IE20060275A1 - A method and aparatus for determining distance between two spaced apart surfaces - Google Patents
A method and aparatus for determining distance between two spaced apart surfacesInfo
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- IE20060275A1 IE20060275A1 IE20060275A IE20060275A IE20060275A1 IE 20060275 A1 IE20060275 A1 IE 20060275A1 IE 20060275 A IE20060275 A IE 20060275A IE 20060275 A IE20060275 A IE 20060275A IE 20060275 A1 IE20060275 A1 IE 20060275A1
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- G—PHYSICS
- G01—MEASURING; TESTING
- G01B—MEASURING LENGTH, THICKNESS OR SIMILAR LINEAR DIMENSIONS; MEASURING ANGLES; MEASURING AREAS; MEASURING IRREGULARITIES OF SURFACES OR CONTOURS
- G01B11/00—Measuring arrangements characterised by the use of optical techniques
- G01B11/02—Measuring arrangements characterised by the use of optical techniques for measuring length, width or thickness
- G01B11/06—Measuring arrangements characterised by the use of optical techniques for measuring length, width or thickness for measuring thickness ; e.g. of sheet material
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- G—PHYSICS
- G01—MEASURING; TESTING
- G01B—MEASURING LENGTH, THICKNESS OR SIMILAR LINEAR DIMENSIONS; MEASURING ANGLES; MEASURING AREAS; MEASURING IRREGULARITIES OF SURFACES OR CONTOURS
- G01B11/00—Measuring arrangements characterised by the use of optical techniques
- G01B11/02—Measuring arrangements characterised by the use of optical techniques for measuring length, width or thickness
- G01B11/06—Measuring arrangements characterised by the use of optical techniques for measuring length, width or thickness for measuring thickness ; e.g. of sheet material
- G01B11/0616—Measuring arrangements characterised by the use of optical techniques for measuring length, width or thickness for measuring thickness ; e.g. of sheet material of coating
- G01B11/0625—Measuring arrangements characterised by the use of optical techniques for measuring length, width or thickness for measuring thickness ; e.g. of sheet material of coating with measurement of absorption or reflection
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- General Physics & Mathematics (AREA)
- Length Measuring Devices By Optical Means (AREA)
Abstract
A method for determining distance between two spaced apart surfaces comprises; sampling values of an interference spectrum produced by reflections of frequency components of white light from the respective surfaces to produce a first array of sampled values of intensity of the interference spectrum and corresponding values of wavelengths of the reflected frequency components of the white light, computing an approximate value of the optical distance (formula) between the surfaces as a function of the wavelengths of two of the reflected frequency components of the white light from the first array based on the equation: <FORMULA> where h1 and h2 are wavelength values at which peaks occur in a waveform (32) derived from the first array, and k is the number of times the waveform (32) derived from the first array crosses a line (33) representative of the average of the intensity values of the first array, transforming the first array into a second array of the sampled values of intensity of the interference spectrum and values of the inverse of the wavelengths of the corresponding reflected frequency components of the white light, computing at least one further approxiamte value of the optical distance between the surfaces, each further approximate optical distance value being computed as a function of the immediately previously computed approximate optical distance value and the phase difference between the waveform (36) of the second array and a corresponding waveform of constant period derived from the immediately previously computed approximate optical distance value, and dividing the last computed optical distance value by the frfractive index of the medium between the surfaces, the distance there between of which is being determining the actual physical distance between the furfaces. <Figures 4 & 5>
Description
The present invention relates to a method for determining distance between two 5 spaced apart surfaces, which may be the opposite surfaces of a medium, or may be the surfaces of respective media spaced apart from each other.
Methods and apparatus for determining distance between spaced apart surfaces are known, and such methods are suitable for determining the distance between opposite surfaces of a medium, for example, glass, a liquid, or a free-standing liquid film, and such known methods are also suitable for determining the distance between two spaced apart surfaces, where one of the surfaces is a surface of one medium, and the other surface is the surface of another medium, and the adjacent surfaces of the respective media are spaced apart from each other by, for example, an air gap, a vacuum, or any other medium. In general, known methods are light based methods, and typically require sampling an interference spectrum produced by the reflected components of white light reflected from the respective surface:,. In such methods an array of sampled values of the intensity of the reflected components of the white light in the interference spectrum against corresponding values of wavelength of the reflected components is prepared, and the distance between the respective surfaces is determined from the following equation:
0 ~ 4η(λ2-λ,) (A)
IE 06 0 275 where do is the distance between the surfaces,
A/ is the wavelength of one of the reflected components of the white light from the first array,
A2 is the wavelength of another one of the reflected components of the white light from the first array, k is the number of times a waveform produced by a plot of the intensity values of the interference spectrum against the corresponding wavelength values of the first array would cross a line, the value of which is approximately equal to the average of the intensity values of the waveform, and n is the refractive index of a medium between the surfaces.
However, since the distance between the surfaces computed from equation (A) is dependent on just two sampled values of the interference spectrum of the reflected light from the two surfaces, the accuracy with which the distance between the surfaces can be computed is limited.
In determining distances between surfaces which are spaced apart by distances of the order of tens of microns, a high resolution method is required. For example, where it is desired to carry out high resolution cone-angle measurements in Tilted Smectic
Liquid Crystal layers of free-standing thin films, a relatively high resolution method is required.
