Method for calculating diffraction efficiency of liquid crystal polarization grating

Abstract

The invention relates to a method for calculating the diffraction efficiency of a liquid crystal polarization grating, belongs to the technical field of non-mechanical beam deflection, and solves the problem that the traditional method cannot accurately calculate the diffraction efficiency of the liquid crystal polarization grating under the influence of a Fresnel effect. The method comprises the following steps: firstly, constructing a corresponding relation between an mth diffraction order electric field component in an observation dielectric layer and a spatial frequency spectrum of a total transmission electric field; obtaining an expression of an m-th diffraction order electric field component by utilizing inverse spatial Fourier transform; deriving a quantitative relational expression between the mth diffraction order electric field component and the total transmission electric field in the observation dielectric layer; extracting an mth diffraction order electric field component in combination with a grating equation and a quantitative relational expression; and calculating the proportion of the light intensity of the mth diffraction order in the total transmission light intensity as the diffraction efficiency. The calculation result is more accurate, the method can be well applied to diffraction efficiency calculation of the multi-layer liquid crystal polarization grating cascade structure, and the current calculation requirement for the diffraction efficiency of a liquid crystal polarization grating system is met.

Description

Technical Field

[0001] The invention belongs to the technical field of non-mechanical light beam deflection, and specifically refers to a method for calculating the diffraction efficiency of a liquid crystal polarization grating. Background Art

[0002] The beam deflection module is one of the core components in many military and civilian fields such as laser radar, deep space exploration, space communication, laser active imaging, etc. The liquid crystal polarization grating, as a new type of geometric phase element, has excellent properties such as large aperture, high efficiency and wide angle. The new non-mechanical beam deflection technology based on liquid crystal polarization grating can take into account the two core requirements of large aperture and large angle at the same time, so it can greatly improve the performance of the beam deflection system, which makes it have good application prospects in many national defense and military fields such as satellite communications, airborne radar, medium-wave infrared staring imaging, laser confrontation, laser guidance, etc.

[0003] At present, most of the beam deflection technologies based on liquid crystal polarization gratings adopt a cascade structure, and its core needs to accurately calculate the diffraction efficiency corresponding to each diffraction order. The traditional method for calculating the diffraction efficiency of liquid crystal polarization gratings requires adding a corresponding refractive index matching medium to the liquid crystal polarization grating, or the diffraction efficiency can only be calculated under the premise that the medium at the output end of the liquid crystal polarization grating is transparent. However, these measures cannot avoid the Fresnel effect generated by the polarization grating system. Therefore, the traditional diffraction efficiency calculation method can only estimate the diffraction efficiency by calculating the light intensity of some low-orders such as 0th order and ±1st order in the liquid crystal polarization grating, but cannot accurately calculate the diffraction efficiency of any order. The disadvantages of this traditional diffraction efficiency calculation method for a single liquid crystal polarization grating are mainly reflected in the fact that the calculated diffraction efficiency result is too large and difficult to verify experimentally. However, for the structure of a cascade of N liquid crystal polarization gratings, the traditional diffraction efficiency calculation method is not applicable, because it is not clear at this time that the energy of the diffracted beam is mainly distributed in the 2nd order emitted from the Nth liquid crystal polarization grating. N Ultimately, the traditional diffraction efficiency calculation method has many limitations because it cannot avoid the Fresnel effect produced by the liquid crystal polarization grating system. Summary of the invention

[0004] The purpose of the present invention is to provide a method for calculating the diffraction efficiency of a liquid crystal polarization grating, so as to solve the problem that the current traditional diffraction efficiency calculation method cannot accurately calculate the diffraction efficiency of the liquid crystal polarization grating under the influence of the Fresnel effect.

