Abstract

We developed an accurate and fast three-dimensional computer model for simulating realistic solid-state laser systems. An iteration of the beam propagation calculation was developed to account for the counterpropagation of the laser beams in the saturated gain medium and eventually obtain the converged solution for the output beam. An analytic method was devised to account for the curved cavity mirror and the surface deformation of the gain medium induced by the temperature gradient due to pump absorption. The temperature gradient induced thermal lensing and stress birefringence is also properly included in the model calculation. This model has been validated and shown to be very accurate and efficient for modeling three-dimensional laser systems in a personal computer.

© 2007 Optical Society of America

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References

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  1. C. L. Xu, W. P. Huang, J. Chrostowski, and S. K. Chaudhuri, "A full-vector beam propagation method for anisotropic waveguide," J. Lightwave Technol. 12, 1926-1931 (1994).
    [CrossRef]
  2. W. P. Huang and C. L. Xu, "Simulation of three-dimensional optical waveguides by a full-vector beam propagation method," IEEE J. Quantum Electron. 29, 2639-2649 (1993).
    [CrossRef]
  3. W. Koechner, Solid-State Laser Engineering (Springer, 1999).
  4. Y. Chen, B. Chen, M. K. R. Patel, and M. Bass, "Calculation of thermal-gradient-induced stress birefringence in slab lasers-I," IEEE J. Quantum Electron. 40, 909-916 (2004).
    [CrossRef]
  5. Y. Chen, B. Chen, M. K. R. Patel, A. Kar, and M. Bass, "Calculation of thermal-gradient-induced stress birefringence in slab lasers-II," IEEE J. Quantum Electron. 40, 917-928 (2004).
    [CrossRef]
  6. B. Chen, Y. Chen, J. Simmons, T. Y. Chung, and M. Bass, "Thermal lensing of edge-pumped slab lasers-I," Appl. Phys. B 82, 413-418 (2006).
    [CrossRef]
  7. comsol multiphysics is a multi-physics modeling program produced by COMSOL Inc.
  8. H. Kogelnik and T. Li, "Laser beams and resonators," Appl. Opt. 5, 1550-1567 (1966).
    [CrossRef] [PubMed]
  9. N. Hodgson and H. Weber, Optical Resonators (Springer, 1997).
  10. A. E. Siegman, Lasers (University Science Books, 1986).
  11. H. Shu and M. Bass, "A computer model for simulating real laser systems, by solving partial differential Eqs. in FEMLAB," Proc. SPIE 5707, 394-402 (2005).
    [CrossRef]
  12. MATLAB is a technical computing language produced by The Mathworks Inc.
  13. H. Kogelnik, "Imaging of optical modes--resonators with internal lenses," Bell Syst. Tech. J. 44, 455-494 (1965).
  14. H. Shu, "Analytic and numeric modeling of diode laser pumped Yb:YAG laser oscillators and amplifiers," Ph.D. dissertation (University of Central Florida, 2003).

2006 (1)

B. Chen, Y. Chen, J. Simmons, T. Y. Chung, and M. Bass, "Thermal lensing of edge-pumped slab lasers-I," Appl. Phys. B 82, 413-418 (2006).
[CrossRef]

2005 (1)

H. Shu and M. Bass, "A computer model for simulating real laser systems, by solving partial differential Eqs. in FEMLAB," Proc. SPIE 5707, 394-402 (2005).
[CrossRef]

2004 (2)

Y. Chen, B. Chen, M. K. R. Patel, and M. Bass, "Calculation of thermal-gradient-induced stress birefringence in slab lasers-I," IEEE J. Quantum Electron. 40, 909-916 (2004).
[CrossRef]

Y. Chen, B. Chen, M. K. R. Patel, A. Kar, and M. Bass, "Calculation of thermal-gradient-induced stress birefringence in slab lasers-II," IEEE J. Quantum Electron. 40, 917-928 (2004).
[CrossRef]

1994 (1)

C. L. Xu, W. P. Huang, J. Chrostowski, and S. K. Chaudhuri, "A full-vector beam propagation method for anisotropic waveguide," J. Lightwave Technol. 12, 1926-1931 (1994).
[CrossRef]

1993 (1)

W. P. Huang and C. L. Xu, "Simulation of three-dimensional optical waveguides by a full-vector beam propagation method," IEEE J. Quantum Electron. 29, 2639-2649 (1993).
[CrossRef]

1966 (1)

1965 (1)

H. Kogelnik, "Imaging of optical modes--resonators with internal lenses," Bell Syst. Tech. J. 44, 455-494 (1965).

Appl. Opt. (1)

Appl. Phys. B (1)

B. Chen, Y. Chen, J. Simmons, T. Y. Chung, and M. Bass, "Thermal lensing of edge-pumped slab lasers-I," Appl. Phys. B 82, 413-418 (2006).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, "Imaging of optical modes--resonators with internal lenses," Bell Syst. Tech. J. 44, 455-494 (1965).