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In Smectic Liquid Crystals (LC) the molecules are organised in layers. The molecules of each layer may tilt such that longitudinal axes of the respective molecules make an angle Θ with the layer normal. Studying the temperature dependence of this angle Θ is one of the primary tasks in investigations of the phase transitions in Smectic LCs. In a Smectic A (SmA) liquid crystalline phase the tilt angle Θ is equal to zero, in other words, the longitudinal axes of the molecules are parallel to the layer normal. In a smectic C (SmC) liquid crystalline phase the tilt angle Θ has a finite non-zero value. Additionally, in a Smectic C (SmC*) liquid crystalline phase, in which the crystals consist of chiral molecules, where the asterisk denotes crystals consisting of chiral molecules, the title angle Θ also has a finite nonzero value. Conventional methods of measuring the tilt angle Θ usually require application of an electric field to the liquid crystal, which disturbs the original structure of the liquid crystalline phases being investigated.
Free-standing films of Smectic LCs have a tendency to keep the number of layers constant regardless of the temperature change. Therefore, by measuring the change in the thickness df of a free-standing film, one can determine the change in the thickness di of the respective layers by assuming the number of layers in the film to be constant, and assuming that the thickness of each of the layers changes by the same amount. By determining the change in the thickness of each layer of the film, the change in the tilt angle Θ of the molecules of the layers can then be determined.
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For example, considering SmA-SmC* phase transition in a crystal with long and narrow molecules one can determine the tilt angle Θ in SmC* phase by a simple formula:
@SmC* = arccos d/SmC*/ dt(SmA) j /
= arccos df(SmC*/ df(SmA) (B)
Accordingly, by determining the change in thickness of a free-standing film of Smectic liquid crystals, a determination of the change of the tilt angle Θ of the molecules of each layer of the film can be obtained.
However, the derivative approaches zero for small angle values of Θ, and practical values of the tilt angle Θ rarely exceed 25°. Thus, to achieve high resolution measurements of the tilt angle Θ, correspondingly high resolution measurements of the film thickness are required.
Accordingly, there is a need for a method for determining the distance between two surfaces with relatively high resolution.
The present invention is directed towards providing such a method.
According to the invention there is provided a method for determining distance 1E 060 275 between two spaced apart surfaces comprising:
sampling values of an interference spectrum produced by reflections of frequency components of a composite multi-frequency signal from the respective surfaces to produce a first array of sampled values of intensity of the interference spectrum and corresponding values of wavelengths of the reflected frequency components of the multi-frequency signal, computing an approximate value indicative of the distance between the surfaces as a function of the wavelengths of two of the reflected frequency components of the multi-frequency signal from the first array, transforming the first array into a second array of the sampled values of intensity of the interference spectrum and values of the inverse of the wavelengths of the corresponding reflected frequency components of the multi-frequency signal, computing at least one further approximate value indicative of the distance between the surfaces, each further approximate distance indicative value being computed as a function of the immediately previously computed approximate distance indicative value and the phase difference between the waveform of the second array and a corresponding waveform of constant period derived from the immediately previously computed approximate distance indicative value.
Preferably, the further approximate distance indicative values are computed until the phase difference between the waveform of the second array and the corresponding waveform of constant period derived from the immediately previously computed •Ε 06 0275 approximate distance indicative value is less than a predetermined phase difference value. Advantageously, the predetennined phase difference value is of negligible value.
In one embodiment of the invention three further approximate distance indicative values are computed.
In one embodiment of the invention the approximate value indicative of the distance between the surfaces computed from the first array as a function of the wavelengths of two of the reflected frequency components of the multi-frequency signal is computed from the following equation:
4(λ2-λ,) where do is the approximate value indicative of the distance between the surfaces, λi is the wavelength value of one of the reflected frequency components of the multi-frequency signal from the first array, /h is the wavelength value of another one of the reflected frequency components of the multi-frequency signal from the first array, and k is the number of times the waveform of the first array crosses an imaginary line, the value of which is approximately equal to the average of the values of the waveform.
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In another embodiment of the invention the phase difference between the waveform of the second array and each corresponding waveform of constant period derived from the immediately previously computed approximate distance indicative value is computed as a function of the difference in the areas under the respective waveforms and a datum value, the datum value being less than or equal to the minimum value of the intensity values of the respective waveforms between predetermined values of the inverse of the wavelength of the reflected components of the multi-frequency signal, and preferably, the predetermined values of the inverse of the wavelengths of the reflected frequency components of the multi-frequency signal are determined by the sampled values of the interference spectrum from which the first array is produced.
In one embodiment of the invention each further approximate distance indicative value is computed as a function of the tan of the phase difference between the waveform of the second array and the corresponding waveform of constant period derived from the immediately previously computed approximate distance indicative value. Preferably, each further approximate distance indicative value is computed as a function of the derivative of the tan of the phase difference between the waveform of the second array and the corresponding waveform of constant period derived from the immediately previously computed approximate distance indicative value.
Advantageously, each further approximate distance indicative value is computed as a function of the quotient of the derivative of the tan of the phase difference between the waveform of the second array and the corresponding waveform of constant period
IE 060275 derived from the immediately previously computed approximate distance indicative value divided by the tan of the phase difference.
In one embodiment of the invention the tan of the phase difference between the 5 waveform of the second array and the corresponding waveform of constant period derived from the immediately previously computed approximate distance indicative value is derived from the following equation:
A, = isin(4itd0xi)(Ri-Raverage)(xi+1 -xs) _O_ tcos(4Tid0xi)(Ri - Raverage)(xM -xt) 0 where
Ai is the tan of the phase difference, do is the immediately previously computed approximate distance indicative value,
Xi is the inverse wavelength value of the ith reflected frequency component of the multi-frequency signal from the second array, x/+j is the inverse wavelength value of the next reflected frequency component after the ith reflected frequency component of the multi-frequency signal from the second array,
R, is the intensity of the interference spectrum corresponding to the ith reflected frequency component of the multi-frequency signal from the second array, and •Ε 06 0 275 ^average is the average of the intensity values of the interference spectrum from the second array.
