Light field mode conversion method

Abstract

The invention discloses a light field mode conversion method, and relates to the technical field of optics. The method aims at solving the problem that in an existing mode conversion technology, efficient conversion between any modes, especially between different space modes is difficult to achieve. The method comprises the following steps: acquiring an intrinsic mode of the elliptical multimode waveguide; applying longitudinal periodic modulation to the waveguide; the coupling strength between the target mode pairs is controlled by adjusting the longitudinal modulation frequency or designing a specific transverse refractive index modulation form, so that the directional conversion of the light field modes is realized. Wherein the refractive index disturbance function is designed into a product form of target conversion mode field distribution, so that coupling and conversion between any two modes can be efficiently excited. The method has the beneficial effects that a flexible and efficient mode conversion implementation way is provided, the conversion rate between specific geometric feature modes can be enhanced in a targeted manner, irrelevant coupling can be inhibited, particularly, conversion between different space balance modes is realized, and the application range of a mode conversion technology is expanded.

Description

Technical Field

[0001] This invention relates to the field of optical technology, and more specifically to a method for converting light field modes. Background Technology

[0002] Rabi oscillations describe the periodic transitions between two energy levels in a quantum system under the influence of an applied alternating field. Commonly observed in atomic, molecular, and Bose-Einstein condensate systems, they have important applications in quantum information processing. Since the paraxial wave equation for light propagation in a medium is similar in form to the Schrödinger equation, relevant theories and methods from quantum mechanics can be applied to optical systems. Compared to quantum systems, optical platforms offer advantages such as easier parameter control and flexible structural design. For example, precise control of the light field can be achieved through waveguide geometry and refractive index distribution, thereby simulating coherent quantum interactions.

[0003] In optics, Rabi oscillations correspond to resonant switching between optical modes. In 2007, researchers proposed a theoretical model based on multimode waveguides, where the waveguide refractive index distribution is analogous to a quantum potential well, and weak periodic refractive index modulation along the propagation direction is equivalent to an alternating electromagnetic field perturbation. Effective mode switching can be achieved when the modulation frequency is approximately equal to the difference between the propagation constants of the two modes. Subsequently, this model has been extended to various systems, such as coupled waveguides, waveguide arrays, and PT-symmetric waveguides, and has been further studied in areas such as subwavelength mode switching and fractional-order effects. In the two-dimensional extended model, novel effects such as optical vortex generation and conversion in helical waveguides, vortex solitons in nonlinear media, and topological edge states have been realized. Simultaneously, spatial periodic refractive index modulation provides a means to realize functions such as parametric amplification, optical tunneling, and optical isolation. Summary of the Invention

[0004] The purpose of this invention is to address the problem that existing mode conversion technologies struggle to achieve efficient conversion between arbitrary modes, especially between different parity modes, and to propose a light field mode conversion method.

[0005] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is as follows:

[0006] A method for converting light field modes includes the following steps:

[0007] S1, set according to formula (1) Parameters characterizing waveguide depth To solve the eigenvalue problem represented by equation (3), the main numerical methods are the finite difference method or the plane wave expansion method, which yields the corresponding eigenvalues. and intrinsic mode The number of supported modes is affected by the waveguide size and refractive index; different modes are named U1, U2...Un in order of their propagation constants; modes with more peaks in the x-axis direction are called long-axis modes, and vice versa; the number of peaks is appended to the sequence number.

[0008] S2, Select intrinsic mode U2 12 As the initial input, weak longitudinal periodic modulation is introduced. Substituting into equation (1) and ignoring nonlinear effects; utilizing the orthogonality of the eigenmodes, we obtain the weight evolution process in the eigenmode conversion process represented by equation (5); when the modulation frequency satisfies At that time, mode U2 12 and U7 14 Transmission in a waveguide can achieve periodic resonant mode conversion; if Then mode U2 12 To Mode U9 32 A conversion occurs; typically, the conversion cycle differs between different modes.

