Abstract

The imaging properties of Gaussian beams of light passing through a nonlinear graded-index rod are studied for the first time to our knowledge. The beam-width parameter, the distance of the focal plane, and the linear magnification in terms of the beam power, diffraction, and the rod dimensions are derived by using the variational approach and the ABCD law of Gaussian beam propagation. A comparison of these results with those derived from the ray equation and the linear case shows that the effects of the beam power on the imaging characteristics are significant.

© 1992 Optical Society of America

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References

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  1. E. W. Marchand, H. Nishihara, “Gradient-index optics: introduction by the feature editors,” Appl. Opt. 29, 3991 (1990).
    [CrossRef] [PubMed]
  2. D. Lin, Z. Yin, S. Zhu, L. Zhang, Fiber Optics (Academic, Beijing, 1987), Chap.4.
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    [CrossRef]
  5. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989).
  6. J. T. Manassah, P. L. Baldeck, R. R. Alfano, “Self-focusing and self-phase modulation in a parabolic graded-index optical fiber,” Opt. Lett. 13, 589–591 (1988).
    [CrossRef] [PubMed]
  7. L. Gagnon, C. Paré, “Nonlinear radiation modes connected to parabolic graded-index profiles by the lens transformation,” J. Opt. Soc. Am. A 8, 601–607 (1991).
    [CrossRef]
  8. R. A. Sammut, C. Pask, “Gaussian and equivalent-step-index approximation for nonlinear waveguides,” J. Opt. Soc. Am. B 8, 395–402 (1991).
    [CrossRef]
  9. A. Yariv, Introduction to Optical Electronics (Holt, Rinehart & Winston, New York, 1976), Chaps. 2 and 3.

1991 (2)

1990 (1)

1989 (1)

1988 (1)

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989).

Alfano, R. R.

Baldeck, P. L.

Gagnon, L.

Lin, D.

D. Lin, Z. Yin, S. Zhu, L. Zhang, Fiber Optics (Academic, Beijing, 1987), Chap.4.

Manassah, J. T.

Marchand, E. W.

Nishihara, H.

Paré, C.

Pask, C.

Sammut, R. A.

Stegeman, G. I.

Stolen, R. H.

Yariv, A.

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart & Winston, New York, 1976), Chaps. 2 and 3.

Yin, Z.

D. Lin, Z. Yin, S. Zhu, L. Zhang, Fiber Optics (Academic, Beijing, 1987), Chap.4.

Zhang, L.

D. Lin, Z. Yin, S. Zhu, L. Zhang, Fiber Optics (Academic, Beijing, 1987), Chap.4.

Zhu, S.

D. Lin, Z. Yin, S. Zhu, L. Zhang, Fiber Optics (Academic, Beijing, 1987), Chap.4.

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Figures (7)

Fig. 1
Fig. 1

Propagation of a Gaussian beam through a nonlinear GRIN rod.

Fig. 2
Fig. 2

Graph of imaging distance L (millimeters) versus object distance L0 (millimeters) for P/Pc = 0 for various values of the beam waist. For curve b, w0 = λ; curve c, w0 = 10λ; curve d, w0 = 100λ. Curve a corresponds to the result of the geometric-optics approximation.

Fig. 3
Fig. 3

Graph of linear magnification |m| versus object distance L0 (millimeters). All other parameters are the same as in Fig. 2.

Fig. 4
Fig. 4

Graph of imaging distance L (millimeters) versus object distance L0 (millimeters) for w0 = 10λ for various values of the normalized power P/Pc. Curve a, 0; curve b, 0.5; curve c, 0.9.

Fig. 5
Fig. 5

Graph of linear magnification |m| versus object distance L0 (millimeters). All other parameters are the same as in Fig. 4.

Fig. 6
Fig. 6

Graph of imaging distance L (millimeters) versus q ( = 2 A Z ) for various values of the normalized power P/Pc. Curve a, 0; curve b, 0.5; curve c, 0.9.

Fig. 7
Fig. 7

Graph of linear magnification |m| versus q ( = 2 A Z ). All other parameters are the same as in Fig. 6.

Equations (26)

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n 2 = n 0 2 ( r ) + α | E | 2 .
q 0 = i π w 0 2 / λ ,
q 1 1 = q 0 + L 0 .
q 1 = n 0 q 1 1 .
1 / q 1 = ρ 1 i λ / π n 0 w 1 2
ρ 1 = n 0 L 0 / [ n 0 2 L 0 2 + ( π n 0 w 0 2 / λ ) 2 ] ,
w 1 2 = w 0 2 [ 1 + ( λ L 0 / π w 0 2 ) 2 ]
i ψ z = 1 2 k ( 2 ψ r 2 + 1 r ψ r ) 1 2 k 2 r 2 ψ + k α 2 n 0 2 | ψ | 2 ψ ,
J = i 2 ( ψ ψ * z ψ * ψ z ) 1 2 k | ψ r | 2 1 2 k 2 r 2 | ψ | 2 + k α 4 n 0 2 | ψ | 4 .
ψ ( r , z ) = ψ 0 w ( z ) exp [ r 2 w 1 2 w ( z ) 2 i k 2 ρ ( z ) r 2 + i k ( z ) ] ,
d 2 w d z 2 + A w + B w 3 ( P P c 1 ) = 0 ,
ρ = 1 w d w d z ,
w 2 = 2 a 0 2 cos ( γ Z ) + ρ 1 A sin ( γ Z ) + a 0 2 ,
ρ = γ 2 w 2 [ ρ 1 A cos ( γ Z ) 2 a 0 2 sin ( γ Z ) ] ,
a 0 = 1 + B A ( 1 P P c ) + ρ 1 2 A ,
γ = 2 A ,
1 / q 2 = ρ i λ / π n 0 w 1 2 w 2 .
q 2 = q 2 / n 0
q 3 = q 2 + L .
1 q 3 = π n 0 w 1 2 w 2 ρ ( π w 1 2 w 2 + π n 0 w 1 2 w 2 ρ L ) + λ 2 L ( π w 1 2 w 2 + π n 0 w 1 2 w 2 ρ L ) 2 + λ 2 L 2 i λ π w 1 2 w 2 ( π w 1 2 w 2 + π n 0 w 1 2 w 2 ρ L ) 2 + λ 2 L 2 = ρ 3 ( Z + L ) i λ / π w 3 2 ( Z + L ) .
L = π 2 n 0 w 1 4 w 4 ρ π 2 n 0 2 w 1 4 w 4 ρ 2 + λ 2
w 3 2 ( Z + L ) = λ 2 L 2 + π 2 w 1 4 w 4 ( 1 + n 0 ρ L ) 2 π 2 w 1 2 w 2
| m | = w 3 / w 0 .
L = 1 n 0 A c tan ( A Z ) ( w 3 0 ) ,
L = n o L 0 A cos ( A Z ) + sin ( A Z ) n 0 A [ n 0 L 0 A sin ( A Z ) cos ( A Z ) ] ,
| m | = 1 / | n 0 L 0 A sin ( A Z ) cos ( A Z ) | ,

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