Abstract

The point spread function of a static fiber bundle is not spatially invariant, and the system cannot therefore be strictly characterized by an optical transