Abstract

We present a theoretical analysis of beam propagation in thick nonlinear media by using the Gaussian decomposition method and considering a thick medium as a stack of thin media. Simple analytic solutions of Z-scan characteristics and optical limiting with thick nonlinear media are obtained. Comparisons of these results with those obtained by use of a distributed-lens model and Gaussian–Laguerre mode decomposition are made. Good agreement is obtained with a distributed-lens model.

© 1995 Optical Society of America

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Corrections

Jian-Guo Tian, Wei-Ping Zang, Cun-Zhou Zhang, and Guangyin Zhang, "Analysis of beam propagation in thick nonlinear media: errata," Appl. Opt. 35, 5331-5331 (1996)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-35-27-5331

References

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  1. M. J. Weber, D. Milam, W. L. Smith, “Nonlinear refractive index of glasses and crystal,” Opt. Eng. 17, 463–469 (1978).
  2. S. R. Friberg, P. W. Smith, “Nonlinear optical glasses for ultrafast optical switches,” IEEE J. Quantum Electron. QE-23, 2089–2094 (1987).
    [CrossRef]
  3. R. Adair, L. L. Chase, S. A. Payne, “Nonlinear refractive-index measurements of glasses using three-wave frequency mixing,” J. Opt. Soc. Am. B 4, 875–881 (1987).
    [CrossRef]
  4. A. Owyoung, “Ellipse rotation studies in laser host materials,” IEEE J. Quantum Electron. QE-9, 1064–1069 (1973).
    [CrossRef]
  5. W. E. Williams, M. J. Soileau, E. W. Van Stryland, “Optical switching and n2 measurement in CS2,” Opt. Commun. 50, 256–260 (1984).
    [CrossRef]
  6. M. Sheik-bahae, A. A. Said, E. W. Van Stryland, “High-sensitivity, single-beam n2 measurements,” Opt. Lett. 14, 955–957 (1989).
    [CrossRef] [PubMed]
  7. J.-G. Tian, C. Zhang, G. Zhang, J. Li, “Position dispersion and optical limiting resulting from induced nonlinearities in Chinese tea liquids,” Appl. Opt. 32, 6628–6632 (1993).
    [CrossRef] [PubMed]
  8. A. A. Said, M. Sheik-bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9, 405–414 (1992).
    [CrossRef]
  9. Q. W. Song, C. Zhang, R. W. Gross, R. Birge, “Optical limiting by chemically enhanced bacteriorhodopsin films,” Opt. Lett. 18, 775–778 (1993).
    [CrossRef] [PubMed]
  10. Q. W. Song, C. Zhang, R. W. Gross, R. Birge, “The intensity-dependent refractive index of chemically enhanced bacteriorhodopsin,” Opt. Commun. (to be published).
  11. M. Sheik-bahae, A. A. Said, T. H. Wei, D. J. Hagan, E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
    [CrossRef]
  12. P. P. Banerjee, R. M. Misra, M. Maghraoui, “Theoretical and experimental studies of propagation of beams through a finite sample of a cubically nonlinear material,” J. Opt. Soc. Am. B 8, 1072–1080 (1991).
    [CrossRef]
  13. M. Sheik-bahae, A. A. Said, D. J. Hagan, E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
    [CrossRef]
  14. J. A. Hermann, R. G. McDuff, “Analysis of spatial scanning with thick optically nonlinear media,” J. Opt. Soc. Am. B 10, 2056–2064 (1993).
    [CrossRef]

1993 (3)

1992 (1)

1991 (2)

P. P. Banerjee, R. M. Misra, M. Maghraoui, “Theoretical and experimental studies of propagation of beams through a finite sample of a cubically nonlinear material,” J. Opt. Soc. Am. B 8, 1072–1080 (1991).
[CrossRef]

M. Sheik-bahae, A. A. Said, D. J. Hagan, E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

1990 (1)

