CN106841110A - Absorbing medium complex refractivity index measurement apparatus and its measuring method - Google Patents

Abstract

The present invention relates to a kind of absorbing medium complex refractivity index measurement apparatus and its measuring method, including generating laser, wedge shape absorbing medium, ccd detector, oscillograph and rotating mechanism, the rotating mechanism includes straight-bar the first platform being fixedly connected with straight-bar one end and the second platform being hinged with the straight-bar other end, the generating laser is placed on the first platform, and wedge shape absorbing medium is placed on the second platform；Ccd detector sets the rear of the second platform, and optical signal is simultaneously delivered to oscillograph and is shown in real time by the light beam after receiving light beam that generating laser directly projects or being reflected through wedge shape absorbing medium.Measuring principle of the present invention is readily appreciated that light path is simple, and instrument cost is low.

Description

Device and method for measuring complex refractive index of absorptive medium

Technical Field

The invention relates to a device and a method for measuring complex refractive index of an absorptive medium.

Background

For the refractive index of the absorbing medium, its value is characterized by the complex refractive index. This change in form also changes the properties of the wave, particularly the imaginary refractive index, not only being a source of the absorption characteristics of the medium, but also affecting the polarization state of the reflected or transmitted light. The research on the complex refractive index not only has theoretical significance, but also has practical application value.

The existing method for measuring the complex refractive index mainly comprises a polarization technology, incident linearly polarized light is changed into elliptically polarized light after being reflected by an absorptive medium, and the real part and the imaginary part of the complex refractive index are measured by measuring the phase and the amplitude of the reflected light, so that the principle is complex, more components are needed for completing the measurement, the optical path is complex, and the price of an instrument is high; also, some measure the complex refractive index by using the surface plasmon resonance technique, and calculate the complex refractive index of the sample by measuring the reflectivity, resonance angle, phase difference, etc., the experimental principle is not understood, and the instrument structure is complicated.

Disclosure of Invention

In view of this, the present invention aims to provide an apparatus and a method for measuring complex refractive index of an absorptive medium, which have the advantages of easily understood measurement principle, simple optical path and low instrument cost.

In order to achieve the purpose, the invention adopts the following technical scheme: an absorptive medium complex refractive index measuring device, characterized in that: the device comprises a laser emitter, a wedge-shaped absorptive medium, a CCD detector, an oscilloscope and a rotating mechanism, wherein the rotating mechanism comprises a straight rod, a first platform fixedly connected with one end of the straight rod and a second platform hinged with the other end of the straight rod; the CCD detector is arranged behind the second platform, receives the light beam directly emitted by the laser emitter or the light beam refracted by the wedge-shaped absorptive medium and transmits the light signal to the oscilloscope for real-time display.

Furthermore, a baffle is arranged on the upper end face of the second platform, an opening is formed in the middle of the baffle, and the hinged point of the straight rod and the second platform is vertically overlapped with the opening.

Furthermore, the downside of second platform sets up the calibrated scale, be provided with on the straight-bar and hug closely the pointer of calibrated scale.

Furthermore, a beam expander is arranged between the laser transmitter and the wedge-shaped absorptive medium to expand the laser.

A measuring method of an absorptive medium complex refractive index measuring device is characterized by comprising the following steps:

step S1: the method comprises the following steps that a laser emitter is placed on a first platform, the angle of the laser emitter is adjusted to enable emitted laser to directly pass through an opening in the middle of a baffle, the direction of the laser is used as the initial direction of the laser, and a CCD detector receives the laser and then transmits corresponding optical signals to an oscilloscope to be displayed;

step S2: recording the corresponding angle of the pointer on the dial at the moment as a reference angle;

step S3: the wedge-shaped absorptive medium is arranged close to the baffle, the first platform and the straight rod are rotated to a first angle, laser emitted by the laser emitter is refracted through the wedge-shaped absorptive medium, and the CCD detector receives the refracted laser and then transmits a corresponding optical signal to the oscilloscope for displaying;

step S5, calculating the distance Y of the light beam moving on the CCD detector collecting window vertical to the laser initial direction through the position movement of the wave crest on the oscilloscope, further calculating the deflection angle β relative to the laser initial direction and the first real refraction angle of the laser when the laser passes through the emergent face of the wedge-shaped absorptive medium

Wherein, L is the distance from an emergent point on the emergent surface to an acquisition window of the CCD detector, and alpha is the vertex angle of the wedge-shaped absorptive medium;

step S6: rotating the first platform communicating straight rod to a second angle, and performing the operations from the step S3 to the step S5 to obtain a second real refraction angle of the laser

Step S7: combining the relationship between the real refractive index and the complex refractive index of the wedge-shaped absorptive medium to obtain the relationship between the real part and the imaginary part of the complex refractive index as follows:

wherein ,θ₁The incident angle at the right-angle side when the laser is obliquely incident on the wedge-shaped absorptive medium for the first time, namely the difference between the first angle and the reference angle, theta₂The refraction angle at the right-angle side when the laser is obliquely incident on the wedge-shaped absorptive medium for the first time; theta'₁Incident angle at the cathetus for the second oblique incidence of the laser light on the wedge-shaped absorptive medium, i.e. the difference between the first angle and the reference angle, theta'₂The refraction angle at the right-angle side when the laser is obliquely incident on the wedge-shaped absorptive medium for the second time; n is_iIs refractive index of air, k₀N and k are respectively the real part and the imaginary part of the wedge-shaped absorptive medium.

Further, the specific estimation process in step S7 is as follows:

converting light wave E (r, t) to E (r) E^-iωt，H(r,t)＝H(r)e^-iωtHuman-substituted maxwell's equationsIt can be obtained that the light wave in the absorbing medium satisfies the following equation:

wherein ,is equivalent complex permittivity, is permittivity, mu is permeability, sigma is conductivity,is the complex refractive index of the absorbing medium, n, k are the real and imaginary parts of the absorbing medium, respectively₀Is a wave vector in the vacuum and,the unit vectors of the isoamplitude and the isophase plane are q and s respectively as complex wave vectors in the absorptive medium, and the included angle between the two unit vectors is ξ -cos^-1(q.s)，k_s and k_qRespectively, the phase constant and the attenuation constant of the wave;

the phase constant and the attenuation constant in the absorptive medium are related to the real part and the imaginary part of the complex refractive index as follows:

since n and k are not zero, as can be seen from the formula (5), ξ ≠ π/2, i.e., the two unit vectors are not perpendicular, it can be calculated:

parameter N_s，N_qIs the effective refractive index of light propagating and attenuating in an absorptive medium, and the size of the effective refractive index is not only related to complex refractive index but also related to the included angle between an isophase surface and an isophase surface;

when the light is at an angle theta₁The right-angle edge of the wedge is incident, and the angle can be known according to the boundary condition,

n_ik₀sinθ₁＝k_ssinθ₂(7)

the relationship between the phase constant and attenuation constant of light wave after entering the absorptive medium via the right-angle edge refraction and the real part and imaginary part of the complex refractive index is as follows

From the formulae (7) and (8)

Similarly, the real refraction angle of the light refracted to the air isSince the phase vector and the attenuated vector of the wave in the absorptive medium both have tangential components at the interface, the phase constant k 'of the refracted wave'_sAttenuation constant k'_qThe relationship between them is as follows:

simultaneously according to boundary conditions

The relation between the real part and the imaginary part of the complex refractive index and the refraction angle can be obtained through the joint type (7), (8), (10) and (11):

in the same way, can be obtained as the angle theta'₁At incidence, let the real angle of refraction be θ 'at the right-angled side'₂Angle of refraction at the hypotenuseThe relationship between the real and imaginary parts of the complex refractive index and the angle of refraction is as follows

Combining the relation between the real refraction angle and the complex refraction index of the absorbing medium, namely equation (12) and equation (13), the relation between the real part and the imaginary part of the complex refraction index is obtained as follows:

wherein ：

the real n and imaginary k complex indices in the above equation can be solved using the function solve () in matlab software.

