A new Technique for Simulating Semiconductor Laser Resonators

A new Technique for Simulating Semiconductor Laser Resonators Britta Heubeck Christoph Pflaum Department of System Simulation (LSS) University of Erlangen-Nuremberg, Germany NUSOD conference, September 1st, 28

Outline 1 Challenge 2 Simulation Technique 3 Simulation Results

Outline 1 Challenge 2 Simulation Technique 3 Simulation Results

Goal Optimal Laser Good beam High output Stabilized quality power wavelength Understanding of the influence of different parameters on the dynamics in the laser device Simulation of the laser device

Distributed Feedback Laser Long resonator with length L injection current Small stripe width of size s of injection current j Layers with different refractive indices (gratings) Internal reflections of the optical wave s Substrate L Contact Active layer lambda Ω j Laser beam n1 n2 n1 n2 n1 n2 n1 n2 n1 n2 n1 n2 n1 n2 n1 n2 s L

Difficulties in the simulation of the optical wave Large-scale simulations of the wave equation require a large number of grid points Internal reflections Propagation in forward and backward direction has to be treated simultaneously Ω j Laser beam n1 n2 n1 n2 n1 n2 n1 n2 n1 n2 n1 n2 n1 n2 n1 n2 s L

Outline 1 Challenge 2 Simulation Technique 3 Simulation Results

Usual Approaches Well-known methods for the simulation of optical waves Beam Propagation Method Finite Difference Time Domain Method Finite Element Method (FEM) with standard Finite Elements But: Internal reflections and large resonators cannot be simulated by these methods 1D Transfer Matrix Method (TMM) But: Tapered lasers need a 2D-simulation

Idea Trigonometric Finite Wave Elements (TFWE) Special Finite Elements Provide the same solution as the TMM for 1D Helmholtz equation Extendable to higher dimensions Extendable to time-dynamic problems TFWE method combines advantages of TMM and FEM

Trigonometric Finite Wave Elements in 1D 1D linear nodal basis functions are multiplied by appropriate sine and cosine functions 1 1 1.5.5,5 P i 1 P i Pi+1 P i 1 Pi P i+1.5 1 x sin(kx) nodal_basis nodal_basis*sin(kx).5 1 cos(kx) nodal_basis nodal_basis*cos(kx) x P i 1 P i P i+1 mix(kx) nodal_basis nodal_basis*mix(kx) x These 3 basis functions per node span the Finite Element space V h. k: wave number

Trigonometric Finite Wave Elements in 2D Construct TFWE in 2D by a tensor product of 1D TFWE in propagation direction and linear nodal basis functions in perpendicular direction Sinusbasisfunction in 2D Cosinusbasisfunction in 2D Mixbasisfunction in 2D.5 1.5.4.2.5 y x y x y x

Oscillation Assumption Oscillation Assumption Let u H 2 (Ω) C(Ω) oscillate with an approximate local wave number k. This means, that u = u + exp(ikx) + u exp( ikx), where u + exp(ikx) H 2 ( Ω h ), u exp( ikx) H 2 ( Ω h ), u + xx L 2 ( Ω h ) u xx L 2 (Ω), and u xx L 2 ( Ω h ) u xx L 2 (Ω). 1D: P i 2 P i 1 P i P i+1 P i+2 P i+3 Ωh 1D: Ωh : Ω without grid points Ωh : Ω without grid lines u xx := d2 u dx 2 u xx := 2 u x 2

Approximation property of TFWE Oscillation Assumption Let u H 2 (Ω) C(Ω) oscillate with local wave number k. Theorem Let u H 2 (Ω) satisfy the Oscillation Assumption. Then, ( ) u Ih osc (u) H 1 (Ω) Ch(k maxh + 1) u + H 2 ( Ω h ) + u H 2 ( Ω h ) where C can be chosen independently of h and k max, if h, k H 1, ( Ω h ), and k H 2, ( Ω h ) are bounded from above. 1D: k max := max 1 j N k j 2D: k max := max 1 j Ny max 1 i Nx k ij k j : discretized wave numbers k ij : discretized wave numbers N: number of grid points N x, N y : number of grid points h := h h := max{h x, h y } Ih osc : H 2 (Ω) V h : interpolation operator, V h : TFWE space

Approximation property of standard Finite Elements Oscillation Assumption Let u H 2 (Ω) C(Ω) oscillate with local wave number k. Theorem Let u H 2 (Ω) satisfy the Oscillation Assumption. Then, we have u I h (u) H 1 (Ω) Ch(k max + 1) 2( u + H 2 ( Ω h ) + u H 2 ( Ω h ) where C can be chosen independently of h and k max, if h, k H 1, ( Ω h ), and k H 2, ( Ω h ) are bounded from above. ) 1D: I h : H 2 (Ω) Vh lin : linear interpolation operator 2D: Vh lin : linear FE space I h : H 2 (Ω) Vh bilin : bilinear interpolation operator Vh bilin : bilinear FE space

Conclusion Conclusion Increasing wave number k max, assuming hk max < C: The upper bound for the approximation error for standard FE increases with k 2 max The upper bound for the approximation error for TFWE keeps constant Remark The assumption hk max < C is true in most applications, e.g. in computational optics one has h < Cλ 2π = C k. λ: wavelength of the considered optical wave C, C > : constants independent of h and k

