Abstract

The dependence of the optical second-harmonic generation (SHG) in a negatively birefringent crystal on the crystal double-refraction angle, the beam divergence, and the diffraction of the focused high-order transverse mode is discussed. Within constant-pump approximation, the SHG efficiency for the TEM11 mode is only one seventh that for the fundamental TEM00 mode and does not increase with the crystal length. Parametric wave mixing of lower-and upper-sideband waves associated with frequency doubling can give a rather high efficiency for the SHG of a broadband laser. As calculated on an example of LiNbO3 crystal, the spectrum linewidth decreases and the acceptance angle for phase matching increases as the fundamental mode number increases.

© 1987 Optical Society of America

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References

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  1. D. A. Kleinman, Phy. Rev. 128, 1761 (1962).
    [Crossref]
  2. G. D. Boyd, A. Ashkin, J. M. Dziedzic, and D. A. Kleinman, Phys. Rev. 137, A1305 (1965).
    [Crossref]
  3. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).
    [Crossref]
  4. J. E. Bjorkholm, Phys. Rev. 142, 126 (1986).
    [Crossref]
  5. R. Asby, Phys. Rev. 187, 1062 (1969).
    [Crossref]
  6. H. S. Kwok and P. H. Chiu, Opt. Lett. 10, 28 (1985).
    [Crossref] [PubMed]
  7. D. A. Kleinman, A. Ashkin, and G. D. Boyd, Phys. Rev. 145, 338 (1966).
    [Crossref]
  8. A. Yariv, in Optical Electronics, 2nd ed. (Wiley, New York, 1985), Chap. 2.
  9. A. Yariv, in Optical Electronics, 2nd ed. (Wiley, New York, 1985), Chap. 5.
  10. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), p. 89.
  11. F. Zernike and J. E. Midwinter, Applied Nonlinear Optics (Wiley, New York, 1973), Chap. 4.
  12. M. V. Mobden and J. Warnec, Phys. Lett. 22, 243 (1966).
    [Crossref]

1986 (1)

J. E. Bjorkholm, Phys. Rev. 142, 126 (1986).
[Crossref]

1985 (1)

1969 (1)

R. Asby, Phys. Rev. 187, 1062 (1969).
[Crossref]

1966 (2)

D. A. Kleinman, A. Ashkin, and G. D. Boyd, Phys. Rev. 145, 338 (1966).
[Crossref]

M. V. Mobden and J. Warnec, Phys. Lett. 22, 243 (1966).
[Crossref]

1965 (1)

G. D. Boyd, A. Ashkin, J. M. Dziedzic, and D. A. Kleinman, Phys. Rev. 137, A1305 (1965).
[Crossref]

1962 (2)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).
[Crossref]

D. A. Kleinman, Phy. Rev. 128, 1761 (1962).
[Crossref]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).
[Crossref]

Asby, R.

R. Asby, Phys. Rev. 187, 1062 (1969).
[Crossref]

Ashkin, A.

D. A. Kleinman, A. Ashkin, and G. D. Boyd, Phys. Rev. 145, 338 (1966).
[Crossref]

G. D. Boyd, A. Ashkin, J. M. Dziedzic, and D. A. Kleinman, Phys. Rev. 137, A1305 (1965).
[Crossref]

Bjorkholm, J. E.

J. E. Bjorkholm, Phys. Rev. 142, 126 (1986).
[Crossref]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).
[Crossref]

Boyd, G. D.

D. A. Kleinman, A. Ashkin, and G. D. Boyd, Phys. Rev. 145, 338 (1966).
[Crossref]

G. D. Boyd, A. Ashkin, J. M. Dziedzic, and D. A. Kleinman, Phys. Rev. 137, A1305 (1965).
[Crossref]

Chiu, P. H.

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).
[Crossref]

Dziedzic, J. M.

G. D. Boyd, A. Ashkin, J. M. Dziedzic, and D. A. Kleinman, Phys. Rev. 137, A1305 (1965).
[Crossref]

Kleinman, D. A.

D. A. Kleinman, A. Ashkin, and G. D. Boyd, Phys. Rev. 145, 338 (1966).
[Crossref]

G. D. Boyd, A. Ashkin, J. M. Dziedzic, and D. A. Kleinman, Phys. Rev. 137, A1305 (1965).
[Crossref]

D. A. Kleinman, Phy. Rev. 128, 1761 (1962).
[Crossref]

Kwok, H. S.