IEEE J. Quantum Electron. (3)

W. P. Huang and C. L. Xu, "Simulation of three-dimensional optical waveguides by a full-vector beam propagation method," IEEE J. Quantum Electron. 29, 2639-2649 (1993).
[CrossRef]

Y. Chen, B. Chen, M. K. R. Patel, and M. Bass, "Calculation of thermal-gradient-induced stress birefringence in slab lasers-I," IEEE J. Quantum Electron. 40, 909-916 (2004).
[CrossRef]

Y. Chen, B. Chen, M. K. R. Patel, A. Kar, and M. Bass, "Calculation of thermal-gradient-induced stress birefringence in slab lasers-II," IEEE J. Quantum Electron. 40, 917-928 (2004).
[CrossRef]

J. Lightwave Technol. (1)

C. L. Xu, W. P. Huang, J. Chrostowski, and S. K. Chaudhuri, "A full-vector beam propagation method for anisotropic waveguide," J. Lightwave Technol. 12, 1926-1931 (1994).
[CrossRef]

Proc. SPIE (1)

H. Shu and M. Bass, "A computer model for simulating real laser systems, by solving partial differential Eqs. in FEMLAB," Proc. SPIE 5707, 394-402 (2005).
[CrossRef]

Other (6)

MATLAB is a technical computing language produced by The Mathworks Inc.

H. Shu, "Analytic and numeric modeling of diode laser pumped Yb:YAG laser oscillators and amplifiers," Ph.D. dissertation (University of Central Florida, 2003).

W. Koechner, Solid-State Laser Engineering (Springer, 1999).

comsol multiphysics is a multi-physics modeling program produced by COMSOL Inc.

N. Hodgson and H. Weber, Optical Resonators (Springer, 1997).

A. E. Siegman, Lasers (University Science Books, 1986).

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Figures (8)

Fig. 1
Fig. 1

(Color online) Flow chart of the iteration process showing inclusion of the saturated gain coefficient.

Fig. 2
Fig. 2

(Color online) Schematic of the transformation of a TEM 00 Gaussian beam by the lenslike medium.

Fig. 3
Fig. 3

Input TEM 00 Gaussian beam at the first beam waist ( P 1 ) and the calculated output beam at the second beam waist ( P 2 ) . (a) Absolute value of the electric field of the input beam on the x axis for y = 0; (b) absolute value of the electric field of the input beam on the whole input plane; (c) absolute value of the electric field of the calculated output beam on the x axis for y = 0, together with the analytic solution; (d) absolute value of the electric field of the calculated output beam on the whole output plane.

Fig. 4
Fig. 4

Schematic of the laser beam passing through two identical Nd:YAG rods.

Fig. 5
Fig. 5

Input and output beams after passing through the first and the second rods. (a) Absolute value of the electric field of the input beam linearly polarized parallel to x in the input plane; (b) absolute value of the x component of the electric field after passing the first rod; (c) absolute value of the y component of the electric field after passing the first rod; (d) absolute value of the x component of the electric field after passing the second rod; (e) absolute value of the y component of the electric field after passing the second rod; (f) absolute value of the y component of the electric field after passing the second rod on the x axis for y = 0, together with the absolute value of the x component of the electric field of the input beam.

Fig. 6
Fig. 6

Schematic of the considered Yb:YAG disk, a stable resonator laser.

Fig. 7
Fig. 7

(a) Calculated output power versus input pump power for the considered Yb:YAG disk laser together with the calculations in [11, 14]. Crosses indicate results obtained with this model, open circles indicate results from [11] and triangles indicate results from [14]. (b) Absolute value of the calculated electric field entering the Yb:YAG disk at the front surface, on the x axis for y = 0, together with the T E M 00 mode of the passive cavity normalized for comparison. (c) Same plots as in (b) except that the light is leaving the Yb:YAG disk. (d) Intensity pattern of the output beam immediately after passing the output mirror. (e) Phase of the output beam.

Fig. 8
Fig. 8

(a) Amplitude of the ratio, A ( n + 1 ) / A ( n ) , versus the number of round trips. (b) Phase of the ratio versus number of round trips. (c) Steady-state saturated gain coefficient in the central transverse plane perpendicular to the laser axis in the gain medium. (d) Steady-state gain coefficient on the x axis for y = 0, in the central transverse plane in the gain medium, together with the total laser intensity.

