Abstract

The objective of this paper is to show that it is possible to transmit a paraxial optical image and transform through a dielectric inhomogeneous medium whose refractive index is given by n2=n12(z)+n02[h1(z)x+h2(z)yg2(z)(x2+y2)], where n0 = n1(0), and n1, g, h1, and h2 are arbitrary functions of z. The optical image transmission, with a scaling factor F = H2(zm), m being an integer, is obtained at planes z = zm such that H1(zm) = 0 (the image condition), and the optical transform transmission is obtained at planes z=z˜m such that H2(z˜m)=0 (the transform condition), where H1(z) and H2(z) are two independent solutions of the paraxial ray equation (z) + g2(z)H(z) = 0 with the initial conditions H1(0) = 0,1(0) = 1,H2(0) = 1, and 2(0) = 0, where the point denotes the derivative with respect to z. Finally, we show that this medium can be represented by a transmittance function similar to the spherical-lens transmittance function and thus can be an element of image-forming systems.

© 1983 Optical Society of America

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  1. K. Iga, Appl. Opt. 19, 1039 (1980).
    [CrossRef] [PubMed]
  2. A. Yariv, J. Opt. Soc. Am. 66, 301 (1976).
    [CrossRef]
  3. S. P. Yukond, B. Bendow, J. Opt. Soc. Am. 70, 172 (1980).
    [CrossRef]
  4. E. W. Marchand, Appl. Opt. 19, 1044 (1980).
    [CrossRef] [PubMed]
  5. J. A. Arnaud, Prog. Opt. 11, 247 (1973).
    [CrossRef]
  6. E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), Chaps. 1 and 10.
  7. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), Chap. 1.
  8. S. G. Krivoshlykov, I. N. Sissakian, Opt. Quant. Electron. 12, 463 (1980).
    [CrossRef]
  9. M. S. Sodha, A. K. Ghatak, Inhomogeneous Optical Waveguides (Plenum, New York, 1977), Chaps. 5 and 8.
    [CrossRef]
  10. J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), Chaps. 2 and 4.
  11. A. Yariv, Quantum Electronics (Wiley, New York, 1975), Chap. 6.
  12. M. D. Feit, E. Maiden, Appl. Phys. Lett. 28, 331 (1976).
    [CrossRef]
  13. J. A. Arnaud, Bell Syst. Tech. J. 49, 2311 (1970).
  14. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Chap. 7, Sec. 9.3.
  15. I. S. Gradsteyn, I. W. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980), Sec. 16.
  16. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3.
  17. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 7.
  18. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Appendix III.

1980

1976

M. D. Feit, E. Maiden, Appl. Phys. Lett. 28, 331 (1976).
[CrossRef]

A. Yariv, J. Opt. Soc. Am. 66, 301 (1976).
[CrossRef]

1973

J. A. Arnaud, Prog. Opt. 11, 247 (1973).
[CrossRef]

1970

J. A. Arnaud, Bell Syst. Tech. J. 49, 2311 (1970).

Arnaud, J. A.

J. A. Arnaud, Prog. Opt. 11, 247 (1973).
[CrossRef]

J. A. Arnaud, Bell Syst. Tech. J. 49, 2311 (1970).

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

Bendow, B.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Appendix III.

Feit, M. D.

M. D. Feit, E. Maiden, Appl. Phys. Lett. 28, 331 (1976).
[CrossRef]

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Chap. 7, Sec. 9.3.

Ghatak, A. K.

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

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3.

Gradsteyn, I. S.

I. S. Gradsteyn, I. W. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980), Sec. 16.

Iga, K.

Krivoshlykov, S. G.

S. G. Krivoshlykov, I. N. Sissakian, Opt. Quant. Electron. 12, 463 (1980).
[CrossRef]

Maiden, E.

M. D. Feit, E. Maiden, Appl. Phys. Lett. 28, 331 (1976).
[CrossRef]

Marchand, E. W.

E. W. Marchand, Appl. Opt. 19, 1044 (1980).
[CrossRef] [PubMed]

E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), Chaps. 1 and 10.

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), Chap. 1.

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Chap. 7, Sec. 9.3.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 7.

Ryzhik, I. W.

I. S. Gradsteyn, I. W. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980), Sec. 16.

Sissakian, I. N.

