CN116183183B - Crystal electro-optic modulation method based on three-dimensional ray tracing - Google Patents
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- 239000013078 crystal Substances 0.000 title claims abstract description 138
- 238000000034 method Methods 0.000 title claims abstract description 87
- 239000013598 vector Substances 0.000 claims abstract description 56
- 230000003287 optical effect Effects 0.000 claims abstract description 34
- 230000010287 polarization Effects 0.000 claims abstract description 28
- 230000000694 effects Effects 0.000 claims description 9
- 230000014509 gene expression Effects 0.000 claims description 9
- GQYHUHYESMUTHG-UHFFFAOYSA-N lithium niobate Chemical compound [Li+].[O-][Nb](=O)=O GQYHUHYESMUTHG-UHFFFAOYSA-N 0.000 claims description 9
- 230000005540 biological transmission Effects 0.000 claims description 4
- 230000005672 electromagnetic field Effects 0.000 claims description 2
- 238000005290 field theory Methods 0.000 claims 1
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- 239000000463 material Substances 0.000 description 3
- 239000011159 matrix material Substances 0.000 description 3
- 230000001902 propagating effect Effects 0.000 description 3
- 238000002474 experimental method Methods 0.000 description 2
- 238000003384 imaging method Methods 0.000 description 2
- XGZVUEUWXADBQD-UHFFFAOYSA-L lithium carbonate Chemical compound [Li+].[Li+].[O-]C([O-])=O XGZVUEUWXADBQD-UHFFFAOYSA-L 0.000 description 2
- 229910052808 lithium carbonate Inorganic materials 0.000 description 2
- 230000005697 Pockels effect Effects 0.000 description 1
- 230000009286 beneficial effect Effects 0.000 description 1
- 230000005684 electric field Effects 0.000 description 1
- 238000012986 modification Methods 0.000 description 1
- 230000004048 modification Effects 0.000 description 1
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Abstract
The invention relates to a crystal electro-optic modulation method based on three-dimensional ray tracing, which comprises the following steps: determining the type of the crystal, the direction of an optical axis, the light passing direction and the direction of an applied voltage according to the main axis refractive index of the crystal; according to the light-passing caliber of the crystal, the voltage at two ends of the crystal, and the electro-optic coefficient of the crystal determines the refractive index change; determining the variation range of an incident angle and an azimuth angle when laser light enters a crystal during electro-optical modulation, and calculating an incident light vector and a refraction light vector according to the incident angle and the azimuth angle; according to the Fresnel equation, calculating the refractive indexes of two refractive rays and the phase difference generated when the two refractive rays propagate in the crystal; when the crystal is electrified, the crystal is changed into a biaxial crystal from a uniaxial crystal, three-dimensional coordinates of two optical axes are calculated respectively, and then the polarization direction vector of the refracted light is calculated according to the refraction light vector so as to determine the included angle between the polarization direction vector and the optical axis surface; and calculating the light intensity of the crystal electro-optic modulation according to the phase difference and the included angle between the polarization direction vector and the optical axis surface.
Description
Technical Field
The invention belongs to the technical field of laser radar imaging, and particularly relates to a crystal electro-optic modulation method based on three-dimensional ray tracing.
Background
Electro-optic modulation is based on the linear electro-optic effect (pockels effect), i.e. the effect in which the refractive index of the optical waveguide is proportional to the change in the applied electric field. The linear change of the refractive index of the optical waveguide in the phase modulator caused by the electro-optic effect causes the optical wave passing through the waveguide to have a phase shift, thereby realizing phase modulation. The spatial and polarization properties of the optical field propagating along its optical axis in the crystal are coupled in a particular way. When crystals are used as solid laser materials, a number of special phenomena can be observed, and if a laser beam is generated along the optical axis, its cone light interference will determine its spatial structure and polarization state, thus providing a possibility to control the spatial and polarization properties of the beam.
