Abstract

We present a realistic method for dynamic simulation of the development of higher order modes in second harmonic generation. The deformation of the wave fronts due to the nonlinear interaction is expressed by expansion in higher order Gauss-Laguerre modes.

© 2008 Optical Society of America

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References

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  1. G. D. Miller, R. G. Batchko, W. M. Tulloch, D. R. Weise, M. M. Fejer, and R. L. Byer, "42%-efficient single-pass cw second-harmonic generation in periodically poled lithium niobate," Opt. Lett. 22, 1834-1836 (1997).
    [CrossRef]
  2. S. V. Tovstonog, S. Kurimura, and K. Kitamura, "High power continuous-wave green light generation by quasiphase matching in Mg stoichiometric lithium tantalite," Appl. Phys. Lett. 90, 051115-1-3 (2007).
    [CrossRef]
  3. E. M. Daly and A. I. Ferguson, "Spatial and temporal dependence of single-pass parametric gain," J. Mod. Opt. 48, 729-744 (2001).
  4. T. Kasamatsu, H. Kubomura, and H. Kan, "Numerical simulation of conversion efficiency and beam quality factor in second harmonic generation with diffraction and pump depletion," Jpn. J. Appl. Phys. 44, 8495-8497 (2005).
    [CrossRef]
  5. V. Magny, "Optimum beams for efficient frequency mixing in crystals with second order nonlinearity," Opt. Commun. 184, 245-255 (2000).
    [CrossRef]
  6. M. Lassen, V. Delaubert, J. Janousek, Wagner, H.-A. Bachor, P. K. Lam, N. Treps, P. Buchhave, C. Fabre, and C. C. Harb, "Tools for Multimode Quantum Information: Modulation, detection and spatial quantum correlations," Phys. Rev. Lett. 98, 083602 (2007).
    [CrossRef] [PubMed]
  7. V. Delaubert, M. Lassen, D. R. N. Pulford, H-A. Bachor, and C. C. Harb, "Spatial mode discrimination using second harmonic generation," Opt. Express 15, 5815-5826 (2007).
    [CrossRef] [PubMed]
  8. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, second edition (John Wiley & Sons, 2007).
  9. B. G. D. Boyd and D. A. Kleinman, "Parametric Interaction of Focused Gaussian Light Beams," J. Appl. Phys. 39, 3597 (1968).
    [CrossRef]

2007 (2)

M. Lassen, V. Delaubert, J. Janousek, Wagner, H.-A. Bachor, P. K. Lam, N. Treps, P. Buchhave, C. Fabre, and C. C. Harb, "Tools for Multimode Quantum Information: Modulation, detection and spatial quantum correlations," Phys. Rev. Lett. 98, 083602 (2007).
[CrossRef] [PubMed]

V. Delaubert, M. Lassen, D. R. N. Pulford, H-A. Bachor, and C. C. Harb, "Spatial mode discrimination using second harmonic generation," Opt. Express 15, 5815-5826 (2007).
[CrossRef] [PubMed]

2005 (1)

T. Kasamatsu, H. Kubomura, and H. Kan, "Numerical simulation of conversion efficiency and beam quality factor in second harmonic generation with diffraction and pump depletion," Jpn. J. Appl. Phys. 44, 8495-8497 (2005).
[CrossRef]

2001 (1)

E. M. Daly and A. I. Ferguson, "Spatial and temporal dependence of single-pass parametric gain," J. Mod. Opt. 48, 729-744 (2001).

2000 (1)

V. Magny, "Optimum beams for efficient frequency mixing in crystals with second order nonlinearity," Opt. Commun. 184, 245-255 (2000).
[CrossRef]

1997 (1)

1968 (1)

B. G. D. Boyd and D. A. Kleinman, "Parametric Interaction of Focused Gaussian Light Beams," J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

Bachor, H-A.

Batchko, R. G.

Boyd, B. G. D.

B. G. D. Boyd and D. A. Kleinman, "Parametric Interaction of Focused Gaussian Light Beams," J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

Byer, R. L.

Daly, E. M.

E. M. Daly and A. I. Ferguson, "Spatial and temporal dependence of single-pass parametric gain," J. Mod. Opt. 48, 729-744 (2001).

Delaubert, V.