M. Sheik-bahae, A. A. Said, T. H. Wei, D. J. Hagan, E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

1989 (1)

1987 (2)

S. R. Friberg, P. W. Smith, “Nonlinear optical glasses for ultrafast optical switches,” IEEE J. Quantum Electron. QE-23, 2089–2094 (1987).
[CrossRef]

R. Adair, L. L. Chase, S. A. Payne, “Nonlinear refractive-index measurements of glasses using three-wave frequency mixing,” J. Opt. Soc. Am. B 4, 875–881 (1987).
[CrossRef]

1984 (1)

W. E. Williams, M. J. Soileau, E. W. Van Stryland, “Optical switching and n2 measurement in CS2,” Opt. Commun. 50, 256–260 (1984).
[CrossRef]

1978 (1)

M. J. Weber, D. Milam, W. L. Smith, “Nonlinear refractive index of glasses and crystal,” Opt. Eng. 17, 463–469 (1978).

1973 (1)

A. Owyoung, “Ellipse rotation studies in laser host materials,” IEEE J. Quantum Electron. QE-9, 1064–1069 (1973).
[CrossRef]

Adair, R.

Banerjee, P. P.

Birge, R.

Q. W. Song, C. Zhang, R. W. Gross, R. Birge, “Optical limiting by chemically enhanced bacteriorhodopsin films,” Opt. Lett. 18, 775–778 (1993).
[CrossRef] [PubMed]

Q. W. Song, C. Zhang, R. W. Gross, R. Birge, “The intensity-dependent refractive index of chemically enhanced bacteriorhodopsin,” Opt. Commun. (to be published).

Chase, L. L.

Friberg, S. R.

S. R. Friberg, P. W. Smith, “Nonlinear optical glasses for ultrafast optical switches,” IEEE J. Quantum Electron. QE-23, 2089–2094 (1987).
[CrossRef]

Gross, R. W.

Q. W. Song, C. Zhang, R. W. Gross, R. Birge, “Optical limiting by chemically enhanced bacteriorhodopsin films,” Opt. Lett. 18, 775–778 (1993).
[CrossRef] [PubMed]

Q. W. Song, C. Zhang, R. W. Gross, R. Birge, “The intensity-dependent refractive index of chemically enhanced bacteriorhodopsin,” Opt. Commun. (to be published).

Hagan, D. J.

A. A. Said, M. Sheik-bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9, 405–414 (1992).
[CrossRef]

M. Sheik-bahae, A. A. Said, D. J. Hagan, E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

M. Sheik-bahae, A. A. Said, T. H. Wei, D. J. Hagan, E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

Hermann, J. A.

Li, J.

Maghraoui, M.

McDuff, R. G.

Milam, D.

M. J. Weber, D. Milam, W. L. Smith, “Nonlinear refractive index of glasses and crystal,” Opt. Eng. 17, 463–469 (1978).

Misra, R. M.

Owyoung, A.

A. Owyoung, “Ellipse rotation studies in laser host materials,” IEEE J. Quantum Electron. QE-9, 1064–1069 (1973).
[CrossRef]

Payne, S. A.

Said, A. A.

A. A. Said, M. Sheik-bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9, 405–414 (1992).
[CrossRef]

M. Sheik-bahae, A. A. Said, D. J. Hagan, E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

M. Sheik-bahae, A. A. Said, T. H. Wei, D. J. Hagan, E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

M. Sheik-bahae, A. A. Said, E. W. Van Stryland, “High-sensitivity, single-beam n2 measurements,” Opt. Lett. 14, 955–957 (1989).
[CrossRef] [PubMed]

Sheik-bahae, M.

A. A. Said, M. Sheik-bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9, 405–414 (1992).
[CrossRef]

M. Sheik-bahae, A. A. Said, D. J. Hagan, E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

M. Sheik-bahae, A. A. Said, T. H. Wei, D. J. Hagan, E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

M. Sheik-bahae, A. A. Said, E. W. Van Stryland, “High-sensitivity, single-beam n2 measurements,” Opt. Lett. 14, 955–957 (1989).
[CrossRef] [PubMed]

Smith, P. W.