In another embodiment of the invention a third array is prepared, the third array comprising the values of the tan of the phase difference and corresponding inverse wavelength values of the reflected frequency components of the multi-frequency signal, and a linear function y = ax + b is fitted to the waveform of the third array for determining the derivative of the tan of the phase difference.
Preferably, the function y = ax + b is fitted to the waveform of the third array in a portion of the waveform of minimum noise. Advantageously, the function y = ax + b is fitted to the waveform of the third array by a less-square method.
Ideally, each further approximate distance value is derived from the equation:
d - do----z2n(l + (ax3/'N+b) ) where d is the approximate distance indicative value currently being computed, do is the immediately previously computed approximate distance indicative value, a is the derivative of the tan of the phase difference between the waveform of the second array and the corresponding waveform of constant period derived from the immediately previously computed approximate distance indicative value, β 060275 the function (αχγ,Ν+b) represents the tan of the phase difference between the waveform of the second array and the corresponding waveform of constant period derived from the immediately previously computed approximate distance indicative value, and xy.N is the value of the inverse wavelength value of the reflected frequency component of the reflected multi-frequency signal corresponding approximately to the mid-value of the reflected frequency components of the multi-frequency signal in the upper half of the inverse wavelength values of the third array.
In one embodiment of the invention the multi-frequency signal is a light signal, and preferably, the multi-frequency signal is white light.
In one embodiment of the invention each approximate value indicative of the distance between the surfaces is computed as an approximate value of the optical distance between the surfaces.
In another embodiment of the invention the actual value of the distance between the surfaces is computed as a function of the last of the computed further approximate distance indicative values and the refractive index of a medium between the surfaces.
Preferably, the multi-frequency signal is directed substantially normal to the respective surfaces.
Ο 6 ο 275
The invention also provides a method for determining a change in an angle a longitudinally extending axis of a molecule makes with a layer normal in a layer of a free-standing Smectic liquid crystal film, the method comprising determining a change in the thickness of the layer resulting from the angle change, the change in the thickness of the layer being determined using the method for determining the distance between two spaced apart surfaces according to the invention by determining the distance between respective surfaces of the Smectic liquid crystal film before the angle change, and after the angle change.
Additionally, the invention provides apparatus for determining distance between two spaced apart surfaces, the apparatus comprising:
a means for sampling values of an interference spectrum produced by reflections of frequency components of a composite multi-frequency signal from the respective surfaces to produce a first array of sampled values of intensity of the interference spectrum and corresponding values of wavelengths of the reflected frequency components of the multi-frequency signal, a means for computing an approximate value indicative of the distance between the surfaces as a function of the wavelengths of two of the reflected frequency components of the multi-frequency signal from the first array, a means for transforming the first array into a second array of the sampled values of intensity of the interference spectrum and values of the inverse of the
IE 060 275 wavelengths of the corresponding reflected frequency components of the multifrequency signal, a means for computing at least one further approximate value indicative of the distance between the surfaces as a function of the previously computed approximate distance indicative value and the phase difference between the waveform of the second array and a corresponding waveform of constant period derived from the immediately previously computed approximate distance indicative value.
In one embodiment of the invention the means for computing each further approximate distance indicative value is adapted for computing further approximate distance indicative values until the phase difference between the waveform of the second array and the corresponding waveform of constant period derived from the immediately previously computed approximate distance indicative value is less than a predetermined phase difference value. Preferably, the predetermined phase difference value is of negligible value.
In one embodiment of the invention the means for computing each further approximate distance indicative value is adapted for computing three further approximate distance indicative values.
In another embodiment of the invention a means is provided for directing the multifrequency signal at the surfaces, the distance therebetween of which is to be
IE 060 275 determined. Preferably, the means for directing the multi-frequency signal at the surfaces, the distance therebetween of which is to be determined comprises a means for projecting light. Ideally, the means for projecting light comprises a means for projecting white light.
The invention will be more clearly understood from the following description of a preferred embodiment thereof, which is given by way of example only, with reference to the accompanying drawings, in which:
Fig. 1 is a side elevational view of a portion of a free-standing multi-layer
Smectic liquid crystal film,
Fig. 2 is a perspective view of apparatus according to the invention for determining the distance between the surfaces of the film, in other words, for determining the thickness of the film of Fig. 1,
Fig. 3 illustrates a first array illustrating reflectance values and corresponding wavelength values of an interference spectrum formed by reflected components of white light reflected from the surfaces of the Smectic liquid crystal film of Fig. 1 prepared by the apparatus of Fig. 2 by sampling the interference spectrum,
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Fig. 4 is an enlarged view of a portion of the first array of Fig. 3,
Fig. 5 illustrates a second array prepared by the apparatus of Fig. 2 from the first array whereby the second array illustrates values of reflectance and corresponding values of the inverse of the wavelengths of the corresponding reflected components of the white light, and
Fig. 6 illustrates a third array prepared by the apparatus of Fig. 2.
Referring to the drawings, a method according to the invention for determining the distance between two spaced apart surfaces will now be described. In this embodiment of the invention the method is used for determining the thickness of a free-standing multi-layer Smectic liquid crystal film, in order that a change in the tilt angle Θ of molecules in layers of the film 1 can be determined. Accordingly, the method according to the invention is used for determining the thickness of the film 1 before and after the change in the tilt angle Θ, so that the change in the tilt angle Θ can be determined. Before describing the method according to the invention in detail, the film 1 will first be described.
