pulse is so short that it can be approximated by a Dirac delta pulse. The consideration of the heat transfer equations begins after the end of the light pulse. The spatial distribution of the absorbed energy in the sample immediately after the laser pulse, which is determined by the optical sample properties, is used as the initial condition.

Analogous to the case of periodically modulated excitation, there is a partial system of differential equations for the sample and the measurement volume. The two homogeneous WLG are subject to the same boundary conditions (2.9) and (2.10) as in the case of periodically modulated excitation radiation. The initial value of the temperature distribution in the sample is given by the heat density (2.6) derived at the beginning of Chapter 2. The initial value of the temperature in the measuring volume is zero. This assumption is justified since the subsequent measurements only take into account relative changes in temperature.

The symbolism corresponds to the case of periodically modulated excitation. The equations for the sample and measurement volumes are (2.38) and (2.39).

A different approach than in the case of periodic modulation is necessary to solve the partial differential equation system, since the time dependence of the temperature cannot be separated. The equations are solved using Laplace transform. Equations (2.38) and (2.39) are transformed into (2.40) and (2.41) after transformation into the laplace space.

The problem has been simplified to an ordinary second order differential equation system in z. A similar system of equations already occurred in the WLG's solution for periodically modulated lighting. The solutions can therefore be written down immediately taking into account the variable names. At the corresponding points of the solutions (2.24) and (2.25) already found, S is to be replaced by Jξ / δ. For the Laplace transform the temperature in the sample space (2.42) and in the measuring volume (2.43) results.

The solutions (2.42) and (2.43) must be transformed back into the time domain. The details of the reverse transformation of the complete solution can be found in the appendix. For further considerations, only the temperature of the sample surface at z = 0 is of interest. The temperature at the sample surface after absorption of a light pulse is described by (2.44).

The surface temperature after an excitation pulse is essentially determined by the exponential term in the integrand and the pre-factor with a 1 / / dependency. The thermal wave from a depth z = x contributes to the temperature at the sample surface with a delay after the so-called thermal transit time τ = x ² / 4D. The finite running time of the thermal waves offers the possibility of separating the heat densities in different sample depths due to a temporally resolved measurement of the surface temperature after the end of the light pulse.

The result of the heat conduction problem for the excitation with a laser pulse leads to a connection between the location dependence of the heat density in the sample and the time dependence of the surface temperature in the form of an integral equation. Equation (2.44), like the result of the WLG under periodically modulated lighting from Section 2.2.1.1, belongs to the class of Fredholm integral equations of the first kind.

To demonstrate the solution properties of (2.43), the example from Section 2.2.1.2 should be recalled. The heat sources are 1 and 50 μm. The resulting surface temperature after pulsed excitation has the course shown in Figure 2.5.

For comparison, the surface temperatures that would result in samples of only one absorbent layer are shown. During the For the first 5 ms, the temperature at the sample surface is determined exclusively by the energy absorbed in the top layer at 1 μm. Only then does the heat source contribute to the temperature on the sample surface at a depth of 50 μm, with a significant delay. The delay is determined by the temperature conductivity of the sample material. The value used in this example corresponds to that of the horny layer of human skin ISennenn /.

Mathematical descriptions for the temperature at the sample surface could be derived from the solutions of the heat conduction equations after optical excitation for the sample and the measurement volume. The relationships found in the form of a Fredholm integral equation of the first kind describe the relationship between the experimentally accessible temperature oscillations of the sample surface and the optically generated heat density in the sample. Unfolding the integral equation leads to the desired link between the spatially changeable optical absorption properties and the surface temperature.

2.5 Relationship between the optical

Absorption coefficient and the temperature of the sample surface

The aim of the work is to determine unknown spatial profiles of the optical absorption coefficient ß (z) from the photoacoustic measurements. In the previous explanations, relationships between the temperature at the sample surface and the spatial distribution of the heat density Q (z) in the sample were derived from the solutions of the heat conduction equations. The relationship between the two variables is physically unambiguous, since heat only arises in the sample due to the light absorption and the coefficient β describing the light absorption represents a measure of the absorbed energy.

The derivation of the mathematical relationship between Q (z) and ß (z) is based on the expression for the heat density in the sample after absorption of the incident light (2.13). Equation (2.13) is to be solved according to the optical absorption coefficient ß (z). The following auxiliary construction is used for this / Afromowitz /

-]><*'> * ^• ₍₂₄₅₎

G (z) = ηl ₀ (l -e °).

Solving the equation (2.45) after e ° and inserting returns the ¹ searched for

Relationship (2.45) between the heat density in the sample and the optical absorption coefficient ß.

Equation (2.46) provides the desired connection between the location-dependent heat density Q (z) created by absorption in the sample material and the optical absorption coefficient ß (z). The heat density Q (z) can be calculated from the experimentally obtained temperature oscillations of the sample surface after solving the Fredholm integral equations of the first kind. The spatial distribution of the optical absorption coefficient can be derived from this. 2.5.1 Limitations for scattering samples

In the derivation of the solutions, an optically homogeneous sample with distributed chromophores was always assumed. This assumption only approximates the optical properties of human skin. Depending on the light wavelength used, the light scattering in the skin must be taken into account, since it leads to an increase in the deposited optical energy in the layers near the surface. The Lambert-Beersche law on which the intensity distribution of light is based is no longer applicable. Formally, this can be remedied by using an effective absorption coefficient ß _eff . This coefficient is the sum of the optical absorption coefficient and the scattering coefficient of the tissue for the wavelength used. The longer the wavelength becomes, the more the scattering part gains influence.

Correct treatment of the scattering problem in human skin leads to a modified Lambert-Beersches law / Prahlt (equation (2.47)).

I (z) = Be ^{~ μeffZ} + Ce ^{~ μtrZ} (2.47)

Without going into the meaning of the constants, only the fundamental difference compared to the Lambert-Beerschen law is given. Instead of the optical absorption coefficient ß, an effective attenuation coefficient μ _e t _{f is} used. It describes the decrease in light intensity in the sample due to absorption and scattering processes. The second term, which is an extension of the Lambert-Beerschen law, describes the increase in light intensity at depth z due to backscattered light from deeper layers of the sample.

Regardless of the use of the simplified approach or the precise derivation of Prahl, the solutions of the heat conduction equations in the previous chapters are only slightly modified. In the general case, knowledge of the scattering coefficient is essential for correctly calculating the depth distribution of the absorbing substances.

In the biological tissues to be examined predominantly forward scattering with a wavelength dependence of ~ λ ^* ° ' ⁴ takes place, therefore in Within the scope of the work it was assumed that the error caused by the scattering of the light in the sample is negligible

Appendix 2

Procedure for solving the Fredholmsche

Integral equations of the first kind

The thermal conduction equations solved in Chapter 2 describe the propagation of the thermal waves that result from the absorption of periodically modulated and pulsed light. The solution represents a relationship between the experimentally accessible surface temperature and the location-dependent heat density inside the sample. The mathematical relationships found belong to the class of Fredholm integral equations of the first kind, the general form of which is described by equation (3.1).

Integral equations of this form occur in many problems in mathematical physics. An analytical solution to equation (3.1) is difficult because there is no mathematical description for Q (z) available. However, at least one approximate solution to the problem is possible using numerical mathematics methods. For this, the integral equation must first be brought into a form suitable for further numerical processing.

3.1 Approximation of the integral equation

The use of numerical methods to solve the integral equation (3.1) makes it necessary to discretize the variables in the space and time domain. The discretization in the time domain is predefined since the temperature T is an experimentally determined variable that is only known at the measuring points. The discretization in the local area can be arbitrary, adapted to the respective problem. A uniform subdivision is the simplest case. The continuous integral equation (3.1) thus turns into a summation over the discrete locations and times.

or in matrix notation T = KQ

The integral equation is approximated by a matrix equation after the discretization of the location variable z and the time t. Each individual column of the matrix K describes the temperature profile over time on the sample surface after excitation of a heat density distribution of the thickness Δz infinitely extended in the x and y directions at a depth of iΔz.

An absorption coefficient that changes continuously with the depth z is approximated by areas of constant absorption. Figure 3.1 illustrates the approximation of a location-dependent distribution using discrete support points. 3.1.1 Properties of the approximated integral equation

The equation (3.2) obtained by the approximation belongs to the group of operator equations, which are divided into correctly and incorrectly put. It is correct if the following applies:

1. For each T, equation (3.2) has a unique solution Q.

2. The solution Q is stable against disturbances of T, ie K ^"1 exists in the entire definition space of T ITikhonovl.

If one of the conditions is violated, the problem is incorrect. That is, there is no unique solution to the integral equation, or the solution is not a continuous function. Even small inaccuracies, such as rounding errors on the left side of the equation, lead to strong oscillations of the solution. The problem is said to be unstable. In the present case, the left side of the equation, the temperature T, represents a measured variable. The measurement errors it contains can only be influenced within very narrow limits. The problem must therefore be formulated in a stable way in order to be able to calculate the location-dependent heat density from the temperature.

The problems with regard to the stability of their solutions are classified using the condition number of the matrix K. This number is formed as the quotient of the largest and smallest singular value of the matrix K. It is 1.66-10 ^{17 in} the case of pulsed excitation. An extremely unstable solution behavior of equation (3.2) must therefore be assumed. The entire problem is referred to in the literature as "severely ill posed" / Power /.

The instability of the solution will be demonstrated using an example. The solution Q of the algebraic equation (3.2) results formally by multiplication from the left by K ^"1 .

K ^ Q (3.3)

This formally performed operation cannot be carried out with the matrices K that result from the solutions of the heat conduction equation. In general, the matrices are not square and not regular. A reverse matrix K ^'1 therefore does not exist. One way out of the problem is offered by the matrix K ⁺ , which has as many properties as possible of the reversed K ^"1. K ^* denotes the Moore-Penropse pseudo-inverse of the matrix K. The pseudo-inverse can be used to solve the equation (3.3) approximately / Hansen /.

The problems that arise when using the sketched solution method will be explained using a simple example. The one-dimensional arrangement from Figure 2.3 is assumed. The absorption of the sample material is strongest on the surface and drops exponentially into the depth of the sample. According to (2.14), the absorption of a light pulse in the sample leads to a spatially distributed heat density. The resulting changes in surface temperature were calculated using the solution (2.43) of the heat conduction equation for pulsed excitation. The result is shown in Figure 3.2. The sample was approximated by 40 layers with a thickness of 1.6 μm each. The data acquisition rate of the measurement was 0.5 ms and the number of measured temperature values was 135. The matrix K describing the problem thus consists of 135 rows and 40 columns.

It was assumed that the conversion of the absorbed optical radiation into thermal energy takes place completely and without delay. Equation (3.2) is solved with the theoretical surface temperature and the pseudo inverse K *. The result is shown in Figure 3.3.

The comparison of the approximate solution Q (z) of the integral equation (3.1) with the actual course of the heat density in the sample shows clear differences. The relative deviations were in the range of MO ^"13. That means that only rounding errors, errors from the discretization or the formation of the pseudo inverses led to the observed deviations. The temperature of the sample surface was included in the calculation with numerical accuracy.

The situation changes fundamentally if experimentally obtained temperature data with measurement errors are assumed. To simulate For real measurement values, the temperature value is replaced by a randomly distributed value at each measurement time. The fluctuation range is ± 0.5% of the maximum theoretical temperature value. The simulated time dependency T (t) calculated in this way does not differ visibly from the org function in the direct graphic comparison. The result of solving Ferdholm's integral equation using pseudo inverses and simulated temperature measurements is shown in Figure 3.4.

The oscillations of the solution that are already visible in Figure 3.3 have increased by several orders of magnitude. There are no longer any similarities with the initial distribution.

An exact solution of equation (3.3) with real temperature measurements therefore seems impossible. However, there are mathematical methods that allow an approximate solution. These methods are called regulatory procedures. They provide efficient and stable algorithms that dampen the solution's oscillations and at the same time keep the approximation error as small as possible. They weight them often physically motivated constraints in order to obtain an exact approximation of the actual solution.

The following sections examine four selected mathematical methods in terms of their applicability to the problem at hand. The quality of the approximation of the individual regulation methods is evaluated on the basis of artificially generated temperature profiles.

3.2 Regulation procedure

An exact solution of equation (3.2), which resulted from the discretization of the integral equation of the heat conduction problems, is not possible in most cases. The solutions found are extremely unstable and therefore cannot be evaluated. Regulation procedures replace the mathematically exact solution with an approximation. The accuracy of the approximation is determined by the so-called residual standard | T-KQ | described. The regulatory procedures minimize this residual standard so that the best possible approximation to the actual solution is achieved. In addition to minimizing the residual standard, the minimization of the solution standard | Q | used as a constraint.

3.2.1 Singular Value Decomposition

Before the actual regulatory procedures are dealt with, the singular value decomposition (SVD) procedure is introduced. It is a universal mathematical tool for solving and evaluating incorrectly posed problems / Hansen /.

Let K e R ^mxn be a rectangular matrix with m> π. SVD decomposition is the factorization of the form (3.4). K = usv ^τ

The matrices U and V are unitary, UU = W = E _n . S is a diagonal matrix that has the non-negative singular values of K in descending order in its diagonal. The vectors u _f and V | are called right or left singular vectors of the matrix K ILawsonl. The condition number of the matrix K mentioned in the previous section results from Sιfs _maX (n). The SVD factorization of the matrix K is clear.

Algebraic solution algorithms to which the matrix K was not accessible can be applied to the product UPS ^T The inverse of the matrix K can now be formed as a so-called Moore-Pennrose pseudoinverse. In the case of a square matrix K, it changes into the inverse K ^"1 .

K ⁺ = V ^τ ST ^l U

With this pseudo inverse, the equation Q = K ^'1 T can be solved.

Examination of the decomposition factors U, V, S gives indications of the properties of the problem. This should be illustrated using the example of the integral core from the solution of the heat conduction equation for pulsed excitation. The matrix K of the problem from section 3.1 is broken down into factors using SVD. The singular values of the matrix K are shown in Figure 3.5.

The number of singular values corresponds to the number of columns of the ¹ matrix K. The ratio of the largest to the smallest singular value is .66-10 ¹⁷ . Figure 3.6 shows the associated right and left singular vectors of matrix K.

Figures 3.5 and 3.6 show two main points of view:

1. The singular values slowly strive towards zero without a special gap.

2. The left and right singular vectors show more sign changes with decreasing singular values.

These properties indicate a very incorrect problem / Hansen /. Usable solutions can therefore not be obtained in the direct way described in section 3.1 using the pseudo-inverses. The properties of the operator equation make it necessary to use regulation methods to calculate a stable solution of the location-dependent heat density. 3.2.2 Tikhonov's regulatory procedures

The oldest and most well-known regulatory procedure goes back to the racial mathematician Tikhonov and also bears his name. The procedure is based on a precise analysis of the situation.

A closer look at the problem (3.2) leads to a changed integral equation. Since the temperature values on the sample surface are not absolutely known, the approach T (t) = T \ t) + λS (t) can be used / Phillips /. S (t) describes the deviations of the measured values from the unknown exact values T t). λ is a weighting factor, which is called the regulation parameter. The matrix approximation of the integral equation changes with this approach to equation (3.5).

T + λS = KQ (3 5)

S disturbance matrix, λ regulation parameters

The Tikhonov method looks for an approximate solution to equation (3.5) depending on the parameter λ. The approximation is achieved by minimizing the residual standard | (T - λS) -KQ | ² and minimization of the solution standard | Q | ² determined. The matrix S is the unit matrix in many cases. Equation (3.5) is solved by the routine of the least squares method implemented in Matlab ^® .

The choice of the parameter λ is the real problem of the solution. He is responsible for damping the solution oscillations, but must be selected so that on the one hand the oscillations are effectively suppressed, on the other hand the solution Q is not smoothed too much.

3.2.3 Expectation Minimum Regulation Procedure

Power has proposed a class of alternative regulation methods that use the stabilizing effect of normally distributed noise. The term expectation minimum (EM) for the class of regulatory process results from the statistical evaluation of a large number of independent attempts at solution.

Compared to the Tikhonov method presented, it is a fundamentally different approach. The matrix K, the kernel of the integral, is "noisy" with well-defined, normally distributed noise of the amplitude n.

