Abstract

An efficient and compact recursive numerical solution of the wave equation is developed and applied to cylindrically symmetric optical systems. Numerical results are given for wave propagation through an aperture and in linear and nonlinear optical fibers. This code is most useful for multiwave mixing and wave propagation in nonlinear media.

© 1986 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  2. A. G. Fox, T. Li, “Resonant Modes in a Maser Interferometer,” Bell Syst. Tech. J. 40, 453 (1961).
  3. G. N. Lawrence, P. N. Wolfe, “Applications of the LOTS Computer Code to Laser Fusion Systems and Other Physical Optics Problems,” Proc. Soc. Photo-Opt. Instrum. Eng. 190, 238 (1979).
  4. W. J. Tomlinson, J. P. Gordon, P. W. Smith, A. E. Kaplan, “Reflection of a Gaussian Beam at a Nonlinear Interface,” Appl. Opt. 21, 2041 (1982) and references therein.
    [CrossRef] [PubMed]
  5. W. J. Tomlinson, R. H. Stolen, A. M. Johnson, “Optical Wave Breaking of Pulses in Nonlinear Optical Fibers,” Opt. Lett. 10, 457 (1985).
    [CrossRef] [PubMed]
  6. J. A. Fleck, R. R. Morris, E. S. Bliss, “Small Scale Self-Focusing Effects in a High Power Glass Laser Amplifier,” IEEE J. Quantum Electron. QE-14, 353 (1978) and references therein.
    [CrossRef]
  7. A. E. Siegman, “Quasi fast Hankel Transform,” Opt. Lett. 1, 13 (1977).
    [CrossRef] [PubMed]
  8. M. Lax et al., “Electromagnetic Field Distribution in Loaded Unstable Resonators,” J. Opt. Soc. Am. A 2, 731 (1985).
    [CrossRef]
  9. A. Goldberg, H. M. Schey, J. L. Schwarz, “Computer-Generated Motion Pictures of One-Dimensional Quantum-Mechanical Transmission and Reflection Phenomena,” Am. J. Phys. 35, 177 (1967).
    [CrossRef]
  10. K. Druhl, S. A. Shakir, M. Yusuf, “Stokes Beam Parameters at Large Gain for Focused Pump Beams,” Opt. Lett. (July1986).
  11. Y. Li, H. Platzer, “An Experimental Investigation of Diffraction Patterns on Low-Fresnel-Number Focusing Systems,” Opt. Acta 30, 1621 (1983).
    [CrossRef]
  12. C. T. Seaton, J. E. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, S. D. Smith, “Calculation of Nonlinear TE Waves Guided by Thin Dielectric Films Bounded by Nonlinear Media,” IEEE J. Quantum Electron. QE-21, 774 (1985).
    [CrossRef]

1986

K. Druhl, S. A. Shakir, M. Yusuf, “Stokes Beam Parameters at Large Gain for Focused Pump Beams,” Opt. Lett. (July1986).

1985

C. T. Seaton, J. E. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, S. D. Smith, “Calculation of Nonlinear TE Waves Guided by Thin Dielectric Films Bounded by Nonlinear Media,” IEEE J. Quantum Electron. QE-21, 774 (1985).
[CrossRef]

M. Lax et al., “Electromagnetic Field Distribution in Loaded Unstable Resonators,” J. Opt. Soc. Am. A 2, 731 (1985).
[CrossRef]

W. J. Tomlinson, R. H. Stolen, A. M. Johnson, “Optical Wave Breaking of Pulses in Nonlinear Optical Fibers,” Opt. Lett. 10, 457 (1985).
[CrossRef] [PubMed]

1983

Y. Li, H. Platzer, “An Experimental Investigation of Diffraction Patterns on Low-Fresnel-Number Focusing Systems,” Opt. Acta 30, 1621 (1983).
[CrossRef]

1982

1979

G. N. Lawrence, P. N. Wolfe, “Applications of the LOTS Computer Code to Laser Fusion Systems and Other Physical Optics Problems,” Proc. Soc. Photo-Opt. Instrum. Eng. 190, 238 (1979).