Referring to Fig. 1, the film 1 comprises opposite first and second surfaces 3 and 4, within which a plurality of layers 5 of molecules 6 of the Smectic liquid crystal are contained. Although only one molecule 6 is illustrated in one of the layers 5, each
IE 0 6 0 27 5 layer 5 comprises a plurality of molecules 6, and additionally, while for convenience only three layers 5 of the film 1 are illustrated, it will be understood by those skilled in the art that the film 1 will comprise many more layers than three, and typically, the number of layers in the film 1 will be in the order of thousands or tens of thousands.
The molecules 6 of each layer 5 in this case are relatively long molecules, and each define a longitudinal axis 7 which extends at a tilt angle Θ to a layer normal 8. In general, the molecules 6 are arranged in the respective layers 5 with their longitudinal axes 7 substantially parallel to each other. The tilt angle Θ of the molecules 6 may range from zero upwards, but typically lies in the range of 0° to 25°. Since the method according to the invention is used for determining a change in the tilt angle Θ of the molecules 6 of the layers 5, the method according to the invention is used to determine two values of the thickness of the film 1, in other words, to determine two values of the distance between the opposite first and second surfaces 3 and 4 of the film 1, one value of the distance between the opposite first and second surfaces 3 and
4 is determined before the change in tilt angle Θ of the molecules 6 occurs, and the other value of the distance between the first and second surfaces 3 and 4 is determined after the change in the tilt angle Θ of the molecules 6 has occurred. By knowing the change in the distance between the first and second surfaces 3 and 4 as a result of the change in the tilt angle Θ of the molecules 6, the actual change in the tilt angle Θ can be determined as will be described below.
Referring now to Fig. 2, apparatus also according to the invention, indicated
IE 060 275 generally by the reference numeral 10, is illustrated for carrying out the method according to the invention for determining the distance between a pair of spaced apart surfaces, in this case the first and second surfaces 3 and 4 of the film 1. The apparatus 10 comprises a stand 11 having a base 12 and a support member 13 extending upwardly from the base 12. A platform 14 is slideably carried on the support member 13 so that the height of the platform 14 above the base 12 is variable. An orifice 15 extends through the platform 14 for accommodating the free-standing film 1 stretched across the orifice 15. The stand 11 may be placed in an oven with the free-standing film 1 stretched across the orifice 15 in the platform 14 for heating the free-standing film 1, so that the change in the tilt angle Θ of the molecules 6 of the film 1 as a result of heating the film 1 can be determined.
A composite multi-frequency signal source, which in this embodiment of the invention is a white light source 16 directs white light through a reflection probe attachment 17 at the free-standing film 1. A bracket 18 slideably carried on the support member 13 of the stand 11 secures a probe 19 of the reflection probe attachment 17 to the stand 11 for directing the white light normal to the first surface 3 of the film 1. Reflected components of the white light reflected from the first and second surfaces 3 and 4 of the film 1, which form an interference spectrum are picked up by the probe 19, and transmitted by the reflection probe attachment 17 to an optical spectrometer 24 where the interference spectrum is analysed. Set screws 21 and 22 secure the platform 14 and the bracket 18 at respective desired positions in the
IE Ο 6 Ο 2 7 5 support member 13.
The optical spectrometer 24 is a fibre optic spectrometer sold under the Trade Mark AVASPEC-2048, and has no moving parts, and is capable of acquiring thousands of reflection spectra with a rate better than one spectrum per minute. Data relating to the sampled values of the interference spectrum are read from the spectrometer 24 by a computer 25 which is programmed to carry out the method according to the invention for determining the distance between the first and second surfaces 3 and 4 of the film 1, as will be described below. Data and waveforms and other relevant data is displayed by the computer 25 on a display screen 26, and a keyboard 27 is provided for inputting commands to the computer 25.
Since the method according to the invention is light based, it may be used for determining the thickness of a transparent medium, and the method according to the invention may also be used for determining the distance between two surfaces of respective media, where the adjacent surfaces of the respective media are spaced apart, and at least one of the media is transparent, and the media are spaced apart by, for example, a vacuum, an air gap or by any other transparent medium between the adjacent surfaces, the distance therebetween is to be determined.
Each value of the distance between the first and second surfaces 3 and 4 is determined using the method according to the invention as follows.
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In accordance with the method of the invention, the spectrometer 24 on sampling the interference spectrum produces a first array of intensity values of the interference spectrum resulting from the reflected components of the white light from the first and second surfaces 3 and 4, with corresponding values of the wavelengths of the reflected components. In this case the first array is prepared with sampled reflectance values and the corresponding values of the wavelengths of the reflected components of the white light. A typical first array of reflectance and corresponding wavelength values is illustrated in Fig. 3, and an enlarged portion of the first array of the values of the reflectance and the corresponding wavelengths is illustrated in Fig. 4. In Figs.
and 4 the values of the first array are illustrated by dots 31 and a waveform 32 is interpolated from the dot values 31 by the spectrometer 24. In Figs. 3 and 4 reflectance is plotted on the Y-axis and wavelength is plotted on the X-axis. Once the first array of sampled values has been prepared by the fibre optic spectrometer 24, an initial approximate value do indicative of the distance between the first and second surfaces 3 and 4 of the film 1 is computed by the computer 25 from the first array.
The initial approximate value do indicative of the distance between the first and second surfaces 3 and 4 of the film 1 is computed in this embodiment of the invention as an approximate optical distance do between the first and second surfaces 3 and 4.
The computer 25 is programmed to compute the initial approximate value of the optical distance do between the first and second surfaces 3 and 4 from the equation:
d0 =
4(+- + )
0)
II 060275 where do is the approximate optical distance between the respective surfaces, λ i is the wavelength of one of the reflected components of the white light derived from the first array,
12 is the wavelength of another one of the reflected components of the white light derived from the first array, k is the number of times the waveform 32 derived from the first array crosses a line 33, the value of which is approximately equal to the average of values of the waveform 32.