K < _Xj , t _k ) k {x _j , t _k ) ³ - ⁶ >

rv random variable which is subject to a normal distribution

In the following, the random sizes that were changed according to regulation (3.6) are marked with a roof over the original symbol. The modified equation to be solved from equation (3.2) is (3.7).

T = KQ ⁽³ ' ⁷⁾

The resulting equation is solved several times with a new random matrix K. The result is a large number of random solutions Q .. The solution to the original problem (3.2) is approximated by forming the arithmetic mean according to (3.8).

The term expectation minimum, coined by Power, was chosen because the solution to the problem (3.2) is the expected value of a randomly changing solution of equation (3.7).

The connection with the Tikhonov regulation method presented in section 3.2.2 results from the variance in the distribution of the noise with which the integral core is modified in matrix form K. Without going into the specific derivation, only the equivalence of the variance of the noise in K and the parameter λ of the Tikhonov regulation method for the problem should be given here / Power /. 3.2.3.1 Expectation Minimum Procedure + Singular Value Decomposition

The "noisy" operator equation (3.7) can be solved with various methods. One possibility is to calculate the Moore-Pennrose pseudo-inverse using SVD and then solve Q = K * T. Instead of the pseudo-inverse calculated from the exact matrix K, The solution of the operator equation thus becomes a randomly changing quantity. Many attempts at solution and their subsequent averaging lead to the desired result according to the EM scheme.

The MATLAB ^® routine pinv was used to carry out the matrix factorization and to calculate the pseudo inverses.

3.2.3.2 Expectation minimum method + method of least non-negative squares

A further possibility of the solution results from the use of a method proposed by awson and Hanson for the first time, which is referred to as the method of the least squares with non-negative constraints.

The following approximate method is called the least squares method. Given a real matrix A _mxn with rank (A) <min (m, n) and a real vector b _m . We are looking for a real vector x „which meets the Euclidean norm | Ax - b | minimized.

The equals sign of the problem Ax = b is replaced by an approximation Ax s b. The accuracy of the approximation is determined by the Euclidean norm | Ax - b | certainly. As an additional constraint, Ax> h can be introduced. In the case of the NNLS method, x ≥ 0 is selected as a secondary condition.

The noisy operator equation is solved according to the EM scheme with the numerical routine NNLS from MATLAB ^® . The formation of the middle value of several independent solution attempts leads to the regularized solution Q.

3.2.4 Conjugate gradient method

As the only representative of the iterative regulation method, the conjugate gradient method (CGM - Conjugate Gradient Method) proposed by Hestenes and Stiefel is used in this work.

An interesting fact in connection with discrete, incorrectly posed problems is the faster convergence of the low-frequency components in the solution compared to the higher-frequency components / Hansen /. The CGM process therefore already has a stabilizing behavior. The number of iteration steps required serves as the regulation parameter. A major advantage of the method is that it avoids the direct calculation of K ^'1 or the corresponding pseudo inverses and instead calculates the product K ^T K. The extent to which this advantage affects a more stable solution behavior depends on the problem and cannot be assessed a priori.

3.3 Methods for choosing the regulation parameter

The mathematical methods presented in the previous sections solve a given problem with the help of an additional parameter, the regulation parameter. He is responsible for the degree of damping of the solution oscillations, but also for the deviation of the approximated from the actual solution. The regulation process is worthless if no methods are available for determining the optimal value of the regulation parameter. The choice of damping parameter is the most critical part of the whole process.

The following sections present three methods for determining the optimal value of the regulation parameter. These are around the L-curve, the general cross evaluation and the quasi-optimum criterion.

3.3.1 L curve

The L-curve is the simplest and intuitively accessible graphic help for choosing the regulation parameter. It was first used by Lawson and H nson. The L curve carries the norm of the regularized solution | Qreg | against the residual standard | KQ _re3 - T | for all regulation parameters. It graphically represents the heart of every regulation, the balance between minimizing the solution and the residual norm.

The curve has the characteristic shape of an L, hence the name L curve. The vertical part of the curve contains solutions to the regulatory problem, which are determined by the error of the measurement, experiment or discretization. The norm of the solutions is sensitive to changes in the regulation parameter. The horizontal part of the curve represents solutions that are determined by the error of the regulatory process. In this part of the curve, the residual standard is sensitive to changes in the parameter.

The best compromise between the increasing damping of the solution oscillations and the increasing error is near the base of the L-shaped plot. The optimal regulation parameter of the problem lies at this point of the L-curve.

Finally, a special feature of the L curve in connection with the Tikhonov regulation process should be pointed out. There is no solution in the plane defined by the solution and residual norm, which lies below the L-curve of the Tikhonov method. The solutions of other regulation methods lie in the drawing plane on the Tikhonov-L curve or above. For a given solution standard, there can therefore be no smaller residual standard than that of the Tikhonov regulation method. Comparisons with other methods are therefore possible simply by comparing the corresponding L-curves / Hansen /.

3.3.2 General cross evaluation

The general cross evaluation (GCV-Generalized Cross Validation) goes back to Golub. It is based on the following considerations. An element t _{k of} the vector T is omitted and the regularized solution Q _k (λ) is sought. If the regulation parameter λ represents an optimal choice, the kth element of KQ must be _k (λ) «t. This means that if λ is optimally selected, the expression becomes

minimal. This condition is also called the simple cross-rating method. The general cross evaluation uses a rotation invariant form of this expression. This results in two arguments for the choice of the regulation parameter. • The absence of an element of the data vector must be taken into account in the solution when the regulation parameter is optimally selected.

• Furthermore, the choice of the regulation parameter must be independent of orthogonal transformations from T / Hansen /.

This leads to a function G, in the minimum of which the optimal regulation parameter can be found.

E _m standard matrix of dimension mxm

K 'denotes the matrix that produces the regularized solution, i.e. Q "" = K * T.

3.3.3 Quasi-optimal criterion

The quasi-optimum criterion assumes that the solution represents a smoothed function at the optimal value of the regulation parameter. Therefore, the relative changes in the solution with the regulation parameter are only slight. The mathematical implementation of this consideration is expressed in the QuasiOptimum function (3.10).

Qλ solution of the regulation problem with regulation parameter λ

The optimal value of the regulation parameter λ is in the minimum of the function 0 (λ). Like the previous methods, this approach strikes a good balance between errors of an experimental nature and the damping introduced by the regulation process / Hansen /. Strictly speaking, the quasi-optimal function is only defined for continuous dependencies of the solution on the regulation parameter. For discrete values of the regulation parameter, the infinitesimal difference d is replaced by the finite difference Δ. 3.4 Comparison of regulatory procedures using numerical test calculations

The presented regulation methods are known methods of numerical mathematics. Their mathematical properties have been examined in detail. The relative strengths and weaknesses of the methods as a function of different spatial dependencies of the absorption coefficient under simulated experimental conditions have not yet been assessed. The assessment of the performance of the different regulatory procedures is to be carried out using model absorption profiles. These profiles must contain the limit trap of possible spatial dependencies of the optical absorption coefficient in order to reveal the essential properties of the method. For this reason, the spatial distributions of the optical absorption coefficient shown in Figure 3.8 were selected.

An essential motivation of the present work is the creation of a possibility for the non-destructive investigation of the spatial concentration distribution of externally applied substances in the human skin. The diffusion of applied substances generally leads to a concentration of the substance that decreases exponentially with depth. For this reason, the two exponential dependencies of the absorption coefficient on test calculations were selected. They differ by a local maximum, which can be caused by a diffusion barrier that may be present in the sample. The monotonically increasing and decreasing absorption coefficient was also included in the ensemble of the test profiles as a limit case of the diffusion processes.

The spatial dependency of the optical absorption coefficient in the form of a step and a delta function was chosen because they provide information about the ability of the regulatory process to resolve large concentration gradients.

Based on the different spatial absorption profiles, the temperature curves over time at the sample surface after pulsed light excitation were calculated using equation (2.44). The temperature data were then subjected to different levels of noise in order to simulate data from an experiment. Rule (3.11) was used for this. t _k = t _k + rv * max (T) (3.li>

T = (_,, t ₂ , ₄ ) exact calculated temperature,

T = (f, t ₂ , .i _t ) noisy temperature, rv random variable of a normal distribution max () maximum value of the temperature vector.

The variance of the normal distribution was varied in steps between 0.001 and 0.3 to simulate experimental situations with different signal / noise ratios.

The spatial distribution of the optical absorption coefficient was calculated from the thus obtained simulated experimental temperature data using the four regulation methods Tikhonov, EM SVD, EM NNLS and CGM. The regulation procedures were created under MATLAB ^® in the form of user programs.

The parameters summarized in the table below were used for the calculation.

For each dependency of the absorption coefficient on the location and each signal-to-noise ratio, the optimal regulation parameter was carried out in the four tested regulation methods by means of visual inspection of the L-curve, general cross evaluation and quasi-optimal criterion. The combinations resulting from this diversity add up to almost SOO different solutions with their own residual and solution standards. Due to the large amount of data, it makes little sense to present all results of the numerical test calculations in detail. Therefore, only the properties of some representative examples are selected and examined in more detail. They don't results shown explicitly are evaluated in verbal assessments in relation to the examples shown.

First, the influence of the method of choosing the regulatory parameter on the regularized solution is discussed. Afterwards, some special features of the EM regulation process are emphasized compared to the Tikhonov and CGM processes. Finally, the calculated profiles are compared for selected values of the signal / noise ratio.

3.4.1 Influence of the methods for the choice of the regulation parameter on the solution

In the context of the numerical tests, the regulation parameter was determined by visual inspection of the L-curve, by means of a general cross evaluation and using the quasi-optimum criterion. Each of the three methods tries to strike a balance between the experimental error and the error introduced by the regulatory process. Various strategies are used for this, which have already been described under point 3.3. It can therefore be assumed that the optimal regulation parameters found differ. In the following, the influence of the method for determining the optimal regulation parameter on its value and the relative error of the solution is evaluated on the basis of the presented test absorption profiles.

A special feature is the direct CGM method, in which the selection of the regulation parameter (iteration steps) was only possible using the graphic aid L-curve.

3.4.1.1 L curve

The visual inspection of the L-curve searches for the optimal value of the regulation parameter near the base of the L-shaped plot. For this purpose, the solution standard is presented for each absorption profile and each regulatory procedure compared to the residual standard. It can be seen that with increasing noise in the simulated experimental temperature data, the shape of the L changes to an obtuse angle. Figure 3.9 illustrates this behavior.

The change in the L curve described is particularly pronounced in the EM NNLS and CGM method. The other two regulatory procedures show a similar, but significantly weaker tendency. The location dependence of the absorption profile had no influence on the shape of the L curve.

The kinking of the horizontal part of the L curve is due to the increasing influence of the experimentally induced error in the solutions of the operator equation. The regulatory process can no longer effectively eliminate these errors. It was shown that the EM NNLS method Even with small experimental errors, the limits of meaningful calculation reach the limits of the other regulation methods.

Changing the L-shaped plot with a pronounced kink to a rounded angle leads to greater uncertainties in the visual choice of the regulation parameter. In the numerical tests, the point given as an example in Figure 3.10 was always assumed to be the optimal value.

The regulation parameter determined in this way was less than or equal to the value found by means of a generalized cross-evaluation for almost all possible combinations of absorption profile, noise and regulation method. 34.1.2 General cross evaluation

The generalized cross validation checks the quality of the selected regulation parameter by omitting a single temperature measurement. In the minimum of the GCV function, the calculated temperature value is a good approximation of the value not considered in the temperature data. Figure 3.11 shows the typical course of the GCV function using the example of the linearly increasing ramp profile, the temperature data of which was noisy with 3%.

The enlargement of the GCV function in Figure 3.11 illustrates a problem that occurs when using the generalized cross-evaluation. The absolute minimum of the function is not always synonymous with the optimal choice of the regulation parameter. It can be seen that there is at least one further minimum at λ = 10 ^"6. It is therefore more advantageous to search for a local minimum. The example from Figure 3.11 shows that A visual check of the regulation parameter is also necessary when using the GCV function.

3.4.1.3 Quasi-optimal criterion

The quasi-optimum function describes the changes in the solution standard relative to the changes in the regulation parameter. The optimum value of the regulation parameter can be found in the minimum of the function (3.10). Figure 3.12 shows an example of the course of such a function.

The absolute minimum of the quasi-optimum function corresponded to a regulation parameter in all cases examined that comes close to the optimum. The absolute value of the regulation parameter found using the quasi-optimum function was the largest for all tested absorption profiles and regulation processes. The solutions appeared to be very smooth. 3.4.1.4 Comparison of the procedures for determining the regulation parameter

The largest value of the regulation parameter is determined by the quasi-optimal criterion in all examined combinations of absorption profile, regulation method and signal / noise ratio. Figures 3.13 and 3.14 illustrate this situation using the example of a step absorption profile, reconstructed using Tikhonov and EM NNLS.

The generally recognizable tendency of a regulation parameter increasing with a decreasing signal-to-noise ratio can be explained by the increase in measurement errors in the signal and the higher degree of regulation required for the solution.

Of the general behavior of the three methods, only the most noisy value is somewhat out of the ordinary. In this case, however, the simulated measurement curve is so noisy that it is impossible to visually derive information from the measurement without knowing the actual course.

The regularized solutions, which were obtained by visual inspection of the L-curve, have the smallest residual standard. So they are accurate from the point of view of the error, but on the other hand they have the greatest solution standard. This behavior proves to be advantageous for absorption profiles that have strong discontinuities, such as the step profile.

The quasi-optimum criterion selects solutions that are characterized by the smallest solution standard. They appear smooth compared to the solutions provided by the L-curve criterion. This behavior is at the expense of the residual standard. It takes the highest values when using the quasi-optimal criterion. The regulation method is therefore well suited for the calculation of slowly and constantly changing absorption profiles

The general cross rating marks the middle and can be used equally for absorption profiles with and without discontinuities.

The outlined properties of the selection process for the optimal value of the regulation parameter are only to be understood as a tendency. In individual cases it is always better to use several procedures in parallel and to visually assess the result. In the following, the selection process was therefore always used which is characterized by a minimal relative error of the solution. I ^Q "* - ^Q l is defined as a relative error. Q _rθk denotes the solution found by means of a regulation process, Q the original.

3.4.2 Special features of the EM regulation procedure

The solution to the operator equation (3.48) found using the EM regulation method is the average of solutions of the individually noisy operator equations that change randomly. There is therefore a dependency of the residual and solution standard on the number of repetitions of the solution attempts. Unless explicitly stated otherwise, 150 repetitions were used in the work. The solution standards that result for a given residual standard are comparable with those from the non-statistical regulatory procedures. However, the associated solutions appear to be less precise, since they show statistical fluctuations due to the principle. Due to the definition of the norm, a statistical fluctuation with a spatially short period around the correct value can have the same solution norm as a smoothed solution. This behavior can be drastically reduced by increasing the number of averages. However, the computing time required then increases extremely. The complete calculation of the solution with 50 different values of the regulation parameter using the EM NNLS method and 300 repetitions takes 1.5 hours on a PC PI 233 MHz.

Using the example of a step absorption profile, the behavior of the solution depending on the number of repetitions in the EM solution process is to be demonstrated. The dimensionless signal / noise ratio of the simulated experimental data was 204. The reconstruction was carried out using the EM SVD method. Figure 3.15 represents the residual and solution norm of the result depending on the number of averages carried out.

After just a few repetitions, the residual standard approaches its final value. The exclusive consideration of this standard says little about the quality of the solution of the regulation process. The solution standard must also be included in the considerations. It falls continuously with the number of averages. After approx. 100 solution attempts, their value approximated the final value with a difference of 0.25%. The error due to the limitation of 150 repetitions for the statistical regulatory procedures in the context of the work is therefore negligible.

