CN103941040B - Based on the device and method of the rear orientation light sense acceleration of nanoparticle detection - Google Patents

Abstract

The invention discloses a kind of device and method of the rear orientation light sense acceleration based on nanoparticle detection, this device comprises reflective mirror, light damping plate, beam splitter, optical filter, the first condenser lens, the first nanoparticle, the second microparticles, grinding core optical fiber, rigid connecting rod, the second condenser lens, pin hole, separated light detector, laser instrument, collimation lens, light intensity modulator, dsp processor, microchannel; The present invention is based on highly sensitive nano-grade displacement to measure, make this structure can obtain high acceleration analysis resolving power, and be only confined to the absolute value of optical pressure restoring force, simultaneously the control loop of closed loop can significantly extended dynamic measurement range; The present invention measures the acceleration of nanometer scale displacement under the condition not having physical contact, and to non-insensitive by other acting forces on measuring acceleration direction, there is low noise, high sensitivity, carry the feature of feedback function, high precision identification and analysis.

Description

Based on the device and method of the rear orientation light sense acceleration of nanoparticle detection

Technical field

The present invention relates to the detection method that a kind of nanometer displacement accelerates, particularly relate to a kind of device and method regulating the nanometer displacement acceleration detection of feedback system based on the detection of nanoparticle rear orientation light and optical pressure.

Background technology

The acting force that the size that newton second law of motion describes object acceleration is subject to object is directly proportional, and the quality with object is inversely proportional to, and the direction of acceleration is identical with the direction of bonding force, just can be realized the measurement of acceleration by power.The contactless support that current high precision acceleration transducer uses the technology such as electrostatic suspension technology or magnetic levitation to realize sensing quality mostly.

A group with the photon stream that the light velocity moves, existing quality has again momentum by the known light beam of quantum theory.When light beam, at dielectric surface, refraction and reflection occurs, the speed of photon and direction change, thus cause the change of its momentum vector, and also namely light beam exists the effect of power to particulate, and be called optical radiation pressure, the minimum particulate yardstick of optical radiation pressure effect is in micron dimension.Along with the widespread use to optical radiation pressure, nowadays also occurred utilizing laser capture technology to realize the non-contact support to sensing quality, these technology greatly reduce the measuring error that conventional contact supports accelerometer.The method that common laser capture technology realizes acceleration analysis is that optical fiber light trap acceleration is measured, it utilizes and produces radiation pressure from many Gaussian beams with the direct outgoing of optical fiber at particle surface, realize the hovering of particle, and observe with the skew of photoelectric image detector to particle, thus realize the adjustment of optical pressure and the measurement of acceleration.The shortcoming of this measurement mechanism is, utilize and as light tweezer, particle is caught from the Gaussian beam of the direct outgoing of optical fiber, the beam waist diameter of Gaussian beam is similar to the mode field diameter of optical fiber, thus its particle size that can catch be restricted can not be too small, and force trapping is not strong.And adopt the skew of photoelectric image detector to particle to measure, can only detect the movement of particle micron dimension, be the principal element limiting this acceleration detection method precision.

To fiber exit end grinding core, can the Gaussian beam of script outgoing be converted to accurate bessel beam, a full width at half maximum magnitude less of Gaussian beam of this light beam, and energy is more concentrated, so less particle can be caught, produce stronger radiation pressure, the sensitivity of detection and investigative range are increased all greatly.

Unmarked (as fluorescence labeling, golden nanometer particle etc.) direct-detection aspect, devote substantial resources detects this problem to nanoparticle yardstick and studies in the world in recent years, representative achievements has the direct optical detection method based on nanoparticle elastic scattering proposed by RochesterUniv. optics institute, the method is the amplitude of the scattered light based on interferometry electromagnetic field, utilize and receive the combination of Flow Control passage and optical interference, and build two adjacent, identical Michelson optical interference circuit, when detecting nano particle by optical interference circuit, nanoparticle scattered light to make a difference signal to two interference optical field amplitudes, its amplitude is associated with the size of nanoparticle, show that nanoparticle can be recorded by its scattered optical field.

Summary of the invention

The object of the invention is to for the deficiencies in the prior art, a kind of device and method of the rear orientation light sense acceleration based on nanoparticle detection is provided.

The object of the invention is by realizing with lower device: a kind of device of the rear orientation light sense acceleration based on nanoparticle detection, comprising: reflective mirror, light damping plate, beam splitter, optical filter, the first condenser lens, the first nanoparticle, the second microparticles, grinding core optical fiber, rigid connecting rod, the second condenser lens, pin hole, separated light detector, laser instrument, collimation lens, light intensity modulator, dsp processor, microchannel; Wherein, one end of rigid connecting rod is fixedly connected with the first nanoparticle, the other end is through after the second microparticles, and be rigidly connected in the fixing turning point bottom microchannel, the first nanoparticle and the second microparticles can be used as an entirety and rotate around this fixing turning point; When rigid connecting rod is vertical, two grinding core optical fiber of the second microparticles and its both sides are on same level line, and two grinding core optical fiber respectively connect a light intensity modulator; Separated light detector is all connected with dsp processor with two light intensity modulators; The laser that laser instrument sends enters beam splitter after being collimated by collimation lens; A part is reflected by beam splitter, through light damping plate decay, mirror reflection, again after light damping plate decay, forms reference light by beam splitter transmission; Another part is focused into a hot spot through optical filter by the first condenser lens by after beam splitter transmission, first nanoparticle is in this hot spot, the rear orientation light of the first nanoparticle is reflected by beam splitter after the first condenser lens is collected and optical filter filters, interfere with reference light, after being focused on by the second condenser lens, through pin hole, received by separated light detector; The light signal of collection is converted into electric signal and is sent to dsp processor by separated light detector; Dsp processor adjusts the light intensity of two grinding core optical fiber outputs by two light intensity modulators, realizes the adjustment of the second microparticles position, thus realizes the adjustment to the first nanoparticle position.

