Nonlinear lens based on transformation material and Kerr effect and design method

Abstract

The invention discloses a nonlinear lens based on a transformation material and a Kerr effect and a design method, and belongs to the field of optical signal processing and transformation materials. The nonlinear lens is a rectangular lens and is made of a nonlinear material. The spatial refractive index distribution is designed by changing the optical principle, the path instability in a dynamic system is combined with the optical Kerr effect, and the circular lens is adjusted into the square lens with the unstable system path by utilizing the spatial distribution of special material parameters; the weak refractive index change is converted into violent offset of a light beam output position under the instability characteristic of a dynamic system, and under the specific refractive index distribution, the lens has a focusing effect on the light beam propagating in the lens, so that the light beam is enhanced in a local area, and then the nonlinear effect is enhanced; and an optical system with higher speed and lower energy consumption is realized on a simpler device. According to the invention, conformal transformation is adopted, the isotropy of a lens material is ensured, and manufacturing of a nonlinear lens is facilitated.

Description

Technical Field

[0001] This invention relates to a nonlinear optical system device and design method based on a transformation medium, belonging to the field of optical signal processing and transformation material technology. Background Technology

[0002] Since 2006, JB Pendry and U. Leonhardt have proposed transformation optics theory. Transformation optics is a method for solving the inverse problem of electromagnetic field distribution and electromagnetic parameters. By utilizing the formal invariance of Maxwell's equations, the problem of solving for the electromagnetic parameters corresponding to the wave field is transformed into a problem of geometric design and calculation of coordinate transformation parameters, thus obtaining the corresponding solution. Materials designed and calculated using transformation optics are called "transformation materials".

[0003] Nonlinear refractive index, also known as the optical Kerr effect, is an important phenomenon in nonlinear optics. Its core principle is that the refractive index of a medium changes with the intensity of incident light, leading to alterations in the phase, propagation characteristics, or polarization state of the light. According to the principles of nonlinear optics, the refractive index change due to the optical Kerr effect can be expressed as… ,in The nonlinear refractive index coefficient of the medium (the unit is usually...) ), The linear refractive index under low light conditions. Incident light intensity (unit: The nonlinear refractive index change caused by the optical Kerr effect can lead to phenomena such as self-focusing, self-phase modulation, and cross-phase modulation in the medium. It also has the characteristics of transient response and is often used in all-optical signal processing problems such as optical switches, all-optical logic gates, and nonlinear activators.

[0004] Currently, the main methods for realizing these nonlinear systems include: using optical resonant structures (such as microring resonators and photonic crystal cavities) to enhance the interaction time between light and matter; using the gain saturation, threshold effect, and mode competition processes of semiconductor lasers to achieve nonlinear activation; and realizing all-optical logic gates based on the combination of semiconductor optical amplifiers and interferometers or based on small changes in the input signal of nonlinear optical fibers. These methods usually require complex material microstructures or are based on complex nonlinear mechanisms, and are often limited in bandwidth and flexibility. Summary of the Invention

[0005] To address the issues of high pump power and complex system structures required in traditional nonlinear optical systems, this invention aims to provide a nonlinear lens and design method based on transforming materials and the Kerr effect. By adjusting the refractive index distribution through transforming optics, a circular lens is transformed into a square lens with an unstable system path. This allows the beam propagation path to be significantly deflected due to the small refractive index change caused by the nonlinear refractive index when the input light power is high. During this process, the lens has a focusing effect on the beam propagating within it, enhancing the beam locally and thus amplifying the nonlinear effect. This enables a higher-speed, lower-energy-consumption optical system on simpler devices.

[0006] The objective of this invention is achieved through the following technical solution.

[0007] This nonlinear lens, based on transforming materials and the Kerr effect, is referred to as this lens, and the design method of this lens is referred to as this method.

[0008] The nonlinear lens disclosed in this invention, based on a transforming material and the Kerr effect, is a rectangular lens made of a nonlinear material. By designing the refractive index distribution of the rectangular lens, the lens exhibits path instability and optical Kerr effect characteristics. This characteristic alters the propagation path of light within the rectangular lens, thereby changing the focusing position of the output surface.