Figure 3.16 shows the solutions of the EM SVD method calculated with a different number of repetitions compared to the solution of the Tikhonov method. The solution obtained by means of the EM regulation method appears more smoothed with increasing number of solution attempts. They differ with a sufficiently large number of attempts to solve only a small amount of the results of the Tikhonov method. A closer look at Figure 3.16 reveals a slight overshoot of the EM solution in the range of 0 - 20 μm. The differences between the two solutions are in deeper areas of the sample negligible. A comparison of Figures 3.15 and 3.16 shows that the reduction in the solution standard is equivalent to the smoothing of the regularized solution

It remains to be seen that the EM method calculates the step profile with less overshoot in the vicinity of the jump point and thus provides a more accurate reconstruction than the Tikhonov method. This property of the EM process will also be noticed in other absorption profiles with discontinuities. 3.4.3 Results of the regulatory procedures with different test profiles

The absorption profiles have been divided into two classes for better evaluation and presentation of the results. In the class of slowly changing absorption profiles, there is the linearly falling and increasing function of the location as well as the two exponentially falling profiles. The delta profile, as well as the ascending and descending stages, have been included in the class of rapidly and continuously changing profiles. The absorption profiles were summarized because they place similar demands on the regulatory process with regard to the resolution of high spatial frequencies in the solution. To compare the regulation procedures, the regulation parameters with the smallest relative error of the solution were used

3.4.3.1 Spatially continuous absorption profiles

In most cases, the diffusion of chromophores into a biological tissue leads to spatially constantly and continuously changing concentration profiles. The solution behavior of the regulatory procedures when calculating the examined continuous absorption profiles therefore provides information about the suitability of the mathematical procedures for application to medical and biological questions.

In principle, each of the four tested regulation methods is able to calculate the slowly and steadily spatially changing curves of the optical absorption coefficient in the sample. Even the temperature data, which was very noisy with 30% of the maximum temperature, tended to lead to correct dependencies of the absorption coefficient on the location in the sample. Such an extremely large measurement error did not occur during the experimental investigations.

The maximum measurement error that allows the absorption profile to be calculated with a relative error of less than 15% depends on the type of the absorption profile. It fluctuates between 5% and 10% of the maximum temperature. From the results of the numerical investigations, no general derive applicable regulations for the selection of the appropriate regulatory procedure. However, there are clear trends. Figure 3.17 shows the relative errors. of the solutions depending on the regulation process used and the noise in the temperature data.

The relative error of the solutions increases with all regulation methods with the noise in the measured values. The regulation procedures are generally not able to compensate for the simulated temperature measurement errors. The results of the individual methods can differ greatly in detail for identical measured values.

Almost without exception, the solutions of the Tikhonov regulatory procedure have the smallest relative error. The results of the EM SVD and EM NNLS methods showed comparably large relative errors. The reason for the differences between the Tikhonov solutions and the solutions of the EM methods essentially lies in the number of averages of the EM methods limited to 150. As already shown in section 3.4.2, it is possible to achieve the results of the Tikhonov regulation procedure by increasing the number of repetitions in the EM procedure. The results summarized in Figures 3.18 - 3.25 show that the graphical mean, based on the bases of the solutions of the EM methods, is almost identical to the Tikhonov solution. The CGM method differs greatly from the general trend. The CGM method is an iterative method in which the number of iteration steps is used as a regulation parameter. Since only whole steps are possible, an optimal choice of the regulation parameter of the solution is not possible for all signal / noise ratios. This behavior is particularly pronounced in Figure 3.25. However, with selected noise values in the temperature data, the CGM method achieves a relative error that is less than or equal to that of the Tikhonov regulation method. The regularizing behavior is obviously good. In the present case, it would have to be revised towards slower convergence. None of the regulatory procedures examined was able to resolve the local maximum of the exponential drop from Figures 3.20 and 3.21. From a relative noise component of 0.1% in the signal, the maximum is smoothed to the point of being unrecognizable in the solution. The Tikhonov regulation procedure calculates the qualitatively correct course, but the position of the maximum is shifted. Smaller local maxima and peculiarities in the location dependence of the absorption coefficient are obviously not resolvable in the experiment with the selected parameters.

The Tikhonov regulation method is the most suitable method for slowly and constantly changing spatial dependencies of the optical absorption coefficient. If the Tikhonov solution shows strong oscillations around zero, the EM NNLS method should be used. The disadvantage of this method is the computing time required. The CGM method is not suitable because the regularization through iterations does not always allow an optimal adaptation to the given experimental situation. The regulation processes that work with a quasi-continuous regulation parameter can adapt much better.

3.4.3.2 Spatially inconsistent absorption profiles

For example, the step profile or the delta profile places high demands on the regulation process. The profiles used are artificial and rarely occur during the investigation of biological materials. Rather, they are intended to show the limits of the regulation process to calculate large spatial absorption gradients. None of the methods tested was able to exactly calculate the original spatial distribution of the optical absorption coefficient. Even the data noisy at 0.1% of the maximum temperature led to a smoothed step. The relative errors of the solutions are therefore larger overall than the comparable errors from section 3.4.3.1. As the noise in the temperature data increases, the relative error of the solution only increases slightly. Only when there is a measurement error of 30% of the maximum temperature does the solution error start to increase disproportionately.

Figure 3.26 shows the relative errors of the solutions as a function of the measurement error in the temperature data. The relative error of the solutions of the EM SVD, CGM and Tikhonov method is similarly large for all examined profiles, the influence of the measurement error being small however, significantly larger than when calculating continuously changing absorption profiles. In particular, the delta profile cannot be recognized from the solutions of the three regulation procedures mentioned. The EM NNLS method is well suited for the reconstruction of discontinuous absorption profiles. It was able to resolve the delta profile as such up to a measurement error of 1% of the maximum temperature. The results of the calculation of the step profile are, without exception, better than those of the other regulation methods. However, none of the methods could reproduce the step function as a step again. The regulatory procedures invariably smoothed the level. However, the spatial position of the abrupt drop remains recognizable up to a measurement error of 5% of the maximum temperature. Figures 3.27-3.32 show the reconstructed solutions compared to the original.

The solution found using EM NNLS most closely follows the original of the ascending level. Not only is the steep rise resolved well, the overshoot for positions> 30 μm in the sample is less than with the other methods.

The reconstruction of the delta function is to solve a test for the regulation process, the degree to which these are able aufzu spatial positions exactly _¬. It turns out that the position of the maximum of the regularized solutions up to a relative error of the data of 5% coincides with the maximum of the original. However, only the EM NNLS method calculates the delta function as such up to a relative error of 0.5%. right. All other methods lead to a smoothing of the delta profile. With increasing noise in the temperature data, the amplitudes of the secondary maxima in the solutions increase, so that a correct determination of the position of the main maximum is no longer possible. The shape of the delta profile can only be seen in the EM NNLS solution anyway. Without knowing the Originals of the solutions of the Tikhonov regulation, the EM SVD and CGM methods have no relation to a delta-shaped absorption profile.

3.5 conclusions

The comparison of four different regulatory procedures based on test profiles with very different properties reveals the performance and the limits of the regulatory procedures. Continuous, slowly changing, spatial profiles of the optical absorption coefficient can be reliably reconstructed by the method up to a relative measurement error of 5% of the maximum temperature. Special smaller maxima or minima cannot be resolved. One way out can be to use a finer discretization of the local area. However, this also significantly increases the computing time required for the reconstruction. In the case of the EM process, a Vz day can be achieved with a Pentium ^® II 230 MHz processor. Alternatively, you can work with a non-uniform subdivision, which divides more finely in the area of the local peculiarity. However, an approximate knowledge of the spatial course of the optical absorption coefficient is necessary for such a procedure.

The reconstruction of non-continuous profiles of the optical absorption coefficient was only possible to a limited extent with the tested regulatory procedures. Only the EM NNLS method is able to reliably reconstruct a delta-shaped course of the optical absorption coefficient up to a measurement error of 1% of the maximum temperature.

The poor suitability of the regulatory procedures for discontinuous absorption profiles is only of minor importance against the background of the topic of the work. In biological materials and tissues, such changes in the absorption properties as in the test profiles used are not to be expected.

For the evaluation of the experimental results within the scope of the thesis, the Tikhonov regulation method or, in the case of higher accuracy requirements, the EM NNLS method should be preferred.

The thermal transport parameters of the layers under consideration are also assumed to be constant in the model.

2.3 Solution of the heat conduction equation for periodically intensity-modulated illumination of the sample

Theoretical treatment of the problem, based on the optical properties of the sample material and the resulting knowledge of the absorbed energy, requires the solution of the equations for heat transport in the sample and in the adjacent gas IRosencwaigl.

Illumination should be perpendicular to the sample surface and homogeneous. It is assumed that the illuminated area is large enough to neglect radial losses. This means that the duration of the lighting should be short compared to the time it takes for the thermal wave to reach the limit of the light spot on the surface. This is the prerequisite for one given approximately one-dimensional treatment of the problem. The model outlined in Figure 2.3 is assumed.

A homogeneous, non-scattering sample according to Figure 2.3 is irradiated with intensity-modulated light with the modulation frequency ω. The wavelength-dependent absorption properties of the sample describes the spectral optical absorption coefficient ß (λ). The light intensity of einfal _¬ lumbar radiation is l (t, λ), wherein only the real part of the complex amplitude is physically relevant

I (t, λ) = I ₀ (W + "") < ^{2 1} >

I light intensity on the sample surface in Wm " ² lo modulation amplitude in Wm ^'2 ω modulation frequency in s" ¹ t time in s λ wavelength in nm

The decrease in light intensity for optically homogeneous, non-scattering samples perpendicular to the surface is described in the Lambert-Beersche law / Bergmann-Schaefert. The proportion of light reflected on the sample surface is taken into account by the reflection coefficient R (λ), which in the general case is also a function of the wavelength. The light intensity in the sample, depending on z, is given by (2.2). Appendix 3

Experimental part

The experimental work was carried out on a laboratory setup. Ei had the necessary degrees of freedom to be able to carry out modifications without major _effort . Two different configurations were used for the experiments. They differ with regard to the light source used and the acquisition of the measurement signals. An NdYag laser was used for the pulsed excitation and a xenon arc lamp for the periodically modulated excitation. The pressure oscillations excited by means of periodically modulated light were measured with a lock-in amplifier. A digital one was used for the experiments with laser excitation. Storage oscilloscope used The measuring cells were identical in both cases. Figure 4.1 shows a basic diagram of the measuring station used.

The change between the two measurement setups is done by adjusting the deflection mirror # 3 in the beam path. The second photoacoustic measuring cell shown in Figure 4.1 served as a reference for recording the intensity fluctuations of the excitation radiation. For this purpose, a soot-blackened glass pane was located in the measuring chamber of the reference cell.

4.1 Optical and mechanical components

4.1.1 Measuring station for periodically modulated excitation

The measuring station for the examination of samples using periodically modulated excitation radiation consisted of the components light source, monochromator, chopper, measuring cell with microphone, amplifier and lock-in amplifier.

4.1.1.1 Light source

An essential motivation of the work was the creation of a possibility for the non-destructive determination of the concentration depth distribution of chromophores in human skin. The light source used would therefore have to cover the wavelength range in which the skin has little or no absorption. This is the case in the range from near ultraviolet (UV, 320 - 400 nm) through visible light (400 - 700 nm) to parts from near infrared (NIR, 700 - 1400 nm). The visible range and the near UV are of particular interest, since the chromophores to be investigated have intense electronic transitions in this spectral range. In the NIR, only the combination and harmonics of the molecules, which are several orders of magnitude weaker, can be excited.

A 150 W xenon arc lamp was therefore chosen as the light source. This type of lamp emits an almost continuous spectrum in the spectral range of interest. The arc lamp was in a lamp house from ORIEL. The light from the plasma arc is reflected by an elliptical mirror and transforms a collimator into almost parallel light. For the experiments in the ultraviolet and visible spectral range, the lamp house was equipped with a water filter that absorbed the intense infrared components of the radiation. The light was focused on the entrance slit of the monochromator through a plano-convex lens.

4.1.1.2 Monochromator

The monochromator is used to split the polychromatic light of the xenon arc lamp into quasi-monochromatic light. The dispersive element of the monochromator is a diffraction grating. The spatial fanning out of the incident polychromatic light through the grating is described by the general grating equation (4.1) / Bergmann-Schaeferl. The angle ß is positive if the incident beam is on the same side relative to the grating normal as the incident beam, otherwise ß is negative.

mλ = g (sin a ± sin ß) ( ⁴ - ^J)

m diffraction order λ wavelength in nm g grating constant α angle of the incident beam relative to the grating standard ß angle of the outgoing beam relative to the grating standard

Incident monochromatic light is diffracted in different diffraction orders and in different spatial directions. By positioning a slit at a suitable position relative to the diffraction grating, sections of quasi-monochromatic light can be filtered out of the diffraction image of the grating. The spectral bandwidth and intensity of the transmitted light is determined by the width of the slit. The monochromator used in the experiment had variable slit widths between 0.05 and 3.16 mm at the entry and exit slits. The spectral bandwidth of the monochromatic light varies depending on the grating used between 0.5 and 20 nm. In the experiments, slit widths between 1.24 and 3.16 mm were used. A compromise between sufficient light intensity and spectral purity of the light could be achieved. The higher diffraction orders from 620 nm were eliminated with so-called edge filters. The division of the light intensity into the individual diffraction orders is independent of the wavelength in the ideal diffraction grating. Clear deviations from the ideal can be observed in the experiment. The efficiency of the grating was introduced to describe real diffraction gratings. It is a measure of the proportion of light that is diffracted into the first order that can be used in the monochromator. The efficiency is a relative quantity and refers to the total intensity of the incident light. Efficiency that is constant for all wavelengths cannot be achieved in reality. It can be influenced by the shape and depth of the furrows in the grid. This process is referred to in the English literature as "blazing". A grating with the blaze wavelength 300 nm works with the highest efficiency at 300 nm. The grids used in the experimental setup had blaze wavelengths of 250 and 500 nm.

1 Figure 4.2 shows the relative efficiency of the two diffraction gratings used as a function of the wavelength. The usable wavelength ranges that result from the mechanics of the monochromator can be seen as an insert in the diagram. The use of different gratings allows an efficient coverage of the spectral range of the near UV and visible light.

The model used is a universal laboratory monochromator from ORIEL (77250) of the Czerny-Tumer type. The monochromator was controlled by a commercially available IBM PC via an IEEE interface. The experiments were carried out using the step method, in which the desired wavelength positions were approached one after the other and a predefined measurement time was waited for in each case.

4.1.1.3 Chopper

The intensity of the continuous light from the xenon arc lamp was periodically modulated by a mechanical chopper from HMS. The chopper consists of a rotating slotted disc, the segments of which alternately interrupt and allow the light beam to pass. The modulation frequency f _m0d results from the number of slots and the _speed of rotation of the disk according to (4.2).

S number of slots

U Speed of rotation of the disk in s ^{~ 1}

A frequency range from 4 Hz to 4 kHz can be covered by using slices of different thicknesses.

The rotating slotted disc produces an almost rectangular intensity modulation of the excitation light. Deviations from the ideal rectangle occur during the entry and exit phase of the pane segments in the light beam.

In contrast to the theoretical approach from Section 2, the type of modulation is not a sinusoidal oscillation. It must be checked which modifications the already derived equations have to undergo in order to be applied to the specific experimental situation.

Figure 4.3 shows that the deviations from the rectangular course are small. In a first approximation it is therefore possible to represent the periodic modulation of the light by a rectangular function of the following form:

for O <t <T / 2 _{γ (t) SQ}

for T / 2 <t <T «Q _^

c constant (results from the maximum intensity of the light) T period of the oscillation in s

The square wave can be developed into an infinite Fourier series (4.3) IBronsteinl.

(4.3) _: ( _{t \} - ¹ i 2 ^» sin ((2ι» + l) ^W '~ 2 ir £ j 2 »+ l

The approximation of the rectangular function takes place through an infinite series of harmonic functions with increasing odd frequencies. The contributions of the higher GUeder to the sum fall like 1 / n. If you insert this approximation into the solution (2.25) of the heat conduction equation for the gas space from section 2.2, you get equation (4.4).

(4.4)

The solution is presented in the form of a sum. It consists of a constant component and an infinite number of summands, the structure of which corresponds to the solution to the heat conduction problem derived in Section 2.2. The direct component cannot be recorded with the measurement technology used and is therefore neglected in the following. The Lock-m measurement technique used filters the portion of a single frequency out of the sum signal described by equation (4.4). The frequency dependency of the temperature after periodically modulated excitation examined in section 2.2 can thus be transferred taking into account a pre-factor. The same applies to the dependence of the phase of the surface temperature on the absorption properties of the sample.