S. R. Friberg, P. W. Smith, “Nonlinear optical glasses for ultrafast optical switches,” IEEE J. Quantum Electron. QE-23, 2089–2094 (1987).
[CrossRef]

Smith, W. L.

M. J. Weber, D. Milam, W. L. Smith, “Nonlinear refractive index of glasses and crystal,” Opt. Eng. 17, 463–469 (1978).

Soileau, M. J.

W. E. Williams, M. J. Soileau, E. W. Van Stryland, “Optical switching and n2 measurement in CS2,” Opt. Commun. 50, 256–260 (1984).
[CrossRef]

Song, Q. W.

Q. W. Song, C. Zhang, R. W. Gross, R. Birge, “Optical limiting by chemically enhanced bacteriorhodopsin films,” Opt. Lett. 18, 775–778 (1993).
[CrossRef] [PubMed]

Q. W. Song, C. Zhang, R. W. Gross, R. Birge, “The intensity-dependent refractive index of chemically enhanced bacteriorhodopsin,” Opt. Commun. (to be published).

Tian, J.-G.

Van Stryland, E. W.

A. A. Said, M. Sheik-bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9, 405–414 (1992).
[CrossRef]

M. Sheik-bahae, A. A. Said, D. J. Hagan, E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

M. Sheik-bahae, A. A. Said, T. H. Wei, D. J. Hagan, E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

M. Sheik-bahae, A. A. Said, E. W. Van Stryland, “High-sensitivity, single-beam n2 measurements,” Opt. Lett. 14, 955–957 (1989).
[CrossRef] [PubMed]

W. E. Williams, M. J. Soileau, E. W. Van Stryland, “Optical switching and n2 measurement in CS2,” Opt. Commun. 50, 256–260 (1984).
[CrossRef]

Wang, J.

Weber, M. J.

M. J. Weber, D. Milam, W. L. Smith, “Nonlinear refractive index of glasses and crystal,” Opt. Eng. 17, 463–469 (1978).

Wei, T. H.

A. A. Said, M. Sheik-bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9, 405–414 (1992).
[CrossRef]

M. Sheik-bahae, A. A. Said, T. H. Wei, D. J. Hagan, E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

Williams, W. E.

W. E. Williams, M. J. Soileau, E. W. Van Stryland, “Optical switching and n2 measurement in CS2,” Opt. Commun. 50, 256–260 (1984).
[CrossRef]

Young, J.

Zhang, C.

Zhang, G.

Appl. Opt. (1)

IEEE J. Quantum Electron. (3)

S. R. Friberg, P. W. Smith, “Nonlinear optical glasses for ultrafast optical switches,” IEEE J. Quantum Electron. QE-23, 2089–2094 (1987).
[CrossRef]

A. Owyoung, “Ellipse rotation studies in laser host materials,” IEEE J. Quantum Electron. QE-9, 1064–1069 (1973).
[CrossRef]

M. Sheik-bahae, A. A. Said, T. H. Wei, D. J. Hagan, E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

J. Opt. Soc. Am. B (4)

Opt. Commun. (1)

W. E. Williams, M. J. Soileau, E. W. Van Stryland, “Optical switching and n2 measurement in CS2,” Opt. Commun. 50, 256–260 (1984).
[CrossRef]

Opt. Eng. (2)

M. J. Weber, D. Milam, W. L. Smith, “Nonlinear refractive index of glasses and crystal,” Opt. Eng. 17, 463–469 (1978).

M. Sheik-bahae, A. A. Said, D. J. Hagan, E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

Opt. Lett. (2)

Other (1)

Q. W. Song, C. Zhang, R. W. Gross, R. Birge, “The intensity-dependent refractive index of chemically enhanced bacteriorhodopsin,” Opt. Commun. (to be published).

