NUSOD 2010 Georgia Tech Atlanta, GA September 6-9, 2010

1. NUSOD 2010 Georgia TechAtlanta, GASeptember 6-9, 2010 Modeling of Vertical External Cavity Semiconductor Laser with MQW Resonant Structure A. Napartovich, N. Elkin, D. Vysotsky SRC RF Troitsk Institute for Innovation and Thermonuclear Research Pushkovykh prop. 12 Troitsk, Moscow province, 142190 Russia

2. Motivation The concept of a vertical-cavity resonant periodic gain surface-emitting laser was suggested in papers [1-3]. The key idea of this design is to create an optimal overlap between the periodic gain medium and the standing wave optical field of the lasing mode. If the period is multiple of half-wavelength the longitudinal confinement factor can be up to twice as large as that for MQW design with chaotic positions of QWs. The upper limit is achieved provided the QWs are located at antinodes of modal wave field. Typically, VECSELs are optically pumped by diode lasers [4]. Electron beam pumping can be implemented, too. Usage of the electron beam is a promising way to extend the laser spectral range into shorter wavelength region [5]. Theoretical modeling of devices with both types of pumping can be unified by introducing of an effective pump current density [6, 7]. In our case, the effective current density J=170 mA/cm2 corresponds to e-beam current density jb =1 mA/cm2.

3. Schematics of simulated device

4. Linear spectral properties of the compound cavity (a) (b) Round trip gain of a plane wave starting from the external mirror for uniform 25-QW structure with g=3400 cm-1. (a) is for coarse wavelength scale, (b) is for fine scale around resonance marked by the arrow on (a). Gain at resonance is about twice as high as other spectral peaks reflecting multiple longitudinal modes associated with QWs. Fine structure of resonance peak (b) is due to external spacing 3 cm between a mirror and chip. The oscillation period is 0.029 nm.

5. Basic equations (linear case) The wave field is taken in a form E(r,φ,z,t)=U(r,φ,z)exp(-iΩt), Ω=ω0+Δω-i, ω0 is the reference frequency, Δω=ω-ω0 is the frequency shift and  is the attenuation factor. The relationship between reference wave number and wavelength: ω0=k0c, k0=2π/0. For cylindrical symmetry of gain and index with fixed radial profiles, the wave field amplitude can be taken in a form U(r,φ,z)=Um(r,z)·exp(imφ), and Um satisfies the equation: here β=(gth+2iΔk) is the complex-valued eigen-number, gth=2/c, Δk=Δω/c, g is gain coefficient, and n is refractive index. Conditiongth=0 means equality of gain and losses, i.e. corresponds to steady state operation of a laser mode.

6. Numerical Method (linear case) In frame of BiBPM a wave field in each horizontal plane is taken as a vector of the upward and downward propagating waves, V+ and V- . The wave fields in two neighbor planes, marked by symbols t and b are coupled by a equation: ,where M is a translation matrix. Transfer matrix for set of layers can be calculated as a product of the elementary interface and propagation matrices [1]: where hk=zk-zk-1, Qk is the operator of longitudinal wave number in the k-th layer: As a result, the boundary problem for the Helmholtz equation can be reduced to solving the closed system of transfer equations and boundary conditions. Reflection of a wave from an apertured external mirror is determined by the function (r) so as (r)= Rm, if r<Dm/2 and (r)=0, if r>Dm/2. Here Dm is the diameter of the mirror, Rm is the reflection coefficient.

7. Numerical Method (linear case) Round trip operator [8] in linear case contains eigenvalue b, which, in turn, is composed of gthand Δk. The specific feature of the problem is non-trivial dependence of round trip operator on the eigenvalue. To achieve our goal (calculation of eigenvalues and optical modes) a subsidiary problem was introduced: Here g is a constant, the value of which depends on b and turns to unity when value of b is the eigenvalue for a given mode. Generally, calculations were organized as follows: an inner iteration procedure solves the equation (*) at fixed value of β to find one or several eigenpairs (u,γ), the external iterative cycle encloses inner cycle and serves to find the value β when γ=1. The fast Hankel transform algorithm was used for effective calculations with the translation matrices. In the wave number space the operator Qk is replaced by the number qk, the operational matrices Tk and Pk are numerical matrices. The calculations in the QW regions were performed in the physical space because of non-uniform transverse gain and index distributions. The approach of locally uniform wave field was used. This approach is applicable since thickness of the QW is far less than the wavelength. Fixed-point method was used in the iteration procedure of the round-trip operator evaluation.

8. Inversion equation and parameters of device D=0.5 cm2s-1,τnr =10-9 s,B=3.5×10-10cm3s-1, Jtr=2.35Acm-2, where α = exp(gmin/g0), g0 and gmin are gain parameters, n0 is the refractive index in the absence of carriers, R is the line enhancement factor.g0=3400 cm-1 , gmin=-1000 cm-1, R=2.5, I=2.35 mA.The pump profile function is f(ρ)=(1+ρ4)-1. The reference wavelength λ0=642.2 nm.Parameters of the external round spherical mirror: curvature radius 3 cm, reflection coefficient is equal to 0.985, transverse diameter 400 μm. Optical path length Lopt of the spacing between the mirror and the heterostructure is a variable parameter. 256 mesh nodes over polar radius were used in calculations.

