Abstract

A new method for computing eigenmodes of a laser resonator by the use of finite element analysis is presented. For this purpose, the scalar wave equation (Δ + k 2)(x, y, z) = 0 is transformed into a solvable three-dimensional eigenvalue problem by the separation of the propagation factor exp(-ikz) from the phasor amplitude (x, y, z) of the time-harmonic electrical field. For standing wave resonators, the beam inside the cavity is represented by a two-wave ansatz. For cavities with parabolic optical elements, the new approach has successfully been verified by the use of the Gaussian mode algorithm. For a diode-pumped solid-state laser with a thermally lensing crystal inside the cavity, the expected deviation between Gaussian approximation and numerical solution could be demonstrated clearly.

© 2004 Optical Society of America

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References

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  1. A. E. Siegman, Lasers (University Science, Mill Valley, 1986).
  2. LASCAD, http://www.las-cad.com .
  3. K. Altmann, “Simulation software tackles design of laser resonators,” Laser Focus World 36, 293–294 (2000).
  4. A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–458 (1961).
  5. M. D. Feit, J. A. Fleck, “Spectral approach to optical resonator theory,” Appl. Opt. 20, 2843–2851 (1981).
    [CrossRef] [PubMed]
  6. A. E. Siegman, H. Y. Miller, “Unstable optical resonator loss calculations using the Prony method,” Appl. Opt. 9, 2729–2736 (1970).
    [CrossRef] [PubMed]
  7. M. Streiff, A. Witzig, W. Fichtner, “Computing optical modes for VCSEL device simulation,” IEE Proc. Optoelectron. 149, 166–173 (2002).
    [CrossRef]
  8. F. Ihlenburg, Finite Element Analysis of Acoustic Scattering (Springer-Verlag, New York, 1998).
    [CrossRef]
  9. H.-G. Roos, M. Stynes, L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations (Springer-Verlag, Berlin, 1996).
    [CrossRef]
  10. K. W. Morton, Numerical Solution of Convection-Diffusion Problems (Chapman & Hall, London, New York, 1996).
  11. Y. Saad, M. H. Schultz, “GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput. 7, 856–869 (1986).
    [CrossRef]

2002 (1)

M. Streiff, A. Witzig, W. Fichtner, “Computing optical modes for VCSEL device simulation,” IEE Proc. Optoelectron. 149, 166–173 (2002).
[CrossRef]

2000 (1)

K. Altmann, “Simulation software tackles design of laser resonators,” Laser Focus World 36, 293–294 (2000).

1986 (1)

Y. Saad, M. H. Schultz, “GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput. 7, 856–869 (1986).
[CrossRef]

1981 (1)

1970 (1)

1961 (1)

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–458 (1961).

Altmann, K.

K. Altmann, “Simulation software tackles design of laser resonators,” Laser Focus World 36, 293–294 (2000).

Feit, M. D.

Fichtner, W.

M. Streiff, A. Witzig, W. Fichtner, “Computing optical modes for VCSEL device simulation,” IEE Proc. Optoelectron. 149, 166–173 (2002).
[CrossRef]

Fleck, J. A.

Fox, A. G.

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–458 (1961).

Ihlenburg, F.

F. Ihlenburg, Finite Element Analysis of Acoustic Scattering (Springer-Verlag, New York, 1998).
[CrossRef]

Li, T.

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–458 (1961).

Miller, H. Y.

Morton, K. W.

K. W. Morton, Numerical Solution of Convection-Diffusion Problems (Chapman & Hall, London, New York, 1996).

Roos, H.-G.

H.-G. Roos, M. Stynes, L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations (Springer-Verlag, Berlin, 1996).
[CrossRef]

Saad, Y.

Y. Saad, M. H. Schultz, “GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput. 7, 856–869 (1986).
[CrossRef]

Schultz, M. H.

Y. Saad, M. H. Schultz, “GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput. 7, 856–869 (1986).
[CrossRef]

Siegman, A. E.

Streiff, M.

M. Streiff, A. Witzig, W. Fichtner, “Computing optical modes for VCSEL device simulation,” IEE Proc. Optoelectron. 149, 166–173 (2002).
[CrossRef]

Stynes, M.

H.-G. Roos, M. Stynes, L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations (Springer-Verlag, Berlin, 1996).
[CrossRef]

Tobiska, L.

H.-G. Roos, M. Stynes, L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations (Springer-Verlag, Berlin, 1996).
[CrossRef]

Witzig, A.

M. Streiff, A. Witzig, W. Fichtner, “Computing optical modes for VCSEL device simulation,” IEE Proc. Optoelectron. 149, 166–173 (2002).
[CrossRef]

Appl. Opt. (2)

Bell Syst. Tech. J. (1)

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–458 (1961).

IEE Proc. Optoelectron. (1)

M. Streiff, A. Witzig, W. Fichtner, “Computing optical modes for VCSEL device simulation,” IEE Proc. Optoelectron. 149, 166–173 (2002).
[CrossRef]

Laser Focus World (1)

K. Altmann, “Simulation software tackles design of laser resonators,” Laser Focus World 36, 293–294 (2000).

SIAM J. Sci. Stat. Comput. (1)

Y. Saad, M. H. Schultz, “GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput. 7, 856–869 (1986).
[CrossRef]

Other (5)

A. E. Siegman, Lasers (University Science, Mill Valley, 1986).

LASCAD, http://www.las-cad.com .

