Abstract

We present a theoretical study, with reference to experimental results, of transverse self-phase modulation effects in the transmission of a laser beam through a nonlinear thin film. The occurrence of interference rings, intensification or dimming of the on-axis beam intensity, and transverse optical bistability in the presence of a feedback can all be systematically documented in terms of geometrical/optical parameter classifications. These studies provide further insights and useful guides for experimental studies.

© 1987 Optical Society of America

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References

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  1. See, for example. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 17 and references therein; J. R. Reintjes, Nonlinear Optical Parametric Processes in Liquids and Gases (Academic, New York, 1984).
  2. A. E. Kaplan, Opt. Lett. 6, 360 (1981).
    [CrossRef] [PubMed]
  3. I. C. Khoo, P. Y. Yan, T. H. Liu, S. Shepard, J. Y. Hou, Phys. Rev. A 29, 2756 (1984).
    [CrossRef]
  4. M. Leberre, E. Ressayre, A. Tallet, K. Tai, H. M. Gibbs, IEEE J. Quantum Electron. QE-21, 1404 (1986).
  5. E. Santamato, Y. R. Shen, Opt. Lett. 9, 564 (1984).
    [CrossRef] [PubMed]
  6. I. C. Khoo, G. M. Finn, R. R. Michael, T. H. Liu, Opt. Lett. 11, 227 (1986); J. A. Hermann, J. Opt. Soc. Am. A 1, 729 (1984).
    [CrossRef]
  7. M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964).
  8. I. C. Khoo, Phys. Rev. A 25, 1637 (1982); I. C. Khoo, T. H. Liu, Mol. Cryst. Liq. Cryst. 131, 315 (1985).
    [CrossRef]

1986

M. Leberre, E. Ressayre, A. Tallet, K. Tai, H. M. Gibbs, IEEE J. Quantum Electron. QE-21, 1404 (1986).

I. C. Khoo, G. M. Finn, R. R. Michael, T. H. Liu, Opt. Lett. 11, 227 (1986); J. A. Hermann, J. Opt. Soc. Am. A 1, 729 (1984).
[CrossRef]

1984

I. C. Khoo, P. Y. Yan, T. H. Liu, S. Shepard, J. Y. Hou, Phys. Rev. A 29, 2756 (1984).
[CrossRef]

E. Santamato, Y. R. Shen, Opt. Lett. 9, 564 (1984).
[CrossRef] [PubMed]

1982

I. C. Khoo, Phys. Rev. A 25, 1637 (1982); I. C. Khoo, T. H. Liu, Mol. Cryst. Liq. Cryst. 131, 315 (1985).
[CrossRef]

1981

Born, M.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964).

Finn, G. M.

Gibbs, H. M.

M. Leberre, E. Ressayre, A. Tallet, K. Tai, H. M. Gibbs, IEEE J. Quantum Electron. QE-21, 1404 (1986).

Hou, J. Y.

I. C. Khoo, P. Y. Yan, T. H. Liu, S. Shepard, J. Y. Hou, Phys. Rev. A 29, 2756 (1984).
[CrossRef]

Kaplan, A. E.

Khoo, I. C.

I. C. Khoo, G. M. Finn, R. R. Michael, T. H. Liu, Opt. Lett. 11, 227 (1986); J. A. Hermann, J. Opt. Soc. Am. A 1, 729 (1984).
[CrossRef]

I. C. Khoo, P. Y. Yan, T. H. Liu, S. Shepard, J. Y. Hou, Phys. Rev. A 29, 2756 (1984).
[CrossRef]

I. C. Khoo, Phys. Rev. A 25, 1637 (1982); I. C. Khoo, T. H. Liu, Mol. Cryst. Liq. Cryst. 131, 315 (1985).
[CrossRef]

Leberre, M.

M. Leberre, E. Ressayre, A. Tallet, K. Tai, H. M. Gibbs, IEEE J. Quantum Electron. QE-21, 1404 (1986).

Liu, T. H.