In this case the computer 25 has taken the values of the wavelength Xj and A2 as being peak value of the waveform 32, namely, the wavelength values 818nm and 855nm, respectively, and thus the value of k is 12, see Fig. 4. The initial approximate value of the optical distance do is stored by the computer 25 from equation (1).
The computer 25 is programmed to prepare a second array from the first array by transforming the first array into an array of the reflectance values against values of the inverse of the wavelengths of the corresponding reflected components of the white light. The second array produces a waveform of substantially constant periods, and if it were ideal, the waveform of the second array would be of constant periods. The second array is illustrated in Fig. 5, where the values of the array are illustrated by dots 35. A waveform 36 interpolated from the dot values 35 is also illustrated in
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Fig. 5. In Fig. 5 reflectance is plotted on the Y-axis, and the inverse values of the wavelengths of the reflected components of the white light are plotted on the X-axis.
The computer 25 is programmed to sequentially determine further approximate values indicative of the distance d between the first and second surfaces 3 and 4, and each further approximate distance indicative value is determined as an approximate optical distance value as a function of the immediately previously computed approximate distance value do and the phase difference between the waveform 36 of the second array and a corresponding ideal waveform of constant periods derived from the immediately previously computed approximate optical distance value. In this case the computer 25 computes each further approximate optical distance value from the sum of the immediately previously computed approximate optical distance value and the quotient of the derivative of the tan of the phase difference between the waveform 36 of the second array and a corresponding ideal waveform of constant periods derived from the immediately previously computed approximate optical distance value, divided by the square of the tan of the phase difference. The further approximate optical distance values are computed until the phase difference between the waveform 36 of the second array and the corresponding ideal waveform of constant periods derived from the immediately previously computed approximate optical distance value is negligible. The computer 25 constructs the ideal waveform of constant periods corresponding to the waveform 36 of the second array and determines the phase difference between the two waveforms by determining the
IE 06 0 275 difference of the areas under the respective waveforms above a datum line of value less than or equal to the minimum value of the reflectance values of the respective waveforms. For convenience, the computer 25 produces the tan of the phase difference between the waveform 36 and the ideal corresponding waveform from the difference of the respective areas. In this embodiment of the invention the tan of the phase difference between the waveform 36 and the ideal corresponding waveform is computed from the following equation:
Σ sin(4 wdgX, )(Rt - Raverage )(xM - x.)
A,. = - (2)
Σ cos^xd^ )(R, - R )(xM - xt) where
A, is the tan of the phase difference between the waveform 36 and the corresponding ideal waveform, do is the immediately previously computed approximate optical distance between the first and second surfaces 3 and 4,
Xi is the value of the inverse of the wavelength of the ith reflected component of the white light from the second array,
Xi+ι is the value of the inverse of the wavelength of the reflected component after the ith reflected component of the white light from the second array,
Ri is the value of the reflectance corresponding to the ith reflected frequency component of the white light from the second array, and
Raverage is the average of the reflectance values from the second array.
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The computer 25 then prepares a third array of the values of the tan of the phase differences Ai and the corresponding values of the inverse of the wavelengths of the reflected components of the white light. The third array is illustrated in Fig. 6. The values of the third array are illustrated by dots 38 in Fig. 6, and a waveform 39 interpolated from the values 38 of the third array is also illustrated in Fig. 6. In Fig. 6 the values of the tan of the phase differences A; are plotted on the Y-axis and the values of the inverse of the wavelength of the reflected components of the white light are plotted on the X-axis.
Once the third array of Fig. 6 has been prepared, each further approximate optical distance value is computed from the third array using the following equation:
d - do----r- (3)
2π(1 + (axy<N + b) ) where d is the newly computed approximate optical distance value, do is the immediately previously computed approximate optical distance value, a represents the derivative of the tan of the phase difference between the waveform of the second array and the corresponding ideal waveform of constant periods derived from the immediately previously computed approximate optical distance value, the function axw+b represents the tan of the phase difference, and
IE 06 0 275 is a specific value as will be described below of the inverse of the wavelengths of the reflected components of the white light.
In order to produce the values of a and (ax%N+b) for equation (3), the computer 25 is programmed to fit a linear function, namely, the function y = ax + b using the lesssquare method to a part of the waveform 39 of the third array of Fig. 6 where the noise in the waveform 39 is at a minimum. In this case, the computer 25 fits the linear function y = ax + b to the waveform 39 of the third array of Fig. 6 intermediate the inverse values of the wavelengths of the reflected components of the white light between the values 0.00110 and 0.11105. The computer 25 from the fitted linear function^ = ax + b determines the derivative of the tan of the phase difference between the two waveforms from the value of the slope a of the linear function y = ax + b, and determines the inverse value of the wavelength at the position 3/
As each new approximate value of the optical distance d between the first and second surfaces 3 and 4 is computed by the computer 25, the new value becomes the value do for use in the computation of the immediately following approximate optical distance
IE 060275 value to be computed. In this case the computer programme 25 is programmed to compute three further approximate optical distance values after the first approximate distance value has been computed. It has been found that the phase difference between the waveform 36 of the second array and the corresponding ideal waveform of constant periods is negligible, and the two waveforms are effectively in phase, and thus, the values of the optical distance d computed stabilises and the optical distance d is thus computed with a resolution better than 0.01%.
Since the distance between the first and second surfaces 3 and 4 of the film 1 is computed by the method according to the invention as the optical distance, the actual physical distance between the first and second surfaces 3 and 4 is determined by dividing the computed optical distance by the effective refractive index of the medium of the film 1. Thus, the thickness d/οϊ the film is obtained by dividing the last of the computed values of the optical thickness d by the effective refractive index of the film 1.
