Optimization method of low-threshold laser generator based on accurate solution of light-substance coupling intensity
The Maxwell-Schrödinger equation is optimized through Helmholtz decomposition and finite element method, and the solution complexity problem of laser generators under different specifications and potential functions is solved, and efficient optimization and precise design of low-threshold laser generators are achieved.
Patent Information
- Application Number
- CN202510341165.3
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-03-21
- Publication Date
- 2025-07-18
AI Technical Summary
The existing Maxwell Schrödinger system solution method is difficult to uniformly use the same finite element method when dealing with nanoscale light-matter interactions, resulting in computational complexity and difficulty in ensuring charge and energy conservation.
The Maxwell-Schrödinger equation was decomposed into two parts: no rotation field and no dispersion field, and fully discrete decomposition was performed through the finite element method, and the P1 coordination element was used for solution to ensure that charge conservation and energy conservation were maintained under different specifications and potential functions, and the laser generator design was optimized.
In the optimization of low-threshold laser generators, the Maxwell-Schrödinger system under different specifications and potential functions can be solved using only the same finite element method, achieving better convergence effect and improving the design accuracy of the laser generator.
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Figure CN120337630A_ABST
Abstract
Description
Technical Field
[0001] The present invention belongs to the field of computational mathematics, and particularly relates to an optimization method for a low-threshold laser generator based on accurately solving the light-matter coupling strength. Background Art
[0002] Maxwell's electromagnetic field theory has played an important role in the study of the optical properties of many materials. However, with the in-depth research and the assumption of weak fields in the theory itself, it cannot be well applied to the study of some nano-scale light-matter interactions. To better describe such research, the concept of probability current density is introduced, and the electromagnetic field is coupled with the quantum system, thus proposing the Maxwell-Schrödinger system, which is used as a model to describe some nano-optical systems. This semi-classical model can describe the states of charged particles in a self-consistent field and an external electromagnetic field, as well as the electron motion in semiconductor nanostructures. The background technology for solving this model covers multiple fields such as computational electromagnetics, quantum mechanics, numerical calculation, and multi-physics field coupling. The core challenge lies in dealing with multi-scale, high-dimensional, and non-linear problems, and at the same time, high-performance computing technology needs to be combined. With the continuous progress of numerical methods, this field shows broad prospects in applications such as quantum optics and nano-optics, and can be applied to the design of photodetectors, the simulation of superconducting quantum circuits, the coupling of electromagnetic fields and quantum states in quantum dots, and the optimization of light absorption and scattering characteristics in the design of sub-wavelength plasmonic antennas, etc.
[0003] For the solution of the Maxwell-Schrödinger system, methods such as the finite-difference time-domain method, the finite element method, and the spectral method are usually used. Among them, the finite element method can better handle complex geometric boundaries, flexibly cope with various boundary conditions, and has certain advantages in the study of inhomogeneous dielectric materials. Since the Maxwell-Schrödinger equation has no unique solution and has the property of gauge invariance, it is necessary to determine the gauge in advance before numerical calculation and then solve it. However, under some gauges, the divergence of the vector potential A is 0, and different finite elements need to be set when solving by the finite element method, which will complicate the solution of the equation. In the present invention, the coupling strength considered can use the same finite element calculation under different gauges, which simplifies the calculation on the basis of ensuring charge conservation and energy conservation. Summary of the Invention
[0004] The present invention provides an optimization method for a low-threshold laser generator based on accurately solving the light-matter coupling strength. In engineering, when designing a microcavity laser based on a quantum dot array, this method can be used to solve the Maxwell-Schrödinger equation to calculate the local electric field strength of the optical field and the dipole moment of the quantum emitter, and finally calculate the light-matter coupling strength, so as to correct the position of the quantum dots and the size of the photonic crystal defect, and optimize the design of the laser generator. This method uses the Helmholtz decomposition to decompose the two vector terms of the Maxwell-Schrödinger system under different gauges and different potential functions into an irrotational field and a solenoidal field respectively, and obtains a new system of equations after decomposition; further determines the finite element space, performs a full discretization decomposition on the new system of equations, and obtains numerical formats under different gauges and different potential functions; numerically solves the above full discretization format and calculates the error convergence order. This method can solve the Maxwell-Schrödinger system under different gauges and different potential functions using only the same finite element, and while ensuring charge conservation and energy conservation in the numerical format, obtains a solution with good convergence effect.