[0009] The present invention discloses a method for designing nonlinear lenses based on transforming materials and the Kerr effect, specifically a method for designing the refractive index distribution of a rectangular lens, comprising the following steps:

[0010] Step 1: Select a circular lens with central focusing and an unstable optical system as the original lens;

[0011] Step 2: Perform the following steps on the circular lens. A conformal transformation yields a rectangular lens;

[0012] Original complex plane before conformal transformation , also known as virtual space, where i is an imaginary number; Let be the real part of the complex plane. The imaginary part of the complex plane, the original complex plane This corresponds to a circular lens;

[0013] Transformed complex plane Also known as physical space Let be the real part of the complex plane. Let be the imaginary part of the complex plane. This corresponds to the transformed rectangular lens;

[0014] According to Fermat's theorem, the refractive index distribution of the lens after conformal transformation is as follows:

[0015] ; (1)

[0016] Among them, symbols The meaning of "represent" is to find the differential. For virtual space refractive index, For physical space refractive index, ; For the Jacobian matrix:

[0017] ; (2)

[0018] Step 3: Design the transformed lens using nonlinear materials. The refractive index distribution of the transformed rectangular lens is then: ,in, It is a nonlinear refractive index coefficient. Input light intensity;

[0019] Step 4: Calculate the Lyapunov index of the rectangular lens made of nonlinear material. The refractive index of the original spatial lens is... , Representing the radial distance to the optical axis, after conformal transformation, we have:

[0020] ; (3)

[0021] Including the nonlinear term, in the final lens design, we have The light comes from the left boundary enter;

[0022] The equation of light is ;in, For the optical path of propagation, The refractive index gradient depends on the light intensity under nonlinear conditions;

[0023] Curvature vector radius of curvature ,in, Let the unit principal normal be at some point on the ray. The formula for calculating the refractive index gradient shows that as the nonlinear effect intensifies, the refractive index gradient decreases and the radius of curvature increases.

[0024] Under these input conditions, as the light intensity increases, the angular deviation... Lyapunov index The initial angular deviation At this point, the designed lens is still an unstable lens for the dynamic system;

[0025] Furthermore, since the original system focuses at the center, while this lens focuses at the output surface, the instability of the system leads to a noticeable shift in the focal point.

[0026] The original lens has a refractive index distribution of A circular lens.

[0027] Beneficial effects:

[0028] 1. The nonlinear lens and design method based on transforming materials and the Kerr effect disclosed in this invention design a spatial refractive index distribution by transforming optical principles. It combines the path instability in the dynamic system with the optical Kerr effect and utilizes the spatial distribution of special material parameters to transform weak nonlinear refractive index changes into drastic shifts in the beam output position under the instability characteristics of the dynamic system. Furthermore, by utilizing the path instability of the dynamic system, the nonlinear effect is amplified under a specific refractive index distribution, resulting in a higher intensity nonlinear response of the output to the input light intensity.

[0029] 2. The nonlinear lens and design method based on transformation materials and Kerr effect disclosed in this invention adjusts the refractive index distribution by transforming optics, transforming a circular lens into a square lens with an unstable system path. This allows the beam propagation path to be significantly deflected due to the small refractive index change caused by the nonlinear refractive index when the input light power is high. During this process, the lens has a focusing effect on the beam propagating within it, enhancing the beam locally and thus enhancing the nonlinear effect. This enables a higher speed and lower energy consumption optical system on a simpler device.

[0030] 3. The nonlinear lens design method based on transformation materials and Kerr effect disclosed in this invention combines the path instability of the dynamic system with the optical Kerr effect. The instability of the system path is equivalent to amplifying the nonlinearity. Therefore, the lens can still produce the required path offset even when the input power is relatively low.

[0031] 4. The nonlinear lens design method based on transforming materials and Kerr effect disclosed in this invention manipulates the spatial distribution of the lens parameters through conformal mapping, combining the path instability of the system with the optical Kerr effect. It does not require a complex structure and only requires a lens to generate the required path offset.

[0032] 5. The nonlinear lens design method based on transforming materials and the Kerr effect disclosed in this invention adopts a conformal transformation method to ensure the isotropy of the lens material, which is beneficial for the fabrication of nonlinear lenses. Attached Figure Description

[0033] Figure 1In Example 1, the composition and specific implementation of the lens before transformation and the lens of this invention, "Nonlinear Lens and Design Method Based on Transformation Material and Kerr Effect," are described. Figure 1 (a) shows the refractive index distribution of the lens before the transformation. Figure 1 (b) shows the refractive index distribution of the lens after the transformation. Figure 1 (c) is the refractive index distribution that limits the refractive index of the transformed lens to a certain range;

[0034] Figure 2 The virtual space W and physical space for the conformal transformation in the method of the present invention "Nonlinear Lens and Design Method Based on Transformation Material and Kerr Effect" are described. A schematic diagram;