Outline 1 Challenge 2 Simulation Technique 3 Simulation Results

System of Coupled Partial Differential Equations Behavior of wave Ẽ is described by the wave equation; assuming Ẽ(x, t) = E(x, t) exp(iωt) leads to 2i k(n A ) E = E + k 2 (n A )E v g t Drift-diffusion equations determine the time-dynamic behavior of the carrier densities n A and n B n A t = (D A n A ) + C 1 n B C 2 n A r rec,a ( E 2, n A ) and n B t = (D B n B ) + C j C 1 n B + C 2 n A r rec,b D A, D B : ambipolar diffusion constants, j: current density, v g : group velocity, r rec,a,r rec,b : recombination densities

System of Coupled Partial Differential Equations r rec,a = A A n A + B A n A 2 + C A n A 3 + r stim, r rec,b = A B n B + B B n B 2, r stim = c n g g nonlin n, g nonlin = { g k = ωn c 1+εn ln( n A n tr ) if n A n tr g 1+εn ( n A n tr 1) if n A < n tr + α g nonlin H + i g nonlin α 2 2 n = ɛ n n g 2 ω Ẽ 2 = ɛ n n g 2 ω E 2. A A, B A, C A, A B, B B : recombination coefficients, n tr : transparency carrier density, α H : Henry factor g : differential gain, ε: gain compression factor, α : absorption

Absorbing Boundary Condition (PML) Wave is considered on domain Ω =], L[ ] W 2, W 2 [ Wave is emitted on left hand side (at x = ) Absorbing boundary condition ike(, y) = E x (, y) Absorbing boundary is simulated by PML Domain Ω is increased to Ω =] δ, L[ ] W 2, W 2 [ Schrödinger equation is transformed to k E 2i v g t = k E x k x + k 2 E k y 2 + k ke k := k iσ(x) := k ñ iσ(x) k := k iσ(x) := k n iσ(x) σ(x) := σ x 2 c δ2 for x <, σ(x) := for x σ c := 2δ 3 log(1/r ) R := 1 4 theoretical reflexion coefficient

Discretization Spatial discretization of the wave equation in propagation direction: at least one grid point per layer for resolving internal reflections Temporal discretization: wave with frequency ω 1.935PHz time step size τ.3ps ωτ 6. In the following examples: 84 layers, 841 grid points in propagation direction, 16/31 grid points in perpendicular direction, total simulation time: 4ns, current: 48mA, stripe width: 2.5µm.

Photon and Carrier density (α H =.) Photon density n Carrier density n A

Frequency Spectrum and Output Power (α H =.) 2e+8,5 1,5e+8,4 l2 (Fouriercoefficient) 1e+8 Output Power [W],3,2 5e+7,1 9,73e-7 9,732e-7 9,734e-7 Wavelength [m] 1e-9 2e-9 3e-9 4e-9 Time [s]

Photon and Carrier density (α H =.5) Photon density n Carrier density n A

Frequency Spectrum and Output Power (α H =.5) 2e+8,5 1,5e+8,4 l2 (Fouriercoefficient) 1e+8 Output Power [W],3,2 5e+7,1 9,73e-7 9,732e-7 9,734e-7 Wavelength [m] 1e-9 2e-9 3e-9 4e-9 Time [s]

Photon and Carrier density (α H = 2.5) Photon density n Carrier density n A

Frequency Spectrum and Output Power (α H = 2.5) 3e+8,3 2,5e+8,25 l2 (Fouriercoefficient) 2e+8 1,5e+8 1e+8 Output Power [W],2,15,1 5e+7,5 9,73e-7 9,732e-7 9,734e-7 Wavelength [m] 1e-9 2e-9 3e-9 4e-9 5e-9 Time [s]

Photon and Carrier density (α H = 3.) Photon density n Carrier density n A

Frequency Spectrum and Output Power (α H = 3.) 1e+8,3 8e+7,25 l2 (Fouriercoefficient) 6e+7 4e+7 Output Power [W],2,15,1 2e+7,5 9,728e-7 9,73e-7 9,732e-7 9,734e-7 Wavelength [m] 1e-9 2e-9 3e-9 4e-9 Time [s]

Photon and Carrier density (α H =., I = 96mA) Photon density n Carrier density n A

Frequency Spectrum and Output Power (α H =., I = 96mA) 3e+8 2,5e+8,8 l2 (Fouriercoefficient) 2e+8 1,5e+8 1e+8 Output Power [W],6,4 5e+7,2 9,73e-7 9,732e-7 9,734e-7 Wavelength [m] 1e-9 2e-9 3e-9 4e-9 Time [s]

Photon and Carrier density (α H = 2., s = 25µm) Photon density n Carrier density n A

Frequency Spectrum and Output Power (α H = 2., s = 25µm) 3e+8,3 2,5e+8,25 l2 (Fouriercoefficient) 2e+8 1,5e+8 1e+8 Output Power [W],2,15,1 5e+7,5 9,73e-7 9,732e-7 9,734e-7 Wavelength [m] 1e-9 2e-9 3e-9 4e-9 Time [s]

Tapering Zoom in photon density n Carrier density n A

Tapering 7e+8,2 6e+8 5e+8,15 l2 (Fouriercoefficient) 4e+8 3e+8 Output Power [W],1 2e+8,5 1e+8 9,73e-7 9,732e-7 9,734e-7 Wavelength [m] 1e-9 2e-9 3e-9 4e-9 Time [s]

Summary TFWE method combines advantages of FEM and TMM TFWE can be applied to time-periodic and time-dynamic wave problems Internal reflections can be simulated in 2D TFWE lead to better performance than standard FE Simulation can support the tuning of DFB lasers Influence of stripe width, current, Henry factor,... on the resulting mode, wavelength, and output power can be examined

Outlook Comparison: Experiment Simulation Simulation of different laser devices (disc lasers, three-section lasers,...) Extension of the simulation to 3D Introduction of multigrid method for solving the wave equation