[0005] The technical solution adopted by the present invention is as follows:
[0006] An optimization method for a low-threshold laser generator based on accurately solving the light-matter coupling strength, the method comprising:
[0007] Step 1, input the initial quantum system of the laser generator to be optimized, as well as the geometric dimensions and material parameters of the optical cavity;
[0008] Step 2, based on the input in Step 1, construct the Maxwell-Schrödinger equation, which describes the state of charged particles in an electromagnetic field by introducing the probability current density;
[0009] Step 3, according to the selected gauge (such as Lorentz gauge, temporal gauge, Coulomb gauge, etc.), obtain the expression form of the Maxwell-Schrödinger equation constructed in Step 2 under the current gauge;
[0010] Step 4, through the Helmholtz decomposition, decompose the two vector fields in the Maxwell-Schrödinger equation obtained in Step 3 into an irrotational field and a solenoidal field, and both parts are 0 at the same time. In this application, the Maxwell-Schrödinger equation obtained in Step 3 is rewritten into a new expression form (decomposed into an irrotational field and a solenoidal field through the Helmholtz decomposition), and it should also satisfy: (1) charge conservation; (2) energy conservation;
[0011] Step 5, through the finite element method (FEM), perform a full discretization transformation on the system of equations composed of the irrotational field and the solenoidal field obtained in Step 4; the full discretization format of this full discretization transformation satisfies: (1) discrete charge conservation; (2) discrete energy conservation;
[0012] Step 6: Solve the system of equations composed of the irrotational field and the divergence-free field according to the fully discrete format in Step 5 to solve the state function, vector potential, and scalar potential of the quantum dots of the laser generator; and calculate the convergence order of the above variables under different norms.
[0013] If the calculated dipole moment of the quantum dot does not converge, adjust the size or position of the quantum dot in the initial quantum system, and re-execute the step of solving the system of equations again according to the fully discrete format in Step 5.
[0014] If the dipole moment of the quantum dot converges, calculate the optical cavity frequency and resonance frequency based on the state function, vector potential, and scalar potential. If the difference between the two is less than or equal to the specified threshold, calculate the current optical cavity geometry size and the light-matter coupling strength of the current quantum dot based on the dipole moment, vector potential, and scalar potential of the quantum dot; if the difference between the two is greater than the specified threshold, adjust the geometry size of the optical cavity and re-execute the step of solving the system of equations again according to the fully discrete format in Step 5.
[0015] Detect whether the currently obtained coupling strength reaches the expected value. If not, move the quantum dot and re-execute the step of solving the system of equations again according to the fully discrete format in Step 5 until the currently obtained coupling strength reaches the expected value.
[0016] Furthermore, the light-matter coupling strength is:
[0017]
[0018] Among them, the superscript n in the parameter is used to identify the iteration number, and d n represents the dipole moment of the quantum dot, represents the scalar potential in the fully discrete format, represents the vector potential in the fully discrete format, represents the gradient operator, represents the reduced Planck constant.
[0019] Furthermore, in Step 6, when it is detected that the currently obtained coupling strength is lower than the expected value, move the position of the quantum dot to the region with a stronger electric field.
[0020] The technical solution provided by this application at least brings the following beneficial effects:
[0021] The optimization method of a low-threshold laser generator based on accurately solving the light-matter coupling strength proposed in this application can be used for the simulation design of microcavity lasers based on quantum dot arrays. This method calculates the local electric field strength of the optical field and the dipole moment of quantum emitters by solving the Maxwell-Schrödinger equation, and finally calculates the light-matter coupling strength to correct the parameters and optimize the design of the laser generator. This method uses the Helmholtz decomposition to decompose the two vector terms of the Maxwell-Schrödinger system under different gauges and different potential functions into an irrotational field and a solenoidal field respectively, and obtains a new set of equations after decomposition. Further, the finite element space is determined, and the new set of equations is fully discretized and decomposed to obtain numerical formats under different gauges and different potential functions. The above fully discretized format is numerically solved and the error convergence order is calculated. In the optimization process of the low-threshold laser generator in this application, only the same finite element is used to solve the Maxwell-Schrödinger system under different gauges and different potential functions. At the same time, while the numerical format ensures charge conservation and energy conservation, a solution with a good convergence effect is obtained, and then the optimized low-threshold laser generator is obtained. Brief Description of the Drawings
[0022] The above and / or additional aspects and advantages of the present application will become apparent and easy to understand from the following description of the embodiments in conjunction with the drawings, where:
[0023] Figure 1 It is a flowchart of an optimization method of a low-threshold laser generator based on accurately solving the light-matter coupling strength provided by an embodiment of the present application. Detailed Embodiments
[0024] In order to enable those skilled in the art to better understand the technical solutions in this specification, the technical solutions in the embodiments of the present application will be described in detail and completely in conjunction with the drawings in the embodiments of the present application. Obviously, the embodiments described by referring to the drawings are exemplary and are intended to explain the present application, and should not be construed as a limitation to the present application.