Compared with the prior art, the invention has the following beneficial effects: the invention only needs to use laser to carry out incidence with different angles for a single sample twice, and the real refraction angle of the laser passing through the single medium twice is respectively measured, so that the complex refraction index of the absorptive medium can be calculated. The measurement principle is easy to understand, the light path is simple, and the measurement is simpler; on the other hand, the device ensures that the laser emitter can still pass through the same incident point of the absorptive medium after rotating at different angles, reduces unstable factors caused by different incident points, such as different thicknesses of the incident points at different positions, and is convenient for reading the rotating angle of the laser emitter through the dial arranged on the lower side surface of the second platform.

Drawings

Fig. 1 is a schematic structural diagram of an apparatus according to an embodiment of the present invention.

Fig. 2 is a top view of a rotating mechanism according to an embodiment of the present invention.

Fig. 3 is a side view of a rotating structure according to an embodiment of the present invention.

Fig. 4 is a schematic view of the laser transmitter of the present invention after rotation.

Fig. 5 is a schematic diagram of refraction of a wedge-shaped absorptive medium.

In the figure: 1-a laser emitter; 11-a beam expander; 2-a wedge-shaped absorbent medium; 3-a CCD detector; 4-an oscilloscope; 5-straight rod; 51-a first platform; 52-a second platform; 53-baffles; 54-an opening; 55-dial scale; 56-pointer.

Detailed Description

The invention is further explained below with reference to the drawings and the embodiments.

Referring to fig. 1, the invention provides an absorptive medium complex refractive index measuring device, which includes a laser emitter 1, a wedge-shaped absorptive medium 2, a CCD detector 3, an oscilloscope 4 and a rotating mechanism, wherein the rotating mechanism shown in fig. 2 and 3 includes a straight rod 5, a first platform 51 fixedly connected with one end of the straight rod 5 and a second platform 52 hinged with the other end of the straight rod 5, the laser emitter 1 is disposed on the first platform 51, and the wedge-shaped absorptive medium 2 is disposed on the second platform 52; the straight rod 5 together with the first platform 51 can rotate around the hinge point, so that the laser emitted from the laser emitter 1 is incident on the wedge-shaped absorbent medium 2 at different angles. The CCD detector 3 is arranged behind the second platform 51, receives the light beam directly emitted by the laser emitter 1 or the light beam refracted by the wedge-shaped absorptive medium 2, and transmits the light signal to the oscilloscope 4 for real-time display.

Referring to fig. 2 and 3, a baffle 53 is arranged on the upper end surface of the second platform 52, an opening 54 is arranged in the middle of the baffle 53, and the hinge point of the straight rod and the second platform 52 is vertically overlapped with the opening 54; the laser emitter is guaranteed to be capable of being incident on the same incident point of the absorptive medium through the opening after rotating at different angles. A dial 55 is arranged on the lower side surface of the second platform 52, and a pointer 56 tightly attached to the dial is arranged on the straight rod 5 and rotates along with the rotation of the straight rod 5 so as to read scales on the dial 55.

Referring to fig. 1, a beam expander 11 is further disposed between the laser emitter 1 and the wedge-shaped absorptive medium 2 to expand the laser.

The embodiment also provides a measuring method of the absorbing medium complex refractive index measuring device, which comprises the following steps:

step S1: referring to fig. 1, a laser emitter 1 is placed on a first platform 51, the angle of the laser emitter is adjusted to enable the emitted laser to directly pass through an opening 54 in the middle of a baffle 53, the direction of the laser is used as the initial direction of the laser, and a CCD detector 3 receives the laser and then transmits a corresponding optical signal to an oscilloscope 4 for display;

step S2: the corresponding angle of the pointer 56 on the dial 55 at this time is recorded as a reference angle;

step S3: referring to fig. 4, the wedge-shaped absorptive medium 2 is arranged to be closely attached to the baffle, the first platform 51 and the straight rod 5 are rotated to a first angle, the laser emitted by the laser emitter 1 is refracted by the wedge-shaped absorptive medium 2, and the CCD detector 3 receives the refracted laser and transmits a corresponding optical signal to the oscilloscope 4 for display;