[0009] S3 modulates the transverse modulation of the waveguide refractive index, making the refractive index change more significantly along the x or y axis while maintaining the longitudinal periodic modulation. This is called x or y partial modulation. The modified refractive index is shown below, taking x partial modulation as an example:

[0010]

[0011] New modulation expression Expanding the series to the first order, we get:

[0012]

[0013] use Replace the formula for calculating the coupling coefficient In W, calculate the coupling coefficient between each mode;

[0014] S4. Arrange the obtained coupling coefficients in index order and write them in matrix form. Eliminate the main diagonal elements and leave only the off-diagonal elements representing the coupling terms. It is easy to see that there is coupling between only a few modes, and the coupling coefficients of the two different modulation methods are significantly different.

[0015] S5. By changing and combining the modulation form and the mode type of the initial input, multiple tests revealed that different combinations of modulation forms and mode types lead to the following results: when the modulation type and mode type are the same, the conversion speed increases while the conversion efficiency remains almost unchanged; when the modulation type and mode type are different, the conversion speed decreases significantly, and in special cases, conversion may fail.

[0016] S6, considering the conversion between modes with different parities, it is necessary to change the transverse modulation form of the waveguide refractive index again; to achieve the conversion between any mode Un and mode Uk, the function describing the minute periodic change of refractive index needs to be changed. Horizontal distribution The following relationship must be satisfied:

[0017]

[0018] At the same time, keep the longitudinal modulation unchanged, Substituting into the formula for calculating the coupling coefficient, we find that the coupling coefficient between mode Un and mode Uk is: All functions in the integral are non-negative. As long as the functions Un and Uk coincide, the integral result is greater than zero, indicating that the two modes are coupled and have an extremely high conversion rate. Through such adjustment, the conversion between any modes can be achieved.

[0019] The propagation of a light beam in a medium with a periodically varying refractive index along the z-axis can be described by the following nonlinear Schrödinger equation:

[0020] (1)

[0021] Here It is the amplitude under dimensionless conditions. The first term on the left side of equation (1) represents envelope propagation, the second term represents transverse diffraction of light, and the third term represents the influence of the refractive index distribution and its small periodic changes on light propagation; where the function describing the refractive index of the waveguide is... , It is the waveguide depth; parameters This represents the depth of the longitudinal periodic modulation, which is typically much smaller than 1. It is the longitudinal modulation frequency. It is the refractive index perturbation function; the last term originates from nonlinear effects. These correspond to the self-focusing (defocusing) effect of Kerr nonlinearity;

[0022] When nonlinear effects and longitudinal periodic modulation are not considered, i.e. and Let the solution to equation (1) have the following form:

[0023] (2)

[0024] Where k is the propagation constant, The lateral distribution of the pattern is described; substituting it into equation (1) yields a linear distribution.

[0025] Eigenvalue equation:

[0026] (3)

[0027] A series of eigenvalues ​​are obtained from equation (3). and intrinsic mode The optical field in the waveguide is a linear superposition of all modes, satisfying: After introducing longitudinal periodic modulation, the weight coefficients It will change with the coordinate z. Substitute ψ into equation (1) and let Considering only the ideal case where there are only two modes in the waveguide, we project equation (1) onto modes U1 and U2, and let... The coupling transformation equation for the resonant mode can be derived:

[0028] (4)

[0029] The high-frequency terms are ignored by using the secular approximation. We can obtain:

[0030] (5)

[0031] In equation (5), The coupling coefficients between different modes of the equation represent the coupling coefficients. It is a tiny change superimposed on the waveguide refractive index V(x,y), and through artificial design, the coupling coefficient is made to meet a specific value.

[0032] The beneficial effects of this invention are:

[0033] This invention proposes a method for efficient periodic conversion of arbitrary optical modes in an elliptical multimode waveguide based on numerical simulation of optical wave transmission using the split-step Fourier algorithm. In this multimode waveguide, due to the non-circular symmetry of the structure, the distribution of optical field modes differs significantly along the x and y directions. By designing refractive index modulation associated with specific coordinates, the conversion rate between optical field modes with corresponding geometric characteristics can be significantly enhanced, while suppressing coupling between other non-target modes and exciting some modes that are difficult to couple under conventional conditions to achieve conversion. Furthermore, this invention proposes a design method for specifically perturbing the transverse distribution of the waveguide refractive index, thereby achieving efficient conversion between different parity (odd-mode and even-mode) modes. Attached Figure Description

[0034] Figure 1 This is the distribution of eigenmodes in an elliptical multimode waveguide (first half).