S. G. Krivoshlykov, I. N. Sissakian, Opt. Quant. Electron. 12, 463 (1980).
[CrossRef]

Sodha, M. S.

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

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Appendix III.

Yariv, A.

A. Yariv, J. Opt. Soc. Am. 66, 301 (1976).
[CrossRef]

A. Yariv, Quantum Electronics (Wiley, New York, 1975), Chap. 6.

Yukond, S. P.

Appl. Opt.

Appl. Phys. Lett.

M. D. Feit, E. Maiden, Appl. Phys. Lett. 28, 331 (1976).
[CrossRef]

Bell Syst. Tech. J.

J. A. Arnaud, Bell Syst. Tech. J. 49, 2311 (1970).

J. Opt. Soc. Am.

Opt. Quant. Electron.

S. G. Krivoshlykov, I. N. Sissakian, Opt. Quant. Electron. 12, 463 (1980).
[CrossRef]

Prog. Opt.

J. A. Arnaud, Prog. Opt. 11, 247 (1973).
[CrossRef]

Other

E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), Chaps. 1 and 10.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), Chap. 1.

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

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

A. Yariv, Quantum Electronics (Wiley, New York, 1975), Chap. 6.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Chap. 7, Sec. 9.3.

I. S. Gradsteyn, I. W. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980), Sec. 16.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 7.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Appendix III.

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Figures (1)

Fig. 1
Fig. 1

Transmittance function.

Equations (51)

