Abstract

The effect on the transmittance function in gradient-index material due to a circular pupil is studied, and we characterize this material by its effective transmittance function.

© 1983 Optical Society of America

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References

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  1. C. Gomez-Reino, E. Larrea, Appl. Opt. 21, 4271 (1982); Appl. Opt. 22, 387 (1983).
    [CrossRef] [PubMed]
  2. C. Gomez-Reino, E. Larrea, M. V. Perez, An. Fis. B. 78, 121 (1982).
  3. C. Gomez-Reino, E. Larrea, Opt. Commun. 44, 8 (1982).
    [CrossRef]
  4. J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), Sec. 2.12.
  5. J. A. Arnaud, Bell Syst. Tech. J. 49, 2311 (1970).
  6. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3.
  7. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975), Sec. 8.8.
  8. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U.P., London, 1922), Chap. 16.
  9. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Eqs. (9.2.1) and (9.1.7).
  10. J. D. Gaskill, Linear Systems, Transforms and Optics (Wiley, New York, 1978), p. 71.

1982

C. Gomez-Reino, E. Larrea, Appl. Opt. 21, 4271 (1982); Appl. Opt. 22, 387 (1983).
[CrossRef] [PubMed]

C. Gomez-Reino, E. Larrea, M. V. Perez, An. Fis. B. 78, 121 (1982).

C. Gomez-Reino, E. Larrea, Opt. Commun. 44, 8 (1982).
[CrossRef]

1970

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

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Eqs. (9.2.1) and (9.1.7).

Arnaud, J. A.

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

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), Sec. 2.12.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975), Sec. 8.8.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Transforms and Optics (Wiley, New York, 1978), p. 71.

Gomez-Reino, C.

C. Gomez-Reino, E. Larrea, Opt. Commun. 44, 8 (1982).
[CrossRef]

C. Gomez-Reino, E. Larrea, M. V. Perez, An. Fis. B. 78, 121 (1982).

C. Gomez-Reino, E. Larrea, Appl. Opt. 21, 4271 (1982); Appl. Opt. 22, 387 (1983).
[CrossRef] [PubMed]

Goodman, J. W.

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

Larrea, E.

C. Gomez-Reino, E. Larrea, M. V. Perez, An. Fis. B. 78, 121 (1982).

C. Gomez-Reino, E. Larrea, Appl. Opt. 21, 4271 (1982); Appl. Opt. 22, 387 (1983).
[CrossRef] [PubMed]

C. Gomez-Reino, E. Larrea, Opt. Commun. 44, 8 (1982).
[CrossRef]

Perez, M. V.

C. Gomez-Reino, E. Larrea, M. V. Perez, An. Fis. B. 78, 121 (1982).

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Eqs. (9.2.1) and (9.1.7).

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U.P., London, 1922), Chap. 16.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975), Sec. 8.8.

An. Fis. B.

C. Gomez-Reino, E. Larrea, M. V. Perez, An. Fis. B. 78, 121 (1982).

Appl. Opt.

Bell Syst. Tech. J.

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

Opt. Commun.

C. Gomez-Reino, E. Larrea, Opt. Commun. 44, 8 (1982).
[CrossRef]

Other

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), Sec. 2.12.

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

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975), Sec. 8.8.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U.P., London, 1922), Chap. 16.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Eqs. (9.2.1) and (9.1.7).

J. D. Gaskill, Linear Systems, Transforms and Optics (Wiley, New York, 1978), p. 71.

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Figures (1)

Fig. 1
Fig. 1

Illustrating the pupil effect in GRIN material.

Equations (53)

