Abstract

A model is presented for analyzing volume Bragg gratings having large diffractive strength that may become distorted upon the passage of high-power laser light. One result of this analysis is that very little distortion in a volume Bragg grating can greatly reduce the reflectivity. This is a critical issue in the design of systems in which these devices are used to combine high-power laser beams.

© 2007 Optical Society of America

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References

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  1. H. Shu, Y. Chen, M. Bass, and M. A. Acharekar, "Modeling a diode pumped Nd:YAG rod laser," in Solid State Lasers XV: Technology and Devices, H. J. Hoffman and R. K. Shori, eds., Proc. SPIE 6100, 61001J-1 (2006).
    [CrossRef]
  2. H. Shu and M. Bass, "A computer model for simulating real laser systems by solving partial differential equations in FEMLAB," in Solid State Lasers XIV: Technology and Devices, H. J. Hoffman and R. K. Shori, eds., Proc. SPIE 5707, 394-402 (2005).
    [CrossRef]
  3. I. Ciapurin, V. Smirnov, G. Venus, L. Glebova, E. Rotari, and L. Glebov, "High-power laser beam control by PTR Bragg gratings," in Proceedings of CLEO/IQEC, San Francisco, Calif. (2004), paper CTuP51.
  4. A. Dergachev, P. F. Moulton, V. Smirnov, and L. Glebov, "High power CW Tm:YLF laser with a holographic outputcoupler," in Proceedings of CLEO/IQEC, San Francisco, Calif. (2004), paper CThZ3.
  5. H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909 (1969).
  6. K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2000).
  7. C. L. Xu, W. P. Huang, J. Chrostowski, and S. K. Chaudhuri, "A full-vectorial beam propagation method for anisotropic waveguides," J. Lightwave Technol. 12, 1926-1931 (1994).
    [CrossRef]

2006 (1)

H. Shu, Y. Chen, M. Bass, and M. A. Acharekar, "Modeling a diode pumped Nd:YAG rod laser," in Solid State Lasers XV: Technology and Devices, H. J. Hoffman and R. K. Shori, eds., Proc. SPIE 6100, 61001J-1 (2006).
[CrossRef]

2005 (1)

H. Shu and M. Bass, "A computer model for simulating real laser systems by solving partial differential equations in FEMLAB," in Solid State Lasers XIV: Technology and Devices, H. J. Hoffman and R. K. Shori, eds., Proc. SPIE 5707, 394-402 (2005).
[CrossRef]

1994 (1)

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

1969 (1)

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909 (1969).

Acharekar, M. A.

H. Shu, Y. Chen, M. Bass, and M. A. Acharekar, "Modeling a diode pumped Nd:YAG rod laser," in Solid State Lasers XV: Technology and Devices, H. J. Hoffman and R. K. Shori, eds., Proc. SPIE 6100, 61001J-1 (2006).
[CrossRef]

Bass, M.

H. Shu, Y. Chen, M. Bass, and M. A. Acharekar, "Modeling a diode pumped Nd:YAG rod laser," in Solid State Lasers XV: Technology and Devices, H. J. Hoffman and R. K. Shori, eds., Proc. SPIE 6100, 61001J-1 (2006).
[CrossRef]

H. Shu and M. Bass, "A computer model for simulating real laser systems by solving partial differential equations in FEMLAB," in Solid State Lasers XIV: Technology and Devices, H. J. Hoffman and R. K. Shori, eds., Proc. SPIE 5707, 394-402 (2005).
[CrossRef]

Chaudhuri, S. K.

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

Chen, Y.

H. Shu, Y. Chen, M. Bass, and M. A. Acharekar, "Modeling a diode pumped Nd:YAG rod laser," in Solid State Lasers XV: Technology and Devices, H. J. Hoffman and R. K. Shori, eds., Proc. SPIE 6100, 61001J-1 (2006).
[CrossRef]

Chrostowski, J.

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

Ciapurin, I.