[0025] In the existing numerical methods for solving the Maxwell - Schrödinger system, the use of the finite - difference method inadequately characterizes the solutions for irregular boundary conditions, while the solution based on the finite - element method requires different finite elements for different gauges. The embodiment of the present application provides an optimization method for a low - threshold laser generator based on accurately solving the light - matter coupling strength, which can be used for the simulation and optimization of microcavity lasers based on quantum - dot arrays. The method proposed in the present application can solve the Maxwell - Schrödinger system under different gauges and different potential functions only by using the same P1 conforming element (the P1 conforming element usually refers to the linear Lagrangian element, whose shape function is linear). It has a good characterization of the solutions for complex boundary conditions, and while ensuring charge conservation and energy conservation in the numerical format, it obtains solutions with good convergence effects, making the optimization of the laser generator more accurate.
[0026] In one embodiment, as Figure 1 shown, it specifically includes the following steps:
[0027] Step 1, given the initial optical - cavity geometric parameters, set the potential term, boundary conditions (including the spatial - position boundary of quantum dots and the time boundary of the evolution time) and initial - value conditions of the Maxwell - Schrödinger equation, and determine the equation form:
[0028]
[0029] where ψ(x,t) is the state function, φ(x,t) is the scalar potential, A(x,t) is the vector potential, and the three respectively satisfy the corresponding boundary conditions and initial values. i is the imaginary unit, x is the quantum - dot coordinate; t is the system - evolution time, is the reduced Planck constant, is the gradient operator, q is the charge of the quantum dots in the laser generator, ∈ is the permittivity, μ is the magnetic permeability, J q (x,t) is the probability - current density, * denotes the conjugate, m is the quantum - dot mass, V0 is the system potential term, that is, the input parameter during the simulation and optimization of the laser generator. In the above formula (1), the first - row equation is the Schrödinger equation, the second - row and third - row equations are the Maxwell equations, and the last - row equation is the probability - current density. Since the equation has no unique solution, it is necessary to use a specific gauge first and then solve it.
[0030] That is, in the embodiment of the present application, based on the optimization requirements of the laser generator, first, it is necessary to input the initial quantum system of the laser generator to be optimized (the relevant parameters that can be included are: the initial coordinates of quantum dots, evolution time, charge of quantum dots, quantum - dot mass, system potential term), as well as the geometric dimensions and material parameters of the optical cavity (mainly including permittivity, magnetic permeability, etc.);
[0031] In the embodiments of the present application, a probability current density is introduced into the constructed Maxwell - Schrödinger equation to describe the state of charged particles in an electromagnetic field, and then the local electric field intensity of the optical field and the dipole moment of the quantum emitter are calculated by solving the constructed Maxwell - Schrödinger equation.
[0032] Step 2: Determine the gauge according to requirements, obtain the equation form under it, and use the Helmholtz decomposition to decompose the Schrödinger equation into a curl - free part and a divergence - free part, and both parts are zero simultaneously.
[0033] The forms of the Maxwell - Schrödinger equation are different under different gauges, and the solutions of the equations can be transformed into each other through gauge transformation. For simplicity, the equations after the embodiments of the present application all use atomic units.
[0034] Under the Lorentz gauge, the transformed equation form of formula (1) is:
[0035]
[0036] Among them, ψ describes the state of the quantum dots in the laser, A describes the vector potential of the quantum dots, that is, ψ and A are the simplified forms of ψ(x, t) and A(x, t) respectively, φ is the simplified form of the scalar potential, Ω is the spatial range of the quantum dots, that is, x ∈ Ω, T is the time boundary, that is, t ∈ (0, T). in Ω×(0, T) means that the corresponding expressions of formula (2) are all within the specified spatial range and time range.