[0035] Figure 2 It represents the distribution of intrinsic modes in an elliptical multimode waveguide (the latter half).

[0036] Figure 3 This is the evolution diagram of the weight coefficients when modes U2 and U7 are transformed into each other.

[0037] Figure 4 It is a transverse cross-sectional view of the light field at different distances when modes U2 and U7 are converted to each other.

[0038] Figure 5 It is an isosurface plot of the square of the amplitude modulus of the time field during the interconversion between modes U2 and U7.

[0039] Figure 6 It is composed of perturbation function A schematic diagram of the obtained coupling coefficients.

[0040] Figure 7 It is composed of perturbation function A schematic diagram of the obtained coupling coefficients.

[0041] Figure 8 This is a graph showing the evolution of the weighting coefficients of short-axis modes U3 and U10 under the modulation of the x-part.

[0042] Figure 9 This is a graph showing the evolution of the weighting coefficients of short-axis modes U3 and U10 under the modulation of the y-part.

[0043] Figure 10 When the perturbation function A schematic diagram of the distribution over time.

[0044] Figure 11 It is composed of perturbation function A schematic diagram of the obtained coupling coefficients.

[0045] Figure 12 It is a transverse cross-sectional view of the light field at different distances when the modes U5 and U9 are converted to each other.

[0046] Figure 13 This is the evolution diagram of the weight coefficients when modes U5 and U9 are transformed into each other.

[0047] Figure 14 When the perturbation function This is a schematic diagram showing the evolution of all modes in the waveguide with transmission distance when modes U5 and U9 are mutually converted. Detailed Implementation

[0048] The present invention will now be described in further detail with reference to the accompanying drawings and embodiments.

[0049] The light field mode conversion method described in this embodiment includes the following steps:

[0050] S1, set according to formula (1) Parameters characterizing waveguide depth To solve the eigenvalue problem represented by equation (3), the main numerical methods include the finite difference method or the plane wave expansion method, etc., to obtain the corresponding eigenvalues. and intrinsic mode The number of supported modes is affected by the waveguide size and refractive index. Different modes are named U1, U2, ... Un in order of their propagation constants. In addition to the sequence number, according to the peak distribution of the light field in the x and y directions of different modes, modes with more peaks in the x-axis direction are called long-axis modes, and vice versa. The number of peaks is appended to the sequence number.

[0051] S2, Select intrinsic mode U2 12 As the initial input, weak longitudinal periodic modulation is introduced. Substituting into equation (1) and ignoring nonlinear effects; utilizing the orthogonality of the eigenmodes, we obtain the weight evolution process in the eigenmode conversion process represented by equation (5); when the modulation frequency satisfies At that time, mode U2 12 and U7 14 Transmission in a waveguide can achieve periodic resonant mode conversion; if Then mode U2 12 To Mode U9 32 A conversion occurs; typically, the conversion cycle differs between different modes.

[0052] S3, change the transverse modulation of the waveguide refractive index to make the refractive index change more significantly along the x (or y) axis, while keeping the longitudinal periodic modulation unchanged. This is called x (or y) partial modulation, and the modified refractive index is (taking x partial modulation as an example):

[0053]

[0054] New modulation expression Expanding the series to the first order, we get:

[0055]

[0056] use Replace the formula for calculating the coupling coefficient In W, calculate the coupling coefficient between each mode;

[0057] S4. Arrange the obtained coupling coefficients in index order. To facilitate observation, write them in matrix form and eliminate the main diagonal elements, leaving only the off-diagonal elements representing the coupling terms. It is easy to see that there is coupling between only a small number of modes, and the coupling coefficients of the two different modulation methods are significantly different.

[0058] S5 changes and combines the modulation form and the mode type of the initial input. After multiple tests, it was found that this corresponds to different results: when the modulation type and the mode type are the same, the conversion speed increases and the conversion efficiency remains almost unchanged; when the modulation type and the mode type are different, the conversion speed decreases significantly, and in special cases, conversion may fail.