I. Ciapurin, V. Smirnov, G. Venus, L. Glebova, E. Rotari, and L. Glebov, "High-power laser beam control by PTR Bragg gratings," in Proceedings of CLEO/IQEC, San Francisco, Calif. (2004), paper CTuP51.

Dergachev, A.

A. Dergachev, P. F. Moulton, V. Smirnov, and L. Glebov, "High power CW Tm:YLF laser with a holographic outputcoupler," in Proceedings of CLEO/IQEC, San Francisco, Calif. (2004), paper CThZ3.

Glebov, L.

A. Dergachev, P. F. Moulton, V. Smirnov, and L. Glebov, "High power CW Tm:YLF laser with a holographic outputcoupler," in Proceedings of CLEO/IQEC, San Francisco, Calif. (2004), paper CThZ3.

I. Ciapurin, V. Smirnov, G. Venus, L. Glebova, E. Rotari, and L. Glebov, "High-power laser beam control by PTR Bragg gratings," in Proceedings of CLEO/IQEC, San Francisco, Calif. (2004), paper CTuP51.

Glebova, L.

I. Ciapurin, V. Smirnov, G. Venus, L. Glebova, E. Rotari, and L. Glebov, "High-power laser beam control by PTR Bragg gratings," in Proceedings of CLEO/IQEC, San Francisco, Calif. (2004), paper CTuP51.

Huang, W. P.

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

Kogelnik, H.

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909 (1969).

Moulton, P. F.

A. Dergachev, P. F. Moulton, V. Smirnov, and L. Glebov, "High power CW Tm:YLF laser with a holographic outputcoupler," in Proceedings of CLEO/IQEC, San Francisco, Calif. (2004), paper CThZ3.

Okamoto, K.

K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2000).

Rotari, E.

I. Ciapurin, V. Smirnov, G. Venus, L. Glebova, E. Rotari, and L. Glebov, "High-power laser beam control by PTR Bragg gratings," in Proceedings of CLEO/IQEC, San Francisco, Calif. (2004), paper CTuP51.

Shu, H.

H. Shu, Y. Chen, M. Bass, and M. A. Acharekar, "Modeling a diode pumped Nd:YAG rod laser," in Solid State Lasers XV: Technology and Devices, H. J. Hoffman and R. K. Shori, eds., Proc. SPIE 6100, 61001J-1 (2006).
[CrossRef]

H. Shu and M. Bass, "A computer model for simulating real laser systems by solving partial differential equations in FEMLAB," in Solid State Lasers XIV: Technology and Devices, H. J. Hoffman and R. K. Shori, eds., Proc. SPIE 5707, 394-402 (2005).
[CrossRef]

Smirnov, V.

A. Dergachev, P. F. Moulton, V. Smirnov, and L. Glebov, "High power CW Tm:YLF laser with a holographic outputcoupler," in Proceedings of CLEO/IQEC, San Francisco, Calif. (2004), paper CThZ3.

I. Ciapurin, V. Smirnov, G. Venus, L. Glebova, E. Rotari, and L. Glebov, "High-power laser beam control by PTR Bragg gratings," in Proceedings of CLEO/IQEC, San Francisco, Calif. (2004), paper CTuP51.

Venus, G.

I. Ciapurin, V. Smirnov, G. Venus, L. Glebova, E. Rotari, and L. Glebov, "High-power laser beam control by PTR Bragg gratings," in Proceedings of CLEO/IQEC, San Francisco, Calif. (2004), paper CTuP51.

Xu, C. L.

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

Bell Syst. Tech. J. (1)

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909 (1969).