Light ray tracing is used in the prior art to describe the propagation of polarized light within a crystal. The polarization state changes of polarized light as it propagates in a two-dimensional coordinate system are typically described by jones matrix and muller matrix, however, both methods are not applicable to three-dimensional imaging scenes where an incident angle exists. The prior art also proposes the use of a three-dimensional coherence matrix to solve the polarization state of polarized light propagating in an anisotropic medium, however the electro-optical properties of biaxial crystals are not analyzed.
Therefore, how to solve the problems of the prior art that the propagation rule and the polarization state change process of the light rays in the electro-optical crystal can not be analyzed under any incident condition is a technical problem to be solved by the technicians in the field.
Disclosure of Invention
The invention provides a crystal electro-optic modulation method based on three-dimensional ray tracing, which comprises the following steps:
step S101: determining the type of the crystal, the direction of an optical axis, the light passing direction and the direction of an applied voltage according to the main axis refractive index of the crystal;
step S102: determining the refractive index change caused by the external voltage according to the light-passing caliber of the crystal, the electro-optic coefficient and the voltages at the two ends of the crystal;
step S103: determining the variation range of an incident angle and an azimuth angle when laser light enters a crystal during electro-optical modulation, and calculating an incident light vector and a refraction light vector according to the incident angle and the azimuth angle;
step S104: according to the refraction light vector, calculating the refractive index of the refraction light generated by the double refraction effect and the phase difference generated when the refraction light propagates in the crystal based on a Fresnel equation in the anisotropic medium;
step S105: when the crystal is electrified, the crystal is changed into a biaxial crystal from a uniaxial crystal, three-dimensional coordinates of two optical axes are calculated respectively, then a polarization direction vector of the refracted light ray is calculated according to the refracted light vector, and an included angle between the polarization direction vector and the optical axis plane is determined;
step S106: and determining the light intensity distribution expression of the crystal electro-optic modulation according to the phase difference obtained in the step S104 and the included angle between the polarization direction vector and the optical axis surface determined in the step S105.
In some embodiments, the crystal is a lithium niobate crystal, and the refractive index ellipsoid equation of the lithium niobate crystal is as follows:
in the method, in the process of the invention,X、Y、Zis the principal refractive index axis of the lithium niobate crystal,n o for the refractive index of the ordinary ray,n e is an extraordinary refractive index of light, wherein,n o =2.2797、n e =2.1969。
in some of these embodiments, the refractive index change in step S102 is determined according to the following equation:
in the method, in the process of the invention,dis the aperture of the light passing through the crystal,Vis the voltage across the crystal,
is the electro-optic coefficient of the crystal,n o for the refractive index of the ordinary ray,n o =2.2797。
in some embodiments, the incident light vector in step S103k i Determining in a three-dimensional coordinate system according to the following formula:
refractive light vectorkAccording to the theory of electromagnetic field, the formula is shown as follows:
in the method, in the process of the invention,
for incident angle, ++>
For the azimuth angle,n i for the refractive index of the incident light,nis the refractive index of the refracted light.
In some of these embodiments, in step S104, a Fresnel Equation (Fresnel edition) in the anisotropic medium is determined according to the following Equation:
in the method, in the process of the invention,k x 、k y 、k z is a refractive light vectorkThe component in the direction of the X, Y, Z axis,nin order to refract the refractive index of the light,n 1 、n 2 、n 3 respectively, the refractive index of the refracted light in the X, Y, Z axial direction.
In some embodiments, in step S104, the refracted ray includes a refracted ray a and a refracted ray B, and the refractive indices of the refracted ray a and the refracted ray B are respectivelyn a 、n b Determined according to the following equation:
wherein,,
in some embodiments, the phase difference in step S104 is determined according to the following equation:
in the method, in the process of the invention,λfor the wavelength of the laser light,Lis the length of the crystal in the light transmission direction,n a in order to refract the refractive index of the light ray a,n b in order to refract the refractive index of the light ray B,
is the refraction angle of the refracted ray A +.>
Is the refraction angle of the refracted ray B +.>
For the refraction angle of the refracted ray A>
Refraction angle with refracted ray B>
Wherein,
in some embodiments, in step S105, two lights are usedThe three-dimensional coordinates of the axes are respectively recorded asN 1 AndN 2 the expression is as follows:
wherein,,
and the included angles between the two optical axes and the Z axis in the three-dimensional coordinate system are respectively.
In some embodiments, in step S105, the polarization direction vector of the refracted ray is determined according to the following equationE：
In the method, in the process of the invention,kis a refracted light vector;k a a refracted light vector that is refracted light ray a;k b a refracted light vector that is refracted light ray B; polarization direction vectorEIncluded angle with optical axis surface
Determined according to the following formula:
in the method, in the process of the invention,
is the normal of the XOY plane, +.>
For incident angle, ++>
Is azimuth.
In some embodiments, in step S106, the intensity of the light is modulated by the crystal electro-optic
Determined according to the following formula:
in the method, in the process of the invention,
is the polarization direction vector of the refracted rayEIncluded angle with optical axis surface, < >>
Is the phase difference that occurs when refracted rays a and B propagate within the crystal.
The invention has the beneficial effects that:
the invention discloses a three-dimensional ray trace-based crystal electro-optic modulation method, which utilizes refractive index ellipsoids and Maxwell equations to systematically study the crystal electro-optic modulation characteristics, and provides the three-dimensional ray trace-based crystal electro-optic modulation method, and the three-dimensional ray trace-based crystal electro-optic modulation method provides expressions of polarization state characteristics, phase delay and interference light intensity distribution of crystal electro-optic modulation under any incidence condition. Compared with the prior art, the research method provided by the application is more scientific and more accurate in effect, the method provided by the application is used for calculating the light intensity of the crystal electro-optic modulation, and the experiment is set, and the calculated result of the method provided by the application is highly consistent with the experimental result.
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In order to more clearly illustrate the technical solutions of the embodiments of the present application, the drawings that are needed in the description of the embodiments will be briefly introduced below, it being obvious that the drawings in the following description are only some embodiments of the present application, and that other drawings may be obtained according to these drawings without inventive effort for a person skilled in the art.
FIG. 1 is a flow chart of some embodiments of a three-dimensional ray trace-based crystal electro-optic modulation method according to the present invention;
FIG. 2 is a schematic diagram of incident light vectors and refracted light vectors in a three-dimensional ray trace-based crystal electro-optic modulation method of the present invention;
FIG. 3 is a schematic diagram of a refractive index ellipsoid of a crystal electro-optic modulation method based on three-dimensional ray tracing in accordance with the present invention;
FIG. 4 is an experimental diagram of laser passing through a crystal under the condition of no voltage applied to the electro-optic crystal in a crystal electro-optic modulation method based on three-dimensional ray tracing;
FIG. 5 is a simulation diagram of laser passing through a crystal under the condition of no voltage applied to the electro-optic crystal in a crystal electro-optic modulation method based on three-dimensional ray tracing;
FIG. 6 is an experimental diagram of laser passing through a crystal when half-wave voltages are applied to two ends of the electro-optic crystal in a crystal electro-optic modulation method based on three-dimensional ray tracing;
FIG. 7 is a simulation diagram of laser passing through a crystal when half-wave voltage is applied across the electro-optic crystal in a three-dimensional ray trace-based crystal electro-optic modulation method of the present invention.
Detailed Description
The following description of the embodiments of the present invention will be made clearly and completely with reference to the accompanying drawings, in which it is apparent that the embodiments described are only some embodiments of the present invention, but not all embodiments.
Examples of embodiments are illustrated in the accompanying drawings, wherein like or similar symbols indicate like or similar elements throughout, or elements having like or similar functions. The embodiments described below by referring to the drawings are illustrative and intended to explain the present invention and should not be construed as limiting the invention.
Referring to fig. 1, a crystal electro-optic modulation method based on three-dimensional ray tracing includes:
step S101: the type of the crystal, the direction of the optical axis, the light passing direction and the direction of the applied voltage are determined according to the principal axis refractive index of the crystal.
In some embodiments of the invention, the crystal is a lithium niobate crystal, and the refractive index ellipsoid equation of the lithium niobate crystal is as follows:
in the method, in the process of the invention,X、Y、Zis the principal refractive index axis of the lithium niobate crystal,n o for the refractive index of the ordinary ray,n e is an extraordinary refractive index of light, wherein,n o =2.2797、n e =2.1969。
step S102: and determining the refractive index change caused by the applied voltage according to the light-passing caliber of the crystal, the electro-optic coefficient and the voltages at the two ends of the crystal.
In some embodiments of the invention, the refractive index change is determined according to the following formula:
in the method, in the process of the invention,dis the aperture of the light passing through the crystal,Vis the voltage across the crystal,
is the electro-optic coefficient of the crystal,n o for the refractive index of the ordinary ray,n o =2.2797。
step S103: and determining the variation range of the incident angle and the azimuth angle when the laser light enters the crystal during electro-optical modulation, and calculating the incident light vector and the refraction light vector according to the incident angle and the azimuth angle.
In some embodiments of the invention, the incident light vectork i Determining in a three-dimensional coordinate system according to the following formula:
refractive light vectorkAccording to theory of electromagnetic fieldThe determination is as follows:
in the method, in the process of the invention,
for incident angle, ++>
For the azimuth angle,n i for the refractive index of the incident light,nis the refractive index of the refracted light.
Step S104: according to the refractive light vectorkBased on the fresnel equation in the anisotropic medium, the refractive index of the refracted ray due to the birefringence effect and the phase difference generated when propagating in the crystal are calculated.
In some embodiments of the present invention, the Fresnel equation in anisotropic media is determined according to the following equation:
in the method, in the process of the invention,k x 、k y 、k z is a refractive light vectorkThe component in the direction of the X, Y, Z axis,nin order to refract the refractive index of the light,n 1 、n 2 、n 3 respectively, the refractive index of the refracted light in the X, Y, Z axial direction.
Specifically, the refractive light includes refractive light A and refractive light B, and refractive indexes of the refractive light A and the refractive light B are respectivelyn a 、n b Determined according to the following equation:
wherein,,
specifically, referring to FIG. 2, the refracted ray includes refracted ray A and refracted ray B, i.e., refracted ray vectorskThere are twok a Andk b ，k a andk b the expressions of (2) are respectively:
specifically, when a voltage is applied to the crystal in the X-axis direction, the crystal forms a phase delay due to the birefringence effect, which can be expressed as:
in the method, in the process of the invention,dis the aperture of the light passing through the crystal,Vis the voltage across the crystal,
is the electro-optic coefficient of the crystal,n o for the refractive index of the ordinary ray,n o =2.2797，Lis the length of the crystal in the light transmission direction,λis the laser wavelength.
In some embodiments of the invention, the phase difference is determined according to the following equation:
in the method, in the process of the invention,λfor the wavelength of the laser light,Lis the length of the crystal in the light transmission direction,n a in order to refract the refractive index of the light ray a,n b in order to refract the refractive index of the light ray B,
is the refraction angle of the refracted ray A +.>
Is the refraction angle of the refracted ray B +.>
For the refraction angle of the refracted ray A>
Refraction angle with refracted ray B>
Wherein,
step S105: when the crystal is electrified, the crystal is changed into biaxial crystal from uniaxial crystal, the three-dimensional coordinates of two optical axes are calculated respectively, and then the refractive light vector is usedkAnd calculating the polarization direction vector of the refracted light ray, and determining the included angle between the polarization direction vector and the optical axis surface.
In some embodiments of the present invention, referring to FIG. 3, three-dimensional coordinates of two optical axes are respectively recorded asN 1 AndN 2 the expression is as follows:
wherein,,
and the included angles between the two optical axes and the Z axis in the three-dimensional coordinate system are respectively.
Specifically, after the crystal is powered on, the refractive index ellipsoid is changed, the crystal is changed from a uniaxial crystal to a biaxial crystal, and the optical axis is inO-XYZOf a coordinate systemXOYThe planes are symmetrically distributed on two sides of the Z axis. Included angles between the two optical axes and the Z axis in the three-dimensional coordinate system
Can be expressed as:
in the method, in the process of the invention,n 1 、n 2 、n 3 is the principal refractive index.