Apply a method for the rear orientation light sense acceleration based on nanoparticle detection of said apparatus, comprise the following steps:

Step 1: device being placed on acceleration is in the environment of zero, regulate the second microparticles and the first nanoparticle entirety in center, two duplicate grinding core optical fiber and the second microparticles center are in the same horizontal line, the optical pressure effect that after the light intensity that grinding core optical fiber is incident equal, second microparticles is balanced, rigidly connected first nanoparticle is also stable at center with it;

Step 2: the optical wavelength of laser instrument injection is different from the light in grinding core optical fiber, is radiated on beam splitter after collimation lens lens; The light beam reflected to return and again by beam splitter, as the reference beam of relevant detection through light damping plate and the former road of reflective mirror; The light transmitted focuses on the second microparticles by the first condenser lens after optical filter, its rear orientation light is collected by the first condenser lens, as detected light beam after beam splitter reflection, collected by the second condenser lens after relevant with reference light, incide after a pin hole elimination high fdrequency component and environment parasitic light on separated light detector; Regulate the position of hot spot on separated light detector, make its hot spot when not having acceleration be in the center of separated light detector, now the differential signal of separated light detector two halves is zero, and feedback system does not regulate the light intensity of grinding core optical fiber;

Step 3: when device has acceleration in grinding core optical fiber direction, the second microparticles departs from center, and rigid connecting rod is that axle drives the second microparticles to deflect thereupon with point of fixity; Owing to being measure in nano level displacement, the circular arc displacement of the first nanoparticle can be similar to regards straight line as; And the displacement of the first nanoparticle is also by the displacement equations of the second microparticles, enlargement factor to be determined to Distance geometry first nanoparticle of point of fixity to the ratio of distances constant of the second microparticles by the second microparticles; Now the hot spot of backscatter signal on separated light detector of the first nanoparticle deviate from center, and offset direction is determined by acceleration direction, and side-play amount is determined by acceleration magnitude;

Step 4: extract and process interference data: after the rear orientation light of the first nanoparticle and reference light are interfered, the light intensity branch be irradiated on separated light detector can be expressed as:

$I={\left|E_r\right|}^2+{\left|E_s\right|}^2+2\mathrm{Re}\left\{E_r^{*}{E}_s\right\}$

E in formula _rfor reference light oscillator intensity, E _sfor scattered light oscillator intensity, I is the light intensity that detector just receives;

The signal P that separated photo-detector records can be expressed as:

$P=\frac{{\∫}_{A1}\mathrm{Ids}-{\∫}_{A2}\mathrm{Ids}}{{\∫}_{A1}\mathrm{Ids}+{\∫}_{A2}\mathrm{Ids}}$

Wherein A1 and A2 represents the two halves up and down of detector surface respectively, and ∫ ds represents the area integral surveyed separated light detector; When the first nanoparticle is in center and system acceleration is zero, reference light and rear orientation light are adjusted to the center of separated light detector, so differential signal P (t) is zero; When the first nanoparticle is not in center and system acceleration is non-vanishing, detectable signal P (t) is expressed as by interference term:

$P\left(t\right)=2\mathrm{Re}\{({\∫}_{A1}{E}_r^{*}{E}_s\mathrm{ds}-{\∫}_{A2}{E}_r^{*}{E}_s\mathrm{ds})/{\∫}_{A1+A2}{\left|E_r\right|}^2\mathrm{ds}\}$

Because much bigger with reference to beam intensity ratio scattered light intensity in this device, the scattered light strong point in molecule can be ignored | E _s| ²; In like manner, the reference light strong point only retained in denominator | E _r| ², and ignore all E _s;

Step 5: carry out feedback regulation and obtain acceleration: differential signal S (t) obtained according to separated light detector; dsp processor starts the light intensity difference regulating two grinding core optical fiber; in the direction of the extraneous acceleration of opposing, larger optical pressure is applied to the second microparticles; second microparticles is recentered, and differential signal S (t) that criterion is the acquisition of separated light detector is zero.Now, the optical pressure acting force be applied on the second microparticles by the direction of measuring acceleration and grinding core optical fiber is contrary, and size is determined by the optical pressure difference of two grinding core optical fiber, is provided by dsp processor; Acceleration calculation process is as follows:

a＝K ₁K ₂I(1+R)/c

Wherein, a is institute's measuring acceleration, and I is the luminous energy unit time impinging perpendicularly on unit area, and R is the energy reflectivity on surface, and c is the light velocity in vacuum, K ₁be a coefficient, K ₂for another coefficient.

A kind of device of the rear orientation light sense acceleration based on nanoparticle detection, comprise: the first laser diode, second laser diode, wavelength-division multiplex wave multiplexer, isolator, power bifurcated device, circulator, collimating apparatus, division dual wavelength catoptron, large-numerical aperture lens, attenuator, first Wave decomposing multiplexer, second Wave decomposing multiplexer, first coupling mechanism, second coupling mechanism, first photodiode, second photodiode, amplifier, analog-digital converter, dsp chip, first nanoparticle, second microparticles, grinding core optical fiber, rigid connecting rod, light intensity modulator, microchannel, wherein, first laser diode is all connected with wavelength-division multiplex wave multiplexer with the second laser diode, wavelength-division multiplex wave multiplexer, isolator are connected successively with power bifurcated device, the output terminal of power bifurcated device connects circulator and attenuator respectively, Second Wave division multiplexer is connected with circulator, circulator connects collimating apparatus, and collimating apparatus is connected with microchannel successively with division dual wavelength catoptron, large-numerical aperture lens, attenuator is connected with first wave division multiplexer, the side of the first coupling mechanism connects the shortwave output terminal of first wave division multiplexer and the input end of the first photodiode respectively, opposite side connects the output terminal of the second Wave decomposing multiplexer, the side of the second coupling mechanism connects the long wave output terminal of the first Wave decomposing multiplexer and the input end of the second photodiode respectively, opposite side connects the output terminal of the second Wave decomposing multiplexer, the output terminal of the first photodiode is all connected with amplifier with the output terminal of the second photodiode, amplifier is connected with dsp chip by analog-digital converter, dsp chip is connected with two light intensity controls respectively.