J. Lightwave Technol. (1)

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

Proc. SPIE (2)

H. Shu, Y. Chen, M. Bass, and M. A. Acharekar, "Modeling a diode pumped Nd:YAG rod laser," in Solid State Lasers XV: Technology and Devices, H. J. Hoffman and R. K. Shori, eds., Proc. SPIE 6100, 61001J-1 (2006).
[CrossRef]

H. Shu and M. Bass, "A computer model for simulating real laser systems by solving partial differential equations in FEMLAB," in Solid State Lasers XIV: Technology and Devices, H. J. Hoffman and R. K. Shori, eds., Proc. SPIE 5707, 394-402 (2005).
[CrossRef]

Other (3)

I. Ciapurin, V. Smirnov, G. Venus, L. Glebova, E. Rotari, and L. Glebov, "High-power laser beam control by PTR Bragg gratings," in Proceedings of CLEO/IQEC, San Francisco, Calif. (2004), paper CTuP51.

A. Dergachev, P. F. Moulton, V. Smirnov, and L. Glebov, "High power CW Tm:YLF laser with a holographic outputcoupler," in Proceedings of CLEO/IQEC, San Francisco, Calif. (2004), paper CThZ3.

K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2000).

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Figures (9)

Fig. 1
Fig. 1

Schematic of the reflection of a laser beam by a VBG.

Fig. 2
Fig. 2

Sketch of the distorted surfaces (heavy outline) and diffracting surfaces in the VBG sustaining a high-power laser beam that is even slightly absorbed.

Fig. 3
Fig. 3

Reflection coefficient as a function of the deviation from the Bragg wavelength calculated by our numeric model together with that calculated using the analytical formula for S = 0.7744 , Δ n = 10 4 , and grating thickness = 2.6 mm . The open circles are the calculation using the analytical formula; the stars are the calculation using our numeric model.

Fig. 4
Fig. 4

Schematic of the division of the grating into two parts.

Fig. 5
Fig. 5

Reflection coefficient as a function of the deviation from the Bragg wavelength calculated by our numeric model together with that calculated using the analytical formula for S = 1.8501 , Δ n = 2.3889 × 10 4 , and grating thickness = 2.6 mm . The open circles are the calculation using the analytical formula; the stars are the calculation using our numeric model.

Fig. 6
Fig. 6

Calculated output beam from a VBG with distortion Δ L = 0.15 μm . (a) Intensity as a function of x for y = 0 , together with that of the input beam normalized for comparison; (b) calculated phase of the electric field as a function of x for y = 0 ; (c) intensity pattern in the xy plane when z = 0 ; (d) phase factor introduced by surface distortion as a function of x for y = 0 .

Fig. 7
Fig. 7

Calculated output beam from a VBG with distortion Δ L = 0.3 μm . (a) Intensity as a function of x for y = 0 , together with that of the input beam normalized for comparison; (b) calculated phase of the electric field as a function of x for y = 0 ; (c) intensity pattern in the xy plane when z = 0 ; (d) phase factor introduced by surface distortion as a function of x for y = 0 .

Fig. 8
Fig. 8

Calculated output beam from a VBG with distortion Δ L = 0.6 μm . (a) Intensity as a function of x for y = 0 , together with that of the input beam normalized for comparison; (b) calculated phase of the electric field as a function of x for y = 0 ; (c) intensity pattern in the xy plane when z = 0 ; (d) phase factor introduced by surface distortion as a function of x for y = 0 .

Fig. 9
Fig. 9

Calculated total intensity reflection coefficient versus Δ L . Open circles: w 0 = 3.5 mm ; solid circles: w 0 = 7 mm ; cross: w 0 = 14 mm .

Equations (34)

