Abstract

The paper considers two questions. The first one is: Is it possible to transmit three-dimensional pictorial information through transparent glass (or other dielectric) fibers? We find that due to modal dispersion, pictorial information is invariably “smeared” in transmission. The second question is: Given nature’s reluctance to transmit pictures through fibers, is there anything we can do about it? We suggest that the answer is yes and point to a class of solutions involving nonlinear optical mixing.

© 1976 Optical Society of America

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References

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  1. A. Yariv, Introduction to Optical Electronics (Holt, Rinehart, and Winston, New York, 1971).
  2. A. Gover, C. P. Lee, and A. Yariv, following paper, J. Opt. Soc. Am. 66, 306 (1976).
    [Crossref]
  3. S. Kawakami and J. Nishizawa, IEEE Trans. MTT-16, 814 (1968).
  4. P. Baues, Opt. Electron. 1, 37 (1969).
    [Crossref]
  5. M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965).
  6. See, for example, J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  7. H. Kogelnik, Appl. Opt. 4, 1562 (1965).
    [Crossref]
  8. A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), p. 120.
  9. D. Marcuse, Bell Syst. Tech. J. 52, 1169 (1973).
    [Crossref]
  10. E. L. Hahn, Phys. Rev. 80, 580 (1950).
    [Crossref]

1976 (1)

1973 (1)

D. Marcuse, Bell Syst. Tech. J. 52, 1169 (1973).
[Crossref]

1969 (1)

P. Baues, Opt. Electron. 1, 37 (1969).
[Crossref]

1968 (1)

S. Kawakami and J. Nishizawa, IEEE Trans. MTT-16, 814 (1968).

1965 (1)

1950 (1)

E. L. Hahn, Phys. Rev. 80, 580 (1950).
[Crossref]

Baues, P.

P. Baues, Opt. Electron. 1, 37 (1969).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965).

Goodman, J. W.

See, for example, J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Gover, A.

Hahn, E. L.

E. L. Hahn, Phys. Rev. 80, 580 (1950).
[Crossref]

Kawakami, S.

S. Kawakami and J. Nishizawa, IEEE Trans. MTT-16, 814 (1968).

Kogelnik, H.

Lee, C. P.

Marcuse, D.

D. Marcuse, Bell Syst. Tech. J. 52, 1169 (1973).
[Crossref]

Nishizawa, J.

S. Kawakami and J. Nishizawa, IEEE Trans. MTT-16, 814 (1968).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965).

Yariv, A.

A. Gover, C. P. Lee, and A. Yariv, following paper, J. Opt. Soc. Am. 66, 306 (1976).
[Crossref]

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart, and Winston, New York, 1971).

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), p. 120.

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

D. Marcuse, Bell Syst. Tech. J. 52, 1169 (1973).
[Crossref]

IEEE Trans. (1)

S. Kawakami and J. Nishizawa, IEEE Trans. MTT-16, 814 (1968).

J. Opt. Soc. Am. (1)

Opt. Electron. (1)

P. Baues, Opt. Electron. 1, 37 (1969).
[Crossref]

Phys. Rev. (1)

E. L. Hahn, Phys. Rev. 80, 580 (1950).
[Crossref]

Other (4)

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965).

See, for example, J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart, and Winston, New York, 1971).

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), p. 120.

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Figures (1)

FIG. 1
FIG. 1

Multimode optical waveguide with nonlinear crystals used for image restoration. The length of the crystal is assumed very short and is neglected in calculating the modal phase dispersion.

Equations (63)