[0005] A method for calculating the diffraction efficiency of a liquid crystal polarization grating, the liquid crystal polarization grating comprising a liquid crystal polarization grating film (2), an observation medium layer (3) respectively plated on both sides of the liquid crystal polarization grating film (2), a transition medium layer (4), and an incident front medium layer (1) plated on the transition medium layer (4), and the components of the incident front medium layer (1) and the observation medium layer (3) are both isotropic media, the method comprising the following steps:

[0006] Step 1: Construct the m-th diffraction order electric field component in the observation medium layer (3) based on Fraunhofer diffraction theory and spatial Fourier transform The spatial spectrum Total transmitted electric field The spatial spectrum The corresponding relationship between them;

[0007] Step 2: Spatial spectrum constructed based on step 1 With spatial spectrum The corresponding relationship between them is used to obtain the m-th diffraction order electric field component by using spatial inverse Fourier transform. Expression of

[0008] Step 3: According to the Floquet-Bloch theorem, the total transmitted electric field After the vector is decomposed, it is expanded into its corresponding Fourier series and the total transmitted electric field is calculated. The Fourier transform is obtained, and the Fourier transform result is substituted into the m-th diffraction order electric field component obtained in step 2. From the expression of , the electric field component of the mth diffraction order in the observation medium layer (3) is derived Total transmitted electric field The quantitative relationship between

[0009] Step 4: Combine the grating equation of the liquid crystal polarization grating and the quantitative relationship derived in step 3 to extract the mth diffraction order electric field component under any incident direction.

[0010] Step 5: Using the optical power density in the observed medium layer (3) as the light intensity for calculating the diffraction efficiency, according to the extracted electric field component of the mth diffraction order Calculate the light intensity I of the mth diffraction order in the observation medium layer (3) m And according to the total transmitted electric field Calculate the total transmitted light intensity I in the observation medium layer (3) total ;

[0011] Step 6: Calculate the light intensity I of the mth diffraction order m The total transmitted light intensity I total The proportion of As the diffraction efficiency of the mth diffraction order emitted from the liquid crystal polarization grating film (2).

[0012] Beneficial effects: The present invention proposes a method for calculating the diffraction efficiency of a liquid crystal polarization grating, which can accurately calculate the diffraction efficiency of any diffraction order of the liquid crystal polarization grating while taking into account the interference of the Fresnel effect generated by the entire liquid crystal polarization grating system. Compared with the shortcomings of the traditional diffraction efficiency calculation method, the diffraction efficiency calculation method proposed in the present invention not only has more accurate calculation results and improves the calculation accuracy of the diffraction efficiency of the liquid crystal polarization grating, but can also be well applied to the calculation of the diffraction efficiency of the multi-layer liquid crystal polarization grating cascade structure, meeting the current calculation requirements for the diffraction efficiency of the liquid crystal polarization grating system, thereby further optimizing the performance of the beam deflection module. BRIEF DESCRIPTION OF THE DRAWINGS

[0013] Figure 1 It is a flow chart of a method for calculating the diffraction efficiency of a liquid crystal polarization grating according to an embodiment of the present invention;

[0014] Figure 2 A schematic diagram of one structure of a liquid crystal polarization grating whose diffraction efficiency can be calculated using the method provided by the present invention;

[0015] Figure 3 The figure is a comparison diagram of the results of calculating the diffraction efficiency of the same liquid crystal polarization grating using the method proposed by the present invention and the traditional method respectively.

[0016] Explanation of the accompanying drawings: 1. Incident front dielectric layer; 2. Liquid crystal polarization grating film; 3. Observation dielectric layer; 4. Transition dielectric layer. DETAILED DESCRIPTION

[0017] In order to enable those skilled in the art to better understand the technical solution of the present invention, the present invention will be further described in detail below in conjunction with the accompanying drawings and preferred embodiments.

[0018] Example 1

[0019] This embodiment provides a method for calculating the diffraction efficiency of a liquid crystal polarization grating, wherein the structure of the liquid crystal polarization grating is as follows: Figure 2As shown, it mainly includes an incident front dielectric layer 1, a liquid crystal polarization grating thin film 2, an observation medium 3 and a transition dielectric layer 4. The observation dielectric layer 3 and the transition dielectric layer 4 are respectively plated on both sides of the liquid crystal polarization grating thin film 2, and the incident front dielectric layer 1 is plated on the transition dielectric layer 4. The components of the incident front dielectric layer 1 and the observation dielectric layer 3 are both isotropic media, such as air. The composition of the transition dielectric layer 4 includes but is not limited to: a single-layer isotropic medium, a single-layer anisotropic medium, a multi-layer isotropic medium, a multi-layer anisotropic medium, or a multi-layer cascade of an isotropic medium and an anisotropic medium, so that it can be well applied to various complex liquid crystal polarization grating cascade systems.