Indeed, compensation for the refractive index of the medium of the film 1 may be made as the approximate optical distance values are being sequentially computed, and additionally, compensation for the refractive index of the medium of the film may be made prior to computing the tan of the phase difference from equation (2) above.
This will be well known to those skilled in the art.
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Once the thickness dp of the film 1 prior to the change of the tilt angle 0 of the molecules of the Smectic liquid crystal film has been determined by the method according to the invention, and the thickness dp of the film after the change in tilt angle 0 has been determined, the change in the tilt angle Θ can readily be determined from the equation:
= arccos dftcosdi dfi (4) where is the value of the angle 0 after the change in the tilt angle,
0i is the value of the tilt angle prior to the change in the tilt angle, dp is the thickness of the film 1 prior to the change in the tilt angle, df2 is the thickness of the film 1 after the change of the tilt angle.
The following is an explanation of the theory behind the method of the invention, 15 which essentially is based on the theory of a phase-locked loop algorithm.
From Fig. 4:
2nd = NX, (5) (it is also called “constructive interference”) where !£ 06 0 275 η is the refractive index of the film medium, d is the actual thickness of the film, λι is the wavelength of one of the peak values of the waveform of Fig. 4, and
N (or N+ % in some cases) is the number of the peak value of the waveform 5 of Fig. 4 at which the value of the wavelength is 1/.
2nd = (N + ηι)λ2 for another maximum (6) where λ2 is the wavelength of the other of the peaks of the waveform of Fig. 4, and m is the number of peaks between the peak where the wavelength is of value 10 λ, and the peak where the wavelength is of value λ2.
Solving the system of equations (5) and (6)
2nd
T
2nd 2nd
T + m A,
2λ{ηά = 2ηάλ2 + ηιλ2λ}
We can easily get the thickness (7)
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Equation (7) can be written as:
where
4η(λ, -+) (8) k is the number of times the waveform of Fig. 4 crosses the line 33 between the peaks corresponding to the values of λι and λ2, respectively.
It is convenient to work with inverse of the wavelength x = —, since the interference A spectrum is an oscillatory function of x with constant frequency as shown in Fig. 5, 10 and if an ideal waveform, would be of constant frequency.
We can approximate the waveform 36 of Fig. 5 with a sine wave:
Rexp = Rosin(4xxd + φ) (9) where Ro and φ are the amplitude and the phase, respectively, of the interference 15 fringes. They are unknown and will have to be cancelled out, d is the thickness of the film which is to be determined, namely, the distance between the first and second
IE Ο 6 Ο 2 7 5 surfaces 3 and 4.
Let us now consider d0 to be the initially computed value of the thickness obtained from equation (8) where d is substituted with do. Then we can represent the real thickness of the film as d = d0 + Ad. And assume Ad <0. Now we can use a method common in telecommunications and construct the following integral:
- jsin(4πxd0)Rexpdx = ^sin(4nxd0)R0sin(4n(d0+Αά)χ+φ)άχ
After integration we obtain:
' sin(4nxAd+(p) sin(4nx(2d0 + Ad) + 0+ Ad) J
Since Ad « d0 we can neglect the second term to simplify the expression, but we still have too many unknowns to find Ad. But we can make one more integral:
Icos - jcos(4nxd0)Rapdx = ^cos(4jtxd0)R0sin(47c(d0 + Ad)x+g cos(4itxAd+φ) cos(47ix(2d0+ Αά) + φ)
4πΑά 4π(2ά0+Ad)
Since Ad « d0 we can neglect the second term to simplify the expression, but we still have too many unknowns to find Ad. Now we can eliminate the unknown Ro by dividing Isin over Icos. This gives us At, namely, the tan of the phase difference between the waveform 36 and the corresponding ideal waveform of constant periods.
_ Bin ~ sin(4nxAd+
Icos cos(4itxAd+(p)
Now we are to remove φ. To do this let us take the derivative dx
- -4 add ' sin(4itxAd+ φ) j:os(4KxAd+(p)
2\
-4icAd(I + A? )
Therefore:
dAjdx
4π(1 + Α?) and d — d0 dAjdx 4π(Ι + Α? ) (10)
In terms of digital computations Isi„ and Icos to be replaced by sums:
i i ^sm^nd^x^Ri-R^^XM-Xi) and £cos(4W0xf)(2f,
Raverage )(^7+1 ^7')
0 (11) respectively. To find the derivative dAjdx. we perform the fitting of the array T,· (x,·) by the linear function y = ax + b using the standard less-square method. A, in the equation (10) can also be obtained as a point of the straight line used for fitting. In order to minimise the error caused by the neglecting the second terms in the Isi„ and
Icos we perform the fitting only for the second half of the array as shown in Figure 6.
,£ 06 0275
It has been found experimentally that the formula with coefficient 'Λ works better then with coefficient % as in equation (10) (more advanced mathematics with less assumptions and approximations are required to explain this). However, the following equation gives adequate results:
d = d0----γ- (12) 0 22)
The updated value of d can be used as the value of do for the next iteration (repeating computations based on equations (11) and (12)). Usually after three to four iterations the calculated thickness is stabilised within the specified resolution.
While the method according to the invention has been described for determining the distance between opposite surfaces of a film of a Smectic liquid crystal, the method may be used for determining the distance between any surfaces, whether the surfaces are the opposite surfaces of a medium, or spaced apart adjacent surfaces of a pair of adjacent media. Furthermore, while the method has been described for determining the change in the tilt angle of the molecules of an SmC* liquid crystalline film, the method could equally well be used for determining the change in the tilt angle of an SmC liquid crystalline film.