M. Lassen, V. Delaubert, J. Janousek, Wagner, H.-A. Bachor, P. K. Lam, N. Treps, P. Buchhave, C. Fabre, and C. C. Harb, "Tools for Multimode Quantum Information: Modulation, detection and spatial quantum correlations," Phys. Rev. Lett. 98, 083602 (2007).
[CrossRef] [PubMed]

V. Delaubert, M. Lassen, D. R. N. Pulford, H-A. Bachor, and C. C. Harb, "Spatial mode discrimination using second harmonic generation," Opt. Express 15, 5815-5826 (2007).
[CrossRef] [PubMed]

Fejer, M. M.

Ferguson, A. I.

E. M. Daly and A. I. Ferguson, "Spatial and temporal dependence of single-pass parametric gain," J. Mod. Opt. 48, 729-744 (2001).

Harb, C. C.

Janousek, J.

M. Lassen, V. Delaubert, J. Janousek, Wagner, H.-A. Bachor, P. K. Lam, N. Treps, P. Buchhave, C. Fabre, and C. C. Harb, "Tools for Multimode Quantum Information: Modulation, detection and spatial quantum correlations," Phys. Rev. Lett. 98, 083602 (2007).
[CrossRef] [PubMed]

Kan, H.

T. Kasamatsu, H. Kubomura, and H. Kan, "Numerical simulation of conversion efficiency and beam quality factor in second harmonic generation with diffraction and pump depletion," Jpn. J. Appl. Phys. 44, 8495-8497 (2005).
[CrossRef]

Kasamatsu, T.

T. Kasamatsu, H. Kubomura, and H. Kan, "Numerical simulation of conversion efficiency and beam quality factor in second harmonic generation with diffraction and pump depletion," Jpn. J. Appl. Phys. 44, 8495-8497 (2005).
[CrossRef]

Kleinman, D. A.

B. G. D. Boyd and D. A. Kleinman, "Parametric Interaction of Focused Gaussian Light Beams," J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

Kubomura, H.

T. Kasamatsu, H. Kubomura, and H. Kan, "Numerical simulation of conversion efficiency and beam quality factor in second harmonic generation with diffraction and pump depletion," Jpn. J. Appl. Phys. 44, 8495-8497 (2005).
[CrossRef]

Lassen, M.

V. Delaubert, M. Lassen, D. R. N. Pulford, H-A. Bachor, and C. C. Harb, "Spatial mode discrimination using second harmonic generation," Opt. Express 15, 5815-5826 (2007).
[CrossRef] [PubMed]

M. Lassen, V. Delaubert, J. Janousek, Wagner, H.-A. Bachor, P. K. Lam, N. Treps, P. Buchhave, C. Fabre, and C. C. Harb, "Tools for Multimode Quantum Information: Modulation, detection and spatial quantum correlations," Phys. Rev. Lett. 98, 083602 (2007).
[CrossRef] [PubMed]

Magny, V.

V. Magny, "Optimum beams for efficient frequency mixing in crystals with second order nonlinearity," Opt. Commun. 184, 245-255 (2000).
[CrossRef]

Miller, G. D.

Pulford, D. R. N.

Tulloch, W. M.

Wagner, J.

M. Lassen, V. Delaubert, J. Janousek, Wagner, H.-A. Bachor, P. K. Lam, N. Treps, P. Buchhave, C. Fabre, and C. C. Harb, "Tools for Multimode Quantum Information: Modulation, detection and spatial quantum correlations," Phys. Rev. Lett. 98, 083602 (2007).
[CrossRef] [PubMed]

Weise, D. R.

J. Appl. Phys. (1)

B. G. D. Boyd and D. A. Kleinman, "Parametric Interaction of Focused Gaussian Light Beams," J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

J. Mod. Opt. (1)

E. M. Daly and A. I. Ferguson, "Spatial and temporal dependence of single-pass parametric gain," J. Mod. Opt. 48, 729-744 (2001).

Jpn. J. Appl. Phys. (1)

T. Kasamatsu, H. Kubomura, and H. Kan, "Numerical simulation of conversion efficiency and beam quality factor in second harmonic generation with diffraction and pump depletion," Jpn. J. Appl. Phys. 44, 8495-8497 (2005).
[CrossRef]

Opt. Commun. (1)

V. Magny, "Optimum beams for efficient frequency mixing in crystals with second order nonlinearity," Opt. Commun. 184, 245-255 (2000).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. Lett. (1)

M. Lassen, V. Delaubert, J. Janousek, Wagner, H.-A. Bachor, P. K. Lam, N. Treps, P. Buchhave, C. Fabre, and C. C. Harb, "Tools for Multimode Quantum Information: Modulation, detection and spatial quantum correlations," Phys. Rev. Lett. 98, 083602 (2007).
[CrossRef] [PubMed]

Other (2)

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, second edition (John Wiley & Sons, 2007).