Michael, R. R.

Ressayre, E.

M. Leberre, E. Ressayre, A. Tallet, K. Tai, H. M. Gibbs, IEEE J. Quantum Electron. QE-21, 1404 (1986).

Santamato, E.

Shen, Y. R.

E. Santamato, Y. R. Shen, Opt. Lett. 9, 564 (1984).
[CrossRef] [PubMed]

See, for example. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 17 and references therein; J. R. Reintjes, Nonlinear Optical Parametric Processes in Liquids and Gases (Academic, New York, 1984).

Shepard, S.

I. C. Khoo, P. Y. Yan, T. H. Liu, S. Shepard, J. Y. Hou, Phys. Rev. A 29, 2756 (1984).
[CrossRef]

Tai, K.

M. Leberre, E. Ressayre, A. Tallet, K. Tai, H. M. Gibbs, IEEE J. Quantum Electron. QE-21, 1404 (1986).

Tallet, A.

M. Leberre, E. Ressayre, A. Tallet, K. Tai, H. M. Gibbs, IEEE J. Quantum Electron. QE-21, 1404 (1986).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964).

Yan, P. Y.

I. C. Khoo, P. Y. Yan, T. H. Liu, S. Shepard, J. Y. Hou, Phys. Rev. A 29, 2756 (1984).
[CrossRef]

IEEE J. Quantum Electron.

M. Leberre, E. Ressayre, A. Tallet, K. Tai, H. M. Gibbs, IEEE J. Quantum Electron. QE-21, 1404 (1986).

Opt. Lett.

Phys. Rev. A

I. C. Khoo, P. Y. Yan, T. H. Liu, S. Shepard, J. Y. Hou, Phys. Rev. A 29, 2756 (1984).
[CrossRef]

I. C. Khoo, Phys. Rev. A 25, 1637 (1982); I. C. Khoo, T. H. Liu, Mol. Cryst. Liq. Cryst. 131, 315 (1985).
[CrossRef]

Other

See, for example. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 17 and references therein; J. R. Reintjes, Nonlinear Optical Parametric Processes in Liquids and Gases (Academic, New York, 1984).

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964).

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Figures (5)

Fig. 1
Fig. 1

Schematic of a laser beam passing through a nonlinear thin film (NL). P is the observation plane. A photodetector (not shown) monitors the intensity at various locations. M is a partially reflecting mirror to be used for providing feedback in transverse optical bistability study.

Fig. 2
Fig. 2

(a) Plot of ϕ, ϕ′ = dϕ/dy, and cos ϕ as a function of y [for y ranging from 0 to 3.00 for Ca = 15, Cb = −4, where Ca and Cb are defined in Eqs. (7)]. (b) Plot of ϕ, ϕ′, and cos ϕ as a function of y for case (b) in Eqs. (7). (c) Plot of ϕ, ϕ′, and cos ϕ for case (c) in Eqs. (7).

Fig. 3
Fig. 3

(a) Plot of the radial intensity distribution (in units of ω) as a function of increasing input intensity corresponding to case (a) in Eqs. (7). (b) Plot of the radial intensity distribution as a function of increasing input intensity for case (b) in Eqs. (7). (c) Plot of the radial intensity distribution as a function of increasing input intensity for case (c) in Eqs. (7).

Fig. 4
Fig. 4

(a) Experimentally observed far-field intensity distribution of a laser beam after its passage through a nonlinear thin film for case (a) defined in Eqs. (7), i.e., for 1/Z + 1/R < 0. The insert shows the geometry of the laser polarization with respect to the nematic-film director axis in a homeotropically aligned nematic film. (b) Experimentally observed far-field intensity for conditions corresponding to that defined in case (b) in Eqs. (7), i.e., 1/Z + 1/R > 0.