All derived results from section 2 can thus be applied to the experimentally obtained measured values.

4.2 Measuring station for pulsed excitation - NdYag laser

The excitation light source for measuring station 2 consists of a Q-switched laser. The active laser element is an approx. 1 cm long rod made of neodymium-ytrium aluminum garnet. The laser is optically pumped using xenon flash lamps. The pump power is 35 J. The pulse duration of the emitted light pulses in Q-switched operation is 8 - ns. In addition to the fundamental emission at 1064 nm, the frequency multiples at 532, 355 and 266 nm were available. The maximum pulse power was 300 mJ at 1064 nm.

The laser power was adjusted to the respective sample situation with the aid of a variable attenuator. Typical pulse powers were in the range of Ö.5 - 2 mJ on the sample. 4.3 Photoacoustic measuring cell

4.3.1 Calculation of the photoacoustic measurement signal

The results of the heat conduction equation with different forms of light excitation derived in Sections 2.2 and 2.3 form the basis for the consideration of the signal generation in the photoacoustic measuring cell.

Strictly speaking, the measuring cell is a closed volume in which the sample to be examined is located. The light enters the volume of the measuring cell through a boundary surface and hits it; put to the test. An optical window or the end of an optical fiber or a fiber bundle serves as the end of the measuring cell. In a further boundary surface there is a measuring instrument for determining the pressure changes due to the absorption of the light in the sample. The pressure is measured using a microphone or piezo element.

It is assumed that the window material used has no absorption in the spectral range under consideration, so that the calculation of the photoacoustic measurement signal can be carried out with the results from the preceding sections.

Under normal experimental conditions, i.e. room temperature and atmospheric air pressure, the gas in the measuring cell can be described by the ideal gas equation / Greiner /, it is one-dimensional case;

R ⁽⁴⁵⁾ p ⁽ z ^, 0 ^« jrTfcO-

R = 8.3143 Jt 'mor ¹ gas constant L length of the gas space in m

Due to the small temperature changes, the density of the gas is regarded as constant in the entire measuring volume. The measuring instrument detects the pressure spatially averaged over the volume of the measuring cell. By integrating equation (4.5) in the gas space within the limits -L and 0, the mathematical formulation results in the one-dimensional case:

Equation (4.6) describes the relationship between the pressure fluctuations in the measuring cell and the temperature profile in the gas and in the sample. The theoretical expression for the pressure corresponds to the one measured in the experiment except for an apparatus function. The explicit use of the temperature in the gas space as a solution to the heat conduction equation leads to a mathematical description for the photoacoustic measurement variable.

4.3.2 Pressure oscillations after periodically modulated excitation

The solution of the heat conduction equation is known from section 2.2. By inserting this solution in (4.6), the pressure in the photoacoustic measuring cell is given by equation (4.7). The influences of the measuring cell and the microphone including amplifier are taken into account by the factor A (ω), which is frequency-dependent in the general case.

(1 + g) Lk _P s _P

Exchanging the order of integration and executing the integration via z in the gas space leads to equation (4.8). ^{JP /} (i + _g ) Lk _pSp s _G

Except for a constant and the term A (ω) taking into account the cell properties, this is the surface temperature of the sample. All results of the theoretical sections are therefore basically applicable to the measured pressure oscillations.

From the prefactor of the integral, some properties can be derived that are decisive for the amplitude of the photoacoustic signal. Figure 4.4 illustrates the behavior of this factor depending on the modulation frequency of the excitation light.

The dependence of the photoacoustic measurement signal on the cell dimension L can be clearly seen.

• The larger the volume, ie the larger L, the smaller the amplitudes of the pressure oscillations.

• The larger the measurement volume, the more the low-frequency components of the pressure oscillations are amplified compared to the high-frequency components.

The specific shape of the photoacoustic measuring chamber changes the frequency dependence of the surface temperature, which was discussed in Section 2.2. The calculation of the heat density in the sample from the photoacoustic signal must therefore be carried out with a modified matrix approximation of the integral core K from 3.1.

4.3.3 Pressure change after pulsed excitation

To calculate the pressure change in the photoacoustic measuring cell after pulsed excitation, the solution of the heat conduction equation in the laplace room (2.42) is used. The Laplace transform of the pressure in the cell can be formally formed, since the integration can be carried out via the location variable z independently of the time t. _» O (4.9)

(p (ξ)) = _j jf _G (z ' ₉ ξ) dz ^<

ξ: Variable in the laplace room

Inserting the specific expression for the temperature dependence in the laplace space leads to equation (4.10).

By swapping the integration, equation (4.10) can be converted into an evaluable form. The integration via the variable z can be carried out and one obtains (4.11).

This expression describes the spatially averaged pressure changes in the laplace room. The corresponding time dependency of the pressure is obtained by reverse transformation. The influence of the pressure amplitude by the properties of the measuring cell and the microphone is taken into account analogously to the case of periodic modulation by the factor A (t). The link in the time domain is a convolution with the time-dependent pressure in the measuring cell.

(p (t)) = A (t) * L- ^l [(p (ξ))]

L ^"1 - inverse Laplace transform operator

Since the integration of the back transformation via ξ is independent of the integration via z ', the integrations can be interchanged.

(4.13)

The back transformation was carried out with the help of correspondence (4.14) and (4.15) ICarslaw andJaeger /.

The evaluation of the mathematical expression for the measurable pressure oscillations is more complicated compared to the results in the case of periodic modulation. The influences of the cell dimensions on the amplitude and the time course of the pressure in the measuring cell can also be shown here.

Two points should be emphasized when evaluating Figure 4.5: • As the volume of the measuring cell increases, the measurable pressure amplitude decreases.

• The gradient of the pressure change in the measuring cell decreases with increasing volume.

These conclusions have already been drawn in a similar form when excited with periodically modulated light. Theoretically, the design of the measuring cell therefore requires the smallest possible volume.

4.3.4 Structure of the measuring cell

The photoacoustic measuring cell is the crucial component of the experimental setup. A so-called open measuring cell of our own design was used. The term open refers to the state without a sample. The gas-tight chamber required for the measurement is formed when it is placed on the sample material. The counterpart to open photoacoustic cells are the closed cells. Here the sample is placed in a pre-vented measurement volume, which is closed during the measurement. These cells have the disadvantage that only smaller samples or fragments of extended samples can be examined. Only open measuring cells can be used for examinations on human skin.

A number of conditions must be taken into account when designing photoacoustic measuring cells:

1. The measuring chamber should be as small as possible in order to make the amplitude of the pressure fluctuations as large as possible. Experimental investigations by Aamodt and Murphy have shown that the minimum size must not be undercut. It is determined by the temperature conductivity of the gas in the measuring chamber and the target lower measuring frequency. The determining variable is the thermal diffusion length of the gas already known from section 2. For example, it is 1 mm for air and 100 Hz modulation frequency. If the distance between the sample surface and the window falls below this value, the dissipation of the thermal energy from the gas into the window material leads to a reduction in the measurable pressure amplitude.

2. The measuring cell should be constructed in such a way that disturbing sound cannot reach the microphone during a measurement. Realizing this requirement is the most difficult task for cell construction.

3. It must be avoided that the excitation light can fall on the microphone diaphragm, since this leads to coherent interference that cannot be eliminated.

4. The measuring cell should not have any pronounced resonances in the desired frequency range, so as not to complicate the evaluation of the measuring signals for the depth information of the optical absorption coefficient.

Not all requirements can be met to the same extent. A balanced compromise between all the listed requirements was found in the constructed measuring cell. Figure 4.6 shows the cell used in the Ouersehnϊ-t

The measuring chamber, the connection channel between it and the microphone and the gas volume in front of the microphone form a classic acoustic resonator. It is generally referred to as a Helmholtz resonator. For this reason, there are signal increases at specific frequencies that must be taken into account when evaluating data. The resonance frequency can be calculated according to the theory of the Hehnholtz resonator for circular cross-sections of the connecting channel between microphone volume and chamber volume according to equation (4.68) / Fereneliusl.

V «volume of the measuring chamber in m ³ V _M volume of the microphone in m ³ d diameter of the channel in ml _k length of the channel in mc speed of sound in the gas in ms ^{" 1}

In the present case there is a resonance frequency of 2.5 kHz. The following dimensions of the measuring cell were used as a basis.

V _M = 2.29 1Cr ⁸ m ³ d = 1.6 10- ³ m

V _κ = 2.55 10 ^"8 m ³ c = 300 m / s lκ = 1.8 10- ³ m

The experimentally determined resonance frequency is 2.73 kHz. The deviations from the theoretical value are due to the simplification of the model. The theory is based on a transmission channel with an ideally smooth surface, so that there is no turbulence in the channel. The channel inlet and outlet openings are not circular, but elliptical. The microphone volume used in the calculation is associated with a large error since it was extremely difficult to determine it without damaging the sensitive membrane.

Despite these limitations, the rough approximation of the Helmholtz resonator describes the cell's transmission behavior well. 4.4 microphone

The absorption of the excitation light creates a heat source that pulsates with the modulation frequency in the sample. The periodic transfer of heat from the sample surface into the adjacent gas space leads to a periodic change in pressure in the volume of the closed measuring cell. This pressure change is measured with a Brüel & Kjaer 4189 microphone.

The measuring microphone used works as a pressure microphone. It consists of a membrane, the front of which faces the sound or pressure source and the back of which is completely insulated against sound effects. The deflections of the membrane are therefore dependent on the pressure in the measuring chamber. For all frequencies whose sound wavelength is large compared to the dimensions of the microphone membrane, the microphone is hit uniformly from all sides by the same pressure / Günther /. Taking into account the dimensions of the photoacoustic measuring cell, which are in the millimeter range and the sound wavelengths, which, depending on the modulation frequency, are between several hundred meters and a few tens of centimeters, uniform pressure fluctuations in the cell can be assumed within the scope of the investigations.

The deflection of the microphone membrane is measured according to the capacitor principle. The membrane forms a capacitor with a counter electrode in the microphone capsule. The movement of the membrane as a result of external pressure changes leads to changes in capacity. From this, voltage profiles can be obtained that are proportional to the pressure profile over a wide range of amplitudes.

So that the microphone does not act as a barometer, a pressure compensation of the diaphragm back volume is made possible via a capillary tube. Adjustment of this pressure equalization by the manufacturer enables almost any lower limit frequency to be implemented. The deviations of the real microphone structure from the idealized pressure microphone lead to a special frequency-dependent transmission behavior, which is shown in Figure 4.7.

The transmission factor of the microphone runs almost smoothly in the range from 7 Hz to 3.S kHz. The 3dB drop in the transmission dimension is referred to as the lower limit frequency. It is 3 Hz for the microphone used. The upper limit frequency is 7 kHz.

4.5 Preamplifier and microphone amplifier

The microphone output signals are in the μV and sub-μV range. The low voltages have to be increased for a stable measurement. At the same time, signal conditioning is necessary to allow forwarding to the actual measuring device. Model 2669 from Brttel & Kjaer was used as a preamplifier. It is optimal on the microphones used matched To avoid electromagnetic interference, the preamplifier is connected directly to the microphone capsule. This model is basically a transimpedance converter with a linear frequency response between 5 Hz and 50 kHz. The supply voltage is provided by the microphone supply type 5935

The signal gain can be regulated in steps between 0 and 50 dB. The amplified measurement signals were taken from the DC-coupled amplifier output of the supply unit without further processing.

4.5.1 Lock-in amplifier

The measurement signal at the output of the microphone preamplifier is a complex frequency mix consisting of the photoacoustic sum signal and various interference frequencies. Since the measuring instrument is a microphone, every oscillation within the frequency band of the microphone leads to a signal at its output. However, these interference signals occur uncorrelated with the modulation frequency of the light. Therefore, they can be largely eliminated from the measurement signal.

The function of the amplifier is based on the multiplication of the frequency mixture by a reference frequency and subsequent low-pass filtering. The multiplication leads to an effective value rectification of the harmonic components with the reference frequency in the input signal. The subsequent low-pass filtering separates the remaining high-frequency components from the direct signal. The output signal is a measure of the harmonic component with the reference frequency in the input signal.

A lock-in from Stanford Research type SR 830 was used as the frequency-selective narrowband amplifier. This model is a two-phase lock-in, which determines the real and imaginary part of the signal at the same time. The bandwidth of the gain was less than 100 mHz with typical measurement settings. 4.5.2 Oscilloscope

The pressure amplitude after pulsed excitation was recorded using a LeCroy 9314A digital storage oscilloscope. The oscilloscope has a bandwidth of 400 MHz. The maximum possible sampling rate is lOOMs / s. Taking the sampling theorem into account, the oscilloscope allows the measurement of signals with a bandwidth up to 50 MHz. The maximum number of measuring points for a waveform is 50,000. The bandwidth of the signals to be examined is clearly below the measurement limits of the oscilloscope. The signal curves p (t) used for the evaluation were obtained by averaging 10 successive laser pulses. The measurement curves were saved on the internal floppy disk drive for further processing.

4.6 Signal processing

4.6.1 Frequency behavior of the measuring system

When describing the measuring station, it became clear that each of the individual components has a pronounced frequency-dependent transmission behavior. Some of the transmission characteristics, such as for the microphone, the preamplifier and the amplifier, are provided by the manufacturer. However, the transmission behavior of the measuring cell is unknown. The approximation by a Helmholtz resonator is only insufficient, as a simple comparison of the resonance frequencies from section 4.3.4 already showed. It is therefore necessary to determine the transmission behavior of the entire measuring cell, including the microphone and amplifier, since mutual interference cannot be ruled out.

The frequency-dependent transmission properties must be determined using a sample material whose optical and thermal properties are known. It should be easy to reproduce a sample from this material under laboratory conditions. The photoacoustic measurement works on a remission principle. Therefore, measured values can only be compared under precisely specified conditions. The general comparability of photoacoustically obtained measurement data requires the data to be standardized to a reference. The reference material should convert all of the incident optical energy into heat. Soot fulfills all of the above requirements in an almost ideal manner. A soot-blackened glass plate is an almost ideal absorber that absorbs all of the incident radiation and converts it to heat. This significantly simplifies the descriptive equations. The spatial distribution of the absorption coefficient in the soot layer can be determined by

ß (z) = ß = const

be approximated.

This simplifies the equations (4.8) for the pressure in the measuring cell. Equation (4.17) results for excitation with periodically intensity-modulated light.

(4-17)

The amplitude and phase of the photoacoustic pressure fluctuations thus have the following form: _{A,. Λ} . CßA (ω) l-2cos (r) e- ^r + e- ^2r

A phtude: p _Amplitude ( ^β >) = y - „2 2, * -: ^~ '< ⁴ - ¹⁸ > ω ² V ß μ _P + 2ßμ _P +2

₊ G # p + y (cos (r) - e -sin (r) phase: »to (^) = arctan (^ - '.), (4.19 ₎ cos (r) - e + (?« + L) sιn (r)

The high absorption of the soot layer allows a further approximation. If ß »1 applies, the denominator of (4.18) can be replaced by ßμ _P. The frequency dependence thus changes from ω ^"3ß to ω ^" \

Figures 4.8 and 4.9 compare the theoretical and experimentally determined frequency dependence of the amplitude and phase of the photoacoustic signal. A soot-blackened glass plate was used as a sample, which also served as a reference for further investigations.

The theoretically described ω ^"1 dependency of the measurement signal is confirmed in a wide frequency interval. Deviations from the linear course in the high and low frequency range are clearly recognizable. The acoustic resonances of the measuring head are responsible for the non-linearity between 1.5 kHz and 4 kHz. The differences in the low-frequency range are based on the one hand on the frequency response of the microphone and can partly be explained by simplifications in the descriptive model, which neglects the influence of the optical window on the pressure oscillations. Works by Korpiun and Büchner, as well as Aamodt and Murphy have shown that taking these effects into account, the deviations at low frequencies can be described in part. However, the complexity of the solutions increases significantly due to the introduction of another differential equation. The measurements within the scope of the work are carried out in the frequency range between 10 Hz and 3 kHz, in which the corrections are minor. However, the mathematical algorithms from Section 3 can also be applied to more complex models without restriction. In this case, a modification of the core matrix K is necessary.

For better comparability, the theoretical phase dependency was additively corrected. The comparison of the phases in theory and experiment shows clear differences. Despite the visible large deviations, these can be explained by arguments analogous to the amplitude comparison. The strong change in the phase above 2 kHz is clearly due to the resonance geometry of the measuring head causes If this influence is eliminated, there is a phase difference that decreases linearly with the frequency and must be attributed to the microphone used.

The resonance peaks that can be clearly seen in Figures 4.8 and 4.9 due to the acoustic resonator geometry of the measuring head are of minor importance for measurements using periodic modulation, since the measuring range does not exceed the 3 kHz limit for technical reasons. The frequency spectrum of the pressure in the measuring cell pulsed excitation, on the other hand, is decisively influenced by the resonances, especially when the sample material has a high optical absorption, since the pressure change in the measuring cell has a large high-frequency component.

When excited with pulsed light, the mathematical description of the photoacoustic measurement signal is simplified if a constant optical absorption coefficient is assumed. The integrations in equation (4.13) can be carried out and we get (4.20).

0 * ») ^«

The RO transform from the laplace room provides the exact expression for the pressure in the photoacoustic measuring cell.

C4.21)

The evaluation of equation (4.21) in this form is difficult. However, some approximations can be introduced for the measurement of soot in the experimentally relevant time range from 0 - 0.1 sec.

erfc (ßjDpl) & 0

eφi- _j J - + ßJÖ s 0

The expression for the pressure in the measuring cell is thus simplified, neglecting the time-independent threshing floor (4.22).

The pressure in the measuring cell behaves similarly to the surface temperature of the sample in the case of a constant optical absorption coefficient. This connection has already been pointed out in the results of the periodic modulation. Assuming an absorption coefficient of 10 ⁶ m ^'1 , the pressure curve shown in Figure 4.10 results. The actually measured pressure curve with a soot sample is shown for comparison.

The resonances in the signal are clearly visible. With pulsed excitation they are not negligible, but must be eliminated mathematically. The comparison of the two signal curves makes it clear that the knowledge of the apparatus function K (t) is of great importance. It can be obtained by unfolding the two curves shown.

4.6.2 Processing the measurement signals

The previous sections dealt with the frequency-dependent transmission properties of the individual components of the photoacoustic measuring stations. Taking these properties into account is of great importance for the evaluation of the measurement data. A precise calculation of the spatial distribution of the absorption coefficient in the sample is only possible after correcting the system-related different weighting of the frequency components in the measurement signal. Apparatus function A was introduced in section 4.6.1. It describes the frequency-dependent transmission behavior of the entire measuring system. A (ω) can be obtained from the time or frequency dependence of the pressure when a soot sample is excited. The measurement signal corresponds in the time domain to a folding of the apparatus function with the theoretical photoacoustic measurement value. In the frequency domain, this relationship corresponds to a simple multiplication. The measured values must therefore be unfolded using the apparatus function A. Different algorithms are to be used for the individual measuring regimes.

Periodic modulation

The measurements are carried out in the frequency domain. At the beginning of the experiments, a so-called reference frequency spectrum is recorded. This reference contains the amplitudes and phase information of the transmission behavior of the entire detection system and is used to correct the measured values.

The corrected photoacoustic measured value results from a simple point-by-point division of the measurement signal p (ωj) / A (<Dj). The phase of the measurement signal is also corrected additively point by point.

Pulsed excitation

The measurement takes place in the time domain. The device function A (t) and the measurement signal p (t) must therefore be developed. Before starting the measurement, a reference spectrum is also recorded here. It is to be recorded with the sampling rates and signal lengths used in the experiment. The further procedure consists of three steps:

1. Transformation of the reference and measurement signal in the frequency domain

R (t) → R (ω) p (t) → p (ω) '

2. Calculate the apparatus function A (ω) by dividing the reference by 1 / Vö>. Formation of the corrected measurement signal by division with the apparatus function A (ω), 3. Reverse transformation of the corrected measurement signal p (ώ) → p (t).

This theoretically correct treatment of the data leads to considerable problems in practice. The reasons for this lie in the numerical implementation of the Fourier transformation, in particular the back transformation. The strict application of the steps outlined above, using the MATLAB ^® routine FFT, leads to strong oscillations in the corrected measurement signal, which make it unusable for further evaluation.

Correcting the measurement data in the frequency domain is equivalent to filtering. The problem is to model the corresponding filter transfer function. The numerical program package MATLAB ^® provides a series of stable filter routines that are suitable for this task. For practical reasons, the task was broken down into two parts. In the first step, the transmission function in the frequency range greater than 200 Hz was formed by parametric modeling as a filter with an infinite impulse response (HR filter) shown. The combination of the two filters approximately describes the transmission behavior of the measuring system.

The measurement data were processed one after the other with the two filters described. Finally, low-pass filtering was carried out to eliminate the high-frequency signal components resulting from the numerical processing. The correction of the measurement signals in this way proved to be far more stable compared to the direct processing in the Fourier space. The procedure described dampens the resonances that are clearly recognizable in the measurement signal. A complete elimination was not possible. However, the reduction in the resonance amplitudes in the signal was sufficient to allow further processing of the measured values using the methods presented in section 3. Figure 4.11 compares the frequency spectra of a photoacoustic measurement signal before and after the numerical correction. The resonance peak at 3 kHz has become significantly smaller after filtering. A complete elimination was not possible. The low-frequency course of the spectrum remains unchanged.

The quality of the correction is sufficient, since a maximum of approximately 150 measuring points can be used for the final calculation using the regulation process. This in turn means a sampling rate of the signal of approx. 0.2 ms - 1 ms. The influence of the measurement signal by resonances in this frequency range is significantly reduced anyway.

The comparison of the transmission functions of the microphone and the approximated 2nd type Cheybyscheff filter shows the quality of the approximation in the frequency range between 1 Hz and 50 Hz. Figure 4.12 compares the two transmission characteristics.

The course of the microphone characteristic could be approximated well by the filter used over a large frequency range. Deviations exist between 6 Hz and 40 Hz. The largest deviation of around 1 dB occurs at 10 Hz. The difference in the survey characteristic curves leads to a damping of the frequency components in the mentioned range by up to 1 dB compared to the actual value when processing the measured values.

The combination of filtering in the high-frequency range to reduce the resonances in the measurement signal with filtering in the low-frequency range to correct the high-pass effect of the measuring system has proven to be an efficient and stable method with which all measurement data could be processed without serious falsifications. 4.7 Sources of error

The temperature changes caused by the absorption of light in the sample material cause pressure oscillations in the photoacoustic measuring cell, which are measured with a microphone. In principle, the microphone responds to any type of pressure change, regardless of its origin. For this reason, dealing with possible sources of error and artifacts in the measurement data is necessary. The main causes of disturbances in the photoacoustic measurement signal are audible sound, relative movements between the sample and the measuring head, structure-borne sound or movements of the measuring head with the sample. In most cases, the amplitude of the interference exceeds that of the photoacoustic pressure oscillations. Precautions must therefore be taken to minimize the interference with the measurement signal. The measures can be both constructive in nature and within the scope of the measured value acquisition.

Reduction of ambient noise

The simplest and most obvious way to reduce external influences on the measurement signal is to reduce laboratory noise. As a rule, this requirement can be met well by the experimenter. The low-frequency components of interference noise, which can no longer be perceived by the human ear, are a problem. These vibrations with frequencies lower than 1 Hz are very difficult to dampen by constructive measures. Their cause mostly lies in construction errors in the construction of the building. As in the author's laboratory, they can assume considerable amplitudes.

The influence of these vibrations can be reduced by an appropriately dimensioned vibration-damped structure. By choosing a microphone with a lower cut-off frequency above the disturbing frequency range, the influence on the measurement signal can be further reduced.

Both measures were taken into account in the work. The setup for measurements on samples was completely on a vibration subdued optical table. During the in-vivo measurements, the measuring head was fixed with an elastic holder. The cutoff frequency of the microphone used is 3.5 Hz.

Reduction of relative movements

This point is of great importance for in-vivo investigations on living objects. Naturally, the body parts accessible to the experiment are constantly in motion, so that a controlled measurement over a longer period of time is difficult. One solution is the relative fixation of the measuring head on the skin by means of a holder. The disadvantage is the limited possibility of examination, since only the arm or leg are accessible. Due to the weight of the measuring head, relative movements could not be completely suppressed even with this solution. A reversal of the principle, the attachment of the body part to the measuring

The resilient suspension of the measuring head reduces the transmission of vibrations from the arrangement and at the same time ensures a constant Contact pressure. By placing the arm on two supports, movements are largely avoided.

structure-borne sound

Structure-borne noise can have a variety of causes and can hardly be eliminated by constructive measures. The use of a vibration-damped measuring arrangement already reduces the influence considerably. However, this solution is only suitable for measurements on dead samples. Studies on living objects are influenced by disorders that are caused by the metabolism of the organism and cannot be influenced. Blood flow in the vessels and muscle contractions cause disturbances in the frequency range below 20 Hz.

Figure 4.14 compares the measurement disturbances due to the activity of the organism with the disturbances caused by external sound sources. The limit of 20 Hz represents a limit for the photoacoustic experiments by means of periodic modulation of the light, below which the electronic data processing can no longer guarantee adequate attenuation.

When using pulsed excitation light, these disturbances can be largely compensated for by averaging a number of measurement cycles.

Movement of the overall system

Movements of the measuring head and sample complex represent a problem only for measurements on living objects. As already described, a partial solution consists in fixing the measuring head relative to the sample. By shortening the measurement time, the probability of movement within the measurement can be further reduced.

Comparison of periodic modulation - pulsed excitation

In the previous sections, the various sources of error in photoacoustic measurement due to external influences were described and discussed. The investigations on the living object are particularly critical. The method of periodic modulation is suitable for all dead samples without restrictions, for which the long measurement duration has no influence on the signal-to-noise ratio. Due to the lower cut-off frequency of approx. 20 Hz specified by the body's own disturbances, this measurement method has a maximum depth 20 μm (temperature conductivity 2.8 10 ^"8 m ² / s). This means that the method remains of minor importance for most medical questions. The method can advantageously be used for the spectral examination of skin areas if chopper frequencies in the range from a few 100 Hz to 2 kHz can be used, which corresponds to a measuring window of approx. 2 - 6 μm depth.

The pulsed excitation can utilize the entire frequency range of the microphone through adequate data acquisition and thus achieve a measurement window of approx. 4 μm to 200 μm. The limitation in depth depends on the upper limit frequency of the microphone and the transmission properties of the measuring head. It can easily be reduced to 1 to 2 μm using a measuring head geometry with other resonance properties,

The pulsed excitation is the more suitable measuring method from the point of view of the disturbance for in-vivo investigations. Due to the lack of Periodically modulated lasers still have to be used for spectral questions.

nec ₁₁₄

Evaluation of the experiments

The experiments should clarify the agreement of the theory presented in sections 2 to 4 with the practical measurement results using a few selected examples. First, samples of non-living material are examined in order to test the general statements of the theory. Subsequently, some measurements on gelatin and pig skin are presented, the thermophysical properties of which come close to the living object. The conclusion would be experiments on living biological tissue. This step-by-step approach was chosen because the experimental difficulties of measuring the living object can overwhelm the degree of over-exposure with theory.

5.1 Colored glass filter with a transparent top layer

The first investigations were carried out on an artificially constructed layer structure made of glasses with different absorbances. An optical colored glass filter of the type RG 610 was covered with a different number of microscope cover glasses. The resulting spatial profile of the optical absorption coefficient differs only slightly from the step profile used in section 3. The position of the step in the spatial absorption profile of the layer structure can be controlled by the number of cover glasses used. The optical and thermal properties of the optical filter material are known. The layered structure of the sample makes it easy to create and reproduce different spatial dependencies of the optical absorption coefficient. A drop of water between the individual glass layers served to improve the thermal contact.

The layer structure of the samples results in deviations from the underlying model of the thermally homogeneous body. The interfaces between the glass layers act as thermal barriers that lead to a change of the heat flow in the structure. Strictly speaking, the use of different types of glass results in a so-called phototheim problem of the 3rd kind / power /.

The structure was selected despite these restrictions, since the layering with water as an adhesion promoter can effectively reduce the thermal barriers and, on the other hand, the temperature conductivity of the two types of glass used differed only slightly. The structure is therefore suitable for the basic investigation of the algorithms and the measurement setup. However, the special features of the system must be taken into account when evaluating and interpreting the results. The physical properties of the filter and the microscope cover glasses are summarized in the following table.

The open photoacoustic measuring cell was sealed on the glass with vacuum grease.

5.1.1 Pulsed excitation

Five different layer systems were examined. The number of coverslips varied between one and five. Figure 5.1 shows the recorded measurement curves after pulsed excitation with a frequency-doubled NdYag laser at 532 nm.

The maximum of the pressure shifts significantly depending on the number of cover glasses used. The amplitude decreases with increasing thickness of the cover glass layer. The cause of the time shift lies in the finite running time of the thermal waves in the sample material excited by the laser pulse. The thicker the cover glass layer to be traversed, the more time the thermal waves generated in the colored glass filter need to reach the surface. The damping of the thermal waves in the crossed cover layer leads to the observed reduction in the pressure amplitude.

The increase in the photoacoustic measurement signal which can be seen immediately after the end of the laser pulse at time t = 0 is caused by the absorption of the laser light in the cover glasses. It is almost independent of the number of coverslips used. This behavior becomes particularly clear on curves 3, 4 and 5. In curves 1 and 2, the absorption in the optical colored glass filter dominates over the proportion of cover glasses. A weakness of the measuring system, which was already discussed in Section 4, can be illustrated using the results of the step absorption profile as an example. The frequency components of the photoacoustic pressure wave are weighted differently by the frequency-dependent transmission behavior of the measurement setup. This results in deviations of the experimental data from the theoretical values. The lower limit frequency of the system is approx. 10 Hz. Therefore, especially low-frequency pressure changes in the measuring cell can no longer be resolved. In the specific case, this leads to the observed early drop in pressure. The filtering of the measurement data in the low-frequency range described in section 4.6 serves to correct this behavior.

After filtering the measurement data, the course of the optical absorption coefficient in the sample was calculated using EM NNLS. The results are summarized in Figure 5.2

The result of the calculations for the absorption profile with a cover glass corresponds to the layer structure used in the spatial position of the step at approx. 140 μm. The error of the solution increases with the number of coverslips used. The cause of this behavior was discussed at the beginning of this section. It lies essentially in the underlying model of heat transport. The structure of the sample deviates from the ideal of the thermally homogeneous body. The resulting interfaces hinder the flow of heat from the deeper layers of the sample. There are reflections. It follows that the error in the calculation of the optical absorption coefficient increases with the number of interfaces.

In addition to this peculiarity, there is a continuous decrease in the optical absorption coefficient with an increasing number of cover glasses. The cause lies in the low-frequency transmission behavior of the measuring system. The correction of the measurement data by the filtering described is only approximately correct. The remaining attenuation of the frequencies less than 10 Hz in the measurement signal leads to an underweighting of the corresponding signal amplitudes in the regulation process. The slow and affected pressure changes that occur only after the laser pulse has ended are caused by thermal waves that originate in deeper layers have the sample. The solution will therefore represent the absorption coefficient in these regions of the sample in a reduced form.

In spite of these limitations, the method discussed succeeds in calculating the location-dependent profile of the optical absorption coefficient, at least as a tendency, correctly. The absorption of the cover glasses, which is 10 ³ times lower than that of the colored glass filter, cannot be resolved. An absorption coefficient falling from the surface was only calculated for samples 3 - 5, for which the contribution of the cover layer was already clearly visible in the measurement data. The necessary regulation process leads to a loss of information. It suppresses details with small amplitudes in the position dependence of the absorption coefficient. The experiment clearly shows the possibilities, but also the problems of practical calculation of the optical properties from the measured photoacoustic data. Concrete results can only be achieved with samples that approximate the model of the thermally homogeneous body. In addition, the temperature conductivity of the sample material must be known. For the measurement setup used, the proportion of frequencies in the measurement signal should be as low as possible below 10 Hz in order to minimize the errors in the solution due to the processing of the measurement signals.