Midwinter, J. E.

F. Zernike and J. E. Midwinter, Applied Nonlinear Optics (Wiley, New York, 1973), Chap. 4.

Mobden, M. V.

M. V. Mobden and J. Warnec, Phys. Lett. 22, 243 (1966).
[Crossref]

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).
[Crossref]

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), p. 89.

Warnec, J.

M. V. Mobden and J. Warnec, Phys. Lett. 22, 243 (1966).
[Crossref]

Yariv, A.

A. Yariv, in Optical Electronics, 2nd ed. (Wiley, New York, 1985), Chap. 2.

A. Yariv, in Optical Electronics, 2nd ed. (Wiley, New York, 1985), Chap. 5.

Zernike, F.

F. Zernike and J. E. Midwinter, Applied Nonlinear Optics (Wiley, New York, 1973), Chap. 4.

Opt. Lett. (1)

Phy. Rev. (1)

D. A. Kleinman, Phy. Rev. 128, 1761 (1962).
[Crossref]

Phys. Lett. (1)

M. V. Mobden and J. Warnec, Phys. Lett. 22, 243 (1966).
[Crossref]

Phys. Rev. (5)

G. D. Boyd, A. Ashkin, J. M. Dziedzic, and D. A. Kleinman, Phys. Rev. 137, A1305 (1965).
[Crossref]

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).
[Crossref]

J. E. Bjorkholm, Phys. Rev. 142, 126 (1986).
[Crossref]

R. Asby, Phys. Rev. 187, 1062 (1969).
[Crossref]

D. A. Kleinman, A. Ashkin, and G. D. Boyd, Phys. Rev. 145, 338 (1966).
[Crossref]

Other (4)

A. Yariv, in Optical Electronics, 2nd ed. (Wiley, New York, 1985), Chap. 2.

A. Yariv, in Optical Electronics, 2nd ed. (Wiley, New York, 1985), Chap. 5.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), p. 89.

F. Zernike and J. E. Midwinter, Applied Nonlinear Optics (Wiley, New York, 1973), Chap. 4.

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Figures (8)

Fig. 1
Fig. 1

Focused high-order Gaussian beam and phase-matching angle.

Fig. 2
Fig. 2

The lot of normalized P2ω versus ω0 at various double-refraction angle ρ values (radians).

Fig. 3
Fig. 3

The plot of normalized P2ω (Ref. 11) versus ω0 at various ρ values.

Fig. 4
Fig. 4

Ratio for the SHG conversion efficiency of multimodes to single mode by a KDP crystal: L, Lorentzian line shape; G, Gaussian line shape; and Δν, the bandwidth.

Fig. 5
Fig. 5

Plot of R versus Δν with LiNbO3 crystal.

Fig. 6
Fig. 6

Plot of acceptance angle ΔθA for single mode and multimode with Δλ = 1.5A and Δλ = 9A.

Fig. 7
Fig. 7

Bandwidth narrowing of the output pulses for Δν = 40 GHz.

Fig. 8
Fig. 8

Bandwidth narrowing of the SHG pulse for Δν = 80 GHz.

Equations (33)

