An external cavity VCSEL laser array mutual injection dynamics modeling and simulation method

Abstract

The application discloses a kind of external cavity VCSEL laser array mutual injection dynamics modeling simulation methods, comprising: setting up the structure parameters of external cavity VCSEL laser array with multiple high-order transverse mode and the working condition of driving external cavity VCSEL laser array normal operation;Based on L-K rate equation model, the dynamics model of external cavity VCSEL laser array mutual injection is established, and the partial differential equation set that carrier density, photon density and phase are related to space and time is constructed;The partial differential equation set is reduced, and the partial differential equation set is converted into only time variable ordinary differential equation set;The ordinary differential equation set is discretized, and numerical solution is calculated using finite difference method;The variation image of carrier density, photon density, phase and wavelength with time is output.The application can analyze the dynamics process of external cavity mutual injection of VCSEL array with multiple high-order transverse mode output, and find the condition that can realize coherent phase locking.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of VCSEL laser, and relates to an external cavity VCSEL laser array mutual injection dynamics modeling simulation method, in particular to an external cavity VCSEL laser array mutual injection dynamics modeling simulation method with multiple high-order transverse modes. BACKGROUND

[0002] At present, the research on VCSEL lasers is mainly based on single VCSEL or VCSEL array with small aperture based on the fundamental transverse mode output. In order to further expand the application field of VCSEL and improve the output power, the aperture size of VCSEL needs to be expanded, which will lead to the weakening of the limitation of high-order transverse modes and the output of multiple high-order transverse modes.

[0003] In addition, a two-dimensional array integration method can also be used to obtain a VCSEL laser array with higher power output. However, in this type of laser array, each VCSEL light emitting unit works independently, and the beam quality cannot be improved, thereby limiting the application range of this type of VCSEL laser array. If the light fields between the VCSEL units in the array can be coupled and the wavelength and phase are locked, a coherent laser light field output can be obtained. Especially in the case of in-phase mode output, the laser energy is concentrated in the axial center, and even a nearly diffraction-limited laser beam can be obtained. The coherent phase locking of the VCSEL array is mainly realized by the external cavity mirror to make the VCSEL array units mutually inject each other, and there is no model for the coherent phase locking of the VCSEL array with multiple high-order transverse modes. SUMMARY

[0004] In view of the deficiencies in the prior art, the application provides an external cavity VCSEL laser array mutual injection dynamics modeling simulation method with multiple high-order transverse modes.

[0005] The application discloses an external cavity VCSEL laser array mutual injection dynamics modeling simulation method, and the external cavity VCSEL laser array has multiple high-order transverse modes.

[0006] The structure parameters of the external cavity VCSEL laser array and the working conditions for driving the external cavity VCSEL laser array to normally operate are set;

[0007] The operation state of the external cavity VCSEL laser array is described by using a polar coordinate system;

[0008] A dynamics model of the external cavity VCSEL laser array mutual injection is established based on an L-K rate equation model, and a partial differential equation group related to the carrier density, the photon density and the phase and space and time is constructed;

[0009] The partial differential equations are reduced to separate the spatial variable and the time variable, and the partial differential equations are converted into ordinary differential equations with only time variable;

[0010] The ordinary differential equations are discretized, and the numerical solution is calculated by using finite difference method;

[0011] The images of the carrier density, the photon density, the phase and the wavelength changing with time are outputted.

[0012] As a further improvement of the application, the external cavity refers to adding a mirror at the Talbot distance of the VCSEL laser array output surface n times Talbot distance to form a Talbot external cavity mirror to realize mutual injection between the VCSEL laser units; wherein, n is an integer or a fraction.

[0013] As a further improvement of the application, the external cavity VCSEL laser array with multiple high-order transverse modes refers to the laser light field output by the VCSEL laser units containing both the fundamental transverse mode and the high-order transverse mode.

