Abstract

By using Gaussian–Hermite-mode expansion, the Raman nonlinear wave equations for the second-order Stokes field generated in the processes of cascade Raman scattering and four-wave mixing are solved for the case of a pump with a focused Gaussian beam. The numerical analysis is made under tight and soft focusing cases, and the calculated results are compared with our recent experiment.

© 1988 Optical Society of America

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References

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  1. X. Cheng, Q. Lou, R. Wang, Z. Wang, “Efficient XeCl/H2Raman shifting to a blue-green region,” Appl. Phys. Lett. 51, 76–77 (1987).
    [CrossRef]
  2. B. N. Perry, P. Rabinowitz, D. S. Bomse, “Stimulated Raman scattering with a tightly focused pump beam,” Opt. Lett. 10, 146–148 (1985).
    [CrossRef] [PubMed]
  3. D. W. Trainor, H. A. Hyman, R. M. Heinrichs, “Stimulated Raman scattering of XeF* laser radiation in H2,” IEEE J. Quantum Electron. QE-18, 1929–1934 (1982).
    [CrossRef]
  4. X. Cheng, R. Wang, Q. Lou, Z. Wang, “Investigation of the Raman wavefront of higher order Gaussian–Hermite mode,” Opt. Commun. 64, 67–71 (1987).
    [CrossRef]
  5. H. Kogelnik, “On the propagation of Gaussian beams of light through lens-like media including those with a loss or gain variation,” Appl. Opt. 4, 1562–1569 (1965).
    [CrossRef]
  6. B. N. Perry, P. Rabinowitz, M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
    [CrossRef]
  7. O. L. Bourne, A. J. Alcock, “A high-power, narrow linewidth XeCl* oscillator,” Appl. Phys. Lett. 42, 777 (1983).
    [CrossRef]
  8. X. Cheng, R. Wang, “Gaussian–Hermite coupling method on solving stimulated Raman scattering wave equation,” Acta Opt. Sin. 7, 443 (1987).

1987 (3)

X. Cheng, Q. Lou, R. Wang, Z. Wang, “Efficient XeCl/H2Raman shifting to a blue-green region,” Appl. Phys. Lett. 51, 76–77 (1987).
[CrossRef]

X. Cheng, R. Wang, Q. Lou, Z. Wang, “Investigation of the Raman wavefront of higher order Gaussian–Hermite mode,” Opt. Commun. 64, 67–71 (1987).
[CrossRef]

X. Cheng, R. Wang, “Gaussian–Hermite coupling method on solving stimulated Raman scattering wave equation,” Acta Opt. Sin. 7, 443 (1987).

1985 (1)

1983 (2)

B. N. Perry, P. Rabinowitz, M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
[CrossRef]

O. L. Bourne, A. J. Alcock, “A high-power, narrow linewidth XeCl* oscillator,” Appl. Phys. Lett. 42, 777 (1983).
[CrossRef]

1982 (1)

D. W. Trainor, H. A. Hyman, R. M. Heinrichs, “Stimulated Raman scattering of XeF* laser radiation in H2,” IEEE J. Quantum Electron. QE-18, 1929–1934 (1982).
[CrossRef]

1965 (1)

Alcock, A. J.

O. L. Bourne, A. J. Alcock, “A high-power, narrow linewidth XeCl* oscillator,” Appl. Phys. Lett. 42, 777 (1983).
[CrossRef]

Bomse, D. S.

Bourne, O. L.

O. L. Bourne, A. J. Alcock, “A high-power, narrow linewidth XeCl* oscillator,” Appl. Phys. Lett. 42, 777 (1983).
[CrossRef]

Cheng, X.

X. Cheng, Q. Lou, R. Wang, Z. Wang, “Efficient XeCl/H2Raman shifting to a blue-green region,” Appl. Phys. Lett. 51, 76–77 (1987).
[CrossRef]

X. Cheng, R. Wang, Q. Lou, Z. Wang, “Investigation of the Raman wavefront of higher order Gaussian–Hermite mode,” Opt. Commun. 64, 67–71 (1987).
[CrossRef]

X. Cheng, R. Wang, “Gaussian–Hermite coupling method on solving stimulated Raman scattering wave equation,” Acta Opt. Sin. 7, 443 (1987).

Heinrichs, R. M.

D. W. Trainor, H. A. Hyman, R. M. Heinrichs, “Stimulated Raman scattering of XeF* laser radiation in H2,” IEEE J. Quantum Electron. QE-18, 1929–1934 (1982).
[CrossRef]

Hyman, H. A.

D. W. Trainor, H. A. Hyman, R. M. Heinrichs, “Stimulated Raman scattering of XeF* laser radiation in H2,” IEEE J. Quantum Electron. QE-18, 1929–1934 (1982).
[CrossRef]

Kogelnik, H.

Lou, Q.

X. Cheng, Q. Lou, R. Wang, Z. Wang, “Efficient XeCl/H2Raman shifting to a blue-green region,” Appl. Phys. Lett. 51, 76–77 (1987).
[CrossRef]

X. Cheng, R. Wang, Q. Lou, Z. Wang, “Investigation of the Raman wavefront of higher order Gaussian–Hermite mode,” Opt. Commun. 64, 67–71 (1987).
[CrossRef]

Newstein, M.

B. N. Perry, P. Rabinowitz, M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
[CrossRef]

Perry, B. N.

B. N. Perry, P. Rabinowitz, D. S. Bomse, “Stimulated Raman scattering with a tightly focused pump beam,” Opt. Lett. 10, 146–148 (1985).
[CrossRef] [PubMed]

B. N. Perry, P. Rabinowitz, M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
[CrossRef]

Rabinowitz, P.

B. N. Perry, P. Rabinowitz, D. S. Bomse, “Stimulated Raman scattering with a tightly focused pump beam,” Opt. Lett. 10, 146–148 (1985).
[CrossRef] [PubMed]

B. N. Perry, P. Rabinowitz, M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
[CrossRef]

Trainor, D. W.

D. W. Trainor, H. A. Hyman, R. M. Heinrichs, “Stimulated Raman scattering of XeF* laser radiation in H2,” IEEE J. Quantum Electron. QE-18, 1929–1934 (1982).
[CrossRef]

Wang, R.

X. Cheng, Q. Lou, R. Wang, Z. Wang, “Efficient XeCl/H2Raman shifting to a blue-green region,” Appl. Phys. Lett. 51, 76–77 (1987).
[CrossRef]

X. Cheng, R. Wang, Q. Lou, Z. Wang, “Investigation of the Raman wavefront of higher order Gaussian–Hermite mode,” Opt. Commun. 64, 67–71 (1987).
[CrossRef]

X. Cheng, R. Wang, “Gaussian–Hermite coupling method on solving stimulated Raman scattering wave equation,” Acta Opt. Sin. 7, 443 (1987).

Wang, Z.

X. Cheng, R. Wang, Q. Lou, Z. Wang, “Investigation of the Raman wavefront of higher order Gaussian–Hermite mode,” Opt. Commun. 64, 67–71 (1987).
[CrossRef]

X. Cheng, Q. Lou, R. Wang, Z. Wang, “Efficient XeCl/H2Raman shifting to a blue-green region,” Appl. Phys. Lett. 51, 76–77 (1987).
[CrossRef]

Acta Opt. Sin. (1)

X. Cheng, R. Wang, “Gaussian–Hermite coupling method on solving stimulated Raman scattering wave equation,” Acta Opt. Sin. 7, 443 (1987).

Appl. Opt. (1)

Appl. Phys. Lett. (2)

O. L. Bourne, A. J. Alcock, “A high-power, narrow linewidth XeCl* oscillator,” Appl. Phys. Lett. 42, 777 (1983).
[CrossRef]

X. Cheng, Q. Lou, R. Wang, Z. Wang, “Efficient XeCl/H2Raman shifting to a blue-green region,” Appl. Phys. Lett. 51, 76–77 (1987).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. W. Trainor, H. A. Hyman, R. M. Heinrichs, “Stimulated Raman scattering of XeF* laser radiation in H2,” IEEE J. Quantum Electron. QE-18, 1929–1934 (1982).
[CrossRef]

Opt. Commun. (1)

X. Cheng, R. Wang, Q. Lou, Z. Wang, “Investigation of the Raman wavefront of higher order Gaussian–Hermite mode,” Opt. Commun. 64, 67–71 (1987).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

B. N. Perry, P. Rabinowitz, M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Normalized intensity distribution of the second-order Stokes field E2s in the four-wave mixing process: solid curve, confocal parameter f = 10L and Gp = 15; dashed curve, f = L/5 and Gp = 20; dotted curve, f = L/50 and Gp = 25.

Fig. 2
Fig. 2

Normalized intensity distribution of the second-order Stokes field E2s initially generated by spontaneous emission: solid curve, f = 10L and Gp = 15; dashed curve, f = L/10 and Gp = 20. The upper solid curve is the transverse distribution of the phase disturbance of the E2s field for the case shown by the dashed curve.

Fig. 3
Fig. 3

Normalized intensity distribution of the second-order Stokes beam E2s, which is initially injected in the form of the zeroth-order Gaussian beam ψ0 with f = L/10 and Gp = 10. The upper curve is the transverse distribution of the phase disturbance.