[0020] In this embodiment Figure 2 The rectangular coordinate system xyz in the figure is established by setting the thickness direction of the liquid crystal polarization grating film 2 and the grating period direction as the x-axis and the y-axis respectively, and setting the normal direction of the xy plane as the z-axis.

[0021] In this embodiment, the liquid crystal polarization grating film 2 is a polarization element that changes the wavefront by generating a geometric phase. It is also a one-dimensional grating that has periodicity in the y direction only. The optical axis of the liquid crystal molecules is Figure 2 The yz plane in the yz plane changes periodically and continuously, and satisfies the following relationship:

[0022]

[0023] In the formula, represents the unit vector in the direction of the optical axis of the liquid crystal molecule, y is the coordinate variable representing the spatial position information in the period direction of the liquid crystal polarization grating film 2, and Λ is the period size of the liquid crystal polarization grating film 2. At the same time, Figure 2 The thickness d of the liquid crystal polarization grating film 2 along the x direction satisfies the half-wave condition:

[0024] d=λ0 / (2Δn) (2)

[0025] Where λ0 is the wavelength in vacuum, Δn = n e -n o , n e and n o They are the principal axis refractive indices of the liquid crystal molecules parallel and perpendicular to the optical axis of the liquid crystal molecules, respectively.

[0026] The method for calculating the diffraction efficiency of the liquid crystal polarization grating of this embodiment specifically includes the following steps:

[0027] Step 1: First, construct the m-th diffraction order electric field component in the observation medium layer 3 based on Fraunhofer diffraction theory and spatial Fourier transform The spatial spectrum Total transmitted electric field The spatial spectrum The corresponding relationship between them.

[0028] Step 1.1: According to Fraunhofer diffraction theory, for a one-dimensional grating with periodicity in the y direction only, observe the total transmitted electric field in the dielectric layer 3 There is a spatial spectrum only in the direction of the grating period, so the total transmitted electric field The spatial spectrum Using spatial Fourier transform, it can be expressed as:

[0029]

[0030] In the formula, f y represents the total transmitted electric field The corresponding spatial frequency, j represents the imaginary unit;

[0031] Step 1.2: According to Fraunhofer diffraction theory, observe the mth diffraction order electric field component in the dielectric layer 3 The spatial angular frequency in the y direction is Here f ym represents the electric field component of the mth diffraction order The corresponding spatial frequency, k0 represents the wave number in vacuum, λ0 represents the wavelength in vacuum, n3 represents the refractive index of the observation medium layer 3, θ m ,φ m denote the diffraction polar angle and diffraction azimuth of the mth diffraction order respectively, then the electric field component of the mth diffraction order is The spatial spectrum It can be expressed as:

[0032]

[0033] Where δ(·) represents the unit impulse function.

[0034] Step 2: Spatial spectrum constructed based on step 1 With spatial spectrum The corresponding relationship between them, using the spatial inverse Fourier transform, the m-th diffraction order electric field component can be obtained Total transmitted electric field The relationship between the Fourier transform of The expression is:

[0035]

[0036] Where F(·) represents the spatial Fourier transform operator.

[0037] Step 3: According to the Floquet-Bloch theorem, the total transmitted electric field can be After the vector is decomposed, it is expanded into its corresponding Fourier series, so that the total transmitted electric field can be calculated The Fourier transform is obtained, and the Fourier transform result is substituted into the m-th diffraction order electric field component obtained in step 2. The m-th diffraction order electric field component in the observation medium layer 3 is derived from the expression Total transmitted electric field The quantitative relationship between them is:

[0038]

[0039] In the formula, Indicates along Figure 2 The unit vector in the positive direction of the a-axis, where a = x, y, z; represents the total transmitted electric field exist The complex amplitude of the component in the direction.

