Liquid crystal-based variable coefficient ordinary differential equation solving method and solver

Abstract

The invention provides a liquid crystal-based variable coefficient ordinary differential equation solving method and solver wherein the liquid crystal-based variable coefficient ordinary differential equation solving method comprises the following steps: constructing a liquid crystal device, and establishing a rectangular plane coordinate system based on the working surface of the liquid crystal device; the liquid crystal device has different directors at different positions; placing the liquid crystal device on a frequency spectrum surface of a 4f system to obtain output signals obtained after solving the input signals under different liquid crystal voltages; the liquid crystal voltage is adjusted, so that the liquid crystal device generates different transfer functions for incident light waves, and ordinary differential equation solving results of different coefficients under different liquid crystal voltages are obtained; the incident light wave is x-ray polarized light. Through the method, the solved ordinary differential equation coefficient can be adjusted, the problem that an existing airspace differential equation solver cannot adjust the equation coefficient is solved, and the calculation speed and density are better.

Description

Technical Field

[0001] The present invention relates to the field of micro-nano optical technologies, and in particular, to a method and a solver for solving variable coefficient ordinary differential equations based on liquid crystals. Background Art

[0002] Ordinary differential equations (ODEs) are a mathematical language used to describe basic physical phenomena and natural laws, and are widely applied in almost all scientific and engineering fields, such as physics, chemistry, sociology, ecology, and medicine. ODEs can be mainly divided into two types. One is ordinary differential equations with constant coefficients, which are classical models in dynamic systems such as signal processing systems, control systems, and electrical systems. The other is ordinary differential equations with variable coefficients, which are usually applied in fields such as complex-valued dynamical systems and complex linear quantum systems.

[0003] For the solution of ordinary differential equations with variable coefficients, existing solution methods need to change and solve the coefficients of ordinary differential equations in the time domain, and the calculation speed and density of the solution process are not ideal enough to meet the usage requirements in fields with high requirements for solution speed, such as photon computing and biological imaging.

[0004] For the problem of poor calculation speed existing in the existing related technologies, no effective solution has been proposed yet. Summary of the Invention

[0005] The present invention provides a method and a solver for solving variable coefficient ordinary differential equations based on liquid crystals to solve the defect of poor calculation speed existing in the existing related technologies and achieve the effect of solving variable coefficient ordinary differential equations in the spatial domain.

[0006] In a first aspect, the present invention provides a method for solving variable coefficient ordinary differential equations based on liquid crystals, including: Construct a liquid crystal device, and establish a plane rectangular coordinate system based on the working surface of the liquid crystal device; the liquid crystal device has different director vectors at different positions; Place the liquid crystal device on the spectrum plane of a 4f system to obtain the output signal after solving the input signal under different liquid crystal voltages; Adjust the liquid crystal voltage to enable the liquid crystal device to generate different transfer functions for the incident light wave, and obtain the solution results of ordinary differential equations with different coefficients under different liquid crystal voltages; the incident light wave is an x-polarized light.

[0007] According to the method for solving variable coefficient ordinary differential equations based on liquid crystals provided by the present invention, constructing a liquid crystal device includes: Determine the material and thickness of the liquid crystal device; Determine the coefficients of the first-order ordinary differential equation corresponding to different liquid crystal voltages according to the phase delay amount added by the liquid crystal device to the incident light wave; Arrange the director of the liquid crystal device.

[0008] According to a method for solving a variable coefficient ordinary differential equation based on liquid crystal provided by the present invention, a plane rectangular coordinate system is established based on the working surface of the liquid crystal device, including: Set the directions of two right-angle sides parallel to the working surface of the liquid crystal device as the x-axis and the y-axis respectively to construct a plane rectangular coordinate system.

[0009] According to a method for solving a variable coefficient ordinary differential equation based on liquid crystal provided by the present invention, the direction of the director of the liquid crystal device is the included angle between the liquid crystal long axis of the liquid crystal device and the x-axis of the plane rectangular coordinate system.

[0010] According to a method for solving a variable coefficient ordinary differential equation based on liquid crystal provided by the present invention, place the liquid crystal device on the spectrum plane of the 4f system to obtain the output signal after solving the input signal under different liquid crystal voltages, including: Place the liquid crystal device on the spectrum plane of the 4f system and irradiate it with an incident light wave of a specific wavelength; Determine the transfer function of the liquid crystal device for modulating the incident light wave; Change the liquid crystal voltage to obtain the output signal after solving the input signal under different coefficients of the ordinary differential equation.