Tables (1)

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Table 1 Parameters of the Yb:YAG Disk Laser Considered

Equations (29)

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× × E μ 0 ω 2 D = 0 ,
D = ε 0 ε E .
2 E + k 0 2 ε E = ( E ) .
ε = [ ε x x ε x y ε x z ε y x ε y y ε y z ε z x ε z y ε z z ] = [ n 2 + Δ ε x x ε x y ε x z ε y x n 2 + Δ ε y y ε y z ε z x ε z y n 2 + Δ ε z z ] ,
E = ( A x e i k 0 n 0 z A y e i k 0 n 0 z A z e i k 0 n 0 z ) .
2 i k 0 n 0 A x ( x , y , z ) z = 2 A x ( x , y , z ) x 2 + 2 A x ( x , y , z ) y 2 + k 0 2 [ i n 0 k 0 g ( x , y , z ) + ε x x ( x , y , z ) n 0 2 ] × A x ( x , y , z ) + k 0 2 ε x y A y ,
2 i k 0 n 0 A y ( x , y , z ) z = 2 A y ( x , y , z ) x 2 + 2 A y ( x , y , z ) y 2 + k 0 2 [ i n 0 k 0 g ( x , y , z ) + ε y y ( x , y , z ) n 0 2 ] × A y ( x , y , z ) + k 0 2 ε y x A x ,
ε x x ( x , y , z ) = n 2 ( x , y , z ) + Δ ε x x ( x , y , z ) ,
ε y y ( x , y , z ) = n 2 ( x , y , z ) + Δ ε y y ( x , y , z ) ,
2 i k 0 n 0 A x ( x , y , z ) z = 2 A x ( x , y , z ) x 2 + 2 A x ( x , y , z ) y 2 + k 0 2 [ i n 0 k 0 g ( x , y , z ) + n 2 ( x , y , z ) n 0 2 ] ×  A x ( x , y , z ) ,
2 i k 0 n 0 A y ( x , y , z ) z = 2 A y ( x , y , z ) x 2 + 2 A y ( x , y , z ) y 2 + k 0 2 [ i n 0 k 0 g ( x , y , z ) + n 2 ( x , y , z ) n 0 2 ] ×  A y ( x , y , z ) .
2 i k 0 n 0 A ( x , y , z ) z = 2 A ( x , y , z ) x 2 + 2 A ( x , y , z ) y 2 + k 0 2 [ i n 0 k 0 g ( x , y , z ) + n 2 ( x , y , z ) n 0 2 ] × A ( x , y , z ) .
g ( x , y , z ) = g 0 ( x , y , z ) 1 + I t o t a l ( x , y , z ) I s a t ,
2 i k 0 A ( x , y , z ) z = 2 A ( x , y , z ) x 2 + 2 A ( x , y , z ) y 2 .
2 i k 0 A ( x , y , z ) z = 0.
Q x = b ,
g ( x , y , z ) = g 0 ( x , y , z ) 1 + | A + ( x , y , z ) + A + ( x , y , z + Δ z ) 2 | 2 + | A ( x , y , z ) + A ( x , y , z + Δ z ) 2 | 2 I s a t 0.5 ε 0 c n 0 ,
n = n 0 ( 1 2 r 2 b 2 ) ,
f = b 2 n 0 sin ( 2 l b ) ,
h = b 2 n 0 tan ( l b ) .
ε = [ n 0 2 + Δ ε x x ε x y ε x z ε y x n 0 2 + Δ ε y y ε y z ε z x ε z y n 0 2 + Δ ε z z ] ,
Δ ε x x = Δ ε y y = 0 ,
ε x y = ε y x = ( 1.3 × 10 6 ) ( 2 x 1 × 10 3 ) ( 2 y 1 × 10 3 ) ,
ε = [ n 0 2 ε x y ε x z ε y x n 0 2 ε y z ε z x ε z y ε z z ] .
E i n = ( E 0 e ( r / w 0 ) 20 e i k 0 n 0 z 0 0 ) ,
r p ( r , z ) = 2 α π w p 2 η a [ exp ( α z α l ) + exp ( α z α l ) ] × exp [ 2 r 2 w p 2 ] .
η a = [ 1 exp ( 2 α l ) ] .
g ( r , z ) = σ τ f P p η a h ν P r p ( r , z ) N 1 0 σ 1 + f I + ( r , z ) + I ( r , z ) I s a t ,
ratio = A ( n + 1 ) A ( n ) ,

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