1978

J. A. Fleck, R. R. Morris, E. S. Bliss, “Small Scale Self-Focusing Effects in a High Power Glass Laser Amplifier,” IEEE J. Quantum Electron. QE-14, 353 (1978) and references therein.
[CrossRef]

1977

1967

A. Goldberg, H. M. Schey, J. L. Schwarz, “Computer-Generated Motion Pictures of One-Dimensional Quantum-Mechanical Transmission and Reflection Phenomena,” Am. J. Phys. 35, 177 (1967).
[CrossRef]

1961

A. G. Fox, T. Li, “Resonant Modes in a Maser Interferometer,” Bell Syst. Tech. J. 40, 453 (1961).

Bliss, E. S.

J. A. Fleck, R. R. Morris, E. S. Bliss, “Small Scale Self-Focusing Effects in a High Power Glass Laser Amplifier,” IEEE J. Quantum Electron. QE-14, 353 (1978) and references therein.
[CrossRef]

Chilwell, J. T.

C. T. Seaton, J. E. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, S. D. Smith, “Calculation of Nonlinear TE Waves Guided by Thin Dielectric Films Bounded by Nonlinear Media,” IEEE J. Quantum Electron. QE-21, 774 (1985).
[CrossRef]

Druhl, K.

K. Druhl, S. A. Shakir, M. Yusuf, “Stokes Beam Parameters at Large Gain for Focused Pump Beams,” Opt. Lett. (July1986).

Fleck, J. A.

J. A. Fleck, R. R. Morris, E. S. Bliss, “Small Scale Self-Focusing Effects in a High Power Glass Laser Amplifier,” IEEE J. Quantum Electron. QE-14, 353 (1978) and references therein.
[CrossRef]

Fox, A. G.

A. G. Fox, T. Li, “Resonant Modes in a Maser Interferometer,” Bell Syst. Tech. J. 40, 453 (1961).

Goldberg, A.

A. Goldberg, H. M. Schey, J. L. Schwarz, “Computer-Generated Motion Pictures of One-Dimensional Quantum-Mechanical Transmission and Reflection Phenomena,” Am. J. Phys. 35, 177 (1967).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Gordon, J. P.

Johnson, A. M.

Kaplan, A. E.

Lawrence, G. N.

G. N. Lawrence, P. N. Wolfe, “Applications of the LOTS Computer Code to Laser Fusion Systems and Other Physical Optics Problems,” Proc. Soc. Photo-Opt. Instrum. Eng. 190, 238 (1979).

Lax, M.

Li, T.

A. G. Fox, T. Li, “Resonant Modes in a Maser Interferometer,” Bell Syst. Tech. J. 40, 453 (1961).

Li, Y.

Y. Li, H. Platzer, “An Experimental Investigation of Diffraction Patterns on Low-Fresnel-Number Focusing Systems,” Opt. Acta 30, 1621 (1983).
[CrossRef]

Morris, R. R.

J. A. Fleck, R. R. Morris, E. S. Bliss, “Small Scale Self-Focusing Effects in a High Power Glass Laser Amplifier,” IEEE J. Quantum Electron. QE-14, 353 (1978) and references therein.
[CrossRef]

Platzer, H.

Y. Li, H. Platzer, “An Experimental Investigation of Diffraction Patterns on Low-Fresnel-Number Focusing Systems,” Opt. Acta 30, 1621 (1983).
[CrossRef]

Schey, H. M.

A. Goldberg, H. M. Schey, J. L. Schwarz, “Computer-Generated Motion Pictures of One-Dimensional Quantum-Mechanical Transmission and Reflection Phenomena,” Am. J. Phys. 35, 177 (1967).
[CrossRef]

Schwarz, J. L.

A. Goldberg, H. M. Schey, J. L. Schwarz, “Computer-Generated Motion Pictures of One-Dimensional Quantum-Mechanical Transmission and Reflection Phenomena,” Am. J. Phys. 35, 177 (1967).
[CrossRef]

Seaton, C. T.