[0014] As a further improvement of the application, the working conditions include the injection current size and the injection current mode, and the structure parameters include the active region volume, the front cavity mirror reflectivity, the Talbot external cavity mirror reflectivity, the array unit aperture diameter, the array spacing, the external cavity length and the array arrangement mode; wherein, the array arrangement mode is a periodic arrangement satisfying the Talbot effect, including but not limited to square arrangement, hexagonal arrangement, honeycomb arrangement and circular arrangement.

[0015] As a further improvement of the application, the specific form of the dynamic model is:

[0016]

[0017]

[0018]

[0019] In the formula, N(r, t) and j(r, t) are respectively the carrier density and the current density in the active region, both of which are related to the spatial variable r and the time variable t, S imn (t) refers to the photon density of the mn-th transverse mode of the i-th laser unit at t time, refers to the phase of the mn-th transverse mode of the i-th laser unit at t time, S jmn (t) refers to the photon density of the mn-th transverse mode of the j-th laser unit at t time, refers to the phase of the mn-th transverse mode of the j-th laser unit at t time, τ N and τS respectively carrier lifetime and photon lifetime, D N is the diffusion coefficient, G mn is the gain coefficient of the mn-order mode, Γ mn is the mn-order transverse mode light field restriction factor, β mn is the spontaneous recombination coefficient, i, j are 1, 2, 3,..., N, N is the number of laser units, mn is 01, 11, 21,..., η is the internal efficiency, v g is the group velocity, N th is the threshold carrier density, α N is the differential gain coefficient, α is the linewidth enhancement factor, κ is the feedback coefficient, θ ijmn is the phase difference generated by the light propagating from the i-th laser unit to the j-th laser unit.

[0020] As a further improvement of the application, the calculation formula of the feedback coefficient κ is:

[0021]

[0022] In the formula, R2, R3 respectively represent the reflectivity of the VCSEL front cavity mirror and the reflectivity of the external cavity Talbot external cavity mirror, C ijmn represents the light field injection coefficient of the i-th light emitting unit and the j-th light emitting unit with the same mn-order transverse mode; its calculation method is:

[0023]

[0024] In the formula, E imn (x, y, 0) is the electric field distribution of the mn-order transverse mode of the i-th light emitting unit at a distance of Z=0, E jmn (x, y, 2L) refers to the electric field distribution of the mn-order transverse mode of the j-th light emitting unit at a distance of Z=2L.

[0025] As a further improvement of the application, the calculation method of the electric field distribution of the mn-order transverse mode of the i-th light emitting unit at a distance of Z=0, comprising:

[0026] The VCSEL is equivalent to a fiber waveguide, starting from the Maxwell equation set, combining the theory of optical waveguide, the distribution function of the transverse mode field is obtained, and the specific form is:

[0027]

[0028]

[0029] In the formula, J m , K m respectively represent the first and second types of Bessel functions of m order, Rcore representing the radius of the core layer, u mn and ω mn are eigenvalues, which are obtained by solving the eigenvalue equation:

[0030]

[0031] As a further improvement of the present application, the method for calculating the electric field distribution of the mn-order transverse mode pattern of the jth light-emitting unit at a distance of Z = 2L includes:

[0032] The diffraction calculation of the electric field distribution at Z = 0 is performed using the convolution method, which is specifically:

[0033]

[0034] In the formula, and respectively represent the Fourier transform and inverse Fourier transform symbols, and h(x, y) is the transfer function, which is specifically:

[0035]

[0036] As a further improvement of the present application, the partial differential equation set is reduced to eliminate the spatial variable r, which specifically includes:

[0037] The carrier density expression is expanded according to the Bessel series and the Fourier series:

[0038]

[0039] The photon number density expression is further expanded:

[0040]

[0041] Equations (11) and (12) are substituted into the rate equation representing the kinetic model, and the orthogonality of the Bessel function and the trigonometric function is used to convert the partial differential equation set into an ordinary differential equation set.