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n 2 ( x , y , z ) = n 1 2 ( z ) n 0 2 g 2 ( z ) ( x 2 + y 2 ) ,
a ( ρ , z ) = n 0 n 1 ( z ) exp [ i k 0 z n 1 ( z ) d z ] 0 2 π 0 r 0 G [ ρ , ρ 0 , τ ( z ) ] × ρ 0 d ρ 0 d θ ,
G [ ρ , ρ 0 , τ ( z ) ] = i k n 0 2 H 1 [ τ ( z ) ] exp ( i k n 0 2 H 1 [ τ ( z ) ] { H ˙ 1 [ τ ( z ) ] ρ 2 + H 2 [ τ ( z ) ] ρ 0 2 2 ρ ρ 0 cos θ } )
ρ = x 2 + y 2 ; ρ 0 = x 0 2 + y 0 2
τ ( z ) = n 0 0 z d z n 1 ( z ) ,
H ¨ 1,2 [ τ ( z ) ] + g 0 2 [ τ ( z ) ] H 1,2 [ τ ( z ) ] = 0
H 1 ( 0 ) = H ˙ 2 ( 0 ) = 0 ,
H ˙ 1 ( 0 ) = H 2 ( 0 ) = 1 ,
H ˙ 1 [ τ ( z ) ] H 2 [ τ ( z ) ] H ˙ 2 [ τ ( z ) ] H 1 [ τ ( z ) ] = 1 ,
t ( ρ , d ) = 1 H 2 [ τ ( d ) ] n 0 n 1 ( d ) × exp { i k 0 d n 1 ( z ) d z } exp { i k n 0 H ˙ 2 [ τ ( d ) ] 2 H 2 [ τ ( d ) ] ρ 2 } .
f = H 2 [ τ ( d ) ] n 0 H ˙ 2 [ τ ( d ) ] .
t = ρ 0 / r 0 ,
u [ τ ( z ) ] = k n 0 H 2 [ τ ( z ) ] H 1 [ τ ( z ) ] r 0 2 ,
υ [ τ ( z ) ] = k n 0 ρ H 1 [ τ ( z ) ] r 0 ,
u [ τ ( z ) ] υ [ τ ( z ) ] = r 0 ρ H 2 [ τ ( z ) ] .
a ( ρ , z ) = i u [ τ ( z ) ] H 2 [ τ ( z ) ] n 0 n 1 ( z ) exp [ i k 0 z n 1 ( z ) d z ] × exp { i H ˙ 1 [ τ ( z ) ] u [ τ ( z ) ] ρ 2 2 H 2 [ τ ( z ) ] r 0 2 } · 0 1 J 0 { υ [ τ ( z ) ] t } exp { i u [ τ ( z ) ] t 2 2 } d t ,
a ( ρ , z ) = i u [ τ ( z ) ] 2 H 2 [ τ ( z ) ] n 0 n 1 ( z ) exp [ i k 0 z n 1 ( z ) d z ] × exp { i H ˙ 1 [ τ ( z ) ] u [ τ ( z ) ] ρ 2 2 H 2 [ τ ( z ) ] r 0 2 } · [ C ( u , υ ) + i S ( u , υ ) ] ,
C ( u , υ ) = 2 u U 1 ( u , υ ) cos u 2 + 2 u U 2 ( u , υ ) sin u 2 S ( u , υ ) = 2 u U 1 ( u , υ ) sin u 2 2 u U 2 ( u , υ ) cos u 2 } for | u υ | < 1
C ( u , υ ) = 2 u sin υ 2 2 u + 2 u V 0 ( u , υ ) sin u 2 2 u V 1 ( u , υ ) cos u 2 S ( u , υ ) = 2 u cos υ 2 2 u 2 u V 0 ( u , υ ) cos u 2 2 u V 1 ( u , υ ) sin u 2 } for | u υ | > 1 ,
U n ( u , υ ) = s = 0 ( 1 ) s ( u υ ) n + 2 s J n + 2 s ( u ) ;
V n ( u , υ ) = s = 0 ( 1 ) s ( υ u ) n + 2 s J n + 2 s ( u ) .
a ( ρ , z ) = t ( ρ , z ) + i H 2 [ τ ( z ) ] n 0 n 1 ( z ) [ i V 0 ( u , υ ) + V 1 ( u , υ ) ] × exp [ i k 0 z n 1 ( z ) d z ] · exp ( i u [ τ ( z ) ] 2 { H ˙ 1 [ τ ( z ) ] ρ 2 H 2 [ τ ( z ) ] r 0 2 + 1 } ) for ρ < r 0 H 2 [ τ ( z ) ] ,