5.1.2 Periodically modulated excitation

The investigations using periodically modulated excitation radiation were carried out with the layer structures already used in 5.1.1. The location-dependent course of the optical absorption coefficient is to be calculated from the frequency dependence of the amplitude. The experiments were carried out with the measuring station described in section 4. The chopper frequency covered the range between 3 Hz and 50 Hz. The chopper was equipped with a 2-slot disc. This was the only way to achieve the necessary frequency stability. The step size varied between 1 and 4 Hz depending on the frequency range. The measurement duration per frequency value was 20 s. The measurement data were standardized on soot as a reference. The investigations as well as the laser experiments took place at 532 nm.

Figure 5.3 shows the results as the frequency-dependent amplitude and phase of the pressure oscillations in the measuring cell. Samples with more than two cover glasses could not be examined because the signal / noise ratio of these measurements was only insufficient. The chopper was unstable in the range of 1-2 Hz required for these measurements. The associated interference cannot be suppressed by the Lόck-In amplifier.

The amplitude and phase are almost congruent for both samples in the frequency range greater than 30 Hz. At lower frequencies there is a local minimum of the amphtude. Depending on the number of cover glasses, it is at approx. 25 Hz or 10 Hz. This is followed by a monotonous increase in the amplitude as the frequency continues to decrease to the maximum value at the lower limit frequency of the experiment. The phase shows a jump near the local minimum of the amplitude.

The behavior observed is largely determined by the frequency-dependent thermal diffusion length μ (Eq. 2.27). It controls the active layer thickness in the sample, from which thermal waves can contribute to the measurement signal. Increasing the chopper frequency reduces the active layer. Above 30 Hz, only thermal waves that are within the am Measuring head adjacent cover glass are generated to the measurement signal. The cover slip absorbs uniformly over the entire material thickness. The photoacoustic pressure therefore does not change with a further increase in frequency. For the same reason, the pressure amplitude and phase for the sample with two cover glasses change only slightly up to a chopper frequency of approx. 10 Hz. The thermal diffusion length is in the frequency range between 10 Hz and 30 Hz within the second cover slip.

The local minimum of the amplitude and the abrupt drop in phase at approx. 25 Hz and 10 Hz is an indication of the passage of the thermal diffusion length through a thermal barrier. On the side facing the color filter, it leads to an increase in the heat density due to partial reflection of the thermal wave. Conversely, the heat density is reduced compared to the theoretical approach on the surface near the surface. The suspicion of thermal inhomogeneities was already expressed when evaluating the results for the pulsed excitation. The results of the periodic modulation confirm the assumption.

As described in the previous sections, the spatial profile of the optical absorption coefficient was calculated from the measurement data. The results are shown in Figure 5.4.

The results largely agree with the actual location dependence of the optical absorption coefficient. The deviations are due to the thermal inhomogeneities already discussed, due to the layer structure. The level in the profile of the absorption coefficient appears shifted into the sample by approx. 10 or 30 μm. The relative error is therefore in the range of 10%.

Compared to the results of the pulsed excitation, the calculated absorption profiles are shifted into the sample in the positive z direction. The overall error is less. This is primarily due to the correction of the measurement data, which works more efficiently in the frequency domain.

The experiments clearly show which frequency range must be used for measurements using periodically modulated excitation radiation in order to be able to achieve depth resolution in the range of 100 μm. The lower limit frequency required for human skin is approx. 1 Hz. This frequency range cannot be reached with the existing measurement setup. The measuring system is therefore not suitable for determining the absorption of human skin up to a maximum depth of 100 μm. The photoacoustic examination with periodically modulated excitation light is therefore only useful for depths less than 100 μm. The duration of the measurement per frequency provides a further argument against the use for in-vivo investigations. Due to the necessary settling time of the lock-in Amplifiers require approx. 10 - 15 s per frequency value. The duration of the entire measurement is thus 5 minutes even with low accuracy requirements with regard to the frequency range and the frequency step size. With such a long measurement duration, disturbances that can be traced back to the movement of the living object can no longer be compensated for. For this reason, the following experiments have only been carried out with the pulsed light of a Q-switched laser.

5.2 Gelatin layers

The thermal properties of gelatin differ little from those of biological tissues. Layers of colorless and colored gelatin correspond to the underlying model of the thermally homogeneous body. The layer structure makes it possible to reproducibly set an initial value β (z), the spatial change of which can be followed using the photoacoustic method due to diffusion processes.

For the preparation, commercially available gelatin was dissolved in distilled water. Part of the solution was stained with gentian violet. Gentian violet is used as the standard dye in microbiology. By staining a bacterial culture with the dye, it is possible to differentiate between gram-positive and gram-negative organisms. By pouring the gelatin into Petri dishes, colorless and colored layers of gelatin were created. After the gelling process had ended, the layer thickness of the colorless layer was approximately 0.25 mm and that of the colored layer was 3 mm. The structures outlined in Figure 5.5 were created by superimposing the semi-solid layers.

The gelatin mainly consists of water. Therefore, the temperature conductivity of water can be assumed approximately. At the beginning of the investigations, a control experiment was carried out, which confirmed the correctness of this assumption. An arrangement for measuring the inverse pyroelectric effect IChirtocl, in which the damping of a heat wave in the material is evaluated, was used for this purpose; During the gelling process over a period of one day, there were no significant changes in the temperature conductivity.

The diffusion of the chromophores from the colored into the colorless gelatin layer begins immediately after the preparation of the layer structure. The aim of the experiment is to track the diffusion process by measuring the photoacoustic pressure response at different times. The spatial delimitation of the colorless from the colored gelatin can no longer be resolved due to the thickness of the colorless layer. The duration of the thermal waves in the colorless gelatin is more than 0.15 s. The resulting pressure amplitudes are no longer detectable due to the frequency-dependent transmission behavior of the measuring system. Rather, the experiments should show how the chromophores in the cover layer diffuse and the spatial profile of the optical absorption coefficient changes over time.

Structure A

Figure 5.6 shows the photoacoustic pressure amplitudes after excitation of structure A with a laser pulse at 532 nm. The entire amplitude increases continuously from the start of the experiment. However, the rise is not uniform. The relative changes decrease over time. Due to the difference in concentration between the colorless and the colored layer, the color molecules of the gentian violet diffuse into the top layer. This increases the concentration of the dye in the top layer, which in turn leads to an increase in the absorbed energy and thus to an increase in the pressure amplitude. The diffusion kinetics follows exponential laws. The relative changes in the chromophore concentration in the top layer are therefore greater at the beginning of the experiment than at the end.

The maximum of the measurement curves from Figure 5.6 moves away more and more from the original of the time axis in the course of the experiment. Figure 5.7 highlights this behavior again. The time difference between the end of the laser pulse and the photoacoustic pressure maximum is shown as a function of the time of the measurement. The end of the sample preparation is assumed to be the starting point t = 0.

The behavior shown is caused by the previously described diffusion process of the color molecules into the top layer.At time t = 0, which could not be resolved in this experiment, the measured photoacoustic signal is only created by the self-absorption of the colorless layer.Diffusion increases the number of chromophores the top layer over time. The spatial course of the optical absorption coefficient changes from a constant value at the beginning of the experiment to a course that increases exponentially to the boundary layer. This changes the energy distribution after absorption of the light in the layer. The amount of energy absorbed in the deeper layers increases, which leads to the observed change in the pressure curve with the duration of the experiment. The measurement curves have not been corrected with regard to the resonant transmission behavior of the measuring cell. The rise time of the pressure was too short to stimulate the acoustic resonances of the measuring cell. The location-dependent heat density in the sample was calculated using the EM NNLS method. The results obtained are shown in Figure 5.8

The spatial dependence of the optical absorber coefficient from wuuuug. . «Is to be equated to the concentration profile of the gentian violet molecules in the top layer. The dye has already passed through a large part of the top layer after only 5 minutes. At this point, the boundary between the colorless and colored section runs at approx. 150 μm. Over time it shifts towards the surface. This is accompanied by an increase in the total concentration in the layer. After 18 minutes there is obviously no area in the previously colorless top layer which is not permeated with dye molecules. This result coincides with Observations during sample preparation experiments. The top layer colored there visibly within a few minutes.

Figure 5.S shows the absorption coefficient only at a depth of 50 μm. Due to the low absorption of light in the colorless top layer, a constant optical absorption coefficient should be close to zero in the layers near the surface. As a result of the calculation, however, a sharp increase can be observed starting from the first point up to a maximum at 20 μm with a subsequent sharp decline (Figure 5.9). The result is independent of the type of regulatory procedure used.

The incomplete correction of the measurement data in the low-frequency range can provide a possible explanation for the behavior. It leads to mispricing decisive frequency components in the measurement signal. A frequency range between 10 Hz and 60 Hz, in which the inaccuracies lie, can be estimated from the solution shown in Figure 5.9. These frequencies result from the consideration of the thermal conductivity of the gelatin and the thermal transit time τ = x ² / 4D. A comparison of the transmission characteristics of the system with the filter from Section 4.6 used for data processing showed the incomplete correction of the measurement data in this frequency range. Another possible explanation is the removal of material from the surface of the structure, which can be both adsorbed molecules and gelatin. Such an ablation leads to an increase in pressure in the measuring cell. It occurs during the duration of the illumination of the laser pulse and then quickly subsides. When evaluating the pressure changes in the measuring cell, this error leads to an overestimation of the absorption of the layers near the surface.

Basically, the found spatial course of the optical absorption coefficient confirms the processes in the sample that were already assumed during the discussion of the pressure signals in Figure 5.6. The increasing concentration of the chromophores in the originally colorless layer leads to an increase in the heat density and a clear shift in the boundary between colorless and colored gelatin to layers near the surface in the sample.

Structure B

Structure B differs from structure A only in the thickness of the colorless gelatin top layer. The preparation took place immediately before the first measurement. The starting point of the diffusion could thus be included. Figure 5.10 shows the measured pressure response depending on the duration of the experiment. The curve shape changes significantly in the course of the experiment. The photoacoustic pressure signal exhibits an exponential drop at the beginning of the experiment. The curve shape indicates an almost uniform distribution of the absorbed optical energy in the cover layer shortly after the sample preparation. At this stage of the experiment, the concentration of the dye in the top layer is low, so that the course of the curve is essentially due to the self-absorption of the gelatin is determined. Over time, the concentration of the dye in the top layer increases due to diffusion. This increases the optical energy absorbed in this layer. The photoacoustic signal changes in a manner similar to that already determined for structure A.

A direct comparison of the photoacoustic pressure changes of the examined gelatin samples from Figures 5.10 and 5.6 shows that the curve shape hardly changes after the first 15 to 30 minutes. In the range between 0 and 0.03 s, apart from the amplitude of the pressure, almost no differences can be determined after 30 minutes. The amplitudes differ almost by a factor of 2. The alignment of the curves indicates a very fast diffusion. This behavior coincides with the experimental observations, which showed a clear discoloration of the top layer after a very short time. The measurement data were treated analogously to the procedure for evaluating the results of structure A. The results of the subsequent EM NNLS regulation procedure are shown in Figure 5.11.

The calculated spatial dependencies of the optical absorption coefficient for structure A and B are very similar. However, the value of the absorption coefficient is only half that. Due to the double-thick top layer, the chromophores are distributed in twice the volume, which must lead to a reduction in the dye concentration in the top layer and thus to a lower amplitude. The boundary between the colorless and the colored area in the top layer hardly differs from that in structure A. The spatial shift of this boundary occurs significantly faster in structure B, since the difference in concentration of the chromophores in the colorless and colored gelatin layer is greater as the driving diffusion force. The experiments with the gelatin layers have shown that the mathematical methods are able to describe the diffusion behavior of absorbent substances in a non-absorbent or poorly absorbent matrix. The solutions should be interpreted with caution, as inadequacies in the data evaluation can have a serious impact on the result. Despite these restrictions, information can be derived from the photoacoustic measurements that cannot be obtained in any other way without destroying the sample.

5.3 In-vitro tests on pig skin

In the following sections, studies on isolated pig skin are presented. The skin came from the inside of the thigh. The removal took place immediately after the animal was killed. The isolated bristles were removed. The skin sample obtained in this way showed a great visual and tactile similarity to that of humans. The measurements took place about 3 hours after the animal's death.

5.3.1 Sunscreen

Sun creams are ideal for testing with photothermal measurement methods. They have the task of absorbing the short-wave UV radiation, which can lead to damage in the organism, and converting it into a form of energy that is harmless to the organism. The class of chemical absorbers under the sunscreen absorbs the corresponding radiation and converts the energy into heat. The fluorescence of the compounds is negligible. The energy absorbed by the substances can be determined by the photothermal measuring methods. There have therefore been numerous attempts in the past to use these measurement methods to characterize the behavior of sunscreen substances on and in the skin. Llmhoff, Kölmel, Foulet /. Sun protection substances are also an interesting object of investigation for another reason. The behavior of the substances in the skin has been well examined by the manufacturers due to various state regulations. It is known in principle in which skin layers the substances are located. They are found exclusively in the stratum corneum, the horny layer, which can be between 2 μm and 20 μm thick, depending on the part of the body. The penetration of the boundary between the horny layer and the living skin underneath does not take place. The experimental situation therefore corresponds to the underlying model of a thermally homogeneous body.

The investigations took place with a light wavelength of 355 nm. With this light energy, the absorption of the sun protection substances acting as UV-A filters is almost at the maximum. The self-absorption in the upper layers of the skin is lower compared to the sunscreen. The penetration depth of the radiation is approx. 50 μm IBruls Weil /. This provides all the necessary experimental requirements for tracking the diffusion processes using the photoacoustic method.

In the following experiment, 3 mg of the product SC8 Suncare from St Etienne Cosmetics, Herbsleben were distributed as uniformly as possible by means of an impression on a 2 x 2 cm skin area. The measurements were carried out before application of the substance, immediately after application, after 5, 15 and 45 minutes. The results are summarized in Fig. 5.12. The measurement data shown have been corrected with regard to acoustic cell resonance.

The absorption of the excitation light by the sunscreen leads to a significant increase in the amplitude of the measurement signal. The time course changes little compared to the situation without cream. The differences in the two curves in the upper third of Figure 5.12 are clearly due to the cream. The substance is immediately on the surface of the skin as a film. Most of the heat generated therefore only comes from the layers directly on the surface of the skin.

The diffusion of the sunscreen substances into the horny layer and their simultaneous decomposition subsequently lead to a reduction in the photoacoustic signal amplitude. The increasing concentration of substances in the deeper layers of the cornea leads to the changes over time that are visible in the middle and lower diagram in Figure 5.12. The measured values shown were corrected with regard to acoustic cell resonance. The local dependence of the optical absorption coefficient was calculated using the Tikhonov regulation method. The negative values of the solution were set to zero. The results of the calculation are shown in Figure 5.13

The solutions of the regulation procedure clearly prove that the behavior of the sunscreen in the skin, which was already described in the discussion of the measurement data. The substances are all still in the horny layer, the extent of which was approx. 10 μm at the point of measurement. After an initial concentration of the cream on the surface of the skin, the substance quickly penetrates into the horny layer. An average depth of approx. 8 μm was reached after only 5 minutes. In the further course of the experiment, the concentration of the active ingredient shifts only slowly into the skin , With 136

the diffusion into deeper layers leads to a reduction in the concentration in the layers near the surface. The changes in concentration at the sample surface and in depth that can be derived from the results in Figure 5.13 are not identical. The substances are diffused in the horny layer without a preferred spatial direction. The measurement only cuts one disk out of the diffusion volume in the form of a hemisphere. Despite this fundamental limitation, valuable information about the spread of the substance in the skin can be obtained. By examining the surface point by point, quasi-three-dimensional diffusion profiles can also be calculated.

5.3.2 Gentian violet

In a further series of experiments, diffusion experiments were carried out with gentian violet, a dye that absorbs at a wavelength of 532 nm. The pig skin was treated as described at the beginning of section 5.3. The fundamental difference to the experiment with sun protection products lies in the significantly lower self-absorption of the skin in the green spectral range. The penetration depth of the light is approx. 2 cm ICheong, Beissertl.