[0059] S6, considering the conversion between modes with different parities, it is necessary to change the transverse modulation form of the waveguide refractive index again; to achieve the conversion between any mode Un and mode Uk, the function describing the minute periodic change of refractive index needs to be changed. Horizontal distribution The following relationship must be satisfied:

[0060]

[0061] At the same time, keep the longitudinal modulation unchanged, Substituting into the formula for calculating the coupling coefficient, we find that the coupling coefficient between mode Un and mode Uk is: All functions in the integral are non-negative. As long as the functions Un and Uk coincide, the integral result will be greater than zero, indicating that the two modes are coupled and have an extremely high conversion rate. Through such adjustment, the conversion between any modes can be achieved.

[0062] The propagation of a light beam in a medium with a periodically varying refractive index along the z-axis can be described by the following nonlinear Schrödinger equation:

[0063] (1)

[0064] Here It is the amplitude under dimensionless conditions. The first term on the left side of equation (1) represents envelope propagation, the second term represents transverse diffraction of light, and the third term represents the influence of the refractive index distribution and its small periodic changes on light propagation; where the function describing the refractive index of the waveguide is... , It is the waveguide depth; parameters This represents the depth of the longitudinal periodic modulation, which is typically much smaller than 1. It is the longitudinal modulation frequency. It is the refractive index perturbation function; the last term originates from nonlinear effects. These correspond to the self-focusing (defocusing) effect of Kerr nonlinearity.

[0065] When nonlinear effects and longitudinal periodic modulation are not considered, i.e. and Let the solution to equation (1) have the following form:

[0066] (2)

[0067] Where k is the propagation constant, The lateral distribution of the pattern is described; substituting it into equation (1) yields a linear distribution.

[0068] Eigenvalue equation:

[0069] (3)

[0070] Solving the eigenvalue problem can be done using the plane wave expansion method. We will use the eigenfunctions... and refractive index distribution function In a limited area , Internal truncation unfolds into a series of superimposed plane waves:

[0071]

[0072]

[0073] in Substituting these expansions into the eigenvalue problem (3) to make the coefficients of the same Fourier mode equal, we can obtain the following information about the coefficients. The eigenvalue system:

[0074]

[0075] in , is an integer representing the cutoff length. The above equation is a two-dimensional matrix equation; solving it yields a series of eigenvalues ​​and plane wave coefficients. Superimposing and restoring these coefficients ultimately yields the corresponding eigenfunctions. .

[0076] Figure 1 , Figure 2 The waveguide refractive index is listed. The supported optical field modes are arranged in descending order of propagation constant. Among them, the parameters... Typically, when the refractive index changes, these light field distributions are no longer eigenfunctions of equation (3) and cannot remain stable during propagation. However, if the refractive index perturbation is small, the eigenmode distributions in this case can be analyzed using perturbation theory, and their weight coefficients will change continuously with the propagation distance.

[0077] The optical field in a waveguide is a linear superposition of all modes: Let's assume there are only two modes, n=1 and 2. Substituting these into equation (1), we can ignore the nonlinear effects when the amplitude is small. After z=0, we introduce a periodic perturbation of the refractive index and project equation (1) onto modes U1 and U2 respectively, and let... The coupling transformation equation for the resonant mode can be derived:

[0078] (4)

[0079] There are some and in equation (4) The relevant high-frequency terms, whose average value over multiple cycles is approximately zero, can be ignored. Those at the resonant frequency are retained. The term that changes slowly over time is called the secular approximation. Simplifying, we get:

[0080] (5)

[0081] Equation (5) under initial conditions The solution at that time is:

[0082]

[0083] It can be seen that the weighting coefficient oscillates periodically with the propagation distance, and the oscillation frequency is affected by the coupling coefficient in equation (5). Size control. When the initial input is mode U2. 12 Time settings Resonant frequency , The fractional Fourier method was used to numerically calculate equation (1), and the weight coefficients were obtained from the orthogonality relationship. The results are as follows. Figure 3 As shown, the weight coefficients roughly conform to the variation of the cosine function, pattern U2 12 The weight gradually decreases, moving towards mode U7. 14 The transformation then gradually reverses (e.g.) Figure 4 , Figure 5 The curve exhibits high-frequency jitter and amplitude fluctuations, which slightly deviates from the solution of equation (5). This is due to neglecting the high-frequency terms in equation (5). Then mode U2 12 To Mode U9 32 A conversion occurs; however, the conversion frequencies are significantly different.