[0011] According to a method for solving a variable coefficient ordinary differential equation based on liquid crystal provided by the present invention, adjust the liquid crystal voltage to make the liquid crystal device generate different transfer functions for the incident light wave, including: Adjust the liquid crystal voltage to change the phase added by the liquid crystal device to the outgoing light wave, so that the liquid crystal device generates different transfer functions for the incident light wave.

[0012] According to a method for solving a variable coefficient ordinary differential equation based on liquid crystal provided by the present invention, the wavelength of the incident light wave is 590 nm.

[0013] In a second aspect, the present invention also provides a solver for a variable coefficient ordinary differential equation based on liquid crystal, including: a liquid crystal device; the liquid crystal device includes a nematic liquid crystal layer, an optically controlled alignment layer is arranged outside the nematic liquid crystal layer, and a conductive glass is arranged outside the optically controlled alignment layer; The liquid crystal device has different directors at different positions.

[0014] According to a solver for a variable coefficient ordinary differential equation based on liquid crystal provided by the present invention, the optically controlled alignment layer includes a first optically controlled alignment layer and a second optically controlled alignment layer arranged on both sides of the nematic liquid crystal layer; The conductive glass includes a first conductive glass layer disposed outside the first photo-controlled alignment layer and a second conductive glass layer disposed outside the second photo-controlled alignment layer.

[0015] A liquid crystal-based variable coefficient ordinary differential equation solver provided by the present invention, the liquid crystal device has a square structure.

[0016] Compared with the prior art, the present invention has the following beneficial effects: The liquid crystal-based variable coefficient ordinary differential equation solving method provided by the present invention selects x-polarized light as the incident light wave, and the polarization state of the outgoing light wave is still the x-polarized light state. By adjusting the director at different positions of the liquid crystal device, and by selecting the liquid crystal voltage to adjust the transfer function added by the liquid crystal device to the incident light wave, the coefficients of the ordinary differential equation to be solved can be adjusted, solving the problem that the existing spatial domain differential equation solver cannot adjust the equation coefficients, and the calculation speed and density are better. Moreover, the designed liquid crystal device has a compact structure, and at the same time has the advantages of small volume and light weight, and has important application prospects in the fields of medical imaging, photon computing, etc. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] In order to more clearly illustrate the technical solutions in the present invention or the prior art, the following will briefly introduce the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings in the following description are some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can also be obtained based on these drawings.

[0018] Figure 1 is a flowchart of the liquid crystal-based variable coefficient ordinary differential equation solving method provided by the present invention; Figure 2 is a schematic diagram of the director distribution of the liquid crystal device in an embodiment of the present invention; Figure 3 is a schematic diagram of the comparison between the simulation input signal and the simulation and theoretical output results in an embodiment of the present invention; Figure 4 is a schematic structural diagram of the liquid crystal-based variable coefficient ordinary differential equation solver provided by the present invention; Figure 5 is a schematic diagram of the optical path for solving the ordinary differential equation in an embodiment of the present invention.

[0019] REFERENCE SIGNS: 110: First conductive glass layer; 120: First photo-controlled alignment layer; 130: Nematic liquid crystal layer; 140: Second photo-controlled alignment layer; 150: Second conductive glass layer. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0020] To make the objectives, technical solutions, and advantages of the present invention clearer, the technical solutions in the present invention will be clearly and completely described below in conjunction with the accompanying drawings in the present invention. Apparently, the described embodiments are some, but not all, of the embodiments of the present invention. All other embodiments obtained by those of ordinary skill in the art based on the embodiments in the present invention without creative efforts shall fall within the protection scope of the present invention.

[0021] The present invention provides a method for solving variable coefficient ordinary differential equations based on liquid crystal. Figure 1 It is a flowchart of the method for solving variable coefficient ordinary differential equations based on liquid crystal provided by the present invention. As Figure 1 shown, the method includes the following steps: Step S101: Construct a liquid crystal device, and based on the working surface of the liquid crystal device, establish a plane rectangular coordinate system (xoy coordinate system); the liquid crystal device has different director vectors at different positions. Step S102: Place the liquid crystal device on the spectrum plane of the 4f system to obtain the output signal after solving the input signal under different liquid crystal voltages. Step S103: Adjust the liquid crystal voltage to make the liquid crystal device generate different transfer functions for the incident light wave, and obtain the solution results of ordinary differential equations with different coefficients under different liquid crystal voltages; the incident light wave is an x-linearly polarized light.