[0040] Step 4: Combine the grating equation of the liquid crystal polarization grating and the quantitative relationship derived in step 3 to extract the mth diffraction order electric field component under any incident direction.

[0041] According to Huygens-Fresnel diffraction theory and Fraunhofer diffraction theory, we can get Figure 2 The grating equation for the liquid crystal polarization grating shown is:

[0042]

[0043] Where n1 represents the refractive index of the incident front medium layer 1, θ0 represents the incident light wave vector and Figure 2 The angle along the positive x direction is usually called the polar angle, and φ0 represents the incident light wave vector in Figure 2 The angle between the projection in the yz plane and the positive direction of the y-axis is usually called the azimuth.

[0044] Substituting the grating equation into formula (6), the mth diffraction order electric field component under any incident direction can be extracted:

[0045] Step 5: Considering the ease and feasibility of experimental measurement in practice, this embodiment chooses to observe the optical power density in the dielectric layer 3 as the light intensity for calculating the diffraction efficiency. The relationship between them is:

[0046]

[0047] Wherein, n3 is the refractive index of the observation medium layer 3, ε0 is the dielectric constant in vacuum, and μ0 is the magnetic permeability in vacuum.

[0048] Based on formula (8), according to the extracted electric field component of the mth diffraction order Calculate the light intensity I of the mth diffraction order in the observation medium layer 3 m , the calculation formula is as follows:

[0049]

[0050] According to the total transmitted electric field Calculate the total transmitted light intensity I in the observation medium layer 3 total , the calculation formula is as follows:

[0051]

[0052] Step 6: Finally, calculate the light intensity I of the mth diffraction order of the observation medium layer 3 m The total transmitted light intensity I total The proportion of As the diffraction efficiency of the mth diffraction order emitted from the liquid crystal polarization grating film 2.

[0053] Specifically, substitute formula (9) and formula (10) into The diffraction efficiency η of the mth diffraction order emitted from the liquid crystal polarization grating can be accurately calculated. m for:

[0054]

[0055] The method for calculating the diffraction efficiency of a liquid crystal polarization grating proposed in this embodiment is different from the traditional method for calculating the diffraction efficiency. This method does not need to add a corresponding refractive index matching medium to the liquid crystal polarization grating film, but directly extracts the electric field component corresponding to each diffraction order in the observation medium layer to calculate its light intensity value, thereby calculating the diffraction efficiency. Compared with the traditional method for calculating the diffraction efficiency, this method for calculating the diffraction efficiency can accurately calculate the diffraction efficiency of any diffraction order of the liquid crystal polarization grating under the interference of the Fresnel effect generated by the entire liquid crystal polarization grating system, thereby improving the calculation accuracy of the diffraction efficiency of the liquid crystal polarization grating. Moreover, compared with the traditional method for calculating the diffraction efficiency, the light intensity used in calculating the diffraction efficiency by this method directly corresponds to the light power density in the observation medium layer, so that the calculation result is easier to verify through experiments. The diffraction efficiency calculation method can be used to accurately calculate the diffraction efficiency of complex diffraction structures such as multi-layer media and multi-layer liquid crystal polarization grating cascades at the incident front end of the liquid crystal polarization grating, which greatly helps to meet the current demand for the accuracy of diffraction efficiency calculation of liquid crystal polarization grating systems, thereby further optimizing the performance of the beam deflection module.

[0056] Example 2

[0057] In this embodiment, the parameters of the liquid crystal polarization grating are set as follows:

[0058] The grating period Λ of the liquid crystal polarization grating film 2 is 15 μm;

[0059] Principal axis refractive index n e is 1.65, n o is 1.55, the vacuum wavelength λ0 is 1064nm, and the thickness d is 5.32μm;

[0060] The composition of the observation medium layer 3 is air;

[0061] The transition medium layer 4 is composed of air;

[0062] The azimuth angle φ0 of the incident light emitted by the observation medium layer 1 is 0°, and the polar angle θ0 is -45° to 45°.