step S5: and calculating the moving distance Y of the light beam on an acquisition window of the CCD detector vertical to the initial direction of the laser through the position movement of the wave crest on the oscilloscope 4. Taking an LM601 type CCD light intensity distribution measuring instrument as an example, its light sensor has 2592 pixels, the center distance of the light sensor is 11 μm, if one period of the waveform corresponds to m large grids on an oscilloscope, the actual space distance corresponding to each large grid on the oscilloscope is 2592 × 11 ÷ m, the corresponding distance of each small grid is 2592 × 11 ÷ m ÷ 5, and if the grid number of the light spot moving on the oscilloscope twice is p small grids, the distance Y corresponding to the movement of the CCD is 2592 × 11 ÷ m ÷ 5 × p.

Then, the deflection angle β of the laser relative to the initial direction of the laser and the first real refraction angle of the laser when the laser passes through the emergent surface of the wedge-shaped absorptive medium are calculated according to the distance Y

Wherein, L is the distance from an emergent point on the emergent surface to an acquisition window of the CCD detector, and alpha is the vertex angle of the wedge-shaped absorptive medium;

step S6: rotating the first platform 51 and the straight rod 5 to a second angle, and performing the operations of the steps S3 to S5 to obtain a second real refraction angle of the laser

Step S7: combining the relationship between the real refractive index and the complex refractive index of the wedge-shaped absorbing medium,

the relationship between the real and imaginary parts of the complex index of refraction is obtained as follows:

wherein ,θ₁The incident angle at the right-angle side when the laser is obliquely incident on the wedge-shaped absorptive medium for the first time, namely the difference between the first angle and the reference angle, theta₂The refraction angle at the right-angle side when the laser is obliquely incident on the wedge-shaped absorptive medium for the first time; theta'₁Incident angle at the cathetus for the second oblique incidence of the laser light on the wedge-shaped absorptive medium, i.e. the difference between the first angle and the reference angle, theta'₂The refraction angle at the right-angle side when the laser is obliquely incident on the wedge-shaped absorptive medium for the second time; n is_iIs refractive index of air, k₀N and k are respectively the real part and the imaginary part of the wedge-shaped absorptive medium.

Further, the specific estimation process in step S7 is as follows:

converting light wave E (r, t) to E (r) E^-iωt，H(r,t)＝H(r)e^-iωtThe following equation of the light wave in the absorbing medium can be obtained instead of the Maxwell equation:

wherein ,is equivalent complex permittivity, is permittivity, mu is permeability, sigma is conductivity,is the complex refractive index of the absorbing medium, n, k are the real and imaginary parts of the absorbing medium, respectively₀Is a wave vector in the vacuum and,the unit vectors of the isoamplitude and the isophase plane are q and s respectively as complex wave vectors in the absorptive medium, and the included angle between the two unit vectors is ξ -cos^-1(q.s)，k_s and k_qRespectively, the phase constant and the attenuation constant of the wave;

the phase constant and the attenuation constant in the absorptive medium are related to the real part and the imaginary part of the complex refractive index as follows:

since n and k are not zero, as can be seen from the formula (5), ξ ≠ π/2, i.e., the two unit vectors are not perpendicular, it can be calculated:

parameter N_s，N_qIs the effective refractive index of light propagating and attenuating in an absorptive medium, and the size of the effective refractive index is not only related to complex refractive index but also related to the included angle between an isophase surface and an isophase surface;

referring to FIG. 5, when the light is at an angle θ₁The right-angle edge of the wedge is incident, and the angle can be known according to the boundary condition,

n_ik₀sinθ₁＝k_ssinθ₂(7)

the relationship between the phase constant and attenuation constant of light wave after entering the absorptive medium via the right-angle edge refraction and the real part and imaginary part of the complex refractive index is as follows

From the formulae (7) and (8)