[0037] Under the Coulomb gauge, the transformed form of formula (1) is:
[0038]
[0039] Under the temporal gauge, the transformed form of formula (1) is:
[0040]
[0041] For different gauges, the decomposition methods are slightly different. The following steps take the Lorentz gauge as an example, but the method proposed in the embodiments of the present application includes and is not limited to being applicable in the Lorentz gauge. Then, use the Helmholtz decomposition to decompose the transformed equation (such as the above formulas (2), (3) or (4)) under the selected gauge.
[0042] It mainly includes the decomposition of the vector potential A and the vectors to be decomposed under each gauge.
[0043] For example, for the transformed equations under the three gauges given in the above examples, the corresponding vectors to be decomposed are specifically as follows:
[0044]
[0045] Among them, F1, F2, and F3 respectively represent the vectors to be decomposed under the Lorenz gauge, Coulomb gauge, and temporal gauge.
[0046] In one embodiment, taking the Lorenz gauge as an example, the specific decomposition process of the vector potential A and F (in this embodiment, corresponding to F1 shown in formula (5), which can represent the probability current density) in this application is as follows:
[0047] The vector potential A is decomposed as follows:
[0048]
[0049] Among them, u and v are scalar functions, which respectively satisfy:
[0050]
[0051] Among them, n is a direction vector.
[0052] Similarly, the probability current density under this gauge can be decomposed as follows:
[0053]
[0054] Among them, p and q are scalar functions, which respectively satisfy:
[0055]
[0056] Substituting into the equation shown in formula (2), the Schrödinger equation is transformed into:
[0057]
[0058] The curl-free part and divergence-free part of this equation are both 0 at the same time, and finally the decomposed Maxwell-Schrödinger equation form is:
[0059]
[0060] Among them is the second-order partial derivative with respect to time. This system satisfies charge conservation and energy conservation.
[0061] Step 3: Split the Schrödinger equation, substitute it into the equation shown in formula (2), and perform full discretization on the obtained new equation, solve it using P1 conforming elements, and calculate the convergence order of each variable under different norms.
[0062] After obtaining the new equation after Helmholtz decomposition, perform finite element discretization on it. Let be the triangulation on the region Ω, and the finite element space and are defined as:
[0063]
[0064] Among them, represents the space of continuous functions from Ω to the real number field , and P1(K) represents the space composed of first-order polynomials defined on the small triangular region K. is the complex number field. To simplify the equation, the following notations are adopted for any variable u next:
[0065]
[0066] Among them, the superscript n represents the number of iterations, and τ represents the time step. At this time, the fully discrete form of the equation shown in formula (11) can be obtained as:
[0067]
[0068] Among them, are the scalar functions p and q in the fully discrete form at the nth iteration respectively; are the scalar functions u and v in the fully discrete form at the nth iteration respectively; and represent the vector potential and the quantum dot state in the fully discrete form at the nth iteration respectively. And Re[] represents the real part operation,
[0069] Iterative solution can be carried out for the above fully discrete form. In the above discrete form, the charge is and satisfies the charge conservation relation
[0070]
[0071] Among them, represents the L 2 norm
[0072] And the energy in the discrete form is:
[0073]
[0074] satisfies the energy conservation relation
[0075]
[0076] That is, in step 3, when solving the system of equations in the fully discrete form, only P1 conforming elements need to be used in this solving process, and the convergence order of each variable under different norms can be calculated under different potential functions, initial value conditions and boundary conditions.
[0077] Step 4: Solve the obtained fully discrete form, and calculate the light-matter coupling strength. Through iterative optimization, stop the iteration when the dipole moment converges and the coupling strength converges to achieve the maximum coupling strength, and obtain the optimized laser generator.
[0078] In iterative optimization, the geometric and material parameters of the initial quantum system and the optical cavity are first set to provide a benchmark for iteration.
[0079] If the dipole moment d of the quantum dot calculated after solving the Schrödinger equation n does not converge, the size or position of the quantum dot needs to be adjusted; if it converges, continue to solve the above fully discrete format, and use the and calculated difference between the optical cavity frequency and the resonance frequency to determine whether to perform the next calculation. If it exceeds the threshold, the geometric size of the optical cavity needs to be adjusted.
[0080] Then, through the dipole moment d n and the electromagnetic potential and calculate the light-matter coupling strength g n :
[0081]
[0082] where d n = ∫ψ n,* xψ n dx.
