Abstract

We have studied the validity of the eikonal equation for an inhomogeneous medium. The propagation of an electromagnetic wave in a medium that is characterized by a parabolic dielectric constant variation in the transverse direction, has been studied in detail. The general path of a ray has been calculated and the irradiance distribution near the focal point has been analytically obtained.

© 1975 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).
  2. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).
  3. L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice–Hall, Englewood Cliffs, N. J., 1973).
  4. M. Kline and I. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, New York, 1965).
  5. H. Goldstein, Classical Mechanics (Addison–Wesley, Reading, Mass., 1968).
  6. E. Wolf (private communication).
  7. J. B. Keller and W. Streifer, J. Opt. Soc. Am. 61, 40 (1971).
    [CrossRef]
  8. K. Koizumi, Y. Ikeda, I. Kitano, M. Furukawa, and T. Sumimoto, Appl. Opt. 13, 255 (1974).
    [CrossRef] [PubMed]
  9. It may be added that this failure is not due to the particular form of ∊(r)chosen, which is unbounded as r→∞. Even for the case in which ∊(x) = ∊e+ ∊2a2/cosh2x/a, which is bounded, the term (1/k02ψ0)∇2ψ0is finite and equal to ∊2a2tanh2x/a.
  10. A. Erdelyi, Higher Transcendental Functions (McGraw–Hill, New York, 1953).
  11. G. P. Agrawal, A. K. Ghatak, and C. L. Mehta, Opt. Comm. 12, 333 (1974).
    [CrossRef]
  12. A two-dimensional analysis is given here for simplicity.
  13. M. S. Sodha, A. K. Ghatak, and D. P. S. Malik, J. Phys. D 4, 1887 (1971).
    [CrossRef]
  14. This implies that the neglect of terms O(1/k0) in βmn and ∂2A/∂z2in Ref. 13, is equivalent to paraxial approximation.

1974 (2)

1971 (2)

M. S. Sodha, A. K. Ghatak, and D. P. S. Malik, J. Phys. D 4, 1887 (1971).
[CrossRef]

J. B. Keller and W. Streifer, J. Opt. Soc. Am. 61, 40 (1971).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, A. K. Ghatak, and C. L. Mehta, Opt. Comm. 12, 333 (1974).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).

Erdelyi, A.

A. Erdelyi, Higher Transcendental Functions (McGraw–Hill, New York, 1953).

Felsen, L. B.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice–Hall, Englewood Cliffs, N. J., 1973).

Furukawa, M.

Ghatak, A. K.

G. P. Agrawal, A. K. Ghatak, and C. L. Mehta, Opt. Comm. 12, 333 (1974).
[CrossRef]

M. S. Sodha, A. K. Ghatak, and D. P. S. Malik, J. Phys. D 4, 1887 (1971).
[CrossRef]

Goldstein, H.

H. Goldstein, Classical Mechanics (Addison–Wesley, Reading, Mass., 1968).

Ikeda, Y.

Kay, I.

M. Kline and I. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, New York, 1965).

Keller, J. B.

Kitano, I.

Kline, M.

M. Kline and I. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, New York, 1965).

Koizumi, K.

Malik, D. P. S.

M. S. Sodha, A. K. Ghatak, and D. P. S. Malik, J. Phys. D 4, 1887 (1971).
[CrossRef]

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).

Marcuvitz, N.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice–Hall, Englewood Cliffs, N. J., 1973).

Mehta, C. L.

G. P. Agrawal, A. K. Ghatak, and C. L. Mehta, Opt. Comm. 12, 333 (1974).
[CrossRef]

Sodha, M. S.

M. S. Sodha, A. K. Ghatak, and D. P. S. Malik, J. Phys. D 4, 1887 (1971).
[CrossRef]

Streifer, W.

Sumimoto, T.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).

E. Wolf (private communication).

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Phys. D (1)

M. S. Sodha, A. K. Ghatak, and D. P. S. Malik, J. Phys. D 4, 1887 (1971).
[CrossRef]

Opt. Comm. (1)

G. P. Agrawal, A. K. Ghatak, and C. L. Mehta, Opt. Comm. 12, 333 (1974).
[CrossRef]

Other (10)

A two-dimensional analysis is given here for simplicity.