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Figures (7)

Fig. 1
Fig. 1

Z-scan geometry arrangement with a thick nonlinear medium.

Fig. 2
Fig. 2

Normalized transmittance T as a function of x: a, l = 5; b, l = 10; c, l = 15.

Fig. 3
Fig. 3

Input–output characteristics of optical limiting for different sample lengths l: a, l = 1; b, l = 3; c, l = 5; d, l = 7.

Fig. 4
Fig. 4

Normalized transmittance T as a function of x(z/z 0), l, and c = 0.46: curve a, our model; curves b and c, distributed-lens model with a = 4 and a = 6, respectively.

Fig. 5
Fig. 5

(a) Peak–valley transmittance difference ΔT p–v as a function of the phase shift c with l = 5. Curves a, b, c, and d, distributed-lens model with a = 1, 4, 5, and 6, respectively; curve e, our model; curve f, GL mode decomposition. (b) Peak–valley transmittance difference ΔT p–v as a function of phase shift c with l = 1. Curves a, b, c, and d, distributed-lens model with a = 1, 4, 5, and 6, respectively; curve e, our model; curve f, GL mode decomposition.

Fig. 6
Fig. 6

(a) Peak–valley transmittance difference ΔT p–v as a function of medium thickness l(l = 0–15) with c = 0.46. Curves a, b, c, and d, distributed-lens model with a = 1, 4, 5, and 6, respectively; curve e, our model; curve f, GL mode decomposition. (b) Peak–valley transmittance difference ΔT p–v as a function of medium thickness l (l = 0–1.0) with c = 0.46. Curves a, b, c, and d, distributed-lens model with a = 1, 4, 5, and 6, respectively; curve e, our model; curve f, GL mode decomposition.

Fig. 7
Fig. 7

(a) Peak–valley transmittance separation ΔZ p–v as a function of medium thickness l(l = 0–15) with c = 0.46. Curve a, distributed-lens model; curve b, our model; curve c, GL mode decomposition. (b) Peak– valley transmittance difference ΔZ p–v as a function of medium thickness l (l = 0–1.0) with c = 0.46. Curve a, distributed-lens model; curve b, our model; curve c, GL mode decomposition.

Equations (33)