[0035] Figure 3 This is a schematic diagram of ray optics simulation of a transformed rectangular lens with different power beams input in the invention "Nonlinear Lens and Design Method Based on Transformation Material and Kerr Effect". Figure 3 (a) Input optical power 0.0001 ( The ray trajectory at that time Figure 3 (b) Input optical power 0.0001 ( The ray trajectory when the focal point of the output surface is magnified locally. Figure 3 (c) is an input optical power of 0.001 ( The ray trajectory when the focal point of the output surface is magnified locally. Figure 3 (d) represents the input optical power of 0.01 ( The ray trajectory when the focal point of the output surface is magnified locally. Figure 3 (e) represents the input optical power of 0.1 ( The ray trajectory when the focal point of the output surface is magnified locally;

[0036] Figure 4 This is a schematic diagram of wave optics simulation for a transformed rectangular lens under different electric field intensities, as described in the present invention, "Nonlinear Lens and Design Method Based on Transformation Material and Kerr Effect". Figure 4 (a) is the wave-like optical field when the input electric field is 2000 V. Figure 4 (b) is the wave-like optical field when the input electric field is 3000 V. Figure 4 (c) shows the wave-like optical field when the input electric field is 4000 V. Figure 4 (d) shows the wave-like optical field when the input electric field is 5000 V. Figure 4 (e) shows the electric field mode distribution on the output surface when the input electric field is 2000V. Figure 4 (f) shows the electric field mode distribution on the output surface when the input electric field is 3000V. Figure 4(g) shows the electric field mode distribution on the output surface when the input electric field is 4000V. Figure 4 (h) represents the electric field mode distribution on the output surface when the input electric field is 5000V;

[0037] Figure 5 This is a schematic diagram illustrating the relationship between the input and output electric field strengths at different detection points in the present invention, "Nonlinear Lens and Design Method Based on Transformation Material and Kerr Effect." Figure 5 (a) shows the relationship between the input and output electric fields when the detection point is 0. Figure 5 (b) shows the relationship between the input and output electric fields when the detection point is 0.027. Figure 5 (c) represents the relationship between the input electric field and the difference between the output electric field at the detection point of 0 and the monitoring point of 0.01. Figure 5 (d) is the relationship between the input electric field and the difference between the output electric field at the detection point of 0.01 and the monitoring point of 0. Detailed Implementation

[0038] To better illustrate the purpose and advantages of the present invention, the invention will be further described below in conjunction with the accompanying drawings and examples.

[0039] Example 1:

[0040] This embodiment illustrates the composition and specific implementation of the lens before transformation and the lens in the present invention, "Nonlinear Lens and Design Method Based on Transformation Material and Kerr Effect".

[0041] Figure 1 (a) shows the refractive index distribution of the lens before the transformation. Figure 1 (b) shows the refractive index distribution of the lens after the transformation. Figure 1 (c) is the refractive index distribution that limits the refractive index of the transformed lens to a certain range.

[0042] The nonlinear lens disclosed in this embodiment, based on a transforming material and the Kerr effect, transforms a circular lens with a uniform refractive index distribution by converting its upper semicircular plane into a rectangular plane, transforming the outer boundary (i.e., the input surface) of the original circular lens into the long side (input surface) of the rectangular lens, and transforming the horizontal axis passing through the center of the original circular lens into the short side (output surface) of the rectangular lens. Figure 1As can be seen, both the left and right sides of the transformed rectangular lens can serve as input surfaces, and the bottom edge of the rectangle is the output surface. Specifically, in this embodiment, the light beam enters from the left side of the rectangle, passes through the lens, and is focused at the output surface, i.e., the bottom edge of the rectangle. Due to the path instability of the system, given the lens refractive index distribution, the focusing position of the beam on the output surface will change due to the accumulation of small system disturbances as the input beam power changes. The design method of this device employs conformal transformation to ensure the isotropy of the lens material, which is beneficial for engineering fabrication.

[0043] The following is proof of the instability of this lens:

[0044] In the original lens, there are two rays: one is a radially propagating ray, and the other is angularly deviated from the radially propagating ray. The offset rays; both rays initially were at... Place;

[0045] Radial transmission of light includes: , The position of a point on the light ray. The optical path length propagating from that point;

[0046] According to Bouguer's formula, the deflected light rays have: , It is a function of refractive index. Position vector With light The angle between the tangents at a point, and the initial position of that point. initial position refractive index , Then there is a constant. That is, at any point during the propagation process, there is:

[0047] , (4)

[0048] Substitute into the original lens refractive index distribution function ,

[0049] , (5)

[0050] , (6)

[0051] For small angles: ,but

[0052] , (7)

[0053] Angular separation of light ,but

[0054] , (8)

[0055] The Lyapunov exponent of the system

[0056] , (9)

[0057] in Let be the angular offset function of the optical path during transmission. For the radial angular offset function during transmission, in the original lens, ;

[0058] The refractive index distribution of the circular lens is larger at the outside and smaller at the inside, with a Lyapunov exponent greater than 0, exhibiting significant instability. When there is a small initial angular separation, the transmission path is significantly affected by the instability. Furthermore, the refractive index gradient of this lens always points towards the center of the circle. With input at the vertical boundary, the beam is always focused at the center of the circle. Choosing this circular lens as the original lens can ensure significant performance even when the focal point is offset.