A drop of 5 ul gentian violet solution was placed on the skin area to be examined. The liquid was completely absorbed by the skin within a few seconds and was no longer visible to the naked eye. Measurements were taken before applying the color solution, after 2 and after 10 min. The measurement results are shown in Figure 5.14. The data were corrected for the acoustic resonance of the measuring cell and the fluctuations in the laser intensity.

The curve recorded before the addition of the dye is determined by the self-absorption of the skin sample at 532 nm. The skin had no visible discoloration. No larger blood vessels were visible. The absorption observed is therefore due to the pigmentation of the skin in the wavelength range under consideration.

After adding the dye, the pressure amplitude is almost doubled and the time dependence changes. The measurement signal drops much more slowly. After 40 ms, it still has not dropped to its initial value. The differences between the measurements after 2 and 10 minutes are small. They are limited to the time interval from 10 to 40 ms. The observed changes in the photoacoustic pressure signal in Figure 5.14 are based on the absorption of the laser light by the added dye. Due to its aqueous consistency, the applied dye solution reaches a stationary distribution in the skin layer after only 2 minutes. The diffusion of the dye differs significantly

on the diffusion behavior of the sun protection substances, which only very slowly reach a greater depth in the skin.

The measurement results shown in Figure 5.14 were processed using the Tikhonov regulation method. The regulation parameter was determined by the quasi-optimum criterion. This selection was made because it is a diffusion process in which a concentration profile that changes slowly and steadily with the depth can be expected. The results of the calculation are shown in Figure 5.15.

The results from Figure 5.15 support the assumptions made when discussing the curve shape. The pig's own absorption takes place in the upper layers of the skin down to approx. 30 μm. No absorption is detectable in deeper layers. The dye leads to an absorption coefficient that changes quadratically with the depth, which had not yet decayed to zero at approx. 160 μm. The change between the first and second measurement is only slight. The concentration of the dye shifts slightly into the depth of the skin. It is noteworthy that the Concentration in the upper layers changes little. The cause could be a dye depot on the border between epidermis and dermis, which slowly empties into the depths.

The results achieved are to be classified as plausible. A check using a destructive method could not be carried out because the corresponding requirements for making tissue sections were missing. In spite of everything, the experiments on isolated skin samples have shown that the use of the photoacoustic measurement method with subsequent calculation of the location dependence of the optical absorption coefficient can be applied to the skin. The remaining work may have to be carried out on a large number of samples on a clinical scale in order to obtain statistical validation. Check-ups using tissue sections are absolutely necessary. This could not be achieved in the course of the work, since the topic was oriented towards the physical and technical side of the problem.

5.4 In vivo measurement with sunscreen

Already when investigating the diffusion of sunscreen on the isolated pig skin, the basic properties of the

Sun protection substances become clear. At the end of the investigations, another measurement with sunscreen, this time on living people, will be presented. The experiment has a demonstrative character, since the concentration distribution in the skin cannot be checked in any way. Examination of a dye in the skin was also considered. Since the chemical reactions that can be triggered by the thermally activated dye in the skin are not known, it has been avoided for health reasons.

The examinations took place on the inner side of the left forearm. The author acted as a subject. During the measurement process, the arm was placed on the holder described in section 4.1. The measurements were carried out before the cream was applied and at specified times afterwards. In the meantime, the skin area was covered with a fleece. 3 mg of Sun protection product SC8 Suncare was distributed as evenly as possible on an arm area of approx. 2 x 2 cm using the impression technique. The first measurement took place 5 minutes after application of the cream. The results of the measurement are summarized in Figure 5.16

The photoacoustic pressure signal increases due to the absorption of the cream after application of the substance compared to the untreated skin. The changes in the amplitude remain small in the first two hours, the fluctuations are still within the measurement accuracy. After approx. 4 hours, the absorption of the cream on the examined skin control is lower, the amplitude of the measurement signal decreases. After 6 hours, however, it was still significantly larger than the skin's own absorption

The time dependence of the measurement signal changed only slightly during the entire investigation. The reason for this behavior lies in the diffusion properties of the sunscreen, which is almost exclusively located in the upper horny layer. The thickness of this skin layer on the inside of the arm is only approx. 5 - 10 μm. Because of the Diffusion of the cream into this layer cannot be recorded due to the transmission properties of the microphone defined measurement window. From Figure 5.16, only information about the total absorption of the cream in the upper horny layer can be derived. A progress of the diffusion of the sun protection substances beyond the border to the living skin would be recognizable in the measurement signal. However, the results, in accordance with the product specification, show that this limit is not overcome.

The amplitude of the pressure in Figure 5.16 decreases with the duration of the experiment. The causes are found in the decomposition and lateral diffusion of the cream's absorbent substances.

The calculation of the spatial dependence of the optical absorption coefficient in the skin confirms the assumptions made when discussing the measurement curves. The results of the Tikhonov regulatory procedure are shown in Figure 5.17. The negative values of the solution were set to zero for the presentation.

The ultraviolet light is only absorbed in the uppermost skin layer, the thickness of which can be estimated at 6 μm according to Figure 5.17. The application of the sunscreen leads to an increase in the absorption, whereby the amount of energy absorbed near the surface increases compared to the amount absorbed in the 4 - 6 μm interval. The film of sunscreen substances, which is still largely on the skin at this time, leads to the observed shift. As a result, the cream penetrates into the horny layer and is distributed in it. Due to this diffusion process, the maximum of the absorption coefficient shifts into the skin. The decrease in absorption is essentially caused by the decomposition and lateral diffusion of the cream.

The results show that studies on living people are possible. The disturbances arising from the examination of an active living sample can be reduced by using the pulsed excitation in combination with a fixing support of the body part.

5.5 Discussion of the results

The experiments carried out with living and dead matter were able to confirm the possibility of calculating the spatial dependence of the optical absorption coefficient from the photoacoustic pressure changes in the basics. The deviations that occurred between the results and the actual course could be explained by the specific properties of the measuring apparatus.

The photoacoustic measuring method is basically a remission method. These processes are influenced by various material parameters. A comparison with the better known fluorescence measurement can clarify this. When the fluorescence of a substance is excited, photons of lower energy are emitted. The spread of the photons in the surrounding medium is determined by the absorption and scattering properties for the respective photon energy. The ideal detector therefore does not register the originally emitted photon, but one that has been changed by absorption and scattering processes. Photoacoustics face a similar problem. The amplitude of the thermal wave generated in the material depends on the incident light intensity, the optical absorption coefficient and the conversion efficiency for non-radiative processes, which can also be referred to as thermal quantum efficiency. The light intensity is easily accessible to the KontroUe. The size sought is the optical absorption coefficient. The thermal quantum yield is unknown and is assumed to be approximately one. The error contained in this approximation is of a fundamental nature and can only be eliminated by an exact determination in a separate experiment. These investigations had gone beyond the scope of the work. Absolute measurements were therefore not possible. The relative results obtained only allow an unrestricted comparison within a material. Since the processes to be investigated are diffusion processes in constant matrices, this procedure is justified and also allows quantitative comparisons.

The thermal waves are influenced on their way to the detector by the thermal conductivity of the medium being traversed. The exact knowledge of this material parameter is important for the calculation of the spatial dependence of the optical absorption coefficient. Values that are too large lead to an overestimation of the absorption in areas close to the surface, whereas values that are too low lead to a displacement of the absorption in deeper areas of the sample. The errors in the temperature conductivity parameter directly lead to misinterpretations in the amplitude and location dependence of the absorption coefficient.

For successful investigations, it must also be ensured that it is a thermally homogeneous material. The problems and ambiguities in the examination of layer structures have become clear when evaluating the tests with colored glass filters.

The influencing factors described so far are of a general physical nature. There is also an experimental difficulty that arises from the frequency-dependent transmission behavior of the measuring apparatus. The resulting falsification of the measurement data must be taken into account in the evaluation. The results show that the experimental difficulties are manageable. However, the usability of the measurement setup used for in vivo human examinations is limited. The detection method with a microphone as a sensitive element leads to a large number of disturbances. One way out could be a non-contact measurement method, such as photothermal radiometry. The necessary light intensities for a sufficient signal / noise ratio are still too high for in-vivo experiments to be feasible.

Appendix 5 145

literature

AAMODT L.C., MURPHY J.C., PARKER J.G. (1977) Size considerations in the design of cells for photoacoustic spectroscopy Journal of Applied Physics 48: pp. 927-933

AFROMOWITZ M.A., YEH P.S., YEH S. (1977) Photoacoustic measurements of spatiaUy varying optical absorption in solids: A theoretical treatment Journal of Applied Physics 48 pp. 209-211

ANJO D.M., MOORE T.A. (1984) A Photoacoustic Depth Profile of Beta-Carotene in Skin Photochemistry andPhotobiology 39: pp. 635-640

ARNAUD F., DELHOMME G., DITTMAR A. (1994)

A micro thermal diffusion sensor for non-invasive skin characterization sensors and actuators A 41-42 pp. 240-243

BAESSO MX., SHEN J., SNOOK R.D. (1994)

Laser-induced Photoacoustic Signal Phase Smdy of Stratum Comeum and

epidermis

Analyst 119 pp. 561-562

BEISSERT S., GRANSTEIN R. D. (1995) UV-Induced Cutaneous Photobiology Critical Reviews in Biochemistry and Molecular Biology 31 pp. 381 - 404

BERGMANN, SCHAEFER (1993)

Textbook of Experimental Physics Volume 3 - Optik de Gruyter Berlin, New York

BOCARRAA.C., FOURNIERD., BADOZ J. (1980)

Thermo-Optical Spectroscopy: Detection by the Mirage Effect Apphed Physics Letters 36 pp. 130-134 BROCKHAUS PHYSIK (1989) FA Brockhaus Verlag Leipzig

BRONSTEIN I. N., SEMENDAJEW K. A. (1985) Taschenbuch der Mathematik BSB B. G. Teubner Verlagsgesellschaft Leipzig

BRULS WEIL A.G., SLAPER H., VAN DER LEUN JAN C, BERRENS L. (1984)

Transmission of Human Epidermis and Stratum Comeum as a Function of Thickness in the Ultraviolett and Visible Wavelengths Photochemistry and Photobiology 4 p.458 - 494

CARSLAW H.S., JAEGER J.C. (1988) Conduction of Heat in Solids Oxford, University Press

CHEONG W-F, PRAHL S.A., WELCH A.J. (1990) A Review of the optical properties of biological tissues IEEE Journal of Quantum Electronics 26: S.2166-2185

CHIRTOC M., BICANIC D., TOSA T. (1991) A versatile inverse photopyroelectric (IPPE) techmque and Instrument for real time observation of the condensation of water vapor in the atmosphere Review of Scientific Instruments 62: S.2257 - 2261

DU H., FANG J.W., ZHENG J.J. (1994) Photoacoustic Phase Spectrum of a Layered Sample Applied Physics A 60 pp. 419 - 423

EL GAMMAL S., HARTWIG R., (1996) nproved resolution of magnetic resonance microscopy in examination of skin tumors Journal of Investigative Dermatology pp. 1287 - 1292

FARROW MM, BURNHAM RK, AUZANNEAU M., OLSEN SL, PURDIE N., EYRING EM (1978) Piezoelectric detection of photoacoustic Signals Applied Optics 17: pp. 1093-1098 FERNELIUS .C. (1979) Helmholtz resonance effect in photoacoustic cells Applied Optics 18 pp. 1784-1787

FUCHS J. (1992) Oxidative Injury in Dermatopathology Springer Verlag Berlin

GOLUB G.H., HEATH M., WHABA G. (1979) Generalized Cross-Validation as a Method for Choosing a Good Ridge Parameter Technometrics 21: 215-223

GÜNTHER B.C., HANSEN K.H., VEIT I. (1994) Technical Acoustics - Selected Chapters Expert Verlag Renningen-Malsheim

GREINER W., NEISE L „STÖCKER H. (1987) Thermodynamics and Statistical Mechanics Verlag Harri Thun Frankfurt

HANSEN P.C. (1992) Numerical tools for analysis and solution of Fredholm integral equations of the first kind Inverse Problems 8: 849-872

HANSEN P.C. (1993) Regularization Tools, a Matlab Package for Analysis and Solution of discrete ill-posed-Problems Report UNIC-92-03

Hestenes M.R., Stiefel E. (1952) Method of Conjugate Gradients for Solving Linear Systems Journal of Research of the National Bureau of Standards 49 pp. 409-437

IMHOF R.E., BIRCH D.J.S., THORNLEY F.R., GILCHRIST J.R., STRTVENS T.A. (1984)

Optothermal transient emission radiometry Journal of Physics E: Scientific Instruments 17 pp. 521-525 IMHOFF RE, WHITTERS CJ, BIRCH DJS (1990) Opto-thermal in vivo monitoring of sunscreens on skin Physics in Medicine and Biology 35 pp. 95-102

JACKSON W., AMERN.M. (1980)

Piezoelectric photoacoustic detection: Theory and experiment Journal of Appüed Physics 51: pp. 3343-3353

KIRKBRIGHT GORDON F., MILLER RICHARD M., SPILLANE DOMINIC E.M. (1984)

Depth Resolved Spectroscopic Analysis of Solid Samples Using Photoacoustic Spectroscopy Analytical Chemistry 12 p. 2043 -2048

KÖLMEL K., NICOLAUS A., SENNHENN B., GIESE K. (1986)

Evaluation of dmg penetration into the skin by photoacoustic measurement Journal of the Society of Cosmetic Chemistry 37 pp. 375-385

KOPP C, NIESSNERR. (1999)

Optoacoustic sensor head for depth profiling Applied Physics B: Lasers and Optics 68 pp. 719-725

KORPIUN P., BÜCHNER B. (1983)

On the Thermodynamics of the Photoacoustic Effect of Condensed Matter in

Gas cells

Applied physics B 30: pp. 121-129

LAWSON C.L., HANSON R.J. (1974) Solving Least Squares Problemes London, Prentice-Hall International

LONG F. H., GERMAN T. F. (1987) Pulsed Photothermal Radiometry of Human Artery IEEE Journal of Quantum Electronics QE23: 1821 - 1826

McDONALD F.ALAN, WETSEL GROVER C. (1978) Generalized theory of the photoacoustic effect Journal of Applied Physics 49 pp. 2313 - 2322 MILNER TE, SMITHIES DJ., GOODMAN DM, LAU A., NELSON JS (1996).