[0084] Consider modifying the refractive index modulation as follows: making the refractive index change more significantly along the x (or y) axis while maintaining its longitudinal periodicity. This is called x (or y) partial modulation. The modified refractive index is (taking x-partial modulation as an example):

[0085]

[0086] Expanding the above formula ,because Values ​​that are often less than or equal to 0.1 can be retained only once, and then... Performing a Taylor expansion, and preserving it to the first order, we get:

[0087]

[0088] This indicates that this modulation method is equivalent to superimposing a waveguide refractive index of the original waveguide onto it. Small perturbations. Similarly, the expression for the modulation of the y-part can be written in a similar form, only the function g has changed. Using Replace the formula for calculating the coupling coefficient In W, the coupling coefficients between the various modes are calculated. Figure 6 and Figure 7 The coupling coefficient matrices for these two cases are shown. For easier observation, color shades are used to represent numerical values, and diagonal elements are removed because only off-diagonal elements represent coupling terms.

[0089] It can be observed that when using y-part modulation, mode U1 11 -U6 31 U2 12 -U9 32 U3 21 -U10 41 U4 13 -U13 33 U5 22 -U14 42 U6 31 -U15 51 The coupling coefficients between modes are significant, while their strength is extremely low during x-part modulation. Furthermore, it's noted that only the number of y-axis peaks increases or decreases during conversion, while the number of x-axis peaks remains constant. Therefore, resonant conversion is more likely to occur between short-axis modes. Changing and combining the modulation scheme and the initial input mode type, multiple tests revealed that when the modulation type and mode type are the same, the conversion speed increases while the conversion efficiency remains almost unchanged. When the modulation type and mode type are different, the conversion speed decreases significantly, and in special cases, conversion may fail. This clearly demonstrates that this type of modulation method makes the mode conversion rate highly dependent on the geometry of the optical field. Figure 8 , Figure 9 It is the initial input short axis mode U3 21 To U10 41 A graph showing the weighting coefficients for the transformation between them. Figure 8 This is the result of x-partial modulation, with a conversion period of 25.8. Figure 9 It is the result of the y-part modulation, with a conversion period of 2.6. The numerical simulation results are in good agreement with the theoretical expectations.

[0090] Previously, resonance transformations could only occur between identical parity modes because of the conventional perturbation function. It is an even function in the x, y plane, which will affect the calculation formula of the coupling coefficient. To achieve the conversion between any mode Un and mode UK, the value must be zero. Horizontal distribution The following relationship must be satisfied:

[0091]

[0092] At the same time, keep the longitudinal modulation unchanged, Substituting into the formula for calculating the coupling coefficient, we find that the coupling coefficient between mode Un and mode Uk is: All functions in the integral are non-negative. As long as the functions Un and Uk coincide, the integral result will be greater than zero, indicating that there is coupling between the two modes, and the transformation between any modes can be achieved.

[0093] In mode U5 22 U9 32 Let's take an example to illustrate this process. Figure 10 It is a function Distribution shape ( (already normalized separately) Figure 11 It is by The calculated coupling coefficient matrix contains matrix elements. Compared to the previous case, it is no longer zero, thus achieving coupling. Figure 12 It is the resonant frequency. , Light field cross sections at different positions during time, Figure 13 This is a schematic diagram of the corresponding weighting coefficient evolution. This modulation method can achieve efficient conversion of arbitrary modes with smaller refractive index perturbations, but it may introduce some irrelevant interference. By discretizing the light field along the propagation direction and performing mode decomposition at different distances, such as... Figure 14 As shown, in addition to modes U5 and U9, weaker mode components U2 and U14 were also found. This phenomenon can be observed by... More precise design eliminates [the need for] elimination.