[0063] Based on all the parameters in this embodiment, taking the +1 diffraction order as an example, the diffraction efficiency calculation method proposed by the present invention and the traditional method are used to calculate the diffraction efficiency of the liquid crystal polarization grating, and the results are as follows: Figure 3 As shown. Figure 3 It can be seen that when the light wave is close to vertical incidence, the difference between the liquid crystal polarization grating diffraction efficiency calculation method proposed in the present invention and the traditional method is small. This is because the energy of the incident light wave in these incident directions is mainly concentrated in the low diffraction order, so the traditional method still has a high accuracy at this time. However, the proportion of light wave energy in higher diffraction orders increases with the increase of the incident polar angle. At this time, the accuracy of the traditional diffraction efficiency calculation method will drop significantly, and the diffraction efficiency calculation error near the incident polar angle of ±45° is as high as 7%. Furthermore, if a multi-stage liquid crystal polarization grating cascade is used, this diffraction efficiency calculation error will be geometrically amplified. Therefore, compared with the obvious defects of the traditional diffraction efficiency calculation method, the diffraction efficiency calculation method of the liquid crystal polarization grating proposed in the present invention can well meet the current demand for the accuracy of the diffraction efficiency calculation of the liquid crystal polarization grating system.

[0064] It can be seen from this embodiment that the method for calculating the diffraction efficiency of a liquid crystal polarization grating proposed in the present invention can fundamentally overcome the defects of traditional calculation methods and accurately calculate the diffraction efficiency of a liquid crystal polarization grating system, which will play an important role especially when used in a cascade structure.

[0065] The technical features of the above-described embodiments may be arbitrarily combined. To make the description concise, not all possible combinations of the technical features in the above-described embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0066] The above-mentioned embodiments only express several implementation methods of the present invention, and the descriptions thereof are relatively specific and detailed, but they cannot be understood as limiting the scope of the invention patent. It should be pointed out that, for ordinary technicians in this field, several variations and improvements can be made without departing from the concept of the present invention, and these all belong to the protection scope of the present invention. Therefore, the protection scope of the patent of the present invention shall be subject to the attached claims.

Claims

1. A method for calculating the diffraction efficiency of a liquid crystal polarization grating, characterized in that: The liquid crystal polarization grating comprises a liquid crystal polarization grating film (2), an observation medium layer (3) respectively plated on both sides of the liquid crystal polarization grating film (2), a transition medium layer (4), and an incident front medium layer (1) plated on the transition medium layer (4), and the incident front medium layer (1) and the observation medium layer (3) are both composed of isotropic media. The method comprises the following steps: Step 1: Construct the m-th diffraction order electric field component in the observation medium layer (3) based on Fraunhofer diffraction theory and spatial Fourier transform The spatial spectrum Total transmitted electric field The spatial spectrum The corresponding relationship between them; Step 2: Spatial spectrum constructed based on step 1 With spatial spectrum The corresponding relationship between them is used to obtain the m-th diffraction order electric field component by using spatial inverse Fourier transform. Expression of Step 3: According to the Floquet-Bloch theorem, the total transmitted electric field After the vector is decomposed, it is expanded into its corresponding Fourier series and the total transmitted electric field is calculated. The Fourier transform is obtained, and the Fourier transform result is substituted into the m-th diffraction order electric field component obtained in step 2. From the expression of , the electric field component of the mth diffraction order in the observation medium layer (3) is derived Total transmitted electric field The quantitative relationship between Step 4: Combine the grating equation of the liquid crystal polarization grating and the quantitative relationship derived in step 3 to extract the mth diffraction order electric field component under any incident direction. Step 5: Using the optical power density in the observed medium layer (3) as the light intensity for calculating the diffraction efficiency, according to the extracted electric field component of the mth diffraction order Calculate the light intensity I of the mth diffraction order in the observation medium layer (3) m And according to the total transmitted electric field Calculate the total transmitted light intensity I in the observation medium layer (3) total ; Step 6: Calculate the light intensity I of the mth diffraction order m The total transmitted light intensity I total The proportion of As the diffraction efficiency of the mth diffraction order emitted from the liquid crystal polarization grating film (2).