F. Ihlenburg, Finite Element Analysis of Acoustic Scattering (Springer-Verlag, New York, 1998).
[CrossRef]

H.-G. Roos, M. Stynes, L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations (Springer-Verlag, Berlin, 1996).
[CrossRef]

K. W. Morton, Numerical Solution of Convection-Diffusion Problems (Chapman & Hall, London, New York, 1996).

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Figures (12)

Fig. 1
Fig. 1

Computational domain.

Fig. 2
Fig. 2

Empty cavity with one concave end mirror (R 1 = 5.0 mm) and one planar end mirror; FEA (solid curve) and Gaussian (dashed curve) mode shapes.

Fig. 3
Fig. 3

Gouy phase shifts for an empty cavity with one concave end mirror (R 1 = 5.0 mm) and one planar end mirror; FEA (solid curve) and Gaussian (dashed curve) mode shifts.

Fig. 4
Fig. 4

Comparison between FEA (solid curve) and Gaussian (dashed curve) results for a long resonator with a short Gaussian duct attached to the left mirror.

Fig. 5
Fig. 5

TEM00 mode in a long resonator.

Fig. 6
Fig. 6

TEM22 mode in a long resonator.

Fig. 7
Fig. 7

Slice and isosurfaces of a refractive-index distribution based on numerical data imported from lascad.

Fig. 8
Fig. 8

Comparison of numerical (dots) and parabolically fitted (triangles) refractive indices. Screen shot of lascad.

Fig. 9
Fig. 9

Comparison of FEA (solid curve) and Gaussian (dashed curve) x-axis spot sizes along the cavity axis for a monolithic laser with thermal effects taken into account.

Fig. 10
Fig. 10

Comparison of FEA (solid curve) and Gaussian (dashed curve) y-axis spot sizes along the cavity axis for a monolithic laser with thermal effects taken into account.

Fig. 11
Fig. 11

Transverse mode profiles along the x axis for a monolithic laser with thermal effects taken into account; FEA (solid curve) and Gaussian (dashed curve) mode results.

Fig. 12
Fig. 12

Coinciding transverse mode profiles along the y axis for a monolithic laser with thermal effects taken into account; FEA (solid curve) and Gaussian (dashed curve) mode results.

Tables (1)

Tables Icon

Table 1 Comparison between FEA and Gaussian Results for a Gaussian Duct with Real Refractive Index

Equations (22)

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Δ+k2E˜x, y, z=0,
γnmE˜nmx, y= K˜x, y, x0, y0×E˜nmx0, y0dx0dy0,
ft=j=1N Kj exppjt
E˜x, y, z=exp-ikfz+iψzv˜x, y, z.
ψz=arctanzzR,
E˜x, y, z=exp-ikf-zũx, y, z.
k=kf+ksx, y, z,
-Δũ+2ikf-ũz-ks2kf+ksũ=2kf-ũ.
-Δũ+2ikfũz-ks2kf+ksũ=2kfũ=:ξũ
-Δũ+2ikfũz+kf2-k2ũ=ξũ.
Δx,yũ+2kf+ksũ=0,
ũx, y, L˜=ũx, y, 0, z ũx, y, L˜=z ũx, y, 0.
ũn-iCbũ=0,
E˜x, y, z=exp-ikf-zũrx, y, z+exp-ikf-L-zũlx, y, z.
ũrx, y, 0=expikfx2R1x+y2R1y-iπϕ0x, y:= ũlx, y, 0, ũlx, y, L=expikfx2R2x+y2R2y-iπϕ1x, y:= ũrx, y, L,
-Δũr+2ikfũrz+kf2-k2ũr=ξũr, -Δũl-2ikfũlz+kf2-k2ũl=ξũl  in Ω,
ũr-ϕ0ũl=0 on Γ0, ũr-ϕ¯1ũl=0 on Γ1, ũrz+ϕ0ũlz=0 on Γ0, ũrz+ϕ¯1ũlz=0 on Γ1, ũrn-iCbũr=0 on Γr, ũln-iCbũl=0 on Γr.
V:=vr, vlVhxy,h×Vhxy,h|vr-ϕ0vl|z=0=0 and vr-ϕ¯1vl|z=L=0
Ωurv¯r+kf2-k2urv¯r+2ikfz urv¯r×dx, y, z-iCbΓr urv¯rdσx, y, z+τh Ω2ikfz ur+kf2-k2urzv¯rdx, y, z+Ωulv¯l+kf2-k2ulv¯l-2ikfz ulv¯l×dx, y, z-iCbΓr ulv¯ldσx, y, z-τh Ω-2ikfz ul+kf2-k2ulzv¯ldx, y, z=ξ Ωurv¯r+τhurzv¯r+ulv¯l-τhulzv¯ldx, y, z
AUr,h, Ul,h=ξhMUr,h, Ul,h,
ũ0, 0, z=exp-iz-ψz|ũ0, 0, z|,
n2z=0.06if 0.0z1.00.0else.

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