[0059] Since the transformation optics method used in this invention is conformal mapping, when the original lens is unstable, the offset of the beam propagation path of this lens under the same disturbance corresponds one-to-one with the offset of the original lens. When there is initial angular separation, the beam propagation path will rapidly separate from the beam without angular separation over time, which is also an unstable system.

[0060] Example 2

[0061] This embodiment illustrates the process and specific implementation of the method in the present invention "Nonlinear Lens and Design Method Based on Transformation Material and Kerr Effect".

[0062] Figure 2 This describes the process and specific implementation of the method in the invention "A Nonlinear Lens Design Method Based on Refractive Index Distribution, Transformation Material, and Kerr Effect".

[0063] Figure 2 As can be seen, this method has the following specific implementation approaches:

[0064] Perform circular lens A conformal transformation yields a rectangular lens;

[0065] in, and These are the original complex plane and the transformed complex plane, respectively.

[0066] The original complex plane before conformal transformation, also known as virtual space. Let be the real part of the complex plane. The imaginary part of the complex plane, the original complex plane This corresponds to a circular lens;

[0067] The transformed complex plane, also known as physical space. Let be the real part of the complex plane. Let be the imaginary part of the complex plane. This corresponds to the transformed rectangular lens;

[0068] Specifically, in this embodiment, such as Figure 2 (a) is a circular lens with a uniform refractive index. Figure 2 (b) is a conformal transformation The subsequent rectangular lens has a length of The original complex plane and The points are transformed into and Points B and D in the original complex plane, and their positions on the transformed plane. The position of the transformed rectangular lens can be set according to the specific example.

[0069] Among them, virtual space and physical space The spatial transformation relationship between them must satisfy the Cauchy-Riemann condition:

[0070] , (10)

[0071] Among them, symbols The meaning represented is to find the partial differential.

[0072] According to Fermat's theorem, physical space Transformation of optical path and virtual space There is a corresponding relationship between the optical paths:

[0073] , (11)

[0074] therefore, ;

[0075] Among them, symbols The meaning of "represent" is to find the differential. For virtual space refractive index, For physical space refractive index, ;

[0076] After the above steps, the nonlinear lens and design method based on the transformation material and Kerr effect in this embodiment are completed.

[0077] Example 3

[0078] Figure 3 This is a schematic diagram of ray optics simulation of inputting beams of different power to a transformed rectangular lens in the present invention, "Nonlinear Lens and Design Method Based on Transformation Material and Kerr Effect". This embodiment utilizes COMSOL Multiphysics for computer simulation.

[0079] The parameters in this embodiment are set as follows: The following parameters ignore the length unit and can be set as needed: wavelength is set to 0.03, lens length is 2.5, and width is... Nonlinear refractive index The original circular lens refractive index distribution is The refractive index distribution of the transformed rectangular lens is as follows: , (Unit is) ) represents the input optical power, and the input window for the beam is 2-2.5.

[0080] In geometric optics simulation, light beams of different powers are input from the input surface of a transformed rectangular lens, and from... Figure 3 (b)- Figure 3 (e) shows the ray trajectories for input optical power of 0.0001, 0.001, 0.01, and 0.1, respectively. As the input optical power increases, the cumulative system disturbance due to nonlinear effects causes a shift in the beam propagation path and focusing position of the unstable system. This shift is observed to the right, due to the nonlinear refractive index reducing the refractive index gradient within the lens and decreasing the radius of curvature. Increase (of which, Let the unit principal normal be at some point on the ray. (This is the gradient calculation formula).

[0081] Example 4

[0082] This embodiment illustrates the fluctuation optical simulation results of the effect of optical nonlinearity on the output of the device in the present invention, "Nonlinear Lens and Design Method Based on Transformation Material and Kerr Effect," under different input power conditions. This embodiment utilizes COMSOL Multiphysics for computer simulation.

[0083] The parameters in this embodiment are set as follows: The following parameters ignore the length unit and can be set as needed: wavelength is set to 0.03, lens length is 4, width is 1, and nonlinear refractive index... The original circular lens refractive index distribution is The refractive index distribution of the transformed rectangular lens is as follows: , (Unit is) ) represents the input electric field, and the input window for the beam is 1-4.