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n 2 = n 1 2 ( z ) + n 0 2 [ h 1 ( z ) x + h 2 ( z ) y g 2 ( z ) ( x 2 + y 2 ) ] ,
2 E + k 2 n 2 E = 0.
θ ( r ) = E ( r ) exp [ i k 0 z n 1 ( z ) d z ]
2 θ + i k 2 n 1 ( θ z + θ n 1 z ) + k 2 ( n 2 n 1 2 ) θ = 0 ,
2 = 2 x 2 + 2 y 2 .
x 1 = x η 1 ( τ ) ,
y 1 = y η 2 ( τ ) ,
τ ( z ) = n 0 0 z d z n 1 ( z ) ,
ψ ( r ) = n 1 ( z ) θ ( r ) exp { i k n 0 [ η ˙ 1 x 1 + η ˙ 2 y 1 + 1 2 0 τ L ( τ ) d τ ] } ,
L ( τ ) = η ˙ 1 2 + η ˙ 2 2 + h 1 0 η 1 + h 2 0 η 2 g 0 2 ( η 1 2 + η 2 2 )
h 1 , 2 ( z ) h 1 , 2 0 [ τ ( z ) ] , g ( z ) g 0 [ τ ( z ) ] ;
η ¨ 1 , 2 ( τ ) + g 0 2 ( τ ) η 1 , 2 ( τ ) = h 1 , 2 0 ( τ ) 2
η 1 , 2 ( 0 ) = η ˙ 1 , 2 ( 0 ) = 0
[ 1 2 + 2 i k n 0 τ k 2 n 0 2 g 0 2 ( τ ) ( x 1 2 + y 1 2 ) ] ψ = 0 ,
1 2 = 2 x 1 2 + 2 y 1 2 .
ψ ( x 1 , y 1 , τ ) = R 2 G ( x 1 , y 1 , x 0 , y 0 , τ ) ψ 0 ( x 0 , y 0 ) d x 0 d y 0 ,
G ( x 1 , y 1 , x 0 , y 0 , 0 ) = δ ( x 0 x 1 , y 0 y 1 ) ,
G ( x 1 , y 1 , x 0 , y 0 , τ ) = i k n 0 2 π H 1 ( τ ) exp { i k n 0 2 H 1 ( τ ) [ H ˙ 1 ( τ ) ( x 1 2 + y 1 2 ) + H 2 ( τ ) ( x 0 2 + y 0 2 ) 2 ( x 1 x 0 + y 1 y 0 ) ] } ,
H ¨ 1 , 2 ( τ ) + g 0 2 ( τ ) H 1 , 2 ( τ ) = 0
H 1 ( 0 ) = H ˙ 2 ( 0 ) = 0 , H ˙ 1 ( 0 ) = H 2 ( 0 ) = 1
H ˙ 1 ( τ ) H 2 ( τ ) H 1 ( τ ) H ˙ 2 ( τ ) = 1.
η 1 , 2 ( τ ) = 1 2 0 τ [ H 1 ( τ ) H 2 ( τ ) H 1 ( τ ) H 2 ( τ ) ] h 1 , 2 ( τ ) d τ .
E ( x , y , z ) = n 0 n 1 ( z ) exp [ i k 0 z n 1 ( z ) d z ] × exp { i k n 0 [ η ˙ 1 ( x η 1 ) + η ˙ 2 ( y η 2 ) + 1 2 0 τ L ( τ ) d τ ] } R 2 G [ x η 1 ( τ ) , y η 2 ( τ ) , x 0 , y 0 , τ ] E 0 ( x 0 , y 0 ) d x 0 d y 0 .
G ( x 1 , y 1 , x 0 , y 0 , τ ) = i k n 0 2 π H 1 ( τ ) · exp { i k n 0 H 2 ( τ ) 2 H 1 ( τ ) [ ( x 0 x 1 μ + ) ( x 0 x 1 μ ) + ( y 0 y 1 μ + ) ( y 0 y 1 μ ) ] }
μ ± ( τ ) = 1 ± i H 1 ( τ ) H ˙ 2 ( τ ) H 2 ( τ ) .
μ + ( τ m ) = μ ( τ m ) = 1 H 2 ( τ m ) ,
G ( x 1 , y 1 , x 0 , y 0 , τ m ) = 1 H 2 ( τ m ) δ [ x 0 x 1 H 2 ( τ m ) , y 0 y 1 H 2 ( τ m ) ] ,
E ( x , y , z m ) = n 0 n 1 ( z m ) 1 H 2 ( τ m ) exp [ i k 0 z m n 1 ( z ) d z ] · exp ( i k n 0 { η ˙ 1 ( τ m ) [ x η 1 ( τ m ) ] + η ˙ 2 ( τ m ) [ y η 2 ( τ m ) ] + 1 2 0 τ m L ( τ ) d τ } ) · E 0 [ x η 1 ( τ m ) H 2 ( τ m ) , y η 2 ( τ m ) H 2 ( τ m ) ] ,
τ m = n 0 0 z m d z n 1 ( z ) ,
I ( x , y , z m ) = n 0 n 1 ( z m ) H 2 2 ( τ m ) I 0 [ x η 1 ( τ m ) H 2 ( τ m ) , y η 2 ( τ m ) H 2 ( τ m ) ] .
I ( x , y , z ˜ m ) = n 0 3 λ 2 n 1 ( z ˜ m ) H 1 2 ( τ ˜ m ) | E ˜ 0 ( u , υ ) | 2 ,
E ˜ 0 ( u , υ ) = R 2 E 0 ( x 0 , y 0 ) exp [ i 2 π ( u x 0 + υ y 0 ) ] d x 0 d y 0 .
u = n 0 λ H 1 ( τ ˜ m ) [ x η 1 ( τ ˜ m ) ] ; υ = n 0 λ H 1 ( τ ˜ m ) [ y η 2 ( τ ˜ m ) ]