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2 E + ω 2 c 2 ε r E = 0 .
E = A e i k 0 n 0 e A r + B e i k 0 n 0 e B r ,
n = n 0 + Δ n cos ( q r + φ ) ,
k 0 n 0 e A + q = k 0 n 0 e B ,
k 0 n 0 e B q = k 0 n 0 e A .
2 ( A e i k 0 n 0 e A r ) + ω 2 c 2 n 0 2 A e i k 0 n 0 e A r + ω 2 c 2 n 0 Δ n B e i k 0 n 0 e A r + i φ = 0 ,
2 ( B e i k 0 n 0 e B r ) + ω 2 c 2 n 0 2 B e i k 0 n 0 e B r + ω 2 c 2 n 0 Δ n A e i k 0 n 0 e B r i φ = 0 .
2 i k 0 n 0 A z = ω 2 c 2 n 0 Δ n B e i φ + 2 A x 2 + 2 A y 2 ,
2 i k 0 n 0 B z = ω 2 c 2 n 0 Δ n A e i φ + 2 B x 2 + 2 B y 2 .
Λ ( x , y ) = Λ 0 L 0 L ( x , y ) ,
n = n 0 + Δ n cos ( 2 π Λ ( x , y , z ) e q r + φ ) ,
Λ 0 = λ 2 n 0 ,
k ( x , y , z ) n 0 e A + 2 π Λ ( x , y , z ) e q = k ( x , y , z ) n 0 e B ,
k ( x , y , z ) n 0 e B 2 π Λ ( x , y , z ) e q = k ( x , y , z ) n 0 e A .
Δ k ( x , y , z ) = k 0 k ( x , y , z ) ,
k 0 = k ( x , y , z ) + Δ k ( x , y , z ) .
2 [ A e i k 0 n 0 e A r + B e i k 0 n 0 e B r ] + ω 2 c 2 n 0 2 [ A e i k 0 n 0 e A r + B e i k 0 n 0 e B r ] + ω 2 c 2 n 0 Δ n { A e i [ k ( x , y , z ) n 0 e B + Δ k ( x , y , z ) n 0 e A ] r i φ + B e i [ k ( x , y , z ) n 0 e A + Δ k ( x , y , z ) n 0 e B ] r + i φ } = 0.
2 i k 0 n 0 A z = ω 2 c 2 n 0 Δ n B e 2 i Δ k ( x , y , z ) n 0 z e i φ + 2 A x 2 + 2 A y 2 ,
2 i k 0 n 0 B z = ω 2 c 2 n 0 Δ n A e 2 i Δ k ( x , y , z ) n 0 z e i φ + 2 B x 2 + 2 B y 2 .
2 i k 0 n 0 A z = ω 2 c 2 ( n 0 + Δ n T ) Δ n B e 2 i Δ k ( x , y , z ) n 0 z e i φ + ω 2 c 2 [ ( n 0 + Δ n T ) 2 n 0 2 ] A + 2 A x 2 + 2 A y 2 ,
2 i k 0 n 0 B z = ω 2 c 2 ( n 0 + Δ n T ) Δ n A e 2 i Δ k ( x , y , z ) n 0 z e i φ + ω 2 c 2 [ ( n 0 + Δ n T ) 2 n 0 2 ] B + 2 B x 2 + 2 B y 2 ,
2 i k 0 n 0 A z = [ ω 2 c 2 n 0 Δ n B A e i φ + 2 x 2 + 2 y 2 ] A ,
2 i k 0 n 0 B z = [ ω 2 c 2 n 0 Δ n A B e i φ + 2 x 2 + 2 y 2 ] B .
A ( z = 0 ) = A input ,
B ( z = L 0 ) = 0.
2 i k 0 n 0 A z = ω 2 c 2 n 0 Δ n B e i φ ,
2 i k 0 n 0 B z = ω 2 c 2 n 0 Δ n A e i φ .
Δ k ( λ ) = k 0 ( λ ) k .
A 1 ( z = 0 ) = A input , B 1 ( z = L 0 2 ) = B 2 ( z = L 0 2 ) ,
A 2 ( z = L 0 2 ) = A 1 ( z = L 0 2 ) , B 2 ( z = L 0 ) = 0.
Δ k ( x , y ) = k 0 k 0 L 0 L ( x , y ) .
A input ( x , y ) = A input ( x , y ) e i k 0 0.5 [ L ( 0 , 0 ) L ( x , y ) ] .
B output ( x , y ) = B output ( x , y ) e i k 0 n 0 [ L ( x , y ) L 0 ] i k 0 0.5 [ L ( 0 , 0 ) L ( x , y ) ] .
L ( x , y ) = L 0 + Δ L e r 2 / w 0 2 ,

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