[0022] In this method, first, a liquid crystal device for realizing the solution of variable coefficient ordinary differential equations is constructed. Then, through theoretical calculation and simulation, the liquid crystal device is placed on the spectrum plane of the 4f system, where f is the focal length of the lens of the 4f system, to obtain the output signal obtained after the input signal passes through the liquid crystal under different liquid crystals. Finally, the solution result of the first-order differential equation of the input signal is observed on the detector plane. Finally, by adjusting the liquid crystal voltage, the liquid crystal device generates different transfer functions for the incident light wave H , and the solution results of ordinary differential equations with different coefficients are observed under different liquid crystal voltages. Since the liquid crystal device cannot be irradiated by blue light, in this method, an incident light wave with a wavelength above 550 nm is selected for operation. Preferably, the wavelength of the incident light wave is 590 nm. In this method, an x-linearly polarized light is selected as the incident light wave, and the polarization state of the outgoing light wave is still the x-linearly polarized state. By adjusting the director vectors at different positions of the liquid crystal device and selecting the liquid crystal voltage to adjust the transfer function added by the liquid crystal device to the incident light wave, the coefficients of the ordinary differential equation to be solved can be adjusted, solving the problem that the existing spatial domain differential equation solver cannot adjust the equation coefficients, and the calculation speed and density are better. Moreover, the designed liquid crystal device has a compact structure and has advantages such as small volume and light weight, and has important application prospects in the fields of medical imaging, photon computing, etc.

[0023] In some of these embodiments, step S101 involves constructing a liquid crystal device, including: determining the material and thickness of the liquid crystal device H ; based on the phase delay amount added to the incident light wave by the liquid crystal device at different liquid crystal voltages when the wavelength is 590 nm φ , determining the coefficients of the first-order ordinary differential equation corresponding to different liquid crystal voltages; arranging the director of the liquid crystal device.

[0024] In this embodiment, the thickness of the liquid crystal device ,..

[0025] On this basis, step S101 involves establishing a plane rectangular coordinate system based on the working surface of the liquid crystal device, including: setting the directions of the two right-angled sides parallel to the working surface of the liquid crystal device as the x-axis and y-axis respectively to construct a plane rectangular coordinate system. In this embodiment, the direction of the director of the liquid crystal device is the angle between the liquid crystal long axis of the liquid crystal device and the x-axis of the plane rectangular coordinate system.

[0026] Specifically Figure 2 is a schematic diagram of the director distribution of the liquid crystal device in the embodiment of the present invention. As Figure 2 shown, the director of the liquid crystal device varies with the coordinate. The director directions at different positions of the liquid crystal device are:[[]]

[0027] Among them is the direction of the director at different positions of the liquid crystal device is the abscissa at different positions of the liquid crystal device is the side length of the liquid crystal device. For a liquid crystal device with a thickness of , the phase added to the incident light wave by it is:[[]]

[0028] Among them is the thickness of the liquid crystal device represents the phase added to the incident light wave by the liquid crystal device is the refractive index of the liquid crystal long axis is the refractive index of the liquid crystal short axis. For different incident wavelengths and show different values.

[0029] In some of these embodiments, in step S102, placing the liquid crystal device on the spectral plane of the 4f system to obtain the output signal after solving the input signal under different liquid crystal voltages includes: placing the liquid crystal device on the spectral plane of the 4f system and injecting an incident light wave of a specific wavelength; determining the transfer function of the liquid crystal device for modulating the incident light wave; changing the liquid crystal voltage to obtain the output signal after solving the input signal under different ordinary differential equation coefficients.

[0030] When an incident light wave with a wavelength of 590 nm is incident, the transfer function of the liquid crystal device for modulating the incident light wave is:

[0031] Wherein, is the transfer function, and are constants, is the spatial frequency, where w = x / D .

[0032] Based on the above embodiments, in step S103, adjusting the liquid crystal voltage to make the liquid crystal device generate different transfer functions for the incident light wave includes: adjusting the liquid crystal voltage to change the phase added to the output light wave by the liquid crystal device, so that the liquid crystal device generates different transfer functions for the incident light wave.

[0033] The liquid crystal device has different angular distributions at different positions, and the liquid crystal can be equivalent to a wave plate. The phase modulation effect at different positions of the liquid crystal can be equivalent to wave plates with different fast axis directions. Therefore, when x-polarized light is incident and x-polarized light is emitted, the complex amplitude modulation amount added to the light wave at a certain position of the liquid crystal device is:

[0034] Wherein, is the complex amplitude modulation amount, θ is the direction of the director at different positions of the liquid crystal device, φ is the phase added to the incident light wave at different positions of the liquid crystal device, i represents the imaginary unit.

