Abstract

Light transmission through a gradient-index medium, whose refractive-index profile can be represented by a one-dimensional hyperbolic secant function, is analyzed by a wave-optics approach. An asymptotic expression of the corresponding optical propagator (the Green’s function) in this medium is evaluated in a such a way that nonparaxial propagation can be studied. Thus optical transformations such as focusing and collimation are obtained, and the pupil effect (diffractive effect) is shown.

© 1994 Optical Society of America

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References

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  1. E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), Chap. 5.
  2. J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), Chaps. 2 and 4.
  3. M. S. Sodha, A. K. Ghatak, Inhomogeneous Optical Waveguides (Plenum, New York, 1977), Chaps. 5 and 8.
  4. C. Gómez-Reino, J. Liniares, “Optical path integrals in graded-index media,” J. Opt. Soc. Am. A 4, 1337–1341 (1987).
    [Crossref]
  5. J. Liniares, “Maxwell paraxial wave optics in inhomogeneous media by path integral formalism,” Phys. Lett. A 141, 207–212 (1989).
    [Crossref]
  6. M. Nazarathy, J. Shamir, “First-order optics, a canonical operator representation: lossless systems,” J. Opt. Soc. Am. A 72, 356–364 (1982).
    [Crossref]
  7. C. Gómez-Reino, E. Larrea, M. V. Perez, “Complex amplitude propagation through a quadratic-index medium: characteristic point function,” Appl. Opt. 22, 2927–2929 (1983).
    [Crossref] [PubMed]
  8. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chaps. 3 and 4.
  9. C. Gómez-Reino, M. V. Pérez, E. Larrea, “Modal propagation in gradient-index (GRIN) material,” Opt. Commun. 45, 372–375 (1983).
    [Crossref]
  10. C. Gómez-Reino, E. Larrea, “Pupil effect in GRIN material,” Appl. Opt. 22, 970–973 (1983).
    [Crossref] [PubMed]
  11. J. Liniares, C. Gómez-Reino, “Pupil effect in a GRIN lens with Gaussian illumination,” J. Mod. Opt. 38, 481–494 (1991).
    [Crossref]

1991 (1)

J. Liniares, C. Gómez-Reino, “Pupil effect in a GRIN lens with Gaussian illumination,” J. Mod. Opt. 38, 481–494 (1991).
[Crossref]

1989 (1)

J. Liniares, “Maxwell paraxial wave optics in inhomogeneous media by path integral formalism,” Phys. Lett. A 141, 207–212 (1989).
[Crossref]

1987 (1)

1983 (3)

1982 (1)

M. Nazarathy, J. Shamir, “First-order optics, a canonical operator representation: lossless systems,” J. Opt. Soc. Am. A 72, 356–364 (1982).
[Crossref]

Arnaud, J. A.

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), Chaps. 2 and 4.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chaps. 3 and 4.

Ghatak, A. K.

M. S. Sodha, A. K. Ghatak, Inhomogeneous Optical Waveguides (Plenum, New York, 1977), Chaps. 5 and 8.

Gómez-Reino, C.

Larrea, E.

Liniares, J.

J. Liniares, C. Gómez-Reino, “Pupil effect in a GRIN lens with Gaussian illumination,” J. Mod. Opt. 38, 481–494 (1991).
[Crossref]

J. Liniares, “Maxwell paraxial wave optics in inhomogeneous media by path integral formalism,” Phys. Lett. A 141, 207–212 (1989).
[Crossref]

C. Gómez-Reino, J. Liniares, “Optical path integrals in graded-index media,” J. Opt. Soc. Am. A 4, 1337–1341 (1987).
[Crossref]

Marchand, E. W.

E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), Chap. 5.

Nazarathy, M.

M. Nazarathy, J. Shamir, “First-order optics, a canonical operator representation: lossless systems,” J. Opt. Soc. Am. A 72, 356–364 (1982).
[Crossref]

Perez, M. V.

Pérez, M. V.

C. Gómez-Reino, M. V. Pérez, E. Larrea, “Modal propagation in gradient-index (GRIN) material,” Opt. Commun. 45, 372–375 (1983).
[Crossref]

Shamir, J.

M. Nazarathy, J. Shamir, “First-order optics, a canonical operator representation: lossless systems,” J. Opt. Soc. Am. A 72, 356–364 (1982).
[Crossref]

Sodha, M. S.

M. S. Sodha, A. K. Ghatak, Inhomogeneous Optical Waveguides (Plenum, New York, 1977), Chaps. 5 and 8.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chaps. 3 and 4.

Appl. Opt. (2)

J. Mod. Opt. (1)

J. Liniares, C. Gómez-Reino, “Pupil effect in a GRIN lens with Gaussian illumination,” J. Mod. Opt. 38, 481–494 (1991).
[Crossref]

J. Opt. Soc. Am. A (2)

M. Nazarathy, J. Shamir, “First-order optics, a canonical operator representation: lossless systems,” J. Opt. Soc. Am. A 72, 356–364 (1982).
[Crossref]

C. Gómez-Reino, J. Liniares, “Optical path integrals in graded-index media,” J. Opt. Soc. Am. A 4, 1337–1341 (1987).
[Crossref]

Opt. Commun. (1)

C. Gómez-Reino, M. V. Pérez, E. Larrea, “Modal propagation in gradient-index (GRIN) material,” Opt. Commun. 45, 372–375 (1983).
[Crossref]

Phys. Lett. A (1)

J. Liniares, “Maxwell paraxial wave optics in inhomogeneous media by path integral formalism,” Phys. Lett. A 141, 207–212 (1989).
[Crossref]

Other (4)

E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), Chap. 5.

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), Chaps. 2 and 4.