Additionally, it is envisaged that while the method has been described based on sampling an interference spectrum of reflected components of white light, the method could be used by sampling the interference spectrum of any type of multi-frequency
IE 06 0275 light signal. Indeed, it is also envisaged that the method according to the invention may be carried out by sampling an interference spectrum of reflected frequency components of any multi-frequency signal, for example, an ultrasonic signal or the like.
The invention is not limited to the embodiment hereinbefore described, which may be varied in construction and detail.
Claims (10)
1. A method for determining distance between two spaced apart surfaces comprising: sampling values of an interference spectrum produced by reflections of
2. A method as claimed in Claim 1 in which the further approximate distance θ 6 Ο 2 7 5 indicative values are computed until the phase difference between the waveform of the second array and the corresponding waveform of constant period derived from the immediately previously computed approximate distance indicative value is less than a predetermined phase difference value.
3. A method as claimed in Claim 2 in which the predetermined phase difference value is of negligible value. 4. = IE 060 275 where A, is the tan of the phase difference, do is the immediately previously computed approximate distance indicative value, 4(+- + ) where do is the approximate value indicative of the distance between the surfaces, Ai is the wavelength value of one of the reflected frequency components of the 20 multi-frequency signal from the first array, λ 2 is the wavelength value of another one of the reflected frequency components of the multi-frequency signal from the first array, and •Ε 0
4. A method as claimed in any preceding claim in which three further 10 approximate distance indicative values are computed. 5. Distance between the surfaces as a function of the previously computed approximate distance indicative value and the phase difference between the waveform of the second array and a corresponding waveform of constant period derived from the immediately previously computed approximate distance indicative value. 10 24. Apparatus as claimed in Claim 23 in which the means for computing each further approximate distance indicative value is adapted for computing further approximate distance indicative values until the phase difference between the waveform of the second array and the corresponding waveform of constant period derived from the immediately previously computed approximate distance indicative 15 value is less than a predetermined phase difference value. 25. Apparatus as claimed in Claim 24 in which the predetermined phase difference value is of negligible value. 20 26. Apparatus as claimed in any of Claims 23 to 25 in which the means for computing each further approximate distance indicative value is adapted for computing three further approximate distance indicative values. IE Ο 6 ο 275 27. Apparatus as claimed in any of Claims 23 to 26 in which a means is provided for directing the multi-frequency signal at the surfaces, the distance therebetween of which is to be determined. 28. Apparatus as claimed in Claim 27 in which the means for directing the multifrequency signal at the surfaces, the distance therebetween of which is to be determined comprises a means for projecting light. 5 liquid crystal film, the method comprising determining a change in the thickness of the layer resulting from the angle change, the change in the thickness of the layer being determined using the method for determining the distance between two spaced apart surfaces as claimed in any preceding claim by determining the distance between respective surfaces of the Smectic liquid crystal film before the angle change, and 10 after the angle change. 23. Apparatus for determining distance between two spaced apart surfaces, the apparatus comprising: a means for sampling values of an interference spectrum produced by 15 reflections of frequency components of a composite multi-frequency signal from the respective surfaces to produce a first array of sampled values of intensity of the interference spectrum and corresponding values of wavelengths of the reflected frequency components of the multi-frequency signal, a means for computing an approximate value indicative of the distance 20 between the surfaces as a function of the wavelengths of two of the reflected frequency components of the multi-frequency signal from the first array, a means for transforming the first array into a second array of the sampled IE Ο 6 Ο 2 7 5 values of intensity of the interference spectrum and values of the inverse of the wavelengths of the corresponding reflected frequency components of the multifrequency signal, a means for computing at least one further approximate value indicative of the 5 Xi is the inverse wavelength value of the ith reflected frequency component of the multi-frequency signal from the second array, Xj+i is the inverse wavelength value of the next reflected frequency component after the ith reflected frequency component of the multi-frequency signal from the second array, 10 Ri is the intensity of the interference spectrum corresponding to the ith reflected frequency component of the multi-frequency signal from the second array, and Raverage is the average of the intensity values of the interference spectrum from the second array. 12. A method as claimed in Claim 11 in which a third array is prepared, the third array comprising the values of the tan of the phase difference and corresponding inverse wavelength values of the reflected frequency components of the multifrequency signal, and a linear function y = ax + b is fitted to the waveform of the 20 third array for determining the derivative of the tan of the phase difference. 13. A method as claimed in Claim 12 in which the function y = ax + b is fitted to IE 06 0275 the waveform of the third array in a portion of the waveform of minimum noise. 14. . A method as claimed in Claim 12 or 13 in which the function y = ax + b is fitted to the waveform of the third array by a less-square method. 15. A method as claimed in any of Claims 12 to 14 in which each further approximate distance value is derived from the equation: d = do----— 2π(1 + (axy N +b) ) where 10 dis the approximate distance indicative value currently being computed, do is the immediately previously computed approximate distance indicative value, a is the derivative of the tan of the phase difference between the waveform of the second array and the corresponding waveform of constant period derived from the 15 immediately previously computed approximate distance indicative value, the function (ax%N+b) represents the tan of the phase difference between the waveform of the second array and the corresponding waveform of constant period derived from the immediately previously computed approximate distance indicative value, and 20 X’/„n is the value of the inverse wavelength value of the reflected frequency component of the reflected multi-frequency signal corresponding approximately to the mid-value of the reflected frequency components of the multi-frequency signal in IE 06 0 275 the upper half of the inverse wavelength values of the third array. 16. A method as claimed in any preceding claim in which the multi-frequency signal is a light signal. 17. A method as claimed in any preceding claim in which the multi-frequency signal is white light. 