Depth determination of chromophores in human skin by pulsed photothermal radiometry Applied Optics 35: pp. 3379-3385

NARAYANAN K., CHANDANI S., RAMAKRISHNA T., MOHAN RAO Ch. (1997)

Depth Profileϊng of Mammalian Cells by Photoacoustic Spectroscopy: Localization of Ligands Biophysical Journal 72: S.2365-2368

PHILLIPS D.L. (1962) A Techmque for the numerical Solution of Certain Integral Equations of the First Kind Journal of the Association of Computing Machines 9 pp. 84-97

POULET P. CHAMBRON J. (1983) In Vivo Photoacoustic-Spectroscopy of the Skin Journal de Physique 44 pp. 413-41

POULET P. CHAMBRON J. (1985) In vivo Cutaneous Spectroscopy by Photoacoustic Detection Medical & Biological Engineering & Computing 23 pp. 585-588

POWER J.F., PRYSTAY M.C. (1995) Expectation Minimum (EM): A New Principle for the Solution of Iü-Posed Problems in Photothermal Science Applied Spectroscopy 49 pp. 709 - 724

POWER JF (1997) Expectation minimum - a new principle of inverse problem theory in the photothermal sciences: theoretical characterization of expectation values Optical Engineering 36 pp. 487-503 PRAHL SA, VITKIN IA, BRUGGEMANN U., WILSON BC, ANDERSON RR (1992)

Determination of optical properties of turbid media using pulsed photothermal radiometry Physics in Medicine and Biology 37 p. 1203-1217

PUCCETTI G., LAHJOMRI F., LEBLANC R.M. (1997) Pulsed photoacoustic spectroscopy applied to the diffusion of sunscreen chromophores in human skin: the weakly absorbent regime Journal of Photochemistry and Photobiology B39 pp. 110-120

PUCCETTI GERMAIN, LEBLANC ROGER M. (1996) A comparative study on chromophore dif usion inside porous filters by pulsed photoacoustic spectroscopy Journal of Membrane Science 119 pp. 213-228

REMANE A., STORCH V., WELSCH U. (1980) Systematic Zoology Gustav Fischer Verlag Stuttgart

ROSENCWAIG A. GERSHO A. (1976) Theory of the photoacoustic effect with solids Journal of Applied Physics 47: 64-69

SANTOS R., MIRANDA L.C.M. (1981) Theory of the Photothermal Radiometry with Solids Journal of Applied Physics 52 p.4194-4198

SCHMIDT K., PABST M., BECKMANN D., LAUCKNER G. (1998) Use of the photoacoustic method for determination of the protection properties of sunscreen ing agents in vivo American Laboratory S. 33C-33F

SENNHENN B., ROHR M., GIESE K. KÖLMEL K. (1992) Penetration of Topically Applied Drugs Through Human Skin Investϊgated by Photoacoustic Spectroscopy

Springer Series in Optical Science 69 III Topical Meeting an Photoacoustic and Photothermal Phenomena pp. 549-551 SHIPKIN YP, UTTS SR, MEGLINSKII IV, PILIPENKO EA (1996) Spectroscopy of Human Skin in vivo: Fluorescence Spectra Optics and Spectroscopy 80: 383-389

SHIPKIN Y.P., UTTS S.R., PILIPENKO E.A. (1996) Spectroscopy of Human Skin in vivo: Reflection Spectra Optics and Spectroscopy 80: 228-234

TAKAMOTO R. NAMBA R. MATSUOKA M. SAWADA T. (1992) Human in vivo Percutaneous Absorptiometry Using the Laser-Photoacoustic Method Analytical Chemistry 64 pp. 2661-2663

TAKAMOTO R. NAMBA R. NAKATA O. SAWADA T. (1990) In Vitro Percutaneous Absorptiometry by Simultaneous Measurement Using the Photoacoustic Method and Absorbance Analytical Chemistry 62 pp. 674-677

TAKAMOTO R. YAMAMOTO S. NAMBA R. MATSUOKA M. SAWADA T. (1995)

New Percutaneous Absorptiometry by a Laser Photoacoustic Method Using an Open-Ended Cell Analytica Chimica Acta 299 pp. 387-391

TAM A.C. (1986) Applications of photoacoustic sensing techniques Review of modern Physics 58: 381-431

TIKHONOV A. N., GONCHARSKY A. V., STEPANOV V. V., YAGOLA A. G.

Numerical Methods for the Solution of 111 Posed Problems Kluwer Academic Publishers

VARGAS H., MIRANDA LCM (1988) Photoacoustic and related photothermal techniques Physics Reports-Review Section of Physics Letters 161: S.43-101 ZHAROV VP, LETOKHOV VS (1986) Laser Optoacoustic Spectroscopy Springer Series in Optical Sciences 37; Springer Verlag Berlin

attachment

8.1 Obtaining the special solution of the inhomogeneous differential equation for the area of the sample using the Cauchy method

The inhomogeneous differential equation to be solved with the symbols from Chapter 2 reads:

the ² Ψ ^~ r p. ι ιωuß _lτ , st,

The solution of the homogeneous equation forms the basis for the determination of a special solution of the inhomogeneous equation. It is:

Ψ (z) = c ^ _! (z) + c ₂ Ψ ₂ (z) ( ⁸ - ² >

With

, , < ^{8 3} >

The constants Ci are determined in such a way that they satisfy the requirement (2.4) for any y.

Ψ (z) = c _l e ^SpZ + c ₂ e ^{~ SpZ} =

Ψ '(z) = c _λ s _p e ^spz - c ₂ s _p e- ^spz = -Q (y) ^(8'4)

The following values result for the constants:

This leads to the special solution of the inhomogeneous differential equation according to the instructions of Cauchy / Bronsteinl

8.2 Reverse transformation of the WLG solution after pulsed lighting

The solution of the differential equation system (2.39) and (2.40) in the laplace room with the symbols from Chapter 2 is:

We are looking for the reverse transformation of the solution in the sample at point z = 0 (sample surface). By Besclu ^* änkung simplified to the surface of the term and the result is Equation (8.8).

-1 operator of the inverse Laplacetraπsformation

The integrations of the reverse transformation via the Laplace variable ξ and the integration via z 'in the area of the sample take place independently of one another and can therefore be interchanged in their order. Using the correspondence (8.9) / Carslaw and Jaeger / the solution is the back-transformed temperature on the sample surface (8.10).

Brief descriptions of the pictures

Figure 2.1 Principle diagram with influencing variables for the creation of a photoacoustic measurement signal

Figure 2.2 Schematic representation of the structure of the mammalian skin / Remane /

Figure 2.3 Model of the photoacoustic measuring cell

Figure 2.4 Oscillations of the temperature on the sample surface after excitation of a layer structure with absorbing layers at lμm and 50μm depth

Figure 2.5 Temperature at the surface of a layered sample with absorbing layers in lμm and 50μm depth after excitation with a laser pulse; Temperature conductivity: 2, 8xl0 ^{~ 8} m ² / s

Figure 3.1 Approximation of an absorption coefficient that changes continuously with the depth z

Figure 3.2 Time dependence of the temperature of the sample surface after pulsed excitation; the inset shows the spatial dependence of the optical absorption coefficient

Figure 3.3 Comparison of the specified heat density distribution with the Q (z) dependency calculated using the pseudo-inverter K ⁺ Figure 3.4 Heat density distribution Q (z) calculated from noisy temperature data (± 0.5% of the temperature maximum) using the pseudo inverse K ⁺

Figure 3.5 Singular values of the matrix approximation of the integral core for pulsed excitation with the parameters from section 3.1

Figure 3.6 Right and left singular vectors of the matrix K from the example from section 3.1 for columns 1, 10 and 20

Figure 3.7 General shape of the L-curve as a graphic aid for the selection of the regulation parameter

Figure 3.8 Location dependencies of the optical absorption coefficient for the comparison of the regulation procedures

Figure 3.9 L curve of the EM NNLS regulation method for the calculation of an exponentially falling absorption profile; Temperature data noisy with 0.1% or 30% of the maximum temperature

Figure 3. 10 Enlargement of the L curve from Figure 3. 9 (30% noise) with optimal regulation parameter Figure 3.11 GCV function depending on the regulation parameter lambda for an ascending ramp profile, reconstructed using EM SVD with 3% noise; Insert: enlarged section of the GCV function

Figure 3.12 Quasi-optimal function of the reconstruction of a step profile using the Tikhonov regulation method; Simulated temperature measurements were noisy with 5% of the maximum temperature

Figure 3.13 Optimal regulation parameter for the reconstruction of a step profile using EMNNLS depending on the noise in the temperature data and the choice of the regulation parameter

Figure 3.14 Optimal regulation parameter for the reconstruction of a step profile by means of Tikhonov regulation depending on the noise in the temperature data and the choice of the regulation parameter

Figure 3.15 Residual and solution standard depending on the number of repetitions for the EM SVD regulation procedure on a step absorption profile (S / N = 204)

Figure 3.16 Regularized solution depending on the number of averages for the EM SVD procedure compared to the solution for the Tikhonov regulation procedure Figure 3.17 Relative error of the solutions for smooth, slowly changing profiles of the optical absorption coefficient depending on the noise in the simulated temperature data; the percentages relate to the maximum of the temperature

Figure 3.18 Comparison of the solutions of the four examined regulatory processes with the original; the temperature values were noisy at 0.5% of the maximum

Figure 3.19 Comparison of the solutions of the four examined regulatory processes with the original; the temperature values were noisy at 10% of the maximum

Figure 3.20 Comparison of the solutions of the four examined regulatory procedures with the original; the temperature values were noisy at 0.5% of the maximum

Figure 3.21 Comparison of the solutions of the four examined regulatory processes in the original; the temperature values were noisy at 10% of the maximum

Figure 3.22 Comparison of the solutions of the four examined regulatory processes with the original; the temperature values were noisy at 0.5% of the maximum Figure 3.23 Comparison of the solutions of the four examined regulatory procedures with the original; the temperature values were noisy at 10% of the maximum

Figure 3.24 Comparison of the solutions of the four examined regulatory procedures with the original; the temperature values were noisy at 0.5% of the maximum

Figure 3.25 Comparison of the solutions of the four examined regulatory processes with the original; the temperature values were noisy at 10% of the maximum

Figure 3.26 Relative error of the solutions calculated with the four investigated regulation methods when excitation profiles are excited as a non-continuous function of the location

Figure 3.27 Comparison of the solutions of the four examined regulatory processes with the original; the temperature values were noisy at 0.5% of the maximum

Figure 3.28 Comparison of the solutions of the four examined regulatory procedures with the original; the temperature values were noisy at 10% of the maximum

Figure 3.29 Comparison of the solutions of the four examined regulatory procedures with the original; the temperature values were noisy at 0.5% of the maximum Figure 3.30 Comparison of the solutions of the four examined regulatory procedures with the original; the temperature values were noisy at 10% of the maximum

Figure 3.31 Comparison of the solutions of the four examined regulatory processes with the original; the temperature values were noisy at 0.5% of the maximum

Figure 3.32 Comparison of the solutions of the four examined regulatory procedures with the original; the temperature values were noisy at 10% of the maximum

Figure 4.4 Block diagram of the measurement setup used for pulsed and periodically modulated excitation of the photoacoustic measurement signal

Figure 4.2 Relative efficiency of the diffraction gratings used depending on the transmitted wavelength

Figure 4.3 Time dependence of the light intensity in the photoacoustic measuring cell, measured with an Si photodiode compared to a rectangular function of the same frequency

Figure 4.4 Photoacoustic pressure amplitude in the measuring cell as a function of the modulation frequency with different lengths of the gas column above the sample Figure 4.5 Measurable pressure in the photoacoustic measuring cell after absorption of a light pulse by a homogeneously absorbing material depending on the length of the gas column above the sample

Figure 4.6 Cross section through the open photoacoustic measuring cell used

Figure 4.7 Frequency response of the transmission dimension of the used 1/2 "type 4189 microphone taking into account the use of the pressure chamber

Figure 4.8 Comparison of the photoacoustic signal amplitude in theory and experiment

Figure 4.9 Comparison of the frequency dependence of the phase of the photoacoustic signal in theory and experiment

Figure 4.10 Comparison of the photoacoustic signal in theory and experiment Illumination of a soot-blackened glass plate with a laser pulse of 9ns pulse duration

Figure 4.11 Frequency spectrum of a typical photoacoustic signal before and after the numerical correction

Figure 4.12 Comparison of the microphone transmission function with the transmission behavior of the filter used for data handling Figure 4. 13 Principle sketch of the in-vivo measuring arrangement with elastic suspension of the measuring cell and support points for the arm

Figure 4.14 Comparison of the Fourier spectra of the interference level for the open measuring cell (without sample) with the in-vivo measuring situation with the arm on

Figure 5.1 Photoacoustic pressure amplitude after excitation of the layer systems consisting of the color filter RG 610 and cover glasses with a laser pulse; Curve index = number of coverslips

Figure 5 .2 Location dependence of the optical absorption coefficient for the layer structures from Figure 5.1

Figure 5.3 Amplitude and phase of the photoacoustic pressure oscillations in the measuring cell as a function of the chopper frequency for a layer structure, consisting of a colored glass filter RG 610 and one or two microscope cover glasses

Figure 5 .4 Location dependencies of the optical absorption coefficient of the investigated layer structures calculated from the pressure oscillations with periodically modulated excitation

Figure 5.5 Schematic structure of the gelatin layers used in the experiments Figure 5.6 Pressure curve in the photoacoustic measuring cell 5, 10, 18, 38 and 60 minutes after preparation of the layers

Figure 5.7 Time difference between the end of the laser pulse and the photoacoustic pressure maximum in the measuring cell depending on the duration of the experiment

Figure 5 .8 Course of the optical absorption coefficient depending on the depth in the gelatin structure at different times after the start of the experiment

Figure 5. 9 Course of the optical absorption coefficient depending on the depth in the gelatin structure in the areas near the surface

Figure 5.10 Pressure curve in the photoacoustic measuring cell as a function of the time after preparation of structure B; Measurements after 0, 10, 15, 30, 45 and 60 minutes

Figure 5. 11 Spatial course of the optical absorption coefficient in the gelatin structure B depending on the duration of the experiment

Figure 5. 12 Time dependence of the photoacoustic pressure signal measured before and after using SC8 Suncare Figure 5. 13 Dependence of the optical absorption coefficient at 355nm in the pig skin before and after the application of sunscreen

Figure 5. 14 Time-dependent photoacoustic measurement signal before and after the addition of gentian violet to pig skin

Figure 5. 15 Calculated curve of the optical absorption coefficient at 532nm before and after adding a gentian violet solution in the pig skin

Figure 5.16 Area of skin on left forearm before and at different times after using SC8 Suncare

Figure 5.17 Profile of the optical absorption coefficient at 355nm before and after the application of sunscreen in human skin

Claims

Expectations

1. Device for measuring a temporal and / or spatial concentration profile of at least one substance in a sample, in particular human skin, with a means for excitation of at least one substance by electromagnetic radiation and with a measuring cell for detection based on the excitation of acoustic waves generated and / or in the sample,

marked by

by means of a computing means for automatically converting the pressure fluctuations caused by temperature changes on a surface of the sample into at least one measure which is dependent on the concentration of at least one of the substances.

2. Norrichtung according to claim 1, characterized in that the computing means automatically determines a time-dependent and / or location-dependent concentration profile of at least one of the substances.

The device according to claim 1 or 2, characterized in that the computing means has a means for automatically solving an integral equation relating to heat conduction by means of a regulation approach.

4. Norrichtung according to at least one of the preceding claims, characterized by a pulsed laser device as a means for excitation, in particular with a Q-switched ΝdYAG laser.

5. Norrichtung according to at least one of the preceding claims, characterized in that the measuring chamber is designed as an acoustic resonator, the frequency tuning depending on a depth of the sample to be examined (1) is adjustable.

6. Norrichtung according to at least one of the preceding claims, characterized by a means for averaging at least two signals of the pressure fluctuation measurement over a predetermined period.

7. Norrichtung according to claim 6, characterized in that the means for averaging is designed as a digital storage oscilloscope.

8. Norrichtung according to at least one of the preceding claims, characterized by a means for calibration of the measuring cell, in particular a reference measuring cell.

9. Norrichtung according to at least one of the preceding claims, characterized by a correction means for frequency-dependent transfer functions of the measuring chamber and / or a measuring head.

10. Norrichtung according to at least one of the preceding claims, characterized in that the measuring cell is arranged in a measuring head which is coupled via a data connection line to a data processing device.

11. ner driving for measuring the concentration of at least one substance in a sample, in particular human skin, in which at least one substance is excited by electromagnetic radiation and the acoustic wave generated by the excitation is detected with a measuring cell, characterized in that

the measured values of the pressure fluctuations on a surface of the sample caused by temperature changes are automatically converted into at least one measure which is dependent on the concentration of at least one of the substances.

12. Ner driving according to claim 11, characterized in that a time-dependent and / or location-dependent concentration profile of at least one substance is automatically determined.

13. Nerfahr according to claim 11 or 12, characterized in that the computing means au to mansch an integral equation regarding heat conduction solves by means of a regulatory approach.

14. Ner driving according to at least one of claims 11 to 13, characterized in that the substance is excited with a pulsed laser device, in particular with a Q-switched ΝdY AG laser.

15. Ner driving according to at least one of claims 11 to 14, characterized in that the measuring chamber is designed as an acoustic resonator and the frequency tuning is set as a function of a depth extension of the sample to be examined.

16. Ner driving according to at least one of claims 11 to 15, characterized in that at least two signals of the pressure fluctuation measurement are averaged over a predetermined period.

17. The method according to at least one of claims 11 to 16, characterized in that for calibration of the measuring cell measurement is carried out on a reference measuring cell.

18. Ner driving according to at least one of claims 11 to 17, characterized in that the frequency-dependent transfer functions of the measuring chamber and / or a measuring head are corrected by a correction.