[0035] For an ordinary differential equation:

[0036] Wherein, and b are both constant coefficients of the first-order ordinary differential equation, g(x) represents the known input signal, f(x) represents the unknown output signal, f’(x) represents the first derivative of the unknown output signal. Its transfer function is:

[0037] Among them, is the transfer function of the ordinary differential equation, i is the imaginary unit, k is the frequency. Performing Taylor expansion on it, we can obtain:

[0038] As can be seen from the above formula, is the same as the complex amplitude modulation amount added to the light wave at a certain position of the liquid crystal device when x-polarized light is incident and x-polarized light is emitted. At this time, let:

[0039] That is, the complex amplitude modulation amount added by the liquid crystal device to the incident light wave with a wavelength of 590 nm is:

[0040] Among them, represents the complex amplitude modulation amount, is the abscissa at different positions of the liquid crystal device, is the side length of the liquid crystal device, φ is the phase added to the incident light wave at different positions of the liquid crystal device.

[0041] As can be seen from the above formula, this function matches the transfer function form of the first-order differentiator. At this time, by changing the liquid crystal voltage, 、, the values of and in the complex amplitude added by the liquid crystal device to the incident light wave with a wavelength of 590 nm can be changed. They respectively correspond to the constants A and B mentioned above. At the same time, let:

[0042]

[0043] Then, by changing the liquid crystal voltage, the change of the coefficients of the ordinary differential equation can be realized, and thus the solution results of the input signal under different coefficients of the ordinary differential equation can be obtained. Figure 3 is a schematic diagram of the comparison between the simulation input signal and the simulation and theoretical output results in the embodiment of the present invention. As Figure 3 shown, it is the simulation result of the output signal received by the detector when the input signal is irradiated by the incident light wave with a wavelength of 590 nm. At this time, =300, =2. It can be seen that the theoretical result F-idea and the simulation result F-actual are in good agreement.

[0044] In summary, in view of the deficiencies of the existing solutions, by utilizing the phase modulation and dispersion characteristics of liquid crystals, the present invention proposes a method for solving variable-coefficient ordinary differential equations based on liquid crystals, and designs a solution for different-coefficient ordinary differential equations of incident signals under different liquid crystals. Through the design of the director at different positions of the liquid crystal, the constructed liquid crystal has the same transfer function as the first-order ordinary differential equation, can realize the solution of the variable-coefficient first-order ordinary differential equation, and has been verified by simulation, and has important application prospects in the fields of photonic computing, biological imaging, etc.

[0045] The present invention also provides a variable-coefficient ordinary differential equation solver based on liquid crystals. Figure 4 FIG. is a schematic structural diagram of the variable-coefficient ordinary differential equation solver based on liquid crystals provided by the present invention. As Figure 4 shown, the solver includes: a liquid crystal device. Preferably, in order to facilitate the use and assembly of the liquid crystal device, the liquid crystal device has a square structure; the liquid crystal device includes a nematic liquid crystal layer 130, and a photo-controlled alignment layer is provided outside the nematic liquid crystal layer 130, and an ITO (Indium Tin Oxide) conductive glass is provided outside the photo-controlled alignment layer; the liquid crystal device has different directors at different positions.

[0046] Specifically, the photo-controlled alignment layer includes a first photo-controlled alignment layer 120 and a second photo-controlled alignment layer 140 provided on both sides of the nematic liquid crystal layer 130; the conductive glass includes a first conductive glass layer 110 provided outside the first photo-controlled alignment layer 120, and a second conductive glass layer 150 provided outside the second photo-controlled alignment layer 140.

[0047] Figure 5 FIG. is a schematic diagram of the optical path for solving ordinary differential equations in an embodiment of the present invention. As Figure 5 shown, applying the above liquid crystal device to the optical path for solving variable-coefficient ordinary differential equations, the optical path mainly consists of a laser, two convex lenses with a focal length of f each, and the liquid crystal device designed by the present invention. The object is placed at the front focal plane of lens 1, the liquid crystal is placed at the rear focal plane of lens 1, the rear focal plane of lens 1 and the front focal plane of lens 2 coincide with each other, and the detector is placed at the rear focal plane of lens 2. By changing the liquid crystal voltage, the solution results of the first-order ordinary differential equation of the input signal under different coefficients can be received on the detector.