Fig. 4
Fig. 4

Intensity distribution of the second-order Stokes field E2s normalized to a total field intensity with f = L and Gp = 15: solid curve, total intensity of the second-order Stokes field; dashed curve, intensity due to the cascade Raman process; dotted curve, intensity due to the four-wave mixing process.

Fig. 5
Fig. 5

Normalized observed intensity distribution of Stokes beams S2 and S3: a, S2 and b, S3 pumped by soft focusing beam (18 MW); c, S2 and d, S3 pumped by tight focusing beam (10 MW).

Equations (29)

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[ ( 2 χ 2 + 2 y 2 ) - 2 i k p z ] E p = - i χ E p E 1 s 2 ,
[ ( 2 χ 2 + 2 y 2 ) - 2 i k 1 s * z ] E 1 s = - i χ E p 2 E 1 s + i χ E p E 1 s * E 2 s × exp ( - i Δ k 01 z ) ,
[ ( 2 χ 2 + 2 y 2 ) - 2 i k 2 s z ] E 2 s = - i χ E 1 s 2 E 2 s - i χ E p * E 1 s 2 × exp ( - i Δ k 01 z ) .
Δ k 01 = k p + k 2 s - 2 k 1 s ,
E 1 s ( x , z ) = ( 1 / w ) ( 2 / π ) 1 / 2 × exp { - ( x 2 / w 2 ) - i [ ( k 1 s x 2 ) / ( 2 R ) + ( β x 2 ) / w 2 ] } × exp { λ [ arctan ( z / f ) - arctan ( z 0 / f ) ] } ,
w 4 ( 0 ) / w p 4 ( 0 ) = { [ 1 + 4 G p 2 w 1 s 4 ( 0 ) / w p 4 ( 0 ) ] 1 / 2 - 1 } / ( 2 G p 2 ) ,
G p = ( χ f ) / [ π w p 2 ( 0 ) k p ] ,
β = G p w 4 ( 0 ) / [ w 1 s 2 ( 0 ) w p 2 ( 0 ) ] ,
Im λ = w 1 s 2 ( 0 ) / w 2 ( 0 ) ,
Re λ = [ 1 - w 2 ( 0 ) / w p 2 ( 0 ) ] G p .
E 2 s ( x , z ) = m = 0 S m ( z ) ψ m ( x , z ) .
d d z S m ( z ) = m g m m ( z ) S m ( z ) + q m ( z ) .
z / f = tan u , S m ( u ) = exp ( - i m u ) f m ( u ) .
d d u f m = m A m m f m ( u ) + h m .
h m = { 0 m = odd [ ( μ - 1 ) m 2 m + 1 m ! ] 1 / 2 m ! ( m / 2 ) ! μ G p m = even ,
μ = 2 / ( w 2 s 2 { ( 2 / w 2 + 1 / w p 2 + 1 / w 2 s 2 ) + i [ 2 β / w 2 - Δ k 01 / ( 2 R ) ] } ) ,
A m m = i m δ m m + exp [ 2 Re λ ( u - u 0 ) ] A m m 0 .
A m m 0 = ( λ 2 s / λ p ) θ w 2 ( 0 ) / w p 2 ( 0 ) I m m .
θ = 1 / [ 1 + w 2 s 2 ( 0 ) / w 2 ( 0 ) ] ,
I m m = 1 ( 2 m + m m ! m ! ) 1 / 2 - + H m [ ( θ ) 1 / 2 τ ] H m [ ( θ ) 1 / 2 τ ] e - τ 2 d τ .
d d u f = A · f + h .
f ( u ) = F ( u ) · F - 1 ( u 0 ) · f ( u 0 ) + F ( u ) · u 0 u F - 1 ( s ) · h ( s ) d s .
d d u f = A · f .
f m ( u 0 ) = 1 ,             m = 0 , 1 , 2 , .
f m ( u 0 ) = δ 0 m ,             m = 0 , 1 , 2 , .
f j ( u j + 1 ) = T j · exp [ ( u j - u j - 1 ) D j ] · ( T j ) - 1 · f j - 1 ( u j - 1 ) + u j - 1 u j T j · exp [ ( u j - u ) D j ] · ( T j ) - 1 · h ( u ) d u ,
D j = ( T j ) - 1 · A j · T j = diag [ λ 0 j , λ 1 j , λ 2 j , ] ,
S ( z ) z = L / 2 = exp [ - i δ arctan ( L / 2 f ) ] · f [ arctan ( L / 2 f ) ] ,
- arctan ( L / 2 f ) u 1 u n arctan ( L / 2 f ) ,

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