It may be added that this failure is not due to the particular form of ∊(r)chosen, which is unbounded as r→∞. Even for the case in which ∊(x) = ∊e+ ∊2a2/cosh2x/a, which is bounded, the term (1/k02ψ0)∇2ψ0is finite and equal to ∊2a2tanh2x/a.

A. Erdelyi, Higher Transcendental Functions (McGraw–Hill, New York, 1953).

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice–Hall, Englewood Cliffs, N. J., 1973).

M. Kline and I. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, New York, 1965).

H. Goldstein, Classical Mechanics (Addison–Wesley, Reading, Mass., 1968).

E. Wolf (private communication).

This implies that the neglect of terms O(1/k0) in βmn and ∂2A/∂z2in Ref. 13, is equivalent to paraxial approximation.

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Figures (1)

FIG. 1
FIG. 1

Relative orientation of the two coordinate systems used in the analysis. The displacement a is not shown here.

Equations (40)

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2 ψ + k 2 ψ = 0 ,
ψ ( r ) = ψ 0 ( r ) e i k 0 S ( r ) ,
k 0 2 { n 2 ( S ) 2 } ψ 0 + i k 0 ( 2 S · ψ 0 + ψ 0 2 S ) + 2 ψ 0 = 0 ,
( S ) 2 = n 2 + 1 k 0 2 ψ 0 2 ψ 0 .
( S ) 2 = n 2 .
= 1 2 ( x 2 + y 2 ) = 1 3 r 2 ,
ψ = m , n A m n ψ m n exp ( i β m n z ) ,
ψ m n = ( 2 m + n m ! n ! π ) 1 / 2 α H m ( x α ) H n ( y α ) exp { x 2 + y 2 2 α 2 } ,
β m n = { 1 k 0 2 2 k 0 2 ( m + n + 1 ) } 1 / 2 , m , n = 0 , 1 , 2 .
1 k 0 2 ψ 0 2 ψ 0 = 2 ( x 2 + y 2 ) 2 2 k 0 ( m + n + 1 ) .
A m n = ψ ( z = 0 ) ψ m n * d ξ d η ,
ψ ( x , y , z ) = d ξ d η ψ ( z = 0 ) K ( ξ , η ; ξ , η ; z ) ,
K ( ξ , η ; ξ , η ; z ) = m , n ψ m n ( ξ , η ) ψ m n ( ξ , η ) exp ( i β m n z )
β m n 1 k 0 δ ( m + n + 1 ) ,
K ( ξ , η ; ξ , η ; z ) = e i 1 k 0 z 2 π i sin δ z exp [ i 2 ( ξ 2 + η 2 ) cot δ z ] × exp [ i sin δ z ( ξ ξ + η η ) + i 2 cot δ z ( ξ 2 + η 2 ) ] .
ψ ( z = 0 ) = 1 π 1 / 4 α 1 / 2 exp { ( x a ) 2 2 α 2 } .
A n = ξ 0 n ( 2 n n ! ) 1 / 2 exp ( ξ 0 2 / 4 ) ,
n 0 = 1 2 k 0 2 a 2 ;
( x y z ) = ( b 11 b 12 b 13 b 21 b 22 b 23 b 31 b 32 b 33 ) ( x a y z ) = ( cos θ sin θ sin φ sin θ cos φ sin ψ sin θ cos ψ cos φ sin ψ cos θ sin φ cos ψ sin φ + sin ψ cos θ cos φ cos ψ sin θ sin ψ cos φ cos ψ cos θ sin φ sin ψ sin φ + cos ψ cos θ cos φ ) ( x a y z )
Ψ ( x , y , z ) = Ψ 0 exp { ( x 2 + y 2 ) 2 ω 0 2 } e i k z .