[0042] As a further improvement of the present application, the finite difference method is used to calculate the numerical solution, which includes:

[0043] Assume the differential equation and the boundary condition of the differential equation are:

[0044]

[0045] y(t0) = y0 (14)

[0046] Then:

[0047]

[0048] Then:

[0049] y(t+Δt) = y0 + f(x)Δt (16)

[0050] Similarly, the numerical solution of the differential equation can be obtained

[0051] In the formula, Δt is the step length of its solution, and the smaller the step length, the more accurate the solution result.

[0052] Compared with the prior art, the present application has the beneficial effects that:

[0053] The present application is based on the theory of semiconductor laser rate equation, and provides a method for modeling and simulating the mutual injection dynamics of an external cavity VCSEL laser array with multiple high-order transverse modes, which can analyze the mutual injection dynamics of the external cavity of the VCSEL array with multiple high-order transverse mode outputs, and find the conditions for realizing coherent phase locking, and has great significance for realizing high-power and high-beam-quality laser output. BRIEF DESCRIPTION OF DRAWINGS

[0054] Figure 1 The flow chart of the mutual injection dynamics modeling method of the external cavity VCSEL array disclosed by the present application.

[0055] Fig. 2(a) is a structural schematic diagram of a square-arranged external cavity VCSEL array disclosed by the present application.

[0056] Fig. 2(b) is a structural schematic diagram of a hexagon-arranged external cavity VCSEL array disclosed by the present application.

[0057] Fig. 2(c) is a structural schematic diagram of a honeycomb-arranged external cavity VCSEL array disclosed by the present application.

[0058] Fig. 2(d) is a schematic diagram of the internal structure of the external cavity VCSEL array disclosed by the present application.

[0059] Figure 3 Fig. 3 is a schematic diagram of the fiber waveguide structure obtained by the present application.

[0060] Figure 4 Fig. 4 is a simulation result curve diagram of the wavelength change over time of all the transverse mode patterns of the VCSEL laser unit 01 obtained by the present application.

[0061] Figure 5 Fig. 5 is a simulation result diagram of the phase change over time of all the transverse mode patterns of the VCSEL laser unit 01 obtained by the present application.

[0062] Figure 6 Fig. 6 is a simulation result curve diagram of the wavelength change over time of all the transverse mode patterns of the VCSEL laser unit 11 obtained by the present application.

[0063] Figure 7 This is a simulation diagram showing the phase change over time for all transverse mode modes of the VCSEL laser unit 11 obtained in this invention.

[0064] Figure 8 The graph shows the simulation results of the average carrier density of all VCSEL laser units obtained in this invention as a function of time.

[0065] In the picture:

[0066] 1. Talbot endoscope

[0067] 2. Anterior laparoscopy

[0068] 3. Active region

[0069] 4. Core layer

[0070] 5. Cladding Detailed Implementation

[0071] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0072] The present invention will now be described in further detail with reference to the accompanying drawings:

[0073] like Figure 1 As shown, this invention provides a dynamic modeling and simulation method for mutual injection of external cavity VCSEL laser arrays with multiple higher-order transverse modes. It establishes a mathematical model of optical field coupling of higher-order transverse mode VCSEL array units under external cavity feedback from the perspective of time dynamics, and simulates the dynamic response under different parameters based on this model.

[0074] An external cavity refers to adding a reflector at n times the Talbot distance from the output surface of a VCSEL laser array to form a Talbot external cavity mirror, so as to realize mutual injection between VCSEL laser units; where n is an integer or fraction; a VCSEL laser unit with multiple higher-order transverse modes means that the laser field output by the VCSEL laser unit simultaneously contains the fundamental transverse mode and higher-order transverse modes, rather than having only a single transverse mode, but having multiple transverse modes simultaneously.

[0075] This simulation method specifically includes:

[0076] S1, set the structure parameters (physical structure and geometric parameters) of the external cavity VCSEL laser array and the working conditions for driving the external cavity VCSEL laser array to operate normally; wherein,

[0077] As shown in Figures 2(a) to 2(c) , the high-order transverse mode VCSEL array is composed of N VCSEL units that can simultaneously output the fundamental mode and high-order modes, where N≥3; as shown in FIG. 2(d), the high-order transverse mode VCSEL array includes a Talbot cavity mirror 1, a front cavity mirror 2 and an active region 3, and the Talbot cavity mirror 1 and the front cavity mirror 2 provide feedback for the VCSEL array by reflecting the output light of the VCSEL.