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E ( x , y , L / 2 ) = y ^ E ω exp [ - ( x 2 + y 2 ) / ω 0 2 ] ,
E ( x , y , z ) = - E k e i k ^ · r ^ d 3 k = y ^ E ω e i k 1 z × exp { - [ ( x 2 + y 2 ) / ω 0 2 ( 1 + ξ 2 ) ] ( i - i ξ ) } 1 + i ξ ,
E 1 U ^ exp [ ( i 2 ω n 2 s ^ · r ^ / c ) ] ,
n σ ^ - n 2 s ^ ψ k N ^ ,
P z 2 ω ( r ) = χ : EE = χ E ω 2 exp ( 2 i k 2 z ) ( 1 + i ξ ) - 2 × exp [ - 2 x 2 + y 2 ω 0 2 ( 1 + i ξ ) ] ,
P k = ( 2 π ) - 3 P 2 ω ( r ) exp ( - i k r ) d 3 r .
P z ( k ) = ( χ E ω 2 / 16 π ) k 1 ω 0 4 exp [ - ( 2 k 1 - k z ) k 1 ω 0 2 / 2 ]
E k 2 ω ( r ) = z g ( 2 i ψ k ) γ k · P ( k ) exp [ i ( K · r - 2 ω t ) ] d 3 k ,
g ( x ) = 0 1 e - p x d p , 2 ψ k = 2 ω c ϕ k z = ( k z - 2 k 1 + ρ k x + k T 2 / 4 k 1 ) z , and γ k = 4 π i ω / n 2 c [ N ^ · σ ^ - ( N ^ · u ^ ) ( σ ^ · u ^ ) ] - 1 u ^ u ^ is a dyadic .
γ k · P ( k ) = 4 π i ω n 2 c u ^ P z ( k ) × sin ( Θ m + ρ + Δ Θ ) cos 2 ρ cos Δ Θ - ½ sin 2 2 ρ sin Δ Θ ,
E 2 ω ( r ) = γ k × P ( r ) G ( r , r ) d r ,
G ( r , r ) = z ( 2 π ) - 3 g ( 2 i ψ k ) exp [ i K · ( r - r ) ] d 3 k = z ( 2 π ) - 3 0 1 d p d 3 k exp [ - i p z ( k z - 2 k 1 + ρ k x ) ] × exp { - i p z ( k x 2 + k y 2 ) / 4 k 1 exp [ i K · ( r - r ) ] } .
G ( r , r ) = exp [ 2 i k 1 ( z - z ) ( 2 π ) - 2 d k x d k y × exp [ i k x ( x - x ) + i k y ( y - y ) - i ρ k x ( z - z ) ] × exp { - i [ ( k x 2 + k y 2 ) / 4 k 1 ] ( z - z ) } ,
G ( r , r ) = ( ω n 1 0 π i c ) 1 Z exp [ 2 i k 1 Z + i k 1 ( X 2 + Y 2 ) / Z ] ,
E 10 ω = E ω ( 1 + i ξ ) 2 2 2 x ω 0 exp [ - ( x 2 + y 2 ) / ω 0 2 ( 1 + i ξ ) ] e i k 1 z ,
P ( r ) = χ E ω 2 ( 1 + i ξ ) 4 8 x 2 ω 0 2 exp [ - 2 ( x 2 + y 2 ) / ω 0 2 ( 1 + i ξ ) ] exp ( 2 i k 1 z ) .
E 10 2 ω ( r ) = 4 π i ω n 1 0 c sin ( θ m + ρ ) cos 2 ρ χ E ω 2 ( 1 + i ξ ) 4 8 x 2 ω 0 2 × exp [ - 2 ( x 2 + y 2 ) / ω 0 2 ( 1 + i ξ ) ] × e 2 i k 1 z ( ω n 1 0 π i c ) 1 Z × exp [ 2 i k 1 Z + i k 1 ( X 2 + Y 2 ) / Z ] d 3 r .
E 10 2 ω ( r ) = 0 1 4 π i ω n 1 0 c sin ( θ m + ρ ) cos 2 ρ 8 χ E ω 2 ω 0 2 exp [ 2 i k 1 z ( ω n 1 0 π i c ) ] × exp { - 2 [ ( x - ρ z p ) 2 + y 2 ] / ω 0 2 ( 1 + i ξ ) } [ 1 + i 2 z b ( 1 - p ) - i L b ] 4 1 p × 1 { 2 ω 0 2 [ 1 + i 2 z b ( 1 - p ) - i L b ] - i k 1 z p } 2 × [ 1 2 - k 1 2 ( x - ρ z p ) 2 z 2 p 2 ( 2 ω 0 2 { 1 + i 2 z b [ ( 1 - p ) - i L b ] } - i k 1 z p ) ] d p .