Ψ ( x , y , z = 0 ) = Ψ 0 exp [ p 2 σ 2 ξ 2 q 2 σ 2 η 2 s σ 2 ξ η + i k α g ξ + i k α h η ] exp [ p 2 ω 0 2 a 2 i k b 31 a ] ,
p = b 11 2 + b 21 2 , q = b 12 2 + b 22 2 , s = b 11 b 12 + b 21 b 22 , g = b 31 i p a k w 0 2 , h = b 32 i s a k w 0 2 σ = w 0 / α .
Ψ ( x , y , z ) = Ψ 0 e i k 0 1 z i sin δ z exp [ p 2 w 0 2 a 2 i k b 31 a ] 1 ( q / σ 2 i cot δ z ) 1 / 2 exp { i 2 ( ξ 2 + η 2 ) cot δ z } / { p σ 2 i cot δ z s 2 σ 4 ( q / σ 2 i cot δ z ) } 1 / 2 × exp [ ( k h α η sin δ z ) 2 / 2 ( q σ 2 i cot δ z ) ] exp [ { k α g ξ sin δ z s σ 2 ( q / σ 2 i cot δ z ) ( k h α η sin δ z ) } 2 / 2 { p σ 2 i cot δ z s 2 σ 4 [ q / σ 2 i cot δ z ] } ] .
Ψ = Ψ 0 exp [ ( x a cos δ z ) 2 + y 2 2 α 2 ] exp [ i ( k 0 1 δ ) z ] × exp [ i 2 α 2 ( 2 x a sin δ z a 2 2 sin 2 δ z ) ] .
x = a cos δ z , y = 0 ,
Ψ ( x , y , z ) = Ψ 0 [ cos 2 δ z + ( sin 2 δ z ) / σ 4 ] 1 / 2 e i χ ( z ) × exp [ ( ξ 2 + η 2 ) 2 σ 2 [ cos 2 δ z + ( sin 2 δ z ) / σ 4 ] ] ,
χ ( z ) = k 0 1 tan 1 ( tan δ z σ 2 ) + 1 4 ( ξ 2 + η 2 ) ( 1 / σ 4 1 ) [ cos 2 δ z + ( sin 2 δ z ) / σ 4 ] sin 2 δ z .
I = | Ψ ( x , y , z ) | 2 = I 0 [ cos 2 δ z + ( sin 2 δ z ) / σ 4 ] exp { ( x 2 + y 2 ) w 0 2 [ cos 2 δ z + ( sin 2 δ z ) / σ 4 ] } = I 0 f 2 ( z ) exp { ( x 2 + y 2 ) w 0 2 f 2 ( z ) } ,
Ψ ( x , y , z ) = Ψ 0 exp { i ( k 0 1 δ / 2 ) z } ( cos 2 δ z + sin 2 δ z cos 4 χ ) 1 / 4 exp { i 2 tan 1 ( cos 2 χ tan δ z ) } × exp ( i 2 η 2 cot δ z ) exp [ ( η + k α sin χ sin δ z ) 2 2 ( sin 2 δ z + cos 2 δ z / cos 2 χ ) 1 2 ( ξ a α cos δ z ) 2 ] exp [ i 2 ( η + k α sin χ sin δ z ) 2 ( cos 2 χ sin 2 δ z + cos 2 δ z ) cot δ z ] × exp [ i 2 ( 2 a ξ α sin δ z a 2 2 α 2 sin 2 δ z ) ] .
x = a cos δ z , y = sin χ δ sin δ z .
r 2 = x 2 + y 2
= a 2 cos 2 δ z + sin 2 χ δ 2 sin 2 δ z ,
sin χ = a δ .
ξ = r α cos θ , η = r α sin θ ,
Ψ ( r , θ , z ) = e i k 0 1 z 2 π i sin δ z exp [ i 2 r 2 α 2 cot δ z ] × 0 d r 0 2 π d θ Ψ ( r , θ , z = 0 ) × exp [ i r r α 2 sin δ z cos ( θ θ ) + i r 2 2 α 2 cot δ z ] r 2 α 2 .
Ψ ( r , θ , z = 0 ) = { Ψ 0 , r a 0 , r > a ,
ρ 2 = ξ α 2 a 2 , υ = a α 2 r sin δ z , u = a 2 α 2 cot δ z .
Ψ ( r , z ) = Ψ 0 a 2 e i k 0 1 z 2 i α 2 sin δ z exp ( i r 2 2 α 2 cot δ z ) × 2 0 1 exp ( i 2 u ρ 2 ) j 0 ( υ ρ ) ρ d ρ .
I ( r ) = | Ψ ( r ) | 2 = I 0 | 2 J 1 ( ( a / α 2 ) r ) ( a / α 2 ) r | 2 ,
I ( r = 0 , z ) = I 0 | sin u / 4 u / 4 | 2 + 16 I 0 α 4 a 4 sin 2 u / 4 .