Claims

1. A method for converting light field modes, characterized by: Includes the following steps: S1, set according to formula (1) Parameters characterizing waveguide depth To solve the eigenvalue problem represented by equation (3), the main numerical methods are the finite difference method or the plane wave expansion method, which yields the corresponding eigenvalues. and intrinsic mode The number of supported modes is affected by the waveguide size and refractive index; different modes are named U1, U2...Un in order of their propagation constants; modes with more peaks in the x-axis direction are called long-axis modes, and vice versa; the number of peaks is appended to the sequence number. S2, Select intrinsic mode U2 12 As the initial input, weak longitudinal periodic modulation is introduced. Substituting into equation (1) and ignoring nonlinear effects; utilizing the orthogonality of the eigenmodes, we obtain the weight evolution process in the eigenmode conversion process represented by equation (5); when the modulation frequency satisfies At that time, mode U2 12 and U7 14 Transmission in a waveguide can achieve periodic resonant mode conversion; if Then mode U2 12 To Mode U9 32 A conversion occurs; typically, the conversion cycle differs between different modes. S3 modulates the transverse modulation of the waveguide refractive index, making the refractive index change more significantly along the x or y axis while maintaining the longitudinal periodic modulation. This is called x or y partial modulation. The modified refractive index is shown below, taking x partial modulation as an example: ， New modulation expression Expanding the series to the first order, we get: ， use Replace the formula for calculating the coupling coefficient In W, calculate the coupling coefficient between each mode; S4. Arrange the obtained coupling coefficients in index order and write them in matrix form. Eliminate the main diagonal elements and leave only the off-diagonal elements representing the coupling terms. It is easy to see that there is coupling between only a few modes, and the coupling coefficients of the two different modulation methods are significantly different. S5. By changing and combining the modulation form and the mode type of the initial input, multiple tests revealed that different combinations of modulation forms and mode types lead to the following results: when the modulation type and mode type are the same, the conversion speed increases while the conversion efficiency remains almost unchanged; when the modulation type and mode type are different, the conversion speed decreases significantly, and in special cases, conversion may fail. S6, considering the conversion between modes with different parities, it is necessary to change the transverse modulation form of the waveguide refractive index again; to achieve the conversion between any mode Un and mode Uk, the function describing the minute periodic change of refractive index needs to be changed. Horizontal distribution The following relationship must be satisfied: ， At the same time, keep the longitudinal modulation unchanged, Substituting into the formula for calculating the coupling coefficient, we find that the coupling coefficient between mode Un and mode Uk is: All functions in the integral are non-negative. As long as the functions Un and Uk coincide, the integral result is greater than zero, indicating that the two modes are coupled and have an extremely high conversion rate. Through such adjustment, the conversion between any modes can be achieved. The propagation of a light beam in a medium with a periodically varying refractive index along the z-axis can be described by the following nonlinear Schrödinger equation: (1) Here It is the amplitude under dimensionless conditions. The first term on the left side of equation (1) represents envelope propagation, the second term represents transverse diffraction of light, and the third term represents the influence of the refractive index distribution and its small periodic changes on light propagation; where the function describing the refractive index of the waveguide is... , It is the waveguide depth; parameters This represents the depth of the longitudinal periodic modulation, which is typically much smaller than 1. It is the longitudinal modulation frequency. It is the refractive index perturbation function; the last term originates from nonlinear effects. These correspond to the self-focusing (defocusing) effect of Kerr nonlinearity; When nonlinear effects and longitudinal periodic modulation are not considered, i.e. and Let the solution to equation (1) have the following form: (2) Where k is the propagation constant, The lateral distribution of the pattern is described; substituting it into equation (1) yields a linear distribution. Eigenvalue equation: (3) A series of eigenvalues ​​are obtained from equation (3). and intrinsic mode The optical field in the waveguide is a linear superposition of all modes, satisfying: After introducing longitudinal periodic modulation, the weight coefficients It will change with the coordinate z. Substitute ψ into equation (1) and let Considering only the ideal case where there are only two modes in the waveguide, we project equation (1) onto modes U1 and U2, and let... The coupling transformation equation for the resonant mode can be derived: (4) The high-frequency terms are ignored by using the secular approximation. We can obtain: (5) In equation (5), The coupling coefficients between different modes of the equation represent the coupling coefficients. It is a tiny change superimposed on the waveguide refractive index V(x,y), and through artificial design, the coupling coefficient is made to meet a specific value.

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