9. Linear spectral properties of the compound cavity To get an idea about properties of the compound cavity loaded with gain located in QW array, calculations were performed of linear modal gains for two lowest modes TEM00 (solid line) and TEM01 as functions of optical path length Lopt. The effective pump current 400 mA, mirror diameter 0.4 mm, and pump beam radius is variable. gth<0 corresponds to above threshold pumping.

10. Linear spectral properties of the compound cavity The results of calculations of linear modal gains for three lowest modes TEM00 (squares), TEM01 (stars), and TEM10 (triangles) as functions of optical path length Lopt. The effective pump current 400 mA, mirror diameter 0.4 mm, and pump beam radius is 25 mm. gth<0 corresponds to above threshold pumping.

11. Linear spectral properties of the compound cavity The results of calculations of wave vector shift for three lowest modes TEM00 (squares), TEM01 (stars), and TEM10 (triangles) as functions of optical path length Lopt. The effective pump current 400 mA, mirror diameter 0.4 mm, and pump beam radius is 25 mm.

12. Results of laser power calculations Laser output power for various pump radii as a function of optical path length Lopt. The effective pump current 400 mA, mirror diameter 0.4 mm.

13. Non-linear effects in laser operation Diagram of laser power and modal stability as a function of optical path length Lopt. The effective pump current 400 mA, mirror diameter 0.4 mm, and pump beam radius is 25 mm. s – stability region, bs – bistability region, ns – no stable solution region.

14. Non-linear effects in laser operation Diagram of wave vector shift and modal stability as a function of optical path length Lopt. The effective pump current 400 mA, mirror diameter 0.4 mm, and pump beam radius is 25 mm. s – stability region, bs – bistability region, ns – no stable solution region.

15. Non-linear effects in laser operation The pump profile function (dashed line) andprofiles of the normalized light intensity IQW in the upper QW for Lopt=2.18 cm (line 1) and Lopt=2.17 cm (line 2).

16. References 1. R. Geels, R. H. Yan, J. W. Scott, S. W. Corzine, R. J. Simes, and L. A. Coldren, "Analysis and design of a novel parallel-driven MQW-DBR surface-emitting diode laser." CLEO'88. Anaheim, CA. Apr. 1988. paper WMI 2. S. W. Corzine. R. S. Geels, J. W. Scott, and L. A. Coldren. "Surface-emitting lasers with periodic gain.'' LEOS.88, Santa Clara, CA, Nov. 1988, paper OE1.2. 3. M. Y. A. Raja, S. R. J . Brueck. M. Osinski. C. F. Schaus. J. G. McInerney. T. M. Brennan, and B. E. Hammons. "Wavelength-resonant enhanced gain/absorption structure for optoelectronic devices." in Post-Deadline Papers. XVI Int. Conf. Quantum Electronics. IQEC’88, Tokyo, Japan. July 1988. paper PD-23. pp. 52-53. 4. C. Troppera, S. Hoogland, “Extended cavity surface-emitting semiconductor lasers,” Progress in Quantum Electronics, vol. 30, pp. 1–43, 2006. 5. V. Yu. Bondarev, V. I. Kozlovsky, A. B. Krysa, Yu. M Popov, Ya. K. Skasyrsky, “Simulation of a longitudinally electron-beam-pumped nanoheterostructure semiconductor laser” Quantum Electron., vol. 34, pp. 919-925, 2004

17. References 6. J. Piprek, S. Björlin, and J. E. Bowers, “Design and Analysis of Vertical-Cavity Semiconductor Optical Amplifiers,” IEEE J. Quantum Electron., vol. 37, no. 1, pp. 127-134, 2001 7. D. V. Vysotsky, N. N. Elkin, A. P. Napartovich, V. I. Kozlovsky, B. M. Lavrushin, “Simulation of a longitudinally electron-beam-pumped nano-heterostructure semiconductor laser,” Quantum Electronics, vol. 39, pp. 1028 – 1032, 2009 8. Elkin N.N., Napartovich A.P., Troshchieva V.N., Vysotsky D.V. “Round-trip operator technique applied for optical resonators with dispersion elements”Lect. Notes Comp. Sci., 4310, 542 (2007).

18. Conclusions & Acknowledgments • The BiBPM for multilayer media is combined with the round-trip operator technique for compound optical resonators including Fox-Li iterations and Krylov subspace methods. • The developed numerical algorithm allows us to calculate the mode spatial profile, output power, exact wavelength and other characteristics of an oscillating mode. • Typical computational time for one variant amounts to several tens of minutes on IBM PC • The developed code package has a subroutine for analyzing single mode operation regime • Various non-linear effects are predicted including laser instabilities and bistable operation Work is partially supported by the RFBR project No. 08-02-00796-a.