Specifically, the polarization direction vector of the refracted ray is determined according to the following formulaE：
In the method, in the process of the invention,kis a refracted light vector;k a a refracted light vector that is refracted light ray a;k b a refracted light vector that is refracted light ray B; polarization direction vectorEIncluded angle with optical axis surface
Determined according to the following formula:
in the method, in the process of the invention,
is the normal of the XOY plane, +.>
For incident angle, ++>
Is azimuth.
Step S106: and determining the light intensity distribution expression of the crystal electro-optic modulation according to the phase difference obtained in the step S104 and the included angle between the polarization direction vector and the optical axis surface determined in the step S105.
In some embodiments of the invention, the intensity of the light is modulated by the crystal electro-optic
Determined according to the following formula:
in the method, in the process of the invention,
is the polarization direction vector of the refracted rayEIncluded angle with optical axis surface, < >>
Is the phase difference that occurs when refracted rays a and B propagate within the crystal.
In order to verify the scientificity of the electro-optic crystal modulation method of the three-dimensional ray tracing, the method provided by the invention is used for carrying out theoretical simulation on the electro-optic modulation result of the lithium carbonate crystal, and the parameters of the lithium carbonate crystal are the same as the experimental setting. Under the condition that no voltage is applied to the electro-optical crystal, an experimental diagram of laser passing through the crystal is shown in FIG. 4; the electro-optic crystal is shown with reference to fig. 5 in a simulated view of laser passing through the crystal without applying a voltage. When half-wave voltage is applied to two ends of the electro-optic crystal, an experimental diagram of laser passing through the crystal is shown in FIG. 6; a simulation of laser passing through the crystal when half-wave voltages are applied across the electro-optic crystal is shown with reference to fig. 7. From the comparison result, the theoretical simulation result is highly consistent with the experimental result, and the scientificity and reliability of the method provided by the invention are verified.
The application utilizes the refractive index ellipsoid to carry out systematic research on the crystal electro-optic modulation characteristic by combining with a Maxwell equation set, and provides a crystal electro-optic modulation method based on three-dimensional ray tracing, which gives expressions of polarization state characteristic, phase delay and interference light intensity distribution of the crystal electro-optic modulation under any incidence condition. Compared with the prior art, the research method provided by the application is more scientific and more accurate in effect, the method provided by the application is used for calculating the light intensity of the crystal electro-optic modulation, and the experiment is set, and the calculated result of the method provided by the application is highly consistent with the experimental result.
In the description of the present specification, reference to the terms "one embodiment," "some embodiments," "examples," "particular examples," "one particular embodiment," or "some examples," etc., means that a particular feature, structure, material, or characteristic described in connection with the embodiment or example is included in at least one embodiment or example of the invention. In this specification, schematic representations of terms do not necessarily refer to the same embodiment or example. Furthermore, the particular features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.
Finally, it should be noted that: the above embodiments are only for illustrating the technical solution of the present application, and are not limiting thereof; although the present application has been described in detail with reference to the foregoing embodiments, one of ordinary skill in the art will appreciate that: the technical scheme described in the foregoing embodiments can be modified or some technical features thereof can be replaced by equivalents; such modifications and substitutions do not drive the essence of the corresponding technical solutions to depart from the spirit and scope of the technical solutions of the embodiments of the present application.