18. A method as claimed in Claim 16 or 17 in which each approximate value 10 indicative of the distance between the surfaces is computed as an approximate value of the optical distance between the surfaces. 19. A method as claimed in any of Claims 16 to 18 in which the actual value of the distance between the surfaces is computed as a function of the last of the 15 computed further approximate distance indicative values and the refractive index of a medium between the surfaces. 20. A method as claimed in any preceding claim in which the multi-frequency signal is directed substantially normal to the respective surfaces. 21. A method for determining distance between two spaced apart surfaces, the method being substantially as described herein with reference to and as illustrated in IE 06 0 275 the accompanying drawings. 22. A method for determining a change in an angle a longitudinally extending axis of a molecule makes with a layer normal in a layer of a free-standing Smectic 5 difference between the waveform of the second array and the corresponding waveform of constant period derived from the immediately previously computed approximate distance indicative value. 10. A method as claimed in Claim 8 or 9 in which each further approximate 10 distance indicative value is computed as a function of the quotient of the derivative of the tan of the phase difference between the waveform of the second array and the corresponding waveform of constant period derived from the immediately previously computed approximate distance indicative value divided by the tan of the phase difference. 11. A method as claimed in of Claims 8 to 10 in which the tan of the phase difference between the waveform of the second array and the corresponding waveform of constant period derived from the immediately previously computed approximate distance indicative value is derived from the following equation: '£sin(4jtd o x l )(Rf - R average )(x M -x.) 0 u average· ^cos(4Ttd 0 x i )(R i -R l average )(x M -Xi) 5 6. A method as claimed in any preceding claim in which the phase difference between the waveform of the second array and each corresponding waveform of constant period derived from the immediately previously computed approximate distance indicative value is computed as a function of the difference in the areas under the respective waveforms and a datum value, the datum value being less than 10 or equal to the minimum value of the intensity values of the respective waveforms between predetermined values of the inverse of the wavelength of the reflected components of the multi-frequency signal.
5. A method as claimed in any preceding claim in which the approximate value indicative of the distance between the surfaces computed from the first array as a function of the wavelengths of two of the reflected frequency components of the 15 multi-frequency signal is computed from the following equation: _ kA,A 2 5 frequency components of a composite multi-frequency signal from the respective surfaces to produce a first array of sampled values of intensity of the interference spectrum and corresponding values of wavelengths of the reflected frequency components of the multi-frequency signal, computing an approximate value indicative of the distance between the 10 surfaces as a function of the wavelengths of two of the reflected frequency components of the multi-frequency signal from the first array, transforming the first array into a second array of the sampled values of intensity of the interference spectrum and values of the inverse of the wavelengths of the corresponding reflected frequency components of the multi-frequency signal, 15 computing at least one further approximate value indicative of the distance between the surfaces, each further approximate distance indicative value being computed as a function of the immediately previously computed approximate distance indicative value and the phase difference between the waveform of the second array and a corresponding waveform of constant period derived from the 20 immediately previously computed approximate distance indicative value.
6. 0 275 k is the number of times the waveform of the first array crosses an imaginary line, the value of which is approximately equal to the average of the values of the waveform.
7. A method as claimed in Claim 6 in which the predetermined values of the 15 inverse of the wavelengths of the reflected frequency components of the multifrequency signal are determined by the sampled values of the interference spectrum from which the first array is produced.
8. A method as claimed in any preceding claim in which each further 20 approximate distance indicative value is computed as a function of the tan of the phase difference between the waveform of the second array and the corresponding waveform of constant period derived from the immediately previously computed IE 060 275 approximate distance indicative value.
9. A method as claimed in Claim 8 in which each further approximate distance indicative value is computed as a function of the derivative of the tan of the phase
10. 29. Apparatus as claimed in Claim 28 in which the means for projecting light comprises a means for projecting white light.
Priority Applications (2)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| IE20060275A IE20060275A1 (en) | 2006-04-10 | 2006-04-10 | A method and aparatus for determining distance between two spaced apart surfaces |
| PCT/IE2007/000046 WO2007116388A1 (en) | 2006-04-10 | 2007-04-05 | A method and apparatus for determining distance between two spaced apart surfaces |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| IE20060275A IE20060275A1 (en) | 2006-04-10 | 2006-04-10 | A method and aparatus for determining distance between two spaced apart surfaces |
Publications (1)
| Publication Number | Publication Date |
|---|---|
| IE20060275A1 true IE20060275A1 (en) | 2007-11-14 |
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ID=38179854
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| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| IE20060275A IE20060275A1 (en) | 2006-04-10 | 2006-04-10 | A method and aparatus for determining distance between two spaced apart surfaces |
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| IE (1) | IE20060275A1 (en) |
| WO (1) | WO2007116388A1 (en) |
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| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN111338332A (en) * | 2018-11-30 | 2020-06-26 | 宝时得科技（中国）有限公司 | Automatic walking equipment and obstacle avoidance method and device thereof |
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| Publication number | Priority date | Publication date | Assignee | Title |
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| JP7206576B2 (en) * | 2019-03-27 | 2023-01-18 | 株式会社東京精密 | Measuring method and equipment |
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| UA74801C2 (en) * | 2002-05-20 | 2006-02-15 | Swedish Lcd Ct | Method for measuring the parameters of a liquid crystal cell |
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- 2006-04-10 IE IE20060275A patent/IE20060275A1/en not_active Application Discontinuation
-
2007
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Cited By (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN111338332A (en) * | 2018-11-30 | 2020-06-26 | 宝时得科技（中国）有限公司 | Automatic walking equipment and obstacle avoidance method and device thereof |
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| Publication number | Publication date |
|---|---|
| WO2007116388A1 (en) | 2007-10-18 |
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