C. T. Seaton, J. E. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, S. D. Smith, “Calculation of Nonlinear TE Waves Guided by Thin Dielectric Films Bounded by Nonlinear Media,” IEEE J. Quantum Electron. QE-21, 774 (1985).
[CrossRef]

Shakir, S. A.

K. Druhl, S. A. Shakir, M. Yusuf, “Stokes Beam Parameters at Large Gain for Focused Pump Beams,” Opt. Lett. (July1986).

Shoemaker, R. L.

C. T. Seaton, J. E. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, S. D. Smith, “Calculation of Nonlinear TE Waves Guided by Thin Dielectric Films Bounded by Nonlinear Media,” IEEE J. Quantum Electron. QE-21, 774 (1985).
[CrossRef]

Siegman, A. E.

Smith, P. W.

Smith, S. D.

C. T. Seaton, J. E. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, S. D. Smith, “Calculation of Nonlinear TE Waves Guided by Thin Dielectric Films Bounded by Nonlinear Media,” IEEE J. Quantum Electron. QE-21, 774 (1985).
[CrossRef]

Stegeman, G. I.

C. T. Seaton, J. E. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, S. D. Smith, “Calculation of Nonlinear TE Waves Guided by Thin Dielectric Films Bounded by Nonlinear Media,” IEEE J. Quantum Electron. QE-21, 774 (1985).
[CrossRef]

Stolen, R. H.

Tomlinson, W. J.

Valera, J. E.

C. T. Seaton, J. E. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, S. D. Smith, “Calculation of Nonlinear TE Waves Guided by Thin Dielectric Films Bounded by Nonlinear Media,” IEEE J. Quantum Electron. QE-21, 774 (1985).
[CrossRef]

Wolfe, P. N.

G. N. Lawrence, P. N. Wolfe, “Applications of the LOTS Computer Code to Laser Fusion Systems and Other Physical Optics Problems,” Proc. Soc. Photo-Opt. Instrum. Eng. 190, 238 (1979).

Yusuf, M.

K. Druhl, S. A. Shakir, M. Yusuf, “Stokes Beam Parameters at Large Gain for Focused Pump Beams,” Opt. Lett. (July1986).

Am. J. Phys.

A. Goldberg, H. M. Schey, J. L. Schwarz, “Computer-Generated Motion Pictures of One-Dimensional Quantum-Mechanical Transmission and Reflection Phenomena,” Am. J. Phys. 35, 177 (1967).
[CrossRef]

Appl. Opt.

Bell Syst. Tech. J.

A. G. Fox, T. Li, “Resonant Modes in a Maser Interferometer,” Bell Syst. Tech. J. 40, 453 (1961).

IEEE J. Quantum Electron.

J. A. Fleck, R. R. Morris, E. S. Bliss, “Small Scale Self-Focusing Effects in a High Power Glass Laser Amplifier,” IEEE J. Quantum Electron. QE-14, 353 (1978) and references therein.
[CrossRef]

C. T. Seaton, J. E. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, S. D. Smith, “Calculation of Nonlinear TE Waves Guided by Thin Dielectric Films Bounded by Nonlinear Media,” IEEE J. Quantum Electron. QE-21, 774 (1985).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Acta

Y. Li, H. Platzer, “An Experimental Investigation of Diffraction Patterns on Low-Fresnel-Number Focusing Systems,” Opt. Acta 30, 1621 (1983).
[CrossRef]

Opt. Lett.

Proc. Soc. Photo-Opt. Instrum. Eng.

G. N. Lawrence, P. N. Wolfe, “Applications of the LOTS Computer Code to Laser Fusion Systems and Other Physical Optics Problems,” Proc. Soc. Photo-Opt. Instrum. Eng. 190, 238 (1979).

Other

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

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Figures (4)

Fig. 1
Fig. 1

Diffraction of a plane wave through a circular aperture of 200λ diameter. The propagation distance is 500λ, and the medium's index is 1. The radial increment is 0.33λ, while the integration interval is 10λ. λ is the field wavelength. The dashed line represents the edge of the aperture.