τ ˜ m = n 0 0 z ˜ m d z n 1 ( z ) .
t ( x , y , d ) = i k n 0 3 / 2 2 π n 1 ( d ) H 1 ( τ d ) exp [ i k 0 d n 1 ( z ) d z ] · exp ( i k n 0 { η ˙ 1 ( τ d ) [ x η 1 ( τ d ) ] + η ˙ 2 ( τ d ) [ y η 2 ( τ d ) ] + 1 2 0 τ d L ( τ ) d τ } ) · R 2 exp [ i k n 0 W ( x , y , x 0 , y 0 , τ d ) ] d x 0 d y 0 ,
W ( x , y , x 0 , y 0 , τ d ) = 1 2 H 1 ( τ d ) ( H ˙ 1 ( τ d ) { [ x η 1 ( τ d ) ] 2 + [ y η 2 ( τ d ) ] 2 } + H 2 ( τ d ) ( x 0 2 + y 0 2 ) 2 { [ x η 1 ( τ d ) ] x 0 + [ y η 2 ( τ d ) ] y 0 } )
E 0 ( x 0 , y 0 ) = 1 , τ d = n 0 0 d d z n 1 ( z ) .
t ( x , y , d ) = n 0 n 1 ( d ) H 2 ( τ d ) exp [ i k 0 d n 1 ( z ) d z ] × exp ( i k n 0 { η ˙ 1 ( τ d ) [ x η 1 ( τ d ) ] + η ˙ 2 ( η d ) [ y η 2 ( τ d ) ] + 1 2 0 τ d L ( τ ) d τ } ) · exp ( i π n 0 H ˙ 2 ( τ d ) λ H 2 ( τ d ) · { [ x η 1 ( τ d ) ] 2 + [ y η 2 ( τ d ) ] 2 } ) ,
t ( x , y , d ) = n 0 n 1 ( d ) H 2 ( τ d ) exp { i k [ 0 d n 1 ( z ) d z + n 0 2 0 τ d L ( τ ) d ] } · exp { i k n 0 H 2 ( τ d ) 2 H ˙ 2 ( τ d ) [ η ˙ 1 2 ( τ d ) + η ˙ 2 2 ( τ d ) ] } · exp ( i π n 0 H ˙ 2 ( τ d ) λ H 2 ( τ d ) · { [ x η 1 ( τ d ) + η ˙ 1 ( τ d ) H 2 ( τ d ) H ˙ 2 ( τ d ) ] 2 + [ y η 2 ( τ d ) + η ˙ 2 ( τ d ) H 2 ( τ d ) H ˙ 2 ( τ d ) ] 2 } ) .
x i = η 1 ( τ d ) η ˙ 1 ( τ d ) H 2 ( τ d ) H ˙ 2 ( τ d ) ,
y i = η 2 ( τ d ) η ˙ 2 ( τ d ) H 2 ( τ d ) H ˙ 2 ( τ d ) ,
z i = H 2 ( τ d ) n 0 H ˙ 2 ( τ d ) ,
E ( x , y , d ) = i k n 0 3 / 2 2 π n 1 ( d ) H 1 ( τ d ) exp [ i k 0 d n 1 ( z ) d z ] × exp ( i k n 0 { η ˙ 1 ( τ d ) [ x η 1 ( τ d ) ] + η ˙ 2 ( τ d ) [ y η 2 ( τ d ) ] + 1 2 0 τ d L ( τ ) d τ } ) · R 2 E 0 ( x 0 , y 0 ) × exp [ i k n 0 W ( x , y , x 0 , y 0 , τ d ) ] · d x 0 d y 0 ,
W x 0 = 0 ; W y 0 = 0 ;
x ˜ 0 = x η 1 ( τ d ) H 2 ( τ d ) ; y ˜ 0 = y η 2 ( τ d ) H 2 ( τ d ) .
W ˜ ( x , y , τ d ) = H ˙ 2 ( τ d ) 2 H 2 ( τ d ) { [ x η 1 ( τ d ) ] 2 + [ y η 2 ( τ d ) ] 2 } ,
W ( x , y , x 0 , y 0 , τ d ) W ˜ ( x , y , τ d ) + H 2 ( τ d ) 2 H 1 ( τ d ) [ ( x 0 x ˜ 0 ) 2 + ( y 0 y ˜ 0 ) 2 ] ,
E 0 ( x ˜ 0 , y ˜ 0 ) exp { i k n 0 W ˜ ( x , y , τ d ) } R 2 exp { i k n 0 H 2 ( τ d ) 2 H 1 ( τ d ) · [ ( x 0 x ˜ 0 ) 2 + ( y 0 y ˜ 0 ) 2 ] } d x 0 d y 0 = i λ H 1 ( τ d ) n 0 H 2 ( τ d ) exp ( i k n 0 H ˙ 2 ( τ d ) 2 H 2 ( τ d ) { [ x η 1 ( τ d ) ] 2 + [ y η 2 ( τ d ) ] 2 } ) · E 0 [ x η 1 ( τ d ) H 2 ( τ d ) , y η 2 ( τ d ) H 2 ( τ d ) ] .
E ( x , y , d ) = t ( x , y , d ) E 0 [ x η 1 ( τ d ) H 2 ( τ d ) , y η 2 ( τ d ) H 2 ( τ d ) ] ,
T ( d ) = t ( x , y , d ) R 2 δ ( x 0 x η 1 ( τ d ) H 2 ( τ d ) , y 0 y η 2 ( τ d ) H 2 ( τ d ) ) d x 0 d y 0
T ( d ) [ E 0 ( x 0 , y 0 ) ] = E ( x , y , d ) ,

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