Similarly, the real refraction angle of the light refracted to the air isSince the phase vector and the attenuated vector of the wave in the absorptive medium both have tangential components at the interface, the phase constant k 'of the refracted wave'_sAttenuation constant k'_qThe relationship between them is as follows:

simultaneously according to boundary conditions

The relation between the real part and the imaginary part of the complex refractive index and the refraction angle can be obtained through the joint type (7), (8), (10) and (11):

in the same way, can be obtained as the angle theta'₁At incidence, let the real angle of refraction be θ 'at the right-angled side'₂Angle of refraction at the hypotenuseThe relationship between the real and imaginary parts of the complex refractive index and the angle of refraction is as follows

Combining the relation between the real refraction angle and the complex refraction index of the absorbing medium, namely equation (12) and equation (13), the relation between the real part and the imaginary part of the complex refraction index is obtained as follows:

wherein ：

the real n and imaginary k complex indices in the above equation can be solved using the function solve () in matlab software.

The above description is only a preferred embodiment of the present invention, and all equivalent changes and modifications made in accordance with the claims of the present invention should be covered by the present invention.

Claims (6)

1. An absorptive medium complex refractive index measuring device, characterized in that: the device comprises a laser emitter, a wedge-shaped absorptive medium, a CCD detector, an oscilloscope and a rotating mechanism, wherein the rotating mechanism comprises a straight rod, a first platform fixedly connected with one end of the straight rod and a second platform hinged with the other end of the straight rod; the CCD detector is arranged behind the second platform, receives the light beam directly emitted by the laser emitter or the light beam refracted by the wedge-shaped absorptive medium and transmits the light signal to the oscilloscope for real-time display.

2. The apparatus according to claim 1, characterized in that: the upper end face of the second platform is provided with a baffle, the middle of the baffle is provided with an opening, and the hinged point of the straight rod and the second platform coincides with the opening in the vertical direction.

3. The apparatus according to claim 2, characterized in that: the downside of second platform sets up the calibrated scale, be provided with on the straight-bar and hug closely the pointer of calibrated scale.

4. The apparatus according to claim 3, characterized in that: and a beam expander is arranged between the laser transmitter and the wedge-shaped absorptive medium to expand the laser.

5. A measuring method based on the absorptive medium complex refractive index measuring device of claim 4, characterized by comprising the following steps:

step S1: the method comprises the following steps that a laser emitter is placed on a first platform, the angle of the laser emitter is adjusted to enable emitted laser to directly pass through an opening in the middle of a baffle, the direction of the laser is used as the initial direction of the laser, and a CCD detector receives the laser and then transmits corresponding optical signals to an oscilloscope to be displayed;

step S2: recording the corresponding angle of the pointer on the dial at the moment as a reference angle;

step S3: the wedge-shaped absorptive medium is arranged close to the baffle, the first platform and the straight rod are rotated to a first angle, laser emitted by the laser emitter is refracted through the wedge-shaped absorptive medium, and the CCD detector receives the refracted laser and then transmits a corresponding optical signal to the oscilloscope for displaying;

step S5: calculating the verticality of light beam on CCD detector by the position shift of wave crest on oscilloscopeThe distance Y of the laser beam moving on the collection window in the initial direction is calculated, and the deflection angle β of the laser beam relative to the initial direction and the first real refraction angle of the laser beam when the laser beam passes through the exit surface of the wedge-shaped absorptive medium are calculated

$\mathrm{\β}=arctan\frac{Y}{L}$

Wherein, L is the distance from an emergent point on the emergent surface to an acquisition window of the CCD detector, and alpha is the vertex angle of the wedge-shaped absorptive medium;

step S6: rotating the first platform communicating straight rod to a second angle, and performing the operations from the step S3 to the step S5 to obtain a second real refraction angle of the laser

Step S7: combining the relationship between the real refractive index and the complex refractive index of the wedge-shaped absorptive medium to obtain the relationship between the real part and the imaginary part of the complex refractive index as follows:

$\left\{\begin{array}{c}{\mathrm{\θ}}_2=\arcsin \left(\frac{n_i{\mathrm{sin\θ}}_1}{\sqrt{\frac{1}{2}\{\sqrt{{(n^2-{\mathrm{\κ}}^2-{n_i}^2{\sin }^2{\mathrm{\θ}}_1)}^2+4n^2{\mathrm{\κ}}^2}+(n^2-{\mathrm{\κ}}^2+{n_i}^2{\sin }^2{\mathrm{\θ}}_1)\}}}\right)\\ {\mathrm{\θ}}_2^{\′}=\arcsin \left(\frac{n_i{\mathrm{sin\θ}}_1^{\′}}{\sqrt{\frac{1}{2}\{\sqrt{{(n^2-{\mathrm{\κ}}^2-{n_i}^2{\sin }^2{\mathrm{\θ}}_1^{\′})}^2+4n^2{\mathrm{\κ}}^2}+(n^2-{\mathrm{\κ}}^2+{n_i}^2{\sin }^2{\mathrm{\θ}}_1^{\′})\}}}\right)\end{array}\right.$

wherein ,θ₁The incident angle at the right-angle side when the laser is obliquely incident on the wedge-shaped absorptive medium for the first time, namely the difference between the first angle and the reference angle, theta₂The refraction angle at the right-angle side when the laser is obliquely incident on the wedge-shaped absorptive medium for the first time; theta'₁Incident angle at the cathetus for the second oblique incidence of the laser light on the wedge-shaped absorptive medium, i.e. the difference between the first angle and the reference angle, theta'₂The refraction angle at the right-angle side when the laser is obliquely incident on the wedge-shaped absorptive medium for the second time; n is_iIs refractive index of air, k₀N and k are respectively the real part and the imaginary part of the wedge-shaped absorptive medium.

6. The measuring method of the absorbing medium complex refractive index measuring apparatus according to claim 5, characterized in that: the specific estimation process in step S7 is as follows:

converting light wave E (r, t) to E (r) E^-iωt，H(r,t)＝H(r)e^-iωtThe following equation of the light wave in the absorbing medium can be obtained instead of the Maxwell equation:

$\left\{\begin{array}{c}\▿\×E\left(r\right)=i\mathrm{\ω}\mathrm{\μ}H\left(r\right)\\ \▿\×H\left(r\right)=-i\mathrm{\ω}\widetilde{\mathrm{\ϵ}}E\left(r\right)\end{array}\right.---\left(3\right)$

$\left\{\begin{array}{c}{\▿}^{2}E\left(r\right)+{\tilde{k}}^{2}E\left(r\right)=0\\ {\▿}^{2}H\left(r\right)+{\tilde{k}}^{2}H\left(r\right)=0\end{array}\right.---\left(4\right)$

wherein ,is equivalent complex permittivity, is permittivity, mu is permeability, sigma is conductivity,is the complex refractive index of the absorbing medium, n, k are the real and imaginary parts of the absorbing medium, respectively₀Is a wave vector in the vacuum and,the unit vectors of the isoamplitude and the isophase plane are q and s respectively as complex wave vectors in the absorptive medium, and the included angle between the two unit vectors is ξ -cos^-1(q.s)，k_s and k_qRespectively, the phase constant and the attenuation constant of the wave;

the phase constant and the attenuation constant in the absorptive medium are related to the real part and the imaginary part of the complex refractive index as follows:

$\left\{\begin{array}{c}{k}_s^2-k_q^2=k_0^2(n^2-{\mathrm{\κ}}^2)\\ 2k_s{k}_{q}cos\mathrm{\ξ}=2{\mathrm{n\κk}}_0\end{array}\right.---\left(5\right)$

since n and k are not zero, as can be seen from the formula (5), ξ ≠ π/2, i.e., the two unit vectors are not perpendicular, it can be calculated:

$\left\{\begin{array}{c}{N}_s=\frac{k_s}{k_0}=\sqrt{\frac{\sqrt{{(n^2-{\mathrm{\κ}}^2)}^2+{\left(\frac{2n\mathrm{\κ}}{\cos \mathrm{\ξ}}\right)}^2}+(n^2-{\mathrm{\κ}}^2)}{2}}\\ N_q=\frac{k_q}{k_0}=\sqrt{\frac{\sqrt{{(n^2-{\mathrm{\κ}}^2)}^2+{\left(\frac{2n\mathrm{\κ}}{\cos \mathrm{\ξ}}\right)}^2}-(n^2-{\mathrm{\κ}}^2)}{2}}\end{array}\right.---\left(6\right)$

parameter N_s，N_qIs the effective refractive index of light propagating and attenuating in an absorptive medium, and the size of the effective refractive index is not only related to complex refractive index but also related to the included angle between an isophase surface and an isophase surface;

when the light is at an angle theta₁The right-angle edge of the incident wedge is arranged at the right-angle edgeAs can be seen from the boundary conditions,

n_ik₀sinθ₁＝k_ssinθ₂(7)

the relationship between the phase constant and attenuation constant of light wave after entering the absorptive medium via the right-angle edge refraction and the real part and imaginary part of the complex refractive index is as follows

$\left\{\begin{array}{c}{k}_s^2-k_q^2=k_0^2(n^2-{\mathrm{\κ}}^2)\\ 2k_s{k}_q{\mathrm{cos\θ}}_2=2{\mathrm{n\κk}}_0\end{array}\right.---\left(8\right)$

From the formulae (7) and (8)

${\mathrm{\θ}}_2=arcsin\left(\frac{n_i{\mathrm{sin\θ}}_1}{\sqrt{\frac{1}{2}\{\sqrt{{(n^2-{\mathrm{\κ}}^2-{n_i}^2{\sin }^2{\mathrm{\θ}}_1)}^2+4n^2{\mathrm{\κ}}^2}+(n^2-{\mathrm{\κ}}^2+n_i^2{\sin }^2{\mathrm{\θ}}_1)\}}}\right)---\left(9\right)$

Similarly, the real refraction angle of the light refracted to the air isSince the phase vector and the attenuation vector of the wave in the absorbing medium both have tangential components at the interface, the phase constant k of the refracted wave_s', attenuation constant k_qThe relationship between' is as follows:

$k_s^{\′2}-k_q^{\′2}=k_0^2{n}_i^2---\left(10\right)$

simultaneously according to boundary conditions

The relation between the real part and the imaginary part of the complex refractive index and the refraction angle can be obtained through the joint type (7), (8), (10) and (11):

by the same token, by the angle θ₁At incidence, the real angle of refraction at the cathetus is assumed to be θ₂', angle of refraction at bevel edgeThe relationship between the real and imaginary parts of the complex refractive index and the angle of refraction is as follows

Combining the relation between the real refraction angle and the complex refraction index of the absorbing medium, namely equation (12) and equation (13), the relation between the real part and the imaginary part of the complex refraction index is obtained as follows:

wherein ：

$\left\{\begin{array}{c}{\mathrm{\θ}}_2=\arcsin \left(\frac{n_i{\mathrm{sin\θ}}_1}{\sqrt{\frac{1}{2}\{\sqrt{{(n^2-{\mathrm{\κ}}^2-{n_i}^2{\sin }^2{\mathrm{\θ}}_1)}^2+4n^2{\mathrm{\κ}}^2}+(n^2-{\mathrm{\κ}}^2+{n_i}^2{\sin }^2{\mathrm{\θ}}_1)\}}}\right)\\ {\mathrm{\θ}}_2^{\′}=\arcsin \left(\frac{n_i{\mathrm{sin\θ}}_1^{\′}}{\sqrt{\frac{1}{2}\{\sqrt{{(n^2-{\mathrm{\κ}}^2-{n_i}^2{\sin }^2{\mathrm{\θ}}_1^{\′})}^2+4n^2{\mathrm{\κ}}^2}+(n^2-{\mathrm{\κ}}^2+{n_i}^2{\sin }^2{\mathrm{\θ}}_1^{\′})\}}}\right)\end{array}\right.$

the real part of the complex refractive index in the above equation can be solved by using the function solve () in matlab software

n and an imaginary part κ.