a ( ρ , z ) = 1 H 2 [ τ ( z ) ] n 0 n 1 ( z ) [ U 2 ( u , υ ) + i U 1 ( u , υ ) ] × exp [ i k 0 z n 1 ( z ) d z ] · exp ( i u [ τ ( z ) ] 2 { H ˙ 1 [ τ ( z ) ] ρ 2 H 2 [ τ ( z ) ] r 0 2 + 1 } ) for ρ > r 0 H 2 [ τ ( z ) ] .
V 0 ( u , υ ) 2 π υ cos ( υ π / 4 ) 1 + ( υ / u ) 2 ,
V 1 ( u , υ ) 2 π υ υ / u 1 + ( υ / u ) 2 cos ( υ 3 π / 4 ) ,
U 1 ( u , υ ) 2 π υ u / υ 1 + ( u / υ ) 2 cos ( υ 3 π / 4 ) ,
U 2 ( u , υ ) 2 π υ ( u / υ ) 2 1 + ( u / υ ) 2 cos ( υ + 3 π / 4 ) .
a ( ρ , d ) t ( ρ , d ) + 0 ( 1 / υ ) for ρ < r 0 H 2 ,
a ( ρ , d ) 0 + 0 ( 1 / υ ) for ρ > r 0 H 2 .
u = υ ,
r = r 0 H 2 [ τ ( d ) ] .
t e ( ρ , d ) = t ( ρ , d ) cyl { ρ 2 r 0 H 2 [ τ ( d ) ] } ,
V 0 ( u , υ ) cos ( υ 2 2 u ) + 0 ( υ 4 ) ,
V 1 ( u , υ ) sin ( υ 2 2 u ) + 0 ( υ 4 ) ,
a ( ρ , d ) t ( ρ , d ) ( 1 exp { i u [ τ ( d ) ] 2 } ) + 0 ( υ 4 ) .
I ( ρ , d ) I 0 sinc 2 u [ τ ( d ) ] 4 ,
I 0 = ( k r 0 2 ) 2 n 0 3 4 H 1 2 [ τ ( z ˜ p ) ] n 1 ( z ˜ p ) .
V 0 ( u , 0 ) = 1 ; V 1 ( u , 0 ) = 0.
a ( 0 , d ) = t ( 0 , d ) ( 1 exp { i u [ τ ( d ) ] 2 } ) ,
t ( 0 , d ) = 1 H 2 [ τ ( d ) ] n 0 n 1 ( d ) exp [ i k 0 d n 1 ( z ) d z ] .
u 4 k n 0 H 2 [ τ ( d ) ] 4 H 1 [ τ ( d ) ] r 0 2 = m π ; ( m = ± 1 , ± 2 , )
H 2 [ τ ( d ) ] = 2 m λ n 0 r 0 2 H 1 [ τ ( d ) ] ,
H 1 [ τ ( d ) ] = 1 g 0 ( 0 ) g 0 [ τ ( d ) ] sin { 0 τ ( d ) g 0 [ τ ( z ) ] d τ } ,
H 2 [ τ ( d ) ] = g 0 ( 0 ) g 0 [ τ ( d ) ] cos { 0 τ ( d ) g 0 [ τ ( z ) ] d τ } + 1 2 g ˙ 0 ( 0 ) g 0 ( 0 ) × 1 g 0 ( 0 ) g 0 [ τ ( d ) ] sin { 0 τ ( d ) g 0 [ τ ( z ) ] d τ } .
cot { 0 τ ( d ) g 0 [ τ ( z ) ] d τ } = 2 m λ n 0 r 0 2 g 0 ( 0 ) g ˙ 0 ( 0 ) 2 g 0 2 ( 0 ) .
0 τ ( d ) g 0 [ τ ( z ) ] d τ π 2 2 m λ n 0 r 0 2 g 0 ( 0 ) + g ˙ 0 ( 0 ) 2 g 0 2 ( 0 ) .
a ( ρ , z p ) = t ( 0 , z p ) cyl { ρ 2 r 0 H 2 [ τ ( z p ) ] } ,
I ( ρ , z ˜ p ) = I 0 ( 2 J 1 { υ [ τ ( z ˜ p ) ] } υ [ τ ( z ˜ p ) ] ) 2 ,
ρ = 0.61 λ H 1 [ τ ( z ˜ p ) ] n 0 r 0 .
τ ( z p ) τ ( d ) g 0 [ τ ( z ) ] d τ = 2 λ n 0 r 0 2 g 0 ( 0 ) ,
g 0 [ τ ( z p ) ] Δ τ = 2 λ n 0 r 0 2 g 0 ( 0 ) ,
| Δ τ | = 2 λ n 0 r 0 2 g 0 ( 0 ) g 0 [ τ ( z p ) ]
| Δ τ ˜ | = 2 λ n 0 r 0 2 g 0 ( 0 ) g 0 [ τ ( z ˜ p ) ]

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