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n 2 ( x , y ) = n 2 [ 1 - ( n 2 / n ) ( x 2 + y 2 ) ] .
f 1 ( x 1 , y 1 ) = ( i k 2 π B ) e - i k L f 0 ( x 0 , y 0 ) exp ( - i k 2 B [ D ( x 1 2 + y 1 2 ) - 2 x 1 x 0 - 2 y 1 y 0 + A ( x 0 2 + y 0 2 ) ] ) d x 0 d y 0 ,
| r 1 d r 1 d z | = | A B C D | | r 0 d r 0 d z | .
A = D = cos [ ( n 2 / n ) 1 / 2 L ] , B = ( n / n 2 ) 1 / 2 sin [ ( n 2 / n ) 1 / 2 L ] , C = - ( n 2 / n ) 1 / 2 sin [ ( n 2 / n ) 1 / 2 L ] .
( n 2 / n ) 1 / 2 L = 2 s π ,             s = 1 , 2 , 3 ,
A = D = 1 ,             C = B = 0.
f 1 ( x 1 , y 1 ) = f 0 ( x 1 , y 1 ) e - i k L ,
2 E + ω 2 μ ( r ) E ( r ) = - ( ( 1 / ) E · ) .
E ( x , y , z ) = ψ ( x , y , z ) e - i k z ,             k = ( ω / c ) n
( r ) = 0 n 2 ( r ) = 0 n 2 [ 1 - ( n 2 / n ) r 2 ] ,             r 2 = x 2 + y 2 .
2 ψ x 2 + 2 ψ y 2 - 2 i k ψ z + 2 ψ z 2 - k 2 n 2 n r 2 ψ = 0.
k ψ z 2 ψ z 2 k 2 ψ ,
2 ψ x 2 + 2 ψ y 2 - 2 i k ψ z - k 2 n 2 n r 2 ψ = 0.
E l m ( x , y , z ) = ψ l m ( x , y , z ) e - i β l m z = H l ( 2 ω x ) H m ( 2 ω y ) exp ( - ( x 2 + y 2 ) ω 2 ) × exp { - i [ k - ( n 2 n ) 1 / 2 ( l + m + 1 ) ] z } ,
ω = ( 2 k ) 1 / 2 ( n n 2 ) 1 / 4 = ( λ π ) 1 / 2 ( 1 n n 2 ) 1 / 4 .
f 0 ( x 0 , y 0 ) = l , m A l m H l ( 2 ω x 0 ) H m ( 2 ω y 0 ) e - ( x 0 2 + y 0 2 ) / ω 2 ,
exp ( - i β l m L ) = exp { - i [ k - ( n 2 / n ) 1 / 2 ( l + m + 1 ) ] L } ,
f 1 ( x 1 , y 1 ) = l , m A l m H l ( 2 ω x 1 ) H m ( 2 ω y 1 ) × e - ( x 1 2 + y 1 2 ) / ω 2 exp { - i [ k - ( n 2 n ) 1 / 2 ( l + m + 1 ) ] L } .
( n 2 / n ) 1 / 2 L = 2 s π ,             s = 1 , 2 , 3 ,
E l m ( x , y , z ) = H l ( 2 ω x ) H m ( 2 ω y ) exp ( - ( x 2 + y 2 ) ω 2 ) × exp { - i k [ 1 - 2 k ( n 2 n ) 1 / 2 ( l + m + 1 ) ] 1 / 2 z } .
β l m = k [ 1 - ( 2 / k ) ( n 2 / n ) 1 / 2 ( l + m + 1 ) ] 1 / 2 .
( β l m - β 00 ) L π ,
L max < ˜ 2 π 2 λ ( n 2 / n ) ( l max + m max + 1 ) 2
ω = ( λ / π ) 1 / 2 ( 1 / n n 2 ) 1 / 4 ,
N max L max π 4 n 2 r 0 4 / 16 λ 3 .
l max , m max = 25             ( i . e . , 25 × 25 resolution elements ) r 0 = 200 μ m ,             n = 1.5 ,             λ = 1 μ m .
E m n ( x , y ) e - i β m n z .
f 0 ( x 0 , y 0 ) = m , n A m n E m n ( x 0 , y 0 ) ,
f 1 ( x 1 , y 1 ) = m , n A m n E m n ( x 1 , y 1 ) e - i β m n L .
f 1 ( compensated ) = m , n A m n E m n ( x 1 , y 1 ) e - i ( β m n - β m n ) L = f 0 ( x 1 , y 1 ) .