Apply a method for the rear orientation light sense acceleration based on nanoparticle detection of said apparatus, comprise the following steps:

Step 1: two wavelength difference are that first laser diode of 10-20nm, the second laser diode are as light source, by wavelength-division multiplex wave multiplexer, two wavelength light beams that two light sources send are incorporated in an optical fiber, prevent echo from affecting its stability to DFB/DBR laser diode by isolator, the light of 90-99% power is separated to collimating device collimation by power bifurcated device, obtain dual wavelength collimated light beam, as detection light; Light two wavelength light after attenuator, the first Wave decomposing multiplexer of residue 1-10% power are separated, as reference light;

Step 2: dual wavelength collimated light beam is by the light beam dividing each reflection wavelength about catoptron, afterwards by large-numerical aperture lens focus two half cone-shaped light beams in microchannel, produce rear orientation light, this rear orientation light, by circulator, after the second Wave decomposing multiplexer, is input to two independently in optical interference circuit respectively by the first coupling mechanism, the second coupling mechanism;

Step 3: the signal of the first photodiode collection short wavelength rear orientation light, is namely incident in the detection light signal kept right in position in the first nanoparticle; The signal of the second photodiode collection long wavelength rear orientation light, is namely incident in the detection light signal that position in the first nanoparticle keeps left; Two photodiodes receive signal after amplifier, analog-digital converter, by dsp chip analyzing and processing data;

Step 4: extract and process interference data: after the rear orientation light of the first nanoparticle and reference light are interfered, the signal that the light intensity of two kinds of wavelength i.e. two photodiodes detect all can be expressed as:

$I={\left|E_r\right|}^2+{\left|E_s\right|}^2+2\mathrm{Re}\left\{E_r^{*}{E}_s\right\}$

E in formula _rfor reference light oscillator intensity, E _sfor scattered light oscillator intensity, I is the light intensity that detector just receives;

Differential signal after being exaggerated can be expressed as:

$S\left(t\right)=\mathrm{\α}(I_{\⋐}-I_{\⊃})$

Wherein with represent left side detection light and the right detection light respectively, α represents amplification coefficient.When the first nanoparticle is in center and system acceleration is zero, the rear orientation light equal and opposite in direction that two wavelength record, so differential signal S (t) is zero; When the first nanoparticle is not in center and system acceleration is non-vanishing, the rear orientation light size that two wavelength record is unequal, that bundle rear orientation light near the first nanoparticle offset direction is comparatively large, and differential signal S (t) detected is non-vanishing;

Step 5: carry out feedback regulation and obtain acceleration: according to differential signal S (t) obtained, dsp chip starts by light intensity modulator the light intensity difference regulating two grinding core optical fiber, in the direction of the extraneous acceleration of opposing, larger optical pressure is applied to the second microparticles, second microparticles is recentered, and it is zero that criterion is differential signal S (t); Now, the optical pressure acting force be applied on the second microparticles by the direction of measuring acceleration and grinding core optical fiber is contrary, and size is determined by the optical pressure difference of two grinding core optical fiber, is provided by dsp chip; Acceleration calculation process is as follows:

a＝K ₁K ₂I(1+R)/c

Wherein, a is institute's measuring acceleration, and I is the luminous energy unit time impinging perpendicularly on unit area, and R is the energy reflectivity on surface, and c is the light velocity in vacuum, K ₁be a coefficient, K ₂for another coefficient.

The invention has the beneficial effects as follows:

1, the present invention has the feature of low noise, high sensitivity, high precision identification and analysis, automatic feedback regulating power.

2, the present invention measures the acceleration of nanometer scale displacement under the condition not having physical contact, and to non-insensitive by other acting forces on measuring acceleration direction.

3, the present invention is based on highly sensitive nano-grade displacement to measure, make this structure can obtain high acceleration analysis resolving power, and be only confined to the absolute value of optical pressure restoring force, simultaneously the control loop of closed loop can significantly extended dynamic measurement range.

Accompanying drawing explanation

Fig. 1 is the rear orientation light sense acceleration scheme light path schematic diagram that the present invention is based on nanoparticle detection;

Fig. 2 is the rear orientation light sense acceleration scheme device schematic diagram that the present invention is based on nanoparticle detection;

Fig. 3 is the relation curve of total scattering light intensity and nanoparticle radius size;

Fig. 4 is the scattered light intensity of nanoparticle and the polar plot of angle;

Fig. 5 is the relative position figure of incident light, microscopic scatterers, scattered light;

Fig. 6 is dsp processor 16 workflow diagram;

Fig. 7 is the side view of rigid connecting rod 9;

Fig. 8 is the rear orientation light sense acceleration Fiber connection scheme light path schematic diagram that the present invention is based on nanoparticle detection;

In figure, reflective mirror 1, light damping plate 2, beam splitter 3, optical filter 4, first condenser lens 5, first nanoparticle 6, second microparticles 7, grinding core optical fiber 8, rigid connecting rod 9, second condenser lens 10, pin hole 11, separated light detector 12, laser instrument 13, collimation lens 14, light intensity modulator 15, dsp processor 16, microchannel 17, first laser diode 18, second laser diode 19, wavelength-division multiplex wave multiplexer 20, isolator 21, power bifurcated device 22, circulator 23, collimating apparatus 24, division dual wavelength catoptron 25, large-numerical aperture lens 26, attenuator 27, first Wave decomposing multiplexer 28, second Wave decomposing multiplexer 29, first coupling mechanism 30, second coupling mechanism 31, first photodiode 32, second photodiode 33, amplifier 34, analog-digital converter 35, dsp chip 36, first nanoparticle 6, second microparticles 7, grinding core optical fiber 8, rigid connecting rod 9, light intensity modulator 15, microchannel 17.

Embodiment

Principle of work of the present invention is as follows:

1, because nanoparticle is more much smaller than optical wavelength size, the back scattering light field sent can use Rayleigh scattering formulae discovery.For uniform dielectric, the scattered field oscillator intensity E of Rayleigh scattering _scan be expressed in matrix as

$\left(\begin{array}{c}{E}_{\mathrm{PS}}\\ E_{\⊥S}\end{array}\right)=\frac{e^{\mathrm{ik}(r-z)}}{-\mathrm{ikr}}\left(\begin{array}{cc}{S}_2& 0\\ 0& S_1\end{array}\right)\left(\begin{array}{c}{E}_{\mathrm{Pi}}\\ E_{\⊥i}\end{array}\right)$

$S_1=-\frac{{\mathrm{ik}}^3}{2\mathrm{\π}}(m-1)\mathrm{\υf}(\mathrm{\θ},\mathrm{\φ})$

$S_2=-\frac{{\mathrm{ik}}^3}{2\mathrm{\π}}(m-1)\mathrm{\υf}(\mathrm{\θ},\mathrm{\φ})\cos \mathrm{\θ}$

Wherein, E _{|| s}and E _{⊥ s}be respectively parallel component and the vertical component of scattered optical field oscillator intensity; E is the nature truth of a matter; I is the imaginary unit in plural number; Wave vector k=2 π N/ λ, N is surrounding medium refractive index, and λ is lambda1-wavelength; R is scattered light vector; Z is incident light vector; E _{|| i}and E _{⊥ i}be respectively parallel component and the vertical component of incident field oscillator intensity; S ₁, S ₂for matrix coefficient; M is relative index of refraction, i.e. the ratio of scattering particle refractive index and surrounding medium refractive index N; υ is particle volume; F (θ, φ) is the function relevant with scattering particle characteristic, and the implication that θ, φ represent as shown in Figure 5.

2, for uniform ball particles, the function relevant with scattering particle characteristic can be expressed as

$f\left(\mathrm{\θ}\right)=\frac{3}{u^3}(\sin u-u\cos u),u=2x\sin \frac{\mathrm{\θ}}{2}$

Wherein θ is the angle of r and z, as shown in Figure 5; X=ka=2 π Na/ λ, a is particle radii, and N is surrounding medium refractive index.

After progressively being substituted into by formula, the parallel and vertical scattered optical field of the nanosphere body particulate of trying to achieve is respectively:

$\mathrm{Es}1=\frac{e^{\mathrm{ik}(r-z)}}{-\mathrm{ik}\·r}\·(-\frac{{\mathrm{ik}}^3}{2\mathrm{\π}}\·\frac{(m-1)\·4}{3}\·\frac{\mathrm{\π}\·a^3\·3}{{(2k\·a\·\sin \left(\frac{\mathrm{\θ}}{2}\right))}^3}\·\cos \mathrm{\θ}\·(\sin (2k\·a\·\sin \left(\frac{\mathrm{\θ}}{2}\right))-(2k\·a\·\sin \left(\frac{\mathrm{\θ}}{2}\right))\·\cos (2k\·a\·\sin \left(\frac{\mathrm{\θ}}{2}\right))))\·\mathrm{Ei}1$

$\mathrm{Es}2=\frac{e^{\mathrm{ik}(r-z)}}{-\mathrm{ik}\·r}\·(-\frac{{\mathrm{ik}}^3}{2\mathrm{\π}}\·\frac{(m-1)\·4}{3}\·\frac{\mathrm{\π}\·a^3\·3}{{(2k\·a\·\sin \left(\frac{\mathrm{\θ}}{2}\right))}^3}\·(\sin (2k\·a\·\sin \left(\frac{\mathrm{\θ}}{2}\right))-(2k\·a\·\sin \left(\frac{\mathrm{\θ}}{2}\right))\·\cos (2k\·a\·\sin \left(\frac{\mathrm{\θ}}{2}\right))))\·\mathrm{Ei}2$

Total scattering light intensity I=E _s ²=E _s1 ²+ E _s2 ², as shown in Figure 3, the upcurve before peak value shows to meet very well much smaller than the nanoparticle of wavelength dimension and Rayleigh scattering formula the result obtained by the relation of software emulation total scattering light intensity and particle size.

As shown in Figure 4, the rear orientation light of nanoparticle is comparatively strong, is easy to measure.

From Fig. 3 and Fig. 4, separated light detector can record hot spot, hot spot produces along with the nano-grade displacement of nanoparticle to move and also can measure.Based on above-mentioned principle, the present invention can detect the micro-displacement change that nanoparticle causes.

The invention provides a kind of device of the rear orientation light sense acceleration based on nanoparticle detection, as depicted in figs. 1 and 2, this device comprises: reflective mirror 1, light damping plate 2, beam splitter 3, optical filter 4, first condenser lens 5, first nanoparticle 6, second microparticles 7, grinding core optical fiber 8, rigid connecting rod 9, second condenser lens 10, pin hole 11, separated light detector 12, laser instrument 13, collimation lens 14, light intensity modulator 15, dsp processor 16, microchannel 17.

Wherein, one end of rigid connecting rod 9 is fixedly connected with the first nanoparticle 6, the other end is through after the second microparticles 7, and be rigidly connected in the fixing turning point bottom microchannel 17, the first nanoparticle 6 and the second microparticles 7 can be used as an entirety and rotate around this fixing turning point.When rigid connecting rod 9 is vertical, the second microparticles 7 is on same level line with two grinding core optical fiber 8 of its both sides, and two grinding core optical fiber 8 respectively connect light intensity modulators 15.Separated light detector 12 is all connected with dsp processor 16 with two light intensity modulators 15.

The laser that laser instrument 13 sends enters beam splitter 3 after being collimated by collimation lens 14; A part is reflected by beam splitter 3, decays, reflective mirror 1 reflects, again after light damping plate 2 is decayed, forms reference light by beam splitter 3 transmission through light damping plate 2.Another part is focused into a hot spot through optical filter 4 by the first condenser lens 5 by after beam splitter 3 transmission, first nanoparticle 6 is in this hot spot, the rear orientation light of the first nanoparticle 6 is reflected by beam splitter 3 after the first condenser lens 5 is collected and optical filter 4 filters, interfere with reference light, after being focused on by the second condenser lens 10, through pin hole 11, received by separated light detector 12.The light signal of collection is converted into electric signal and is sent to dsp processor 16 by separated light detector 12.Dsp processor 16 adjusts the light intensity of two grinding core optical fiber 8 outputs by two light intensity modulators 15, realizes the adjustment of the second microparticles 7 position, thus realizes the adjustment to the first nanoparticle 6 position.

Above-mentioned laser instrument 13 and grinding core optical fiber 8 adopt the light of different wave length.The above-mentioned intensity modulation to grinding core optical fiber 8 adopts the feedback system be made up of separated light detector 12, dsp processor 16, light intensity modulator 15 automatically to complete.Particulate is fixed on the plane determined by grinding core optical fiber 8 and position sensing laser by the rigid connecting rod 9 of above-mentioned fixing turning point, make it can not have any component motion perpendicular on the direction of this plane, its side view as shown in Figure 7, because the thrust of light is very little, the material of three-legged structure is made up of nano-filaments.The aperture of above-mentioned pin hole 11 is 1 to 5 microns, and this pin hole is positioned at the focus place of the second condenser lens 10.

The workflow of above-mentioned dsp processor 16 as shown in Figure 6, can sketch and be by principle of work:

1, extract and process interference data: after the rear orientation light of the first nanoparticle 6 and reference light are interfered, the light intensity branch be irradiated on separated light detector 12 can be expressed as

$I={\left|E_r\right|}^2+{\left|E_s\right|}^2+2\mathrm{Re}\left\{E_r^{*}{E}_s\right\}$

E in formula _rfor reference light oscillator intensity, E _sfor scattered light oscillator intensity, I is the light intensity that detector just receives.

Separated photo-detector is surveyed the 12 signal P obtained and can be expressed as

$P=\frac{{\∫}_{A1}\mathrm{Ids}-{\∫}_{A2}\mathrm{Ids}}{{\∫}_{A1}\mathrm{Ids}+{\∫}_{A2}\mathrm{Ids}}$

Wherein A1 and A2 represents the two halves up and down of detector surface respectively, as shown in Figure 1; ∫ ds represents the area integral to separated light detector survey 12.When the first nanoparticle 6 is in center and system acceleration is zero, reference light and rear orientation light are adjusted to the center of separated light detector 12, so differential signal P (t) is zero.When the first nanoparticle 6 is not in center and system acceleration is non-vanishing, detectable signal P (t) is expressed as by interference term

$P\left(t\right)=2\mathrm{Re}\{({\∫}_{A1}{E}_r^{*}{E}_s\mathrm{ds}-{\∫}_{A2}{E}_r^{*}{E}_s\mathrm{ds})/{\∫}_{A1+A2}{\left|E_r\right|}^2\mathrm{ds}\}$

Because much bigger with reference to beam intensity ratio scattered light intensity in this device, the scattered light strong point in molecule can be ignored | E _s| ².In like manner, the reference light strong point only retained in denominator | E _r| ², and ignore all E _s.

2, carry out feedback regulation and obtain acceleration: differential signal S (t) obtained according to separated light detector 12, dsp processor 16 starts the light intensity difference regulating two grinding core optical fiber 8, in the direction of the extraneous acceleration of opposing, larger optical pressure is applied to the second microparticles 7, second microparticles 7 is recentered, and differential signal S (t) that criterion is separated light detector 12 acquisition is zero.Now, the optical pressure acting force be applied on the second microparticles 7 by the direction of measuring acceleration and grinding core optical fiber 8 is contrary, and size is determined by the optical pressure difference of two grinding core optical fiber 8, is provided by dsp processor 16.Acceleration calculation process is as follows:

a＝K ₁K ₂I(1+R)/c

Wherein, a is institute's measuring acceleration, and I is the luminous energy unit time impinging perpendicularly on unit area, and R is the energy reflectivity on surface, and c is the light velocity in vacuum, K ₁be a coefficient, K ₂for another coefficient.

The present invention is based on the method for the rear orientation light sense acceleration of nanoparticle detection, comprise the following steps:

Step 1: device being placed on acceleration is in the environment of zero, regulate the second microparticles 7 and the first nanoparticle 6 entirety in center, two duplicate grinding core optical fiber 8 and the second microparticles 7 center are in the same horizontal line, the optical pressure effect that after the light intensity that grinding core optical fiber 8 is incident equal, second microparticles 7 is balanced, rigidly connected first nanoparticle 6 is also stable at center with it.

Step 2: the optical wavelength that laser instrument 13 penetrates is different from the light in grinding core optical fiber 8, is radiated on beam splitter 3 after collimation lens 14 lens.The light beam reflected to return and again by beam splitter 3, as the reference beam of relevant detection through light damping plate 2 and the former road of reflective mirror 1; The light transmitted focuses on the second microparticles 7 by the first condenser lens 5 after optical filter 4, its rear orientation light is collected by the first condenser lens 5, as detected light beam after beam splitter 3 reflects, collected by the second condenser lens 10 after relevant with reference light, incide on separated light detector 12 after a pin hole 11 elimination high fdrequency component and environment parasitic light.Regulate the position of hot spot on separated light detector 12, make its hot spot when not having acceleration be in the center of separated light detector 12, now the differential signal of separated light detector 12 two halves is zero, and feedback system does not regulate the light intensity of grinding core optical fiber 8.

Step 3: when device has acceleration in grinding core optical fiber 8 direction, the second microparticles 7 departs from center, and rigid connecting rod 9 is that axle drives the second microparticles 7 to deflect thereupon with point of fixity.Owing to being measure in nano level displacement, the circular arc displacement of the first nanoparticle 6 can be similar to regards straight line as.And the displacement of the first nanoparticle 6 is also by the displacement equations of the second microparticles 7, and enlargement factor determines by the ratio of distances constant of the second microparticles 7 to Distance geometry first nanoparticle 6 to the second microparticles 7 of point of fixity.Now the hot spot of backscatter signal on separated light detector 12 of the first nanoparticle 6 deviate from center, and offset direction is determined by acceleration direction, and side-play amount is determined by acceleration magnitude.

Step 4: extract and process interference data: after the rear orientation light of the first nanoparticle 6 and reference light are interfered, the light intensity branch be irradiated on separated light detector 12 can be expressed as

$I={\left|E_r\right|}^2+{\left|E_s\right|}^2+2\mathrm{Re}\left\{E_r^{*}{E}_s\right\}$

E in formula _rfor reference light oscillator intensity, E _sfor scattered light oscillator intensity, I is the light intensity that detector just receives.

Separated photo-detector is surveyed the 12 signal P obtained and can be expressed as

$P=\frac{{\∫}_{A1}\mathrm{Ids}-{\∫}_{A2}\mathrm{Ids}}{{\∫}_{A1}\mathrm{Ids}+{\∫}_{A2}\mathrm{Ids}}$

Wherein A1 and A2 represents the two halves up and down of detector surface respectively, as shown in Figure 1; ∫ ds represents the area integral to separated light detector survey 12.When the first nanoparticle 6 is in center and system acceleration is zero, reference light and rear orientation light are adjusted to the center of separated light detector 12, so differential signal P (t) is zero.When the first nanoparticle 6 is not in center and system acceleration is non-vanishing, detectable signal P (t) is expressed as by interference term

$P\left(t\right)=2\mathrm{Re}\{({\∫}_{A1}{E}_r^{*}{E}_s\mathrm{ds}-{\∫}_{A2}{E}_r^{*}{E}_s\mathrm{ds})/{\∫}_{A1+A2}{\left|E_r\right|}^2\mathrm{ds}\}$

Because much bigger with reference to beam intensity ratio scattered light intensity in this device, the scattered light strong point in molecule can be ignored | E _s| ².In like manner, the reference light strong point only retained in denominator | E _r| ², and ignore all E _s.