[0048] Through the description of the above embodiments, those skilled in the art can clearly understand that each embodiment can be implemented by means of software plus a necessary general hardware platform, and of course, it can also be implemented by hardware. Based on such an understanding, the essence of the above technical solution, or the part that contributes to the prior art, can be embodied in the form of a software product. This computer software product can be stored in a computer-readable storage medium, such as ROM / RAM, magnetic disk, optical disk, etc., and includes several instructions to enable a computer device (which can be a personal computer, a server, or a network device, etc.) to execute the methods described in each embodiment or some parts of the embodiments.

[0049] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that they can still modify the technical solutions described in the foregoing embodiments, or perform equivalent replacements for some of the technical features; and these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the spirit and scope of the technical solutions of each embodiment of the present invention.

Claims

1. A method for solving variable coefficient ordinary differential equations based on liquid crystals, characterized in that, Comprising: Construct a liquid crystal device, and based on the working surface of the liquid crystal device, establish a rectangular coordinate system; the liquid crystal device has different director vectors at different positions; Place the liquid crystal device on the spectral plane of a 4f system to obtain the output signal after solving the input signal at different liquid crystal voltages; Adjust the liquid crystal voltage so that the liquid crystal device generates different transfer functions for the incident light wave, and obtain the solution results of ordinary differential equations with different coefficients at different liquid crystal voltages; the incident light wave is an x-linearly polarized light.

2. The method for solving variable coefficient ordinary differential equations based on liquid crystals according to claim 1, wherein Construct a liquid crystal device, including: Determine the material and thickness of the liquid crystal device; According to the phase delay amount added by the liquid crystal device to the incident light wave, determine the coefficients of the corresponding first-order ordinary differential equations at different liquid crystal voltages; Arrange the director vectors of the liquid crystal device.

3. The method for solving variable coefficient ordinary differential equations based on liquid crystal according to claim 1, characterized in that, Based on the working surface of the liquid crystal device, establish a rectangular coordinate system, including: Set the directions of two right-angled sides parallel to the working surface of the liquid crystal device as the x-axis and y-axis respectively to construct a rectangular coordinate system.

4. The method for solving variable coefficient ordinary differential equations based on liquid crystal according to claim 3, wherein The direction of the director vector of the liquid crystal device is the angle between the liquid crystal long axis of the liquid crystal device and the x-axis of the rectangular coordinate system.

5. The method for solving variable coefficient ordinary differential equations based on liquid crystals according to claim 1, characterized in that, Place the liquid crystal device on the spectral plane of a 4f system to obtain the output signal after solving the input signal at different liquid crystal voltages, including: Place the liquid crystal device on the spectral plane of a 4f system and inject an incident light wave of a specific wavelength; Determine the transfer function of the liquid crystal device for modulating the incident light wave; Change the liquid crystal voltage to obtain the output signal after solving the input signal under different coefficients of ordinary differential equations.

6. The method for solving variable coefficient ordinary differential equations based on liquid crystal according to claim 1, characterized in that Adjust the liquid crystal voltage so that the liquid crystal device generates different transfer functions for the incident light wave, including: Adjust the liquid crystal voltage to change the phase added by the liquid crystal device to the outgoing light wave, so that the liquid crystal device generates different transfer functions for the incident light wave.

7. The method for solving variable coefficient ordinary differential equations based on liquid crystals according to claim 1, characterized in that The wavelength of the incident light wave is 590 nm.

8. A liquid crystal-based variable coefficient ordinary differential equation solver, characterized in that, Comprising: A liquid crystal device; the liquid crystal device includes a nematic liquid crystal layer (130), an optically controlled alignment layer is provided outside the nematic liquid crystal layer (130), and a conductive glass is provided outside the optically controlled alignment layer; The liquid crystal device has different director vectors at different positions.

9. The method for solving variable coefficient ordinary differential equations based on liquid crystals according to claim 8, characterized in that, The optically controlled alignment layer includes a first optically controlled alignment layer (120) and a second optically controlled alignment layer (140) provided on both sides of the nematic liquid crystal layer (130); The conductive glass includes a first conductive glass layer (110) provided outside the first optically controlled alignment layer (120) and a second conductive glass layer (150) provided outside the second optically controlled alignment layer (140).

10. The method for solving variable coefficient ordinary differential equations based on liquid crystal according to claim 8, characterized in that, The liquid crystal device is of a square structure.

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