( f 1 ) c . c . = m , n A m n * E m n * e i β m n L .
f 2 ( x 2 , y 2 ) = A m n * E m n * ,
E 1 = m , n A m n E m n ( x , y ) e - ( ω 1 t - β m n z ) + c . c . ,
E 3 e i ( ω 3 t - β 3 L ) + c . c .
E 3 E 1 * = m , n E 3 e i ( ω 3 t - β 3 L ) A m n * E m n * e - i ( ω 1 t - β m n L ) = E 3 e - i β 3 L m , n A m n * E m n * e i ( ω 2 t + β m n L ) ,
E 3 e - i β 3 L m , n A m n * E m n * e i ω 2 t ,
A m n E m n e i ω 1 t = f 0 ( x , y ) .
× E ¯ = - μ H t , × H = t ( 0 E + P ) ,
2 E = μ 0 2 E t 2 + μ 2 t 2 P ( r , t ) ,
2 E - μ ( r ) 2 E t 2 = 2 t 2 P NL ( r , t ) ,
E 3 ( z , t ) = 1 2 E 3 e i ( ω 3 t - β 3 z ) + c . c .
E 2 ( r , t ) = 1 2 m B m ( z ) E 2 ( m ) ( x , y ) e i ( ω 2 t - β m 2 ) z .
( 2 x 2 + 2 y 2 - β m 2 2 ) E 2 ( m ) + ω 2 2 μ ( r ) E 2 ( m ) = 0.
e i ω 2 t m [ ( - i β m 2 d B m d z + 1 2 d 2 B m d z 2 ) E 2 ( m ) e - i β m 2 z ] + c . c . = μ 2 t 2 P NL ( r , t ) ,
| d 2 B m d z 2 | β m 2 | d B m d z | ,
m ( - i β m 2 d B m d z E 2 ( m ) ( x , y ) e i ( ω 2 t - β m 2 z ) ) + c . c . = μ 2 t 2 P NL ( r , t ) .
E 1 ( r , t ) = 1 2 l A 1 ( z ) E 1 ( l ) ( x , y ) e i ( ω 1 t - β l 1 z )
P 2 ( r , t ) = l d E 3 A l * ( z ) 2 E 1 ( l ) e - i ( β 3 - β l 1 ) z e i ( ω 3 - ω 1 ) t ,
m - i β m 2 d B m d z E 2 ( m ) e - i β m 2 z e i ω 2 t + c . c . = - μ ω 2 2 2 d E 3 l A l * E 1 ( l ) e - i ( β 3 - β l 1 ) z e i ω 2 t + c . c .
- E 2 ( s ) E 2 ( m ) d x d y = δ s m .
- i β s 2 d B s d z = - ω 2 2 μ 2 d E 3 l A l * e - i ( β 3 - β l 1 - β s 2 ) z × - E 2 ( l ) E 2 ( s ) d x d y .
- E 2 ( l ) E 2 ( s ) d x d y = δ l s ,
d B s d z = - i ω 2 2 μ 2 β s 2 d E 3 A s * e - i ( Δ β s ) z ,
d A s * d z = i ω 1 2 μ 2 β s 1 d E 3 B s e i ( Δ β s ) z ,
Δ β s β 3 - β s 1 - β s 2 .
d B s d z = - i g 2 A s * e - i ( Δ β s ) z , d A s * d z = i g 2 B s e i ( Δ β s ) z ;
g ω ( μ / ) 1 / 2 d E 3 .
A s ( 0 ) A m n ,
A s * ( L ) = A s * ( 0 ) e + i β s 1 L .
B s ( z ) = - i g 2 b A s * ( 0 ) e i β s 1 L sinh [ b ( z - L ) ] e i ( Δ β s / 2 ) ( z - L ) , b = 1 2 [ g 2 - ( Δ β s ) 2 ] 1 / 2
B s ( z ) = - i A s * ( 0 ) e i β s 1 L sinh [ 1 2 g ( z - L ) ] e i ( Δ β s / 2 ) ( z - L ) .
E 2 ( z = L + l ) = - i 2 sinh ( g l 2 ) m A m * ( 0 ) E 2 ( m ) × e + i β m 1 L e i ( Δ β s ) 1 / 2 e i ω 2 t .
E 1 ( z = L ) = m A m ( 0 ) E 1 ( m ) e - i β m 1 L e i ω 1 t .