[0078] The working conditions include the injection current size and the injection current mode, and the structure parameters include the active region volume, the front cavity mirror reflectivity, the Talbot external cavity mirror reflectivity, the array unit aperture diameter, the array spacing, the external cavity length and the array arrangement mode; wherein the array arrangement mode is a periodic arrangement that satisfies the Talbot effect, including but not limited to square arrangement, hexagonal arrangement, honeycomb arrangement and circular arrangement.

[0079] Taking the structure parameters and simulation calculation conditions of the VCSEL units and arrays in Table 1 as an example:

[0080] Table 1

[0081]

[0082]

[0083] S2, describe the operating state of the external cavity VCSEL laser array using a polar coordinate system;

[0084] S3, establish a dynamic model of mutual injection of the external cavity VCSEL laser array based on the L-K rate equation model, and construct a partial differential equation set related to space and time for carrier density, photon density and phase; wherein,

[0085] The specific form of the dynamic model is:

[0086]

[0087]

[0088]

[0089] In the formula, N(r, t) and j(r, t) are respectively the carrier density and current density in the active region, both of which are related to the spatial variable r and the time variable t, S imn (t) refers to the photon density of the mn-th transverse mode of the i-th laser unit at time t, denotes the phase of the mn-th transverse mode of the i-th laser unit at time t, S jmn (t) denotes the photon density of the mn-th transverse mode of the j-th laser unit at time t, denotes the phase of the mn-th transverse mode of the j-th laser unit at time t, τ N and τ S are the carrier lifetime and the photon lifetime, respectively, D N is the diffusion coefficient, G mn is the gain coefficient of the mn-th mode, Γ mn is the mn-th transverse mode light field restriction factor, β mn is the spontaneous recombination coefficient, i, j can be 1, 2, 3,..., N, N is the number of laser units, mn can be 01, 11, 21,..., η is the internal efficiency, v g is the group velocity, N th is the threshold carrier density, α N is the differential gain coefficient, α is the linewidth enhancement factor, κ is the feedback coefficient, θ ijmn is the phase difference generated by the light propagating from the i-th laser unit to the j-th laser unit.

[0090] The calculation formula of the feedback coefficient κ is:

[0091]

[0092] In the formula, R2 and R3 represent the reflectivity of the VCSEL front cavity mirror and the reflectivity of the Talbot external cavity mirror, respectively, c ijmn represents the light field injection coefficient of the i-th light emitting unit and the j-th light emitting unit with the same mn-th transverse mode; the calculation method is as follows:

[0093]

[0094] In the formula, E imn (x, y, 0) is the electric field distribution of the mn-th transverse mode of the i-th light emitting unit at a distance of Z=0, E jmn (x, y, 2L) refers to the electric field distribution of the mn-th transverse mode of the j-th light emitting unit at a distance of Z=2L.

[0095] The calculation method of the electric field distribution of the mn-th transverse mode of the i-th light emitting unit at a distance of Z=0, includes:

[0096] The VCSEL is equivalent to a fiber waveguide, as shown in Figure 3 The circularly symmetric step-index optical waveguide includes a core layer 4 and a cladding layer 5; starting from the Maxwell equations, combined with the theory of optical waveguide, the distribution function of the transverse mode field is obtained, and the specific form is as follows:

[0097]

[0098]

[0099] where J m , K m represent the first and second kind Bessel functions of order m, respectively, R core represents the core radius, u mn and ω mn are eigenvalues, which are obtained by solving the eigenvalue equation:

[0100]

[0101] The calculation method of the electric field distribution of the mn-order transverse mode of the jth light-emitting unit at a distance of Z = 2L includes:

[0102] The diffraction calculation of the electric field distribution at Z = 0 is performed by using the convolution method, and the specific form is:

[0103]

[0104] where and represent the Fourier transform and inverse Fourier transform symbols, respectively, and h(x, y) is a transfer function, and the specific form is:

[0105]

[0106] S4, the partial differential equation set is reduced to separate the spatial variable and the time variable, so that the partial differential equation set is converted into a common differential equation set with only the time variable; wherein

[0107] The partial differential equation set is reduced to eliminate the spatial variable r, and specifically includes:

[0108] The carrier density expression is expanded according to the Bessel series and the Fourier series:

[0109]

[0110] The photon number density expression is expanded again:

[0111]

[0112] Equations (11) and (12) are substituted into the rate equation representing the kinetic model, and the orthogonality of the Bessel function and the trigonometric function is used to convert the partial differential equation set into a common differential equation set.

[0113] S5, discretize the system of ordinary differential equations and calculate the numerical solution by using finite difference method; wherein, the calculation of the numerical solution by using finite difference method comprises:

[0114] Suppose the differential equation and the boundary condition of the differential equation are:

[0115]

[0116] y(t0)=y0 (14) then:

[0117]

[0118] then:

[0119] y(t+Δt)=y0+f(x)Δt (16)

[0120] Similarly, the numerical solution of the differential equation can be obtained

[0121] In the formula, Δt is the step length of the solution, and the smaller the step length, the more accurate the solution result.

[0122] S6, output the images of the changes of carrier density, photon density, phase and wavelength with time, and obtain the complete process of the external cavity VCSEL laser array optical field mutual injection dynamics of the multiple high-order transverse modes. Figures 4 to 8 Output the images of the changes of the wavelengths and phases of different order transverse modes and the average carrier density with time.

[0123] The above only for the preferred embodiments of the present application and not for limiting the present application, for those skilled in the art, the present application can have various changes and variations. Any modification, equivalent replacement, improvement, etc. within the spirit and principles of the present application shall be included in the protection scope of the present application.

Claims

1. A method for modeling and simulating the mutual injection dynamics of an external cavity VCSEL laser array, wherein the external cavity VCSEL laser array has multiple higher-order transverse modes; characterized in that, The simulation method includes: Set the structural parameters of the external cavity VCSEL laser array and the working conditions for driving the external cavity VCSEL laser array to operate normally. The operating state of the external cavity VCSEL laser array is described using polar coordinates; A dynamic model for mutual injection of external cavity VCSEL laser arrays is established based on the LK rate equation model, and a set of partial differential equations relating carrier density, photon density, and phase with spatial and temporal dependencies are constructed. The specific form of the dynamic model is as follows: In the formula, N(r,t) and j(r,t) are the carrier density and current density in the active region, respectively, both related to the spatial variable r and the time variable t. imn (t) refers to the photon density of the mn-th transverse mode of the i-th laser unit at time t. This refers to the phase of the mn-th transverse mode of the i-th laser unit at time t, S jmn (t) refers to the photon density of the mn-th transverse mode of the j-th laser unit at time t. This refers to the phase of the mn-th transverse mode of the j-th laser unit at time t, τ N With τ S These are the carrier lifetime and photon lifetime, respectively, D N G is the diffusion coefficient. mn For the mn-th order mode, Γ is the gain coefficient. mn β is the confinement factor for the transverse mode optical field of order mn. mn The spontaneous recombination coefficient is denoted by i, j, which are 1, 2, 3...N, where N is the number of laser units, mn is 0, 1, 1, 2, ..., η is the internal efficiency, and ν is the internal efficiency. g For group velocity, N th For the threshold carrier density, α N α is the differential gain coefficient, κ is the linewidth enhancement factor, and θ is the feedback coefficient. ijmn It is the phase difference generated when light propagates from the i-th laser unit to the j-th laser unit; The partial differential equation system is reduced by separating the spatial and time variables, thus transforming the partial differential equation system into an ordinary differential equation system with only time variables. The ordinary differential equation system is discretized, and the numerical solution is calculated using the finite difference method. Output images showing the changes in carrier density, photon density, phase, and wavelength over time.

2. The method for modeling and simulating the mutual injection dynamics of an external cavity VCSEL laser array as described in claim 1, characterized in that, An external cavity refers to a mirror placed at n times the Talbot distance from the output surface of a VCSEL laser array, forming a Talbot external cavity mirror to achieve mutual injection between VCSEL laser units; where n is an integer or fraction.

3. The method for modeling and simulating the mutual injection dynamics of an external cavity VCSEL laser array as described in claim 1, characterized in that, An external cavity VCSEL laser array with multiple higher-order transverse modes refers to a VCSEL laser unit whose output laser field simultaneously contains both the fundamental transverse mode and higher-order transverse modes.

4. The method for modeling and simulating the mutual injection dynamics of an external cavity VCSEL laser array as described in claim 1, characterized in that, The operating conditions include the magnitude and method of the injected current, and the structural parameters include the volume of the active region, the reflectivity of the front cavity mirror, the reflectivity of the Talbot external cavity mirror, the aperture diameter of the array unit, the array spacing, the length of the external cavity, and the array arrangement; wherein, the array arrangement is a periodic arrangement that satisfies the Talbot effect, including but not limited to square arrangement, hexagonal arrangement, honeycomb arrangement, and circular arrangement.

5. The method for modeling and simulating the mutual injection dynamics of an external cavity VCSEL laser array as described in claim 1, characterized in that, The formula for calculating the feedback coefficient κ is: In the formula, R2 and R3 represent the reflectivity of the VCSEL front endoscope and the reflectivity of the Talbot external endoscope, respectively, and c ijmn This represents the light field injection coefficient of the i-th and j-th emitting units having the same mn-th transverse mode; its calculation method is as follows: In the formula, E imn (x,y,0) is the electric field distribution of the mn-th order transverse mode of the i-th emitting unit at a distance Z=0, E jmn (x,y,2L) refers to the electric field distribution of the mn-th transverse mode of the j-th emitting unit at a distance of Z = 2L.

6. The method for modeling and simulating the mutual injection dynamics of an external cavity VCSEL laser array as described in claim 5, characterized in that, The method for calculating the electric field distribution of the mn-th order transverse mode of the i-th emitting unit at a distance of Z=0 includes: By equating the VCSEL to an optical fiber waveguide, and starting from Maxwell's equations and combining them with the theory of optical waveguides, the distribution function of the transverse mode field is obtained, which has the following specific form: In the formula, J m K m R represents the first and second kind of Bessel functions of order m, respectively. core Represents the core radius, u mn and ω mn These are eigenvalues, which are obtained by solving the following eigenvalue equations:

7. The method for modeling and simulating the mutual injection dynamics of an external cavity VCSEL laser array as described in claim 5, characterized in that, The method for calculating the electric field distribution of the mn-th transverse mode of the j-th emitting unit at a distance of Z = 2L includes: The electric field distribution at Z=0 is calculated using the convolution method, and its specific form is as follows: In the formula, and ...

8. The method for modeling and simulating the mutual injection dynamics of an external cavity VCSEL laser array as described in claim 1, characterized in that, The partial differential equation system is reduced by eliminating the spatial variable r, specifically including: Expand the carrier density expression using Bessel and Fourier series: Expanding the expression for photon number density: Substituting equations (11) and (12) into the rate equations representing the dynamic model, we can use the orthogonal properties of Bessel functions and trigonometric functions to transform the partial differential equations into ordinary differential equations.

9. The method for modeling and simulating the mutual injection dynamics of an external cavity VCSEL laser array as described in claim 1, characterized in that, Numerical solutions are calculated using the finite difference method, including: Assume the differential equation and its boundary conditions are as follows: y(t0)=y0 (14) but: but: y(t+Δt)=y0+f(x)Δt (16) By analogy, the numerical solution of the differential equation can be obtained. In the formula, Δt is the step size for solving the problem; the smaller the step size, the more accurate the solution.

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