M. S. Sodha, A. K. Ghatak, Inhomogeneous Optical Waveguides (Plenum, New York, 1977), Chaps. 5 and 8.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chaps. 3 and 4.

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Equations (31)

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n ( x ) = n 0 sech ( α x ) ,
ϕ ( x , z ) = R K ( x , x 0 ; z ) ϕ 0 ( x 0 ) d x 0 ,
K ( x , x 0 ; z ) = F 0 exp ( - 1 2 m 0 x 2 S d z ) exp ( i k S ) .
S = 1 m 0 0 z 1 n 2 d z = n 0 2 m 0 0 z 1 sech 2 [ α x ( z ) ] d z .
u ( x ) = sinh ( α x ) = A sin ( τ )
A 2 = n 0 2 m 0 2 - 1 , τ = sin - 1 ( u 0 A ) + α z ,
S = n 0 α { tan - 1 [ n 0 m 0 tan ( τ 1 ) ] - tan - 1 [ n 0 m 0 tan ( τ 0 ) ] } ,
u ( z ) = u 0 cos ( α z ) + [ u 1 - u 0 cos ( α z 1 ) ] sin ( α z 1 ) sin ( α z ) ,
u ˙ = α ( A 2 - u 2 ) 1 / 2 ,
n 2 m 0 2 = 1 + x ˙ 2 ,
tan ( τ ) = u u ˙ α ,
S ( u 1 , u ˙ 1 , u 0 , u ˙ 0 ) = n 0 α ( tan - 1 { u 1 [ 1 + α 2 ( 1 + u 1 2 ) u ˙ 1 2 ] 1 / 2 } - tan - 1 { u 0 [ 1 + α 2 ( 1 + u 0 2 ) u ˙ 0 2 ] 1 / 2 } ) .
S x 1 = m 0 x ˙ 1 ,             2 S x 1 2 = u ˙ 1 u 1 ,
K ( u 1 , u 0 ; z 1 ) = F 0 H 1 1 / 2 ( z 1 ) × exp [ i k n 0 α ( tan - 1 { u 1 [ 1 + α 2 ( 1 + u 1 2 ) H 1 2 ( u 1 H ˙ 1 - u 0 ) 2 ] 1 / 2 } - tan - 1 { u 0 [ 1 + α 2 ( 1 + u 0 2 ) H 1 2 ( u 1 - u 0 H 2 ) 2 ] 1 / 2 } ) ] ,
ϕ s [ u 1 ( x 1 ) ; z 1 ] = K ( u 1 , 0 ; z 1 ) = F s ( z 1 ) exp ( i θ s ) = α F 0 sin ( α z 1 ) × exp [ i k n 0 α ( tan - 1 { u 1 [ 1 + α 2 ( 1 + u 1 2 ) H 1 2 u 1 2 H ˙ 1 2 ] 1 / 2 } ) ] .
ϕ c [ u 1 , z 1 ] = F c ( z 1 ) exp ( i θ c ) = - i F 0 H 2 1 / 2 ( z 1 ) × exp [ i k n 0 α ( tan - 1 { [ u 1 2 + α 2 ( 1 + u 1 2 ) H 2 2 H ˙ 2 2 ] 1 / 2 } ) ] ,
θ = θ ( 0 , z 1 ) + α u 1 + β 2 ! u 1 2 + γ 3 ! u 1 3 + , ,
θ s ( 0 , z 1 ) = k n 0 z 1 ,             α = 0 , β = k n 0 H ˙ 1 ( z 1 ) H 1 ( z 1 ) ,             γ = 0 , ,
θ c ( 0 , z 1 ) = k n 0 z 1 ,             α = 0 , β = k n 0 H ˙ 2 ( z 1 ) H 2 ( z 1 ) ,             γ = 0 , .
ϕ s ( x 1 , z 1 ) F 0 H 1 1 / 2 ( z 1 ) exp ( i k n 0 z 1 ) exp { i [ k n 0 H ˙ 1 ( z 1 ) 2 H 1 ( z 1 ) u 1 2 ] }
ϕ c ( x 1 , z 1 ) F 0 H 2 1 / 2 ( z 1 ) exp ( i k n 0 z 1 ) exp { i [ k n 0 H ˙ 2 ( z 1 ) H 2 ( z 1 ) u 1 2 ] } .
ϕ s ( x 1 , z 1 ) = exp ( i k n 0 z m ) δ ( u ) ,
S p n 0 α [ τ - τ 0 + 1 2 α ( u 1 u ˙ 1 - u 0 u ˙ 0 ) ] + higher - order terms .
K p ( u 1 , u 0 ; z 1 ) F 0 H 1 1 / 2 ( z 1 ) exp ( i k n 0 z ) exp [ i k n 0 2 α 2 H 1 ( z 1 ) × ( u 1 2 H ˙ 1 + u 0 2 H 2 - 2 u 1 u 0 ) ] .
S g n 0 z + n 0 2 ( x ˙ 1 x 1 - x ˙ 0 x 0 ) ,
S h = lim α 0 S = n 0 2 z m 0 ,
S p = 0 z 1 [ n 0 x ˙ 2 2 + n ( x ) ] d z .
S p = 0 z 1 [ n 0 u ˙ 2 2 α 2 ( 1 + u 2 ) + n 0 ( 1 + u 2 ) 1 / 2 ] d z .
S p n 0 z 1 + 0 z 1 ( n 0 u ˙ 2 2 α 2 + n 0 u 2 2 ) d z .
i k K p z 1 [ - 1 2 n 0 k 2 2 u 1 2 - n 0 ( 1 + u 2 ) ] K p .
ϕ ( u 1 , z 1 ) = - a a F 0 α exp [ - i k ( u 1 u 0 ) α ] ( 1 + u 0 2 ) 1 / 2 d u 0 ,

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