[0083] Detect whether the currently obtained coupling strength g n reaches the expected value. If not, move the quantum dot and re-solve the fully discrete format equations (i.e., formula (14)) until the currently obtained coupling strength reaches the expected value to obtain the optimization result of the laser generator. In one embodiment, if the currently calculated coupling strength g n is low, move the position of the quantum dot to a region with a stronger electric field and perform iterative optimization until convergence to achieve the maximum coupling strength, thereby realizing the optimization of the laser generator.
[0084] In one possible implementation, in step 3, under different gauges (such as the Coulomb gauge or the temporal gauge), the form of the vector field is different, and there are also differences in the equation forms after the decomposition of the vector field. However, all can be decomposed by this method so that the solenoidal part and the divergence-free part of the equation are both 0, thereby giving a new equation form.
[0085] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the application and are not intended to limit them; although the present application 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 recorded 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 the various embodiments of the present application.
Claims
1. An optimization method for a low-threshold laser generator based on accurately solving the light-matter coupling strength, characterized in that, It includes the following steps: Step 1: Input the initial quantum system of the laser generator to be optimized, as well as the geometric dimensions and material parameters of the optical cavity; Step 2: Based on the input in Step 1, construct the Maxwell-Schrödinger equation, which describes the state of charged particles in an electromagnetic field by introducing the probability current density; Step 3: According to the selected gauge, obtain the expression form of the Maxwell-Schrödinger equation constructed in Step 2 under the current gauge; Step 4: Through Helmholtz decomposition, decompose the two vector fields in the Maxwell-Schrödinger equation obtained in Step 3 into an irrotational field and a solenoidal field, and both parts are 0 at the same time; the decomposed irrotational field and solenoidal field both satisfy charge conservation and energy conservation; Step 5: Perform a full discretization transformation on the system of equations composed of the irrotational field and solenoidal field obtained in Step 4 by the finite element method; The full discretization format of this full discretization transformation satisfies: (1) discrete form of charge conservation; (2) discrete form of energy conservation; Step 6: Resolve the system of equations again according to the full discretization format in Step 5 to solve the state function, electromagnetic potential, and scalar potential of the quantum dots of the laser generator; and calculate the convergence order of the state function, vector potential, and scalar potential under different norms; If the calculated dipole moment of the quantum dots does not converge, then adjust the size or position of the quantum dots in the initial quantum system, and re-execute the step of resolving the system of equations again according to the full discretization format in Step 5; If the dipole moment of the quantum dots converges, then calculate the optical cavity frequency and resonance frequency based on the state function, electromagnetic potential, and scalar potential. If the difference between the two is less than or equal to the specified threshold, then calculate the current optical cavity geometric dimensions and the light-matter coupling strength of the current quantum dots based on the dipole moment, electromagnetic potential, and scalar potential of the quantum dots; if the difference between the two is greater than the specified threshold, then adjust the geometric dimensions of the optical cavity, and re-execute the step of resolving the system of equations again according to the full discretization format in Step 5; Detect whether the currently obtained coupling strength reaches the expected value. If not, then move the quantum dots, and re-execute the step of resolving the system of equations again according to the full discretization format in Step 5 until the currently obtained coupling strength reaches the expected value.
2. The method according to claim 1, characterized in that, The specific form of the Maxwell-Schrödinger equation constructed in Step 2 is: Among them, ψ(x,t) represents the state function, φ(x,t) represents the scalar potential, and A(x,t) represents the vector potential. All three terms satisfy the corresponding boundary conditions and initial values. i is the imaginary unit, x is the quantum dot coordinate, and t is the evolution time. is the reduced Planck constant. is the gradient operator, q is the charge of the quantum dot, ∈ is the dielectric constant, μ is the magnetic permeability, and J q (x,t) is the probability current density, m is the mass of the quantum dot, and V0 is the input potential energy term. * represents the conjugate.
3. The method according to claim 1 or 2, characterized in that, In Step 3, the selected gauge is the Lorentz gauge, the temporal gauge, or the Coulomb gauge.
4. The method according to claim 1 or 2, characterized in that, In Step 6, the light-matter coupling strength is: Among them, the superscript n in the parameters is used to identify the iteration number, represents the scalar potential in the fully discrete format, the intermediate quantity τ represents the time step, represents the vector potential in the fully discrete format, represents the gradient operator, represents the reduced Planck constant.
5. The method according to claim 1 or 2, characterized in that, In Step 6, when it is detected that the currently obtained coupling strength is lower than the expected value, move the position of the quantum dots to the region with a stronger electric field.