Step 5: carry out feedback regulation and obtain acceleration: differential signal S (t) obtained according to separated light detector 12, dsp processor 16 starts the light intensity difference regulating two grinding core optical fiber 8, in the direction of the extraneous acceleration of opposing, larger optical pressure is applied to the second microparticles 7, second microparticles 7 is recentered, and differential signal S (t) that criterion is separated light detector 12 acquisition is zero.Now, the optical pressure acting force be applied on the second microparticles 7 by the direction of measuring acceleration and grinding core optical fiber 8 is contrary, and size is determined by the optical pressure difference of two grinding core optical fiber 8, is provided by dsp processor 16.Acceleration calculation process is as follows:

a＝K ₁K ₂I(1+R)/c

Wherein, a is institute's measuring acceleration, and I is the luminous energy unit time impinging perpendicularly on unit area, and R is the energy reflectivity on surface, and c is the light velocity in vacuum, K ₁be a coefficient, K ₂for another coefficient.

In addition, except realizing except this method by space optics device, can also be realized by optical fiber splicing device, as shown in Figure 8, this device comprises: the first laser diode 18, second laser diode 19, wavelength-division multiplex wave multiplexer 20, isolator 21, power bifurcated device 22, circulator 23, collimating apparatus 24, division dual wavelength catoptron 25, large-numerical aperture lens 26, attenuator 27, first Wave decomposing multiplexer 28, second Wave decomposing multiplexer 29, first coupling mechanism 30, second coupling mechanism 31, first photodiode 32, second photodiode 33, amplifier 34, analog-digital converter 35, dsp chip 36, first nanoparticle 6, second microparticles 7, grinding core optical fiber 8, rigid connecting rod 9, light intensity modulator 15, microchannel 17.

Wherein, first laser diode 18 is all connected with wavelength-division multiplex wave multiplexer 20 with the second laser diode 19, wavelength-division multiplex wave multiplexer 20, isolator 21 are connected successively with power bifurcated device 22, the output terminal of power bifurcated device 22 connects circulator 23 and attenuator 27 respectively, Second Wave division multiplexer 29 is connected with circulator 23, circulator 23 connects collimating apparatus 24, and collimating apparatus 24 is connected with microchannel 17 successively with division dual wavelength catoptron 25, large-numerical aperture lens 26, attenuator 27 is connected with first wave division multiplexer 28, the side of the first coupling mechanism 30 connects the shortwave output terminal of first wave division multiplexer 28 and the input end of the first photodiode 32 respectively, opposite side connects the output terminal of the second Wave decomposing multiplexer 29, the side of the second coupling mechanism 31 connects the long wave output terminal of the first Wave decomposing multiplexer 28 and the input end of the second photodiode 33 respectively, opposite side connects the output terminal of the second Wave decomposing multiplexer 29, the output terminal of the first photodiode 32 is all connected with amplifier 34 with the output terminal of the second photodiode 33, amplifier 34 is connected with dsp chip 36 by analog-digital converter 35, dsp chip 36 is connected with two light intensity controls 15 respectively.

Above-mentioned first laser diode 18 and the second laser diode 19 can be DFB laser diode or DBR laser diode.The numerical aperture of above-mentioned large-numerical aperture lens 26 is greater than 0.65.Structure in above-mentioned microchannel 17 is identical with a kind of front method.

For said apparatus, the present invention realizes the method for the rear orientation light sense acceleration based on nanoparticle detection, comprises the following steps:

Step 1: two wavelength difference are that first laser diode 18, second laser diode 19 of 10-20nm is as light source, by wavelength-division multiplex wave multiplexer 20, two wavelength light beams that two light sources send are incorporated in an optical fiber, prevent echo from affecting its stability to DFB/DBR laser diode by isolator 21, the light being separated 90-99% power by power bifurcated device 22 collimates to collimating apparatus 24, obtain dual wavelength collimated light beam, as detection light.Light two wavelength light after attenuator 27, first Wave decomposing multiplexer 28 of residue 1-10% power are separated, as reference light.

Step 2: dual wavelength collimated light beam respectively reflects the light beam of a wavelength by dividing catoptron about 25, two half cone-shaped light beams are focused in microchannel 17 afterwards by large-numerical aperture lens 26, produce rear orientation light, this rear orientation light, by after circulator 23, second Wave decomposing multiplexer 29, is input to two independently in optical interference circuit respectively by the first coupling mechanism 30, second coupling mechanism 31.

Step 3: the signal of short wavelength's rear orientation light collected by the first photodiode 32, is namely incident in the detection light signal that in the first nanoparticle 6, keep right in position; The signal of long wavelength's rear orientation light collected by second photodiode 33, is namely incident in the detection light signal that in the first nanoparticle 6, position keeps left.Two photodiodes receive signal after amplifier 34, analog-digital converter 35, by dsp chip 36 analyzing and processing data.

Step 4: extract and process interference data: after the rear orientation light of the first nanoparticle 6 and reference light are interfered, the signal that the light intensity of two kinds of wavelength i.e. two photodiodes detect all can be expressed as

$I={\left|E_r\right|}^2+{\left|E_s\right|}^2+2\mathrm{Re}\left\{E_r^{*}{E}_s\right\}$

E in formula _rfor reference light oscillator intensity, E _sfor scattered light oscillator intensity, I is the light intensity that detector just receives.

Differential signal after being exaggerated can be expressed as

$S\left(t\right)=\mathrm{\α}(I_{\⋐}-I_{\⊃})$

Wherein with represent left side detection light and the right detection light respectively, α represents amplification coefficient.When the first nanoparticle 6 is in center and system acceleration is zero, the rear orientation light equal and opposite in direction that two wavelength record, so differential signal S (t) is zero.When the first nanoparticle 6 is not in center and system acceleration is non-vanishing, the rear orientation light size that two wavelength record is unequal, that bundle rear orientation light near the first nanoparticle 6 offset direction is comparatively large, and differential signal S (t) detected is non-vanishing.

Step 5: carry out feedback regulation and obtain acceleration: according to differential signal S (t) obtained, dsp chip 36 starts by light intensity modulator 15 the light intensity difference regulating two grinding core optical fiber 8, in the direction of the extraneous acceleration of opposing, larger optical pressure is applied to the second microparticles 7, second microparticles 7 is recentered, and it is zero that criterion is differential signal S (t).Now, the optical pressure acting force be applied on the second microparticles 7 by the direction of measuring acceleration and grinding core optical fiber 8 is contrary, and size is determined by the optical pressure difference of two grinding core optical fiber 8, is provided by dsp chip 36.Acceleration calculation process is as follows:

a＝K ₁K ₂I(1+R)/c

Wherein, a is institute's measuring acceleration, and I is the luminous energy unit time impinging perpendicularly on unit area, and R is the energy reflectivity on surface, and c is the light velocity in vacuum, K ₁be a coefficient, K ₂for another coefficient.