S. V. Tovstonog, S. Kurimura, and K. Kitamura, "High power continuous-wave green light generation by quasiphase matching in Mg stoichiometric lithium tantalite," Appl. Phys. Lett. 90, 051115-1-3 (2007).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Quasi-plane wave case, low power and short interaction length (1 mm); a-b: Fundamental and SH power (1 mm nonlinear crystal) and c-d: Fundamental and SH power (10 mm nonlinear crystal). Movie shows the field developing through the crystal.

Fig. 2.
Fig. 2.

The most powerful higher order G-L-modes at the output of the 10 mm long nonlinear crystal, using the same parameters as in Fig.1© and 1(d).

Fig. 3.
Fig. 3.

Development of power in F and SH beams through the crystal with optimal B-K focusing parameter ξ=2.84. Pump power is kept low to avoid back conversion.

Fig. 4.
Fig. 4.

Development of power in F and SH beams through the crystal with optimal B-K focusing parameter ξ=2.84. High pump power is considered to allow for back conversion.

Fig. 5.
Fig. 5.

The transverse field distribution of a) the fundamental input pulse and b) the generated SH in the middle of the pulse, ξ=0.10. Movie shows time development of these fields. c) shows the development of power in pump input pulse, transmitted pump pulse and generated SH output as a function of time. The peak pump power is 400 W.

Fig. 6.
Fig. 6.

The transverse field distribution of a) the fundamental input pulse and b) the generated SH in the middle of the pulse, ξ=2.84. Movie shows time development of these fields. c) shows the development of power in pump input pulse, transmitted pump pulse, and generated SH output as a function of time. The peak pump power is 200 W.

Fig. 7.
Fig. 7.

Comparison of SHG conversion efficiency using Boyd-Kleinman theory and the G-L mode expansion. Same parameters as in previous figure.

Equations (16)

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Δ a 1 ( x , y , z ) = i g ( a 2 ( x , y , z ) * a 3 ( x , y , z ) exp [ i Δ k z ] ) Δ z Δ a 2 ( x , y , z ) = i g ( a 1 ( x , y , z ) * a 3 ( x , y , z ) exp [ i Δ k z ] ) Δ z Δ a 3 ( x , y , z ) = i g ( a 1 ( x , y , z ) a 2 ( x , y , z ) exp [ i Δ k z ] ) Δ z
a j ( x , y , z + Δ z ) = a j ( x , y , z ) + Δ a j ( x , y , z )
u j , p , l G L ( r , θ , z , t ) = c p , l w 0 , j w ( z ) j e r 2 w ( z ) j 2 ( 2 r 2 w ( z ) j 2 ) 1 2 L p l ( 2 r 2 w ( z ) j 2 ) cos ( l θ )
× exp [ i k j r 2 2 R ( z ) j + i l θ + i k j z i ω j t + i ( 2 p + l + 1 ) arctan ( z z 0 , j ) ]
c p , l = 2 p ! ( 1 + δ ( l ) ) π ( l + p ) !
c p , 0 = 2 π
c j , p , 0 ( n ) ( r ) = 2 π c p , 0 r = 0 r a j ( n ) ( r ) 1 w ( z ) j exp [ r 2 w ( z n ) j 2 ] L p 0 ( 2 r 2 w ( z n ) j 2 )
× exp [ i k j r 2 2 R ( z n ) j + i k j z n i ω j t + i ( 2 p + 1 ) arctan ( z n z 0 , j ) ] dr
a j , p , 0 ( n ) ( r , z ) = c j , p , 0 ( n ) ( r ) u j , p , 0 G L ( r , z , t )
a j ( n ) ( r ) = p = 0 p max a j , p , 0 ( n ) ( r )
a j ( out ) ( r ) = a j ( n ) ( r )
P j , p , 0 ( n ) = 2 π r = 0 r a j , p , 0 ( n ) ( r ) a j , p , 0 ( n ) ( r ) * dr
P j ( out ) = 2 π r = 0 r a j ( out ) ( r ) a j ( out ) ( r ) * dr
η P SH P F = 2 ω 1 2 d eff 2 π n 1 2 n 3 ε 0 c 0 3 P 1 l c k 1 h ( σ , B , ξ )
u j , 0 , 0 G L ( r ) = E j , 0 , 0 w 0 , j w ( z i ) j e r 2 w ( z i ) j 2
× exp [ i k j r 2 2 R ( z i ) j + i k j z i i ω j t + i arctan ( z i z 0 , j ) ]

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