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n = n 0 + n 2 E 2 ,
E ( z , r , t ) = E 0 ( t ) ω 0 ω ( z ) exp [ - r 2 ω 2 ( z ) - i k r 2 2 R ( z ) ] × exp [ - i Φ ( z , t ) ] ,
E a ( z , r , t ) = E al ( z , r , t ) + E an ( z , r , t ) Δ Φ 0 + O ( Δ Φ 0 2 ) ,
E a ( z , r , t ) = E al ( z , r , t ) + E an ( z , r , t ) Δ Φ 0 .
E a ( z , r , t ) = E ( z , r = 0 , t ) exp ( - α L / 2 ) m = 0 [ - i Δ Φ 0 ( z , t ) ] m m ! × ω m 0 ω m exp ( - r 2 ω m 2 - i k r 2 2 R m + i θ m ) ,
T ( z , Δ Φ 0 ) 1 + 4 Δ Φ 0 ( t ) x ( x 2 + 9 ) ( x 2 + 1 ) ,
E an ( z , r , t ) E al ( z , r , t ) = i ( ω 10 ω 1 ) ( ω 00 ω 0 ) exp ( - r 2 ω 1 2 + r 2 ω 0 2 - i k r 2 2 R 1 + i k r 2 2 R 0 + i θ 1 + i θ 0 ) = - i a 10 exp ( - b 10 Y a 2 - i c 10 Y a 2 + i d 10 ) ,
Y a = r D ω 0 = r z 0 d ω 0 ,             a 10 = x 2 + 1 x 2 + 9 , b 10 = 2 ( x 2 - 3 ) x 2 + 9 ,             c 10 = 8 x x 2 + 9 , d 10 = tan - 1 ( 2 x x 2 + 3 ) .
E a ( z , r , t ) = E al ( z , r , t ) [ 1 + i = 1 n E ani ( z i , r , t ) E ali ( z i , r , t ) Δ Φ 0 i ] ,
T ( z , r = 0 , t ) = | [ 1 + i = 1 n E ani ( z i , r , t ) E ali ( z i , r , t ) Δ Φ 0 i ] | 2 exp [ i = 1 n 4 Δ Φ 0 i x i ( x i 2 + 1 ) ( x i 2 + 9 ) ] ,
T ( z , r = 0 , t ) = exp [ x - l / 2 x + l / 2 4 Δ Φ b ( t ) x d x ( x 2 + 1 ) ( x 2 + 9 ) ] = { [ ( x + l 2 ) 2 + 1 ] [ ( x - l 2 ) 2 + 9 ] [ ( x - l 2 ) 2 + 1 ] [ ( x + l 2 ) 2 + 9 ] } Δ Φ b ( t ) / 4 ,
T ( z , Δ Φ 0 , t ) 1 + 4 Δ Φ 0 ( t ) x ( x 2 + 9 ) ( x 2 + 1 ) ,
X p , v = ± { ( l 2 2 - 10 ) + [ ( l 2 + 10 ) 2 + 108 ] 1 / 2 6 } 1 / 2 .
Δ Z p - v = 2 { ( l 2 2 - 10 ) + [ ( l 2 + 10 ) 2 + 108 ] 1 / 2 6 } 1 / 2 z 0 .
Δ Z p - v l z 0 = L .
Δ Z p - v 1.7 z 0 .
Δ T p - v = | 2 sinh { c 4 ln [ ( x p , v + l 2 ) 2 + 1 ] [ ( x p , v - l 2 ) 2 + 9 ] [ ( x p , v - l 2 ) 2 + 1 ] [ ( x p , v + l 2 ) 2 + 9 ] } | ,
d 2 ω ( z ) d z 2 = 1 ω 3 ( z ) [ λ 0 2 π 2 n 0 2 - 4 n 2 E 2 ( z ) ω 2 ( z ) n 0 ] ,
P = c 0 ɛ n 0 π 0 E ( z , r , t ) 2 r d r = π E 2 ( z ) ω 2 ( z ) / 4 η ,
η = 1 n 0 ( μ 0 ɛ 0 ) 1 / 2 .
d 2 ω ( z ) d z 2 = 1 ω 3 ( z ) ( λ 0 2 π 2 n 0 2 - 16 n 2 η P n 0 π ) .
d 2 ω ( z ) d z 2 = 1 ω 3 ( z ) ( λ 0 2 π 2 n 0 2 - 16 n 2 η P n 0 π a ) .
Δ n ( r ) = Δ n ( 0 ) exp ( - 2 r 2 / ω 2 ) Δ n ( 0 ) ( 1 - 2 r 2 / a ω 2 ) ,
Δ T p - v = ¼ β l n [ h ( l ) ] ,
Δ Z p - v 2 ( thick ) = Δ Z p - v 2 ( thin ) + U ( l ) ,
h ( l ) = 9 + Ω - Γ 1 + Ω - Γ × 1 + Ω + Γ 9 + Ω + Γ ,
Ω = η + 1 / 3 l 2 ,
Γ = l ( η + 1 12 l 2 ) 1 / 2 ,
η = - 5 3 + [ 3 + 1 9 ( 5 + 1 / 2 l 2 ) 2 ] 1 / 2 ,
β = ( 4 π ) k 2 P η n 2 ,
U ( l ) = 1 3 l 2 + 4 { [ 3 + 1 / 9 ( 5 + 1 / 2 l 2 ) 2 ] 1 / 2 - ( 3 + 25 9 ) 1 / 2 } .
Δ Z p - v ( thin ) = 1.717.
Δ Φ b = ( 2 / π ) k 2 P η n 2 = β / 2.

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