Fig. 2
Fig. 2

Reflection and refraction of a Gaussian doughnut wave packet in an optical fiber. The core index is 1.5, and the cladding index is 1.32. The wave angle of incidence with the normal to the cladding is 60°. The fiber diameter is 40λ; axial and radial increments are 0.8λ and 0.067λ, respectively. The frames are for propagation distances of (a) z = 0, (b) z = 6.4λ, (c) z = 9.6λ, (d) z = 12.8λ, (e) z = 16λ, (f) z = 19.2λ, (g) z = 22.4λ, (h) z = 28.8λ. The dashed line represents the boundary between the core and cladding. The horizontal axis is in units of λ.

Fig. 3
Fig. 3

Axial propagation of a Gaussian wave in a nonlinear medium. The axial and radial increments are 20λ and 0.08λ, respectively. The intial wave at z = 0 is (a), while (b)–(d) are the propagated waves at z = 600λ. The nonlinear index of the medium is (b) n = 1.5 + I(r), (c) n = 1.5, (d) n = 1.5-I(r), where I is the intensity of the wave at the radial distance r. The point r = 1 corresponds to 20λ.

Fig. 4
Fig. 4

Reflection of a Gaussian doughnut wave by a nonlinear interface. The cladding index is 1.0 + I(r), and the fiber's diameter is 15λ. The axial and radial increments are 0.5λ and 0.01667λ, respectively. Other parameters are as in Fig. 2. Propagation distances are (a) 0 and 3λ, (b) 6λ, (c) 9λ, (d) 12λ, (e) 21λ, (f) 102λ. The radial distance of 20 corresponds to 10λ.

Equations (22)

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Ĥ ( r , ϕ , E ) E = i E z ,
( r , ϕ , z , t ) = E ( r , ϕ , z ) exp [ i ( k z z wt ) ] .
{ 2 r 2 + 1 r r + 1 r 2 2 ϕ 2 + [ k 0 2 n 2 k z 2 + f ( E ) ] } E = 2 ik z E z ,
E ( r , z + Δ z ) = exp ( i Δ z Ĥ ) E ( r , z ) ,
exp ( i Δ z Ĥ ) 1 ̂ + i ½ Δ z Ĥ 1 ̂ i ½ Δ z Ĥ .
[ 1 ̂ i Δ z 2 Ĥ ] E n + 1 = [ 1 ̂ + i Δ z 2 Ĥ ] E n .
a j E j + 1 n + 1 + b j E j n + 1 + c j E j 1 n + 1 = F j n .
E j n + 1 = α j + 1 n E j + 1 n + 1 + β j + 1 n ,
E j n + 1 = ( a j b j + c j α j n ) E j + 1 n + 1 + F j n c j β j b j + c j α j n .
α j n = b j c j a j c j α j + 1 n ;
β j n = F j n c j + a j c j β j + 1 n α j + 1 n .
E ( r , z ) = rE ( r , z ) .
E 0 n = E J n = 0 ,
E J 2 n + 1 = b J 1 c J 1 E J 1 n + 1 + F J 1 n c J 1 .
α J 1 n = b J 1 c J 1 , β J 1 n = F J 1 n c J 1 , E i n = β 1 n α 1 n .
H ̂ E = { 2 r 2 1 r r + [ k 0 2 n 2 ( r ) k z 2 + 1 r 2 + f ( E ) r ] } E ( r , z ) = i E z .
a j = 1 1 2 j , c j = 1 + 1 2 j , b j = Δ r 2 [ k 0 2 n 2 ( r ) k z 2 + f ( E / r ) ] + 1 j 2 2 + i γ , F j n = a j E j + 1 n ( b j 2 i γ ) E j n c j E j 1 n ,
Δ z 8 π 2 λ .
E j n + 1 = 4 E j 1 n + 1 E j 2 n + 1 3 + O ( Δ z 4 ) .
E ( r , ϕ , z ) = l = 0 R l ( r , z ) cos ϕ l .
2 R m r 2 + 1 r R m r + [ n 2 k 0 2 k z 2 m 2 r 2 + V NL ] R m = i 2 k z R m z ,
V NL = 1 π 0 2 π k 0 2 n NL 2 ( r , ϕ ) cos l ϕ cos m ϕ ,

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