Claims (7)
1. The crystal electro-optic modulation method based on three-dimensional ray tracing is characterized by comprising the following steps of:
step S101: determining the type of the crystal, the direction of an optical axis, the light passing direction and the direction of an applied voltage according to the main axis refractive index of the crystal;
step S102: determining the refractive index change caused by the external voltage according to the light-passing caliber of the crystal, the electro-optic coefficient and the voltages at the two ends of the crystal;
step S103: determining the variation range of an incident angle and an azimuth angle when laser light enters a crystal during electro-optical modulation, and calculating an incident light vector and a refraction light vector according to the incident angle and the azimuth angle;
step S104: according to the refraction light vector, calculating the refractive index of the refraction light generated by the double refraction effect and the phase difference generated when the refraction light propagates in the crystal based on a Fresnel equation in the anisotropic medium;
step S105: when the crystal is electrified, the crystal is changed into a biaxial crystal from a uniaxial crystal, three-dimensional coordinates of two optical axes are calculated respectively, the two optical axes are symmetrically distributed on two sides of a Z axis, and the three-dimensional coordinates are respectively recorded asN 1 AndN 2 the expression is as follows:
wherein,,
the included angles between the two optical axes and a Z axis in a three-dimensional coordinate system are respectively; and then calculating the polarization direction vector of the refracted light according to the refraction light vector, determining the included angle between the polarization direction vector and the optical axis surface, and determining the polarization direction vector of the refracted light according to the following formulaE：
In the method, in the process of the invention,kis a refracted light vector;k a a refracted light vector that is refracted light ray a;k b a refracted light vector that is refracted light ray B; the polarization direction vectorEIncluded angle with optical axis surface
Determined according to the following formula:
in the method, in the process of the invention,xis the normal to the XOY plane,
for incident angle, ++>
Is azimuth;
step S106: determining an expression of light intensity distribution of crystal electro-optical modulation according to the phase difference obtained in the step S104 and the included angle between the polarization direction vector and the optical axis surface determined in the step S105, wherein the light intensity of the crystal electro-optical modulation
Determined according to the following formula:
in the method, in the process of the invention,
is a phase difference.
2. The three-dimensional ray trace-based crystal electro-optic modulation method according to claim 1, wherein the crystal is a lithium niobate crystal, and the refractive index ellipsoid equation of the lithium niobate crystal is as follows:
in the method, in the process of the invention,X、Y、Zis the principal refractive index axis of the lithium niobate crystal,n o for the refractive index of the ordinary ray,n e is an extraordinary refractive index.
3. The three-dimensional ray trace-based crystal electro-optic modulation method according to claim 2, wherein the refractive index change in step S102 is determined according to the following equation:
in the method, in the process of the invention,dis the aperture of the light passing through the crystal,Vis the voltage across the crystal,
is the electro-optic coefficient of the crystal,n o is the ordinary refractive index.
4. A three-dimensional ray trace based crystal electro-optic modulation method according to claim 3 wherein in step S103 the incident light vector is determined in three-dimensional coordinate system according to the following equationk i ：
Determination of refractive light vector based on electromagnetic field theorykThe following formula is shown:
in the method, in the process of the invention,
for incident angle, ++>
For the azimuth angle,n i for the refractive index of the incident light,nis the refractive index of the refracted light.
5. The three-dimensional ray trace-based crystal electro-optic modulation method according to claim 1, wherein in step S104, the fresnel equation in the anisotropic medium is determined according to the following formula:
in the method, in the process of the invention,k x 、k y 、k z is a refractive light vectorkThe component in the direction of the X, Y, Z axis,nin order to refract the refractive index of the light,n 1 、n 2 、n 3 respectively, the refractive index of the refracted light in the X, Y, Z axial direction.
6. The three-dimensional ray tracing based crystal electro-optical modulation method according to claim 4, wherein in step S104, the refractive ray comprises refractive ray a and refractive ray B, and refractive indexes of the refractive ray a and the refractive ray B are respectivelyn a 、n b And is determined according to the following formula:
wherein,,
7. the three-dimensional ray trace-based crystal electro-optic modulation method according to claim 6, wherein in the step S104, the phase difference is determined according to the following formula:
in the method, in the process of the invention,λfor the wavelength of the laser light,Lis the length of the crystal in the light transmission direction,n a in order to refract the refractive index of the light ray a,n b in order to refract the refractive index of the light ray B,
is the refraction angle of the refracted ray A +.>
Is the refraction angle of the refracted ray B +.>
For the refraction angle of the refracted ray A>
Refraction angle with refracted ray B>
Average value of (2), wherein%>
。
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