Fig. 5
Fig. 5

Plot of the functions B1B2U and (1 + U2)−2 for solving U.

Equations (34)

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I ( laser ) = I 0 exp ( - 2 r 2 ω 2 ) ,
I ( r 1 , Z ) = ( 2 π λ Z ) 2 I 0 | 0 r d r J 0 ( 2 π r r 1 / λ z ) × exp ( - 2 r 2 ω 2 ) exp [ - i ( ϕ D + ϕ NL ) ] | 2 ,
ϕ D = k ( r 2 2 Z + r 2 2 R ) ,
ϕ NL = k n ¯ 2 d I 0 exp ( - 2 r 2 ω 2 ) ,
y = r / ω ,
C 1 = 4 π ( λ z ) 2 ω 4 ,
C 2 = 2 π λ ω tan α 0 ,             α 0 = λ π ω 0 ,
C a = 2 π λ n ¯ 2 d I 0 ,
C b = π λ ω 2 ( 1 Z + 1 R ) ,
θ = r 1 Z tan α 0 ,
I ( r 1 , z ) = C 1 I 0 | 0 exp ( - 2 y 2 ) y exp { - i [ C a exp ( - 2 y 2 ) + C b y 2 ] } J 0 ( C 2 θ y ) d y | 2 .
ϕ ( y ) = C a exp ( - 2 y 2 ) + C b y 2
( case a )             C a = 15 ,             C b = - 4 ,
( case b )             C a = 15 ,             C b = 4 ,
( case c )             C a = 15 ,             C b = 0.
n ¯ 2 < 0 , C a = - 15 ,             C b = - 4             ( bright on - axis intensity ) ,
C a = - 15 ,             C b = 4             ( dark on - axis intensity ) ,
C a = - 15 ,             C b 0             ( transition region ) .
ϕ N L = k n ¯ 2 d [ I 0 exp ( - 2 r 2 ω 2 ) + R m I ( r 1 , Z ) ] ,
π λ ω 2 ( 1 Z + 1 R ) > 0.
I ( r 1 , z ) = n = 0 ( - 1 ) n A 2 n r 1 2 n .
B 1 - B 2 U = ( 1 + U 2 ) - 2 ,
B 1 = 8 z 4 ( 1 / 2 Z + 1 / 2 R - 2 n ¯ 2 I 0 d / ω 2 ) R m n ¯ 2 I 0 d ω 6 k 4 ,
B 2 = 8 Z 4 R m n 2 I 0 d 8 k 5 ,
U = ω 2 k ( 1 / 2 Z + 1 / 2 R - 2 n 2 I 0 d / ω 2 - n ¯ 2 d R m A 2 ) ,
A 0 = I 0 4 Z 2 [ ( ω 2 k ) - 2 + ( 1 / 2 Z + 1 / 2 R - 2 n ¯ 2 I 0 d ω 2 - n ¯ 2 d R m A 2 ) 2 ] .
m ( I c ) > - 1.04 ,
b x ( I c ) = 1.12.
I c = 1 2 n ¯ 2 d k [ ω 2 k 2 ( 1 Z + 1 R ) - 1.12 ] .
[ ( 1 R + 1 Z ) ] ω 2 k > 30.8 Z 4 R m ( ω 2 k ) 4 + 2.24
π ω 2 λ ( 1 R + 1 Z ) > 15.4 Z 4 R m ( ω 2 k ) 4 + 1.12 > 0.
π ω 2 λ ( 1 R + 1 Z ) < - [ 15.4 Z 4 R m ( ω 2 k ) 4 + 1.12 ] < 0.
π ω 2 λ ( 1 R + 1 Z ) > [ 0.96 R m ( Z Z ) 4 + 1.12 ]             ( for n ¯ 2 > 0 )
π ω 2 λ ( 1 R + 1 Z ) < - [ 0.96 R m ( Z Z ) 4 + 1.12 ]             ( for n ¯ 2 < 0 ) .

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