Claims (4)

1. the device based on the rear orientation light sense acceleration of nanoparticle detection, it is characterized in that, this device comprises: reflective mirror (1), light damping plate (2), beam splitter (3), optical filter (4), first condenser lens (5), first nanoparticle (6), second microparticles (7), grinding core optical fiber (8), rigid connecting rod (9), second condenser lens (10), pin hole (11), separated light detector (12), laser instrument (13), collimation lens (14), light intensity modulator (15), dsp processor (16), microchannel (17), wherein, one end of rigid connecting rod (9) is fixedly connected with the first nanoparticle (6), the other end is through after the second microparticles (7), be rigidly connected in the fixing turning point of microchannel (17) bottom, the first nanoparticle (6) and the second microparticles (7) can be used as an entirety and rotate around this fixing turning point, when rigid connecting rod (9) is vertical, second microparticles (7) is on same level line with two grinding core optical fiber (8) of its both sides, and two grinding core optical fiber (8) respectively connect a light intensity modulator (15), separated light detector (12) is all connected with dsp processor (16) with two light intensity modulators (15), the laser that laser instrument (13) sends enters beam splitter (3) after being collimated by collimation lens (14), a part is reflected by beam splitter (3), through light damping plate (2) decay, reflective mirror (1) reflection, again after light damping plate (2) decay, forms reference light by beam splitter (3) transmission, another part is focused into a hot spot through optical filter (4) by the first condenser lens (5) by after beam splitter (3) transmission, first nanoparticle (6) is in this hot spot, the rear orientation light of the first nanoparticle (6) is reflected by beam splitter (3) after the first condenser lens (5) is collected and optical filter (4) filters, interfere with reference light, after being focused on by the second condenser lens (10), through pin hole (11), received by separated light detector (12), the light signal of collection is converted into electric signal and is sent to dsp processor (16) by separated light detector (12), the light intensity that dsp processor (16) is exported by two light intensity modulator (15) adjustment, two grinding core optical fiber (8), realize the adjustment of the second microparticles (7) position, thus realize the adjustment to the first nanoparticle (6) position.

2. application rights requires a method for the rear orientation light sense acceleration based on nanoparticle detection of device described in 1, and it is characterized in that, the method comprises the following steps:

Step 1: device being placed on acceleration is in the environment of zero, regulate the second microparticles (7) and the first nanoparticle (6) entirety in center, two duplicate grinding core optical fiber (8) and the second microparticles (7) center are in the same horizontal line, the optical pressure effect that after the light intensity that grinding core optical fiber (8) is incident equal, second microparticles (7) is balanced, rigidly connected first nanoparticle (6) is also stable at center with it;

Step 2: the optical wavelength that laser instrument (13) penetrates is different from the light in grinding core optical fiber (8), is radiated on beam splitter (3) after collimation lens (14) lens; The light beam reflected to return and again by beam splitter (3), as the reference beam of relevant detection through light damping plate (2) and reflective mirror (1) former road; The light transmitted focuses on the second microparticles (7) by the first condenser lens (5) after optical filter (4), its rear orientation light is collected by the first condenser lens (5), as detected light beam after beam splitter (3) reflection, collected by the second condenser lens (10) after relevant with reference light, incide on separated light detector (12) after a pin hole (11) elimination high fdrequency component and environment parasitic light; Regulate the position of hot spot on separated light detector (12), its hot spot when not having acceleration is made to be in the center of separated light detector (12), now the differential signal of separated light detector (12) two halves is zero, and feedback system does not regulate the light intensity of grinding core optical fiber (8);

Step 3: when device has acceleration in grinding core optical fiber (8) direction, the second microparticles (7) departs from center, and rigid connecting rod (9) is that axle drives the second microparticles (7) to deflect thereupon with point of fixity; Owing to being measure in nano level displacement, the circular arc displacement of the first nanoparticle (6) can be similar to regards straight line as; And the displacement of the first nanoparticle (6) is also by the displacement equations of the second microparticles (7), enlargement factor to be determined to Distance geometry first nanoparticle (6) of point of fixity to the ratio of distances constant of the second microparticles (7) by the second microparticles (7); Now the hot spot of backscatter signal on separated light detector (12) of the first nanoparticle (6) deviate from center, and offset direction is determined by acceleration direction, and side-play amount is determined by acceleration magnitude;

Step 4: extract and process interference data: after the rear orientation light of the first nanoparticle (6) and reference light are interfered, the light intensity branch be irradiated on separated light detector (12) can be expressed as:

$I=|E_r{|}^2+|E_s{|}^2+2\mathrm{Re}\left\{E_r^{*}{E}_s\right\}$

E in formula _rfor reference light oscillator intensity, E _sfor scattered light oscillator intensity, I is the light intensity that detector just receives;

The signal P that separated photo-detector survey (12) obtains can be expressed as:

$P=\frac{{\∫}_{A1}Ids-{\∫}_{A2}Ids}{{\∫}_{A1}Ids+{\∫}_{A2}Ids}$

Wherein A1 and A2 represents the two halves up and down of detector surface respectively, and ∫ ds represents area integral separated light detector being surveyed to (12); When the first nanoparticle (6) is in center and system acceleration is zero, reference light and rear orientation light are adjusted to the center of separated light detector (12), so differential signal P (t) is zero; When the first nanoparticle (6) is not in center and system acceleration is non-vanishing, detectable signal P (t) is expressed as by interference term:

$P\left(t\right)=2\mathrm{Re}\{({\∫}_{A1}{E}_r^{*}{E}_{s}ds-{\∫}_{A2}{E}_r^{*}{E}_{s}ds)/{{\∫}_{A1+A2}\left|E_r\right|}^{2}ds\};$

Step 5: carry out feedback regulation and obtain acceleration: differential signal S (t) obtained according to separated light detector (12), dsp processor (16) starts the light intensity difference regulating two grinding core optical fiber (8), in the direction of the extraneous acceleration of opposing, larger optical pressure is applied to the second microparticles (7), second microparticles (7) is recentered, and it is zero that criterion is differential signal S (t) that separated light detector (12) obtains; Now, contrary with the optical pressure acting force that grinding core optical fiber (8) is applied on the second microparticles (7) by the direction of measuring acceleration, size is determined by the optical pressure difference of two grinding core optical fiber (8), is provided by dsp processor (16); Acceleration calculation process is as follows:

a＝K ₁K ₂I(1+R)/c

Wherein, a is institute's measuring acceleration, and I is the luminous energy unit time impinging perpendicularly on unit area, and R is the energy reflectivity on surface, and c is the light velocity in vacuum, K ₁be a coefficient, K ₂for another coefficient.