2. The method for calculating the diffraction efficiency of a liquid crystal polarization grating according to claim 1, characterized in that: Step 1 includes the following steps: Step 1.1: Observe the total transmitted electric field in the dielectric layer (3) according to Fraunhofer diffraction theory The spatial spectrum Using spatial Fourier transform, it can be expressed as: In the formula, f y represents the total transmitted electric field The corresponding spatial frequency, j represents the imaginary unit; Step 1.2: According to Fraunhofer diffraction theory, the electric field component of the mth diffraction order is obtained The spatial spectrum It is expressed as: Where δ(·) represents the unit impulse function; f ym represents the electric field component of the mth diffraction order The corresponding spatial frequency satisfies: Among them, k ym is the spatial angular frequency of the electric field component of the mth diffraction order in the y direction, k0 represents the wave number in vacuum, λ0 represents the wavelength in vacuum, n3 represents the refractive index of the observation medium layer (3), θ m ,φ m They respectively represent the diffraction polar angle and diffraction azimuth angle of the mth diffraction order.

3. The method for calculating the diffraction efficiency of a liquid crystal polarization grating according to claim 2, characterized in that: The electric field component of the mth diffraction order in step 2 The expression is: Where F(·) represents the spatial Fourier transform operator.

4. The method for calculating the diffraction efficiency of a liquid crystal polarization grating according to claim 3, characterized in that: The quantitative relationship derived in step 3 is: In the formula, represents the unit vector along the positive direction of the a-axis, where a = x, y, z; represents the total transmitted electric field exist The complex amplitude of the component in the direction.

5. The method for calculating the diffraction efficiency of a liquid crystal polarization grating according to claim 4, characterized in that: The grating equation of the liquid crystal polarization grating is: Wherein, n1 represents the refractive index of the incident front medium layer (1), θ0 represents the polar angle, and φ0 represents the azimuth angle.

6. The method for calculating the diffraction efficiency of a liquid crystal polarization grating according to claim 5, characterized in that: Observe the light intensity I of the mth diffraction order in the medium layer (3) m for: Observe the total transmitted light intensity I in the dielectric layer (3) total for: Where ε0 is the dielectric constant in vacuum and μ0 is the magnetic permeability in vacuum.

7. The method for calculating the diffraction efficiency of a liquid crystal polarization grating according to claim 6, characterized in that: The diffraction efficiency η of the mth diffraction order m The calculation formula is:

8. A method for calculating the diffraction efficiency of a liquid crystal polarization grating according to any one of claims 1 to 7, characterized in that: The optical axes of the liquid crystal molecules of the liquid crystal polarization grating film (2) change periodically and continuously in the yz plane and satisfy the following relationship: In the formula, represents a unit vector in the direction of the optical axis of the liquid crystal molecule, y is a coordinate variable representing spatial position information in the period direction of the liquid crystal polarization grating film (2), and Λ is the period size of the liquid crystal polarization grating film (2); At the same time, the thickness d of the liquid crystal polarization grating film (2) along the x direction satisfies the half-wave condition: d=λ0 / (2Δn) (2) Where λ0 is the wavelength in vacuum, Δn = n e -n o , n e and n o They are the principal axis refractive indices of the liquid crystal molecules parallel and perpendicular to the optical axis of the liquid crystal molecules, respectively.

9. A method for calculating the diffraction efficiency of a liquid crystal polarization grating according to any one of claims 1 to 7, characterized in that: The transition medium layer (4) is composed of any one of a single-layer isotropic medium, a single-layer anisotropic medium, a multi-layer isotropic medium, a multi-layer anisotropic medium, or a multi-layer cascade of an isotropic medium and an anisotropic medium.

10. A method for calculating the diffraction efficiency of a liquid crystal polarization grating according to any one of claims 1 to 7, characterized in that: The incident front medium layer (1) and the observation medium layer (3) are both composed of air.

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