[0084] In wave optics simulation, electric fields of different intensities are input from the input surface of the transformed rectangular lens. Figure 4 (a), (b), (c), and (d) respectively demonstrate the wave-like optical fields with input electric fields of 2000, 3000, 4000, and 5000 V. Figure 4 Figures (e), (f), (g), and (h) respectively illustrate the electric field mode distribution on the output surface. As the input optical power increases, the cumulative system disturbance due to nonlinear effects causes a shift in the wave field propagation path of this path-unstable system, resulting in a shift in the focusing position. This shift is observed to the right, due to the nonlinear refractive index reducing the refractive index gradient within the lens and decreasing the radius of curvature. Increase (of which, Let the unit principal normal be at some point on the ray. (This is the gradient calculation formula), which is consistent with the results from X-ray optics.

[0085] Example 5

[0086] This embodiment illustrates the relationship between the input and output electric field strengths at different detection points in the present invention, "Nonlinear Lens and Design Method Based on Transformation Material and Kerr Effect".

[0087] To verify that the present invention can be used as a nonlinear activation function, this embodiment selects different detection points, and under the condition of changing input electric field, the input electric field and output electric field exhibit different nonlinear activation functions.

[0088] in, Figure 5 (a) shows the relationship between the input and output electric fields when the detection point is 0, expressed in the form of a sigmoid function. Figure 5 (b) shows the relationship between the input and output electric fields when the detection point is 0.027, expressed in ReLU function form. Figure 5 (c) represents the relationship between the input electric field and the difference between the output electric field at the detection point of 0 and the monitoring point of 0.01. Figure 5 (d) is the relationship between the input electric field and the difference between the output electric field at the detection point of 0.01 and the monitoring point of 0.

[0089] The above detailed description further illustrates the purpose, technical solution, and beneficial effects of the invention. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A nonlinear lens based on transforming materials and the Kerr effect, characterized in that: The rectangular lens is made of a nonlinear material. By designing the refractive index distribution of the rectangular lens, the lens exhibits path instability and the optical Kerr effect. This characteristic alters the propagation path of light within the rectangular lens, thereby changing the focusing position of the output surface.

2. A design method for nonlinear lenses based on transforming materials and the Kerr effect, characterized in that: The method for designing the refractive index distribution of a rectangular lens includes the following steps: Step 1: Select a circular lens with central focusing and an unstable optical system as the original lens; Step 2: Perform the following steps on the circular lens. A conformal transformation yields a rectangular lens; Original complex plane before conformal transformation , also known as virtual space, where i is an imaginary number; Let be the real part of the complex plane. The imaginary part of the complex plane, the original complex plane This corresponds to a circular lens; Transformed complex plane Also known as physical space Let be the real part of the complex plane. Let be the imaginary part of the complex plane. This corresponds to the transformed rectangular lens; According to Fermat's theorem, the refractive index distribution of the lens after conformal transformation is: ； （1） Among them, symbols The symbol represents the process of finding the differential. The meaning represented is to find the partial differential. For virtual space refractive index, For physical space refractive index, , For the Jacobian matrix: ； （2） Step 3: Design the transformed lens using nonlinear materials. The refractive index distribution of the transformed rectangular lens is then: ,in, It is a nonlinear refractive index coefficient. The input light intensity.

3. The method as described in claim 2, characterized in that: The original lens has a refractive index distribution of A circular lens, This indicates the radial distance to the optical axis.

4. The method as described in claim 2, characterized in that: The rectangular lens made of nonlinear material obtained in step three is verified using the following method. Calculate the Lyapunov exponent for a rectangular lens made of a nonlinear material, given that the refractive index of the original spatial lens is... After conformal transformation, we have ； （3） Including the nonlinear term, in the final lens design, we have The light comes from the left boundary enter; The equation of light is ;in, For the optical path of propagation, The refractive index gradient depends on the light intensity under nonlinear conditions; Curvature vector radius of curvature ,in, Let the unit principal normal be at some point on the ray. The formula for calculating the refractive index gradient shows that as the nonlinear effect intensifies, the refractive index gradient decreases and the radius of curvature increases. Under these input conditions, as the light intensity increases, the angular deviation... Lyapunov index The initial angular deviation At this point, the designed lens is still an unstable lens for the dynamic system; Furthermore, since the original system focuses at the center, while this lens focuses at the output surface, the instability of the nonlinear lens leads to a significant shift in the focal point, thus enabling the verification of rectangular lenses for nonlinear materials.

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