P 2 = 96 π 2 ω 2 ( n 1 0 ) 3 c 3 P ω 2 sin 2 ( θ m + ρ ) cos 2 ρ χ 2 L 2 ω 0 2 ,
P 2 = 96 π 5 / 2 ω 2 ( n 1 0 ) 3 c 3 P ω 2 sin 2 ( θ m + π ) cos 2 ρ χ 2 L ω 0 ρ .
E 11 ω ( r ) = E ω ( 1 + i ξ ) 3 8 x y ω 0 2 exp [ - ( x 2 + y 2 ) / ω 0 2 ( 1 + i ξ ) ] × exp ( i k 1 z ) .
E 2 ω 11 ( r ) = 4 π i ω n 1 0 c sin ( θ m + ρ ) cos 2 ρ χ E ω 2 ( 1 + i ξ ) 6 64 x 2 y 2 ω 0 4 × exp [ - 2 ( x 2 + y 2 ) / ω 0 2 ( 1 + i ξ ) ] × exp ( 2 i k 1 z ) ( ω n 1 0 π i c ) 1 Z × exp [ 2 i k 1 Z + i k 1 ( X 2 + Y 2 ) / Z ] d x d y d z = 0 1 4 π i ω n 1 0 c sin ( θ m + ρ ) cos 2 ρ 64 χ E ω 2 ω o 4 ( ω n 1 0 π i c ) × exp { - 2 [ ( x - ρ z p ) 2 + y 2 ] } [ 1 + i 2 z b ( 1 - p ) - i L b ] 1 p × 1 { 2 ω 0 2 [ 1 + i 2 z b ( 1 - p ) - i L b ] - i k 1 z p } 3 × ( 1 2 - k 1 2 ( x - ρ z p ) 2 z 2 p 2 { 2 ω 0 2 [ 1 + i 2 z b ( 1 - p ) - i L b ] - i k 1 z p } ) × { 1 2 - k 1 2 y 2 2 ω 0 2 [ 1 + i 2 z b ( 1 - p ) - i L b ] - i k 1 z P } d p .
P 2 = 18 π 2 ω 2 ( n 1 0 ) 3 c 3 P ω 2 sin 2 ( θ m + ρ ) cos 2 ρ χ 2 L 2 ω 0 2 .
P 2 ω 11 = 18 π 5 / 2 ω 2 ( n 1 0 ) 3 c 3 ( P ω 11 ) 2 sin 2 ( θ m + ρ ) cos 2 ρ χ 2 L ω 0 ρ .
E 2 ω 11 ( r ) = 0 1 4 π i ω n 1 0 c sin ( θ m + ρ ) cos 2 ρ 64 χ E ω 2 ω 0 4 ( ω n 1 0 π i c ) × exp [ - 2 ( ρ z p ) 2 ρ z 10 p 6 y 2 d p ] 2 L ω 0 9 k 1 5 .
E n + m = - i μ 0 0 ω n + m 2 n ( ω n + m ) χ ( 2 ) E n E m exp ( i Δ k n m L - 1 ) i Δ k n m × exp [ i ( ω n + m t - k n + m L ) ] ,
g ( ν n ) = exp [ - 2 ( ν n - ν 0 2 Δ ν ) 2 ]
g ( ν n ) = ( Δ ν ) 2 Δ ν 2 + ( ν n - ν 0 ) 2 ,
η 2 ω = 2 ( μ 0 0 ) 3 / 2 χ ( 2 ) 2 L 2 I o | n , m = - N / 2 N / 2 ( ω n + m 2 ) 2 × 1 n ( ω n + m ) n ( ω n ) n ( ω m ) sinc 2 ( Δ k n m L 2 ) × g 2 ( ν n ) g 2 ( ν m ) | / m , n = N / 2 N / 2 g ( ν m ) g ( ν n ) .
R ω 2 n 0 3 | n = - N / 2 N / 2 ( ω n + m 2 ) 2 1 n ( ω n + m ) n ( ω n ) n ( ω m ) × g ( ν n ) g ( ν m ) sinc 2 | ( Δ k n m L 2 ) × / | m , n = N / 2 N / 2 g ( ν n ) | 2
Δ θ A = 2 π k ω L [ n e 2 ( 2 ω ) n o 2 ( 2 ω ) - n e 2 ( 2 ω ) ] 1 sin 2 θ m .
n o 2 = a 1 + b 1 + c 1 T 2 λ 2 - ( d 1 + e 1 T 2 ) 2 - f 1 λ 2 ,
n e 2 = a 2 + b 2 T 2 + c 2 + d 2 T 2 λ 2 - ( e 2 + f 2 T 2 ) 2 - g 2 λ 2 ,

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