3. the device based on the rear orientation light sense acceleration of nanoparticle detection, it is characterized in that, this device comprises: the first laser diode (18), second laser diode (19), wavelength-division multiplex wave multiplexer (20), isolator (21), power bifurcated device (22), circulator (23), collimating apparatus (24), division dual wavelength catoptron (25), large-numerical aperture lens (26), attenuator (27), first Wave decomposing multiplexer (28), second Wave decomposing multiplexer (29), first coupling mechanism (30), second coupling mechanism (31), first photodiode (32), second photodiode (33), amplifier (34), analog-digital converter (35), dsp chip (36), first nanoparticle (6), second microparticles (7), grinding core optical fiber (8), rigid connecting rod (9), light intensity modulator (15), microchannel (17), wherein, first laser diode (18) is all connected with wavelength-division multiplex wave multiplexer (20) with the second laser diode (19), wavelength-division multiplex wave multiplexer (20), isolator (21) is connected successively with power bifurcated device (22), the output terminal of power bifurcated device (22) connects circulator (23) and attenuator (27) respectively, Second Wave division multiplexer (29) is connected with circulator (23), circulator (23) connects collimating apparatus (24), collimating apparatus (24) and division dual wavelength catoptron (25), large-numerical aperture lens (26) are connected successively with microchannel (17), attenuator (27) is connected with first wave division multiplexer (28), the side of the first coupling mechanism (30) connects the shortwave output terminal of first wave division multiplexer (28) and the input end of the first photodiode (32) respectively, opposite side connects the output terminal of the second Wave decomposing multiplexer (29), the side of the second coupling mechanism (31) connects the long wave output terminal of the first Wave decomposing multiplexer (28) and the input end of the second photodiode (33) respectively, opposite side connects the output terminal of the second Wave decomposing multiplexer (29), the output terminal of the first photodiode (32) is all connected with amplifier (34) with the output terminal of the second photodiode (33), amplifier (34) is connected with dsp chip (36) by analog-digital converter (35), dsp chip (36) is connected with two light intensity controls (15) respectively.

4. application rights requires a method for the rear orientation light sense acceleration based on nanoparticle detection of device described in 3, and it is characterized in that, the method comprises the following steps:

Step 1: two wavelength difference are that first laser diode (18) of 10-20nm, the second laser diode (19) are as light source, by wavelength-division multiplex wave multiplexer (20), two wavelength light beams that two light sources send are incorporated in an optical fiber, prevent echo from affecting its stability to DFB/DBR laser diode by isolator (21), the light being separated 90-99% power by power bifurcated device (22) collimates to collimating apparatus (24), obtain dual wavelength collimated light beam, as detection light; Light two wavelength light after attenuator (27), the first Wave decomposing multiplexer (28) of residue 1-10% power are separated, as reference light;

Step 2: dual wavelength collimated light beam is by the light beam dividing each reflection in catoptron (25) left and right wavelength, two half cone-shaped light beams are focused in microchannel (17) afterwards by large-numerical aperture lens (26), produce rear orientation light, this rear orientation light, by circulator (23), after the second Wave decomposing multiplexer (29), is input to two independently in optical interference circuit respectively by the first coupling mechanism (30), the second coupling mechanism (31);

Step 3: the signal of short wavelength's rear orientation light collected by the first photodiode (32), is namely incident in the detection light signal kept right in the upper position of the first nanoparticle (6); The signal of long wavelength's rear orientation light collected by second photodiode (33), is namely incident in the detection light signal that the upper position of the first nanoparticle (6) keeps left; Two photodiodes receive signal after amplifier (34), analog-digital converter (35), by dsp chip (36) analyzing and processing data;

Step 4: extract and process interference data: after the rear orientation light of the first nanoparticle (6) and reference light are interfered, the signal that the light intensity of two kinds of wavelength i.e. two photodiodes detect all can be expressed as:

$I=|E_r{|}^2+|E_s{|}^2+2\mathrm{Re}\left\{E_r^{*}{E}_s\right\}$

E in formula _rfor reference light oscillator intensity, E _sfor scattered light oscillator intensity, I is the light intensity that detector just receives;

Differential signal after being exaggerated can be expressed as:

$S\left(t\right)=\mathrm{\α}\left(I_{\⋐}-I_{\⊃}\right)$

Wherein with represent left side detection light and the right detection light respectively, α represents amplification coefficient; When the first nanoparticle (6) is in center and system acceleration is zero, the rear orientation light equal and opposite in direction that two wavelength record, so differential signal S (t) is zero; When the first nanoparticle (6) is not in center and system acceleration is non-vanishing, the rear orientation light size that two wavelength record is unequal, that bundle rear orientation light near the first nanoparticle (6) offset direction is comparatively large, and differential signal S (t) detected is non-vanishing;

Step 5: carry out feedback regulation and obtain acceleration: according to differential signal S (t) obtained, dsp chip (36) starts by light intensity modulator (15) the light intensity difference regulating two grinding core optical fiber (8), in the direction of the extraneous acceleration of opposing, larger optical pressure is applied to the second microparticles (7), second microparticles (7) is recentered, and it is zero that criterion is differential signal S (t); Now, contrary with the optical pressure acting force that grinding core optical fiber (8) is applied on the second microparticles (7) by the direction of measuring acceleration, size is determined by the optical pressure difference of two grinding core optical fiber (8), is provided by dsp chip (36); Acceleration calculation process is as follows:

a＝K ₁K ₂I(1+R)/c

Wherein, a is institute's measuring acceleration, and I is the luminous energy unit time impinging perpendicularly on unit area, and R is the energy reflectivity on surface, and c is the light velocity in vacuum, K ₁be a coefficient, K ₂for another coefficient.