Abstract

The problem of coherent image transmission through a single multimode optical fiber is discussed. A scheme is presented for recovering the transmitted image after distortions brought about by the fiber modes dispersion. Realization of this scheme by holographic techniques and with lens systems is proposed, and its limitations pointed out. The application of this scheme in canceling out temporal signal dispersion in a multimode fiber transmission line is also discussed briefly.

© 1976 Optical Society of America

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References

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  1. L. G. Cohen and S. D. Personick, Appl. Opt. 14, 1361 (1975).
    [Crossref] [PubMed]
  2. A. Yariv, preceding paper, J. Opt. Soc. Am. 66, 301 (1976).
    [Crossref]
  3. H. Kogelnik and K. S. Pennington, J. Opt. Soc. Am. 58, 273 (1968).
    [Crossref]
  4. For example, O. Bryngdahl, J. Opt. Soc. Am. 64, 1092 (1974); D. C. Chu and J. R. Fineup, Opt. Eng. 13, 189 (1974).
    [Crossref]
  5. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  6. J. Upatnieks, A. Vanderlugt, and E. Leith, Appl. Opt. 5, 589 (1966).
    [Crossref] [PubMed]

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Figures (3)

FIG. 1
FIG. 1

Scheme for image restoration composed of (a) optical channel carrying dispersive modes; (b) optical transformer T functioning as mode spatial analyser which projects each mode to an isolated spot on the transform plane x3, y3; (c) mode phase compensator plate t(x3, y3); and (d) inverse transformer T−1 transforming the phased mode projections from plane x3, y3 and synthesizing them on the reconstruction plane x4, y4.

FIG. 2
FIG. 2

Image u 1 ( x 1 , y 1 ) which excites all the operational channel modes, is transmitted through the channel, and its transform (T) is recorded on a hologram. This hologram can be used as the phase compensator t(x3, y3).

FIG. 3
FIG. 3

Fourier transforming lens may function as the optical transformer T or T−4.

Equations (35)

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E p ( x , y , z ) = p ( x , y ) exp ( - i β p z ) ,
cross section p ( x , y ) p ( x , y ) d x d y = δ p p .
u 1 ( x , y ) = p = 1 N A p p ( x , y ) ,
A p = cross section u ( x , y , 0 ) p ( x , y ) d x d y .
u 2 ( x , y , L ) = p = 1 N A p p ( x , y ) e - α p L exp ( - i β p L ) ,
u corrected = p = 1 N A p p ( x , y ) e - α p L ,
u 1 ( x 1 , y 1 ) = A p p ( x 1 , y 1 )
u 2 ( x 2 , y 2 ) = p A p p ( x 2 , y 2 ) e i ϕ p ,
ũ 2 ( x 3 , y 3 ) = T { u 2 ( x 2 , y 2 ) } = p A p e i ϕ p ˜ p ( x 3 , y 3 ) ,
˜ p ( x 3 , y 3 ) T { p ( x 2 , y 2 ) }
˜ p ( x 3 , y 3 ) ˜ p ( x 3 , y 3 ) = 0             ( p p ) ,
t ( x 3 , y 3 ) ũ 2 ( x 3 , y 3 ) = t ( x 3 , y 3 ) p A p e i ϕ p ˜ p ( x 3 , y 3 ) = p A p ˜ p ( x 3 , y 3 ) .
u 4 ( x 4 , y 4 ) = T - 1 p A p ˜ p ( x 3 , y 3 ) = p A p p ( x 4 , y 4 ) = u 1 ( x 4 , y 4 ) ,
u 1 ( x 1 , y 1 ) = p p ( x 1 , y 1 ) .
u 2 ( x 2 , y 2 ) = p e i ϕ p p ( x 2 , y 2 ) .
t ( x 3 , y 3 ) = α ( [ ũ 2 ( x 3 , y 3 ) ] * R ( x 3 , y 3 ) + α ũ 2 ( x 3 , y 3 ) R * ( x 3 , y 3 ) + dc terms ,
t ( x 3 , y 3 ) = α e - i ( β x x 3 + β y y 3 ) p ˜ p ( x 3 , y 3 ) e - i ϕ p + c . c . ,
t ( x 3 , y 3 ) ũ 2 ( x 3 , y 3 ) = α e - i ( β x x 3 + β y y 3 ) p ˜ p e - i ϕ p p A p ˜ p e i ϕ p = α e - i ( β x x 3 + β y y 3 ) p A p [ ˜ p ( x 3 , y 3 ) ] 2 ,
˜ p ( x 3 , y 3 ) ˜ p , ( x 3 , y 3 ) ˜ p ( x 3 , y 3 ) δ p p ,
t ũ 2 = α e - i ( β x x 3 + β y y 3 ) p A p ˜ p ( x 3 , y 3 ) .
T { u 2 } ũ 2 ( x 3 , y 3 ) = F { u 2 ( x 2 , y 2 ) } f x = x 3 / λ f 1 , f y = y 3 / λ f 1 ,
F { u } = - u ( x , y ) e i 2 π ( f x x + f y y ) d x d y .
u 4 ( x 4 , y 4 ) = F { ũ 2 ( x 3 , y 3 ) } f x = x 4 / λ f 2 , f y = x 4 / λ f 2 = F F { u 2 } = u 2 ( - f 1 f 2 x 4 , - f 1 f 2 y 4 ) .
t ( x 3 , y 3 ) ũ 2 ( x 3 , y 3 ) = α e - i ( β x x 3 + B y y 3 ) p A p ˜ p e - i ϕ p p A p p e i ϕ p = α e - i ( β x x 3 + β y y 3 ) p A p ˜ p p A p ˜ p α e - i ( β x x 3 + β y y 3 ) ũ 1 ( x 3 , y 3 ) ũ 1 ( x 3 , y 3 ) ,
u 4 ( x 4 , y 4 ) = α δ ( x 4 λ f 2 - β x 2 π , y 4 λ f 2 - β y 2 π ) * u 1 ( - f 1 f 2 x 4 , - f 1 f 2 y 4 ) * u 1 ( - f 1 f 2 x 4 , - f 1 f 2 y 4 ) = α u 1 ( - f 1 f 2 x 4 , - f 1 f 2 y 4 ) * u 1 ( - f 1 f 2 x 4 + β x 2 π / λ f 1 , - f 1 f 2 y 4 + β x 2 π / λ f 1 ) ,
u 4 ( x 4 , y 4 ) = α u 1 ( - f 1 f 2 x 4 + β x 2 π / λ f 1 , - f 1 f 2 y 4 + β x 2 π / λ f 1 ) .
m n ( x , y ) = N m n { cos m π x w sin m π x w } { cos n π y t sin n π y t } rect ( x w ) rect ( y t ) ,
˜ m n ( x 3 , y 3 ) = N m n i λ f 1 w t 4 × ( sin π ( t x 3 / λ f 1 - 1 2 m ) π ( t x 3 / λ f 1 - 1 2 m ) ± sin π ( t x 3 / λ f 1 + 1 2 m ) π ( t x 3 / λ f 1 + 1 2 m ) ) × ( sin π ( w y 3 / λ f 1 - 1 2 n ) π ( w y 3 / λ f 1 - 1 2 n ) ± sin π ( w y 3 / λ f 1 + 1 2 n ) π ( w y 3 / λ f 1 + 1 2 n ) ) ,
u 2 ( x 2 , y 2 ) = h ( x 2 - x 1 , y 2 - y 1 ) * u 1 ( x 1 , y 1 ) ,
ũ 2 ( x 3 , y 3 ) = ũ 1 · H ,
ũ 2 ( x 3 , y 3 ) t ( x 3 , y 3 ) = α H 2 ũ 1 R = α H ( x 3 , y 3 ) 2 ũ 1 ( x 3 , y 3 ) e - i ( β x x 3 + β y y 3 ) .
H ( x 3 , y 3 ) 2 = const ,
h = 1 m δ ( x 2 - x 1 m , y 2 - y 1 m )
H ( f x , f y ) = e i ϕ ( f x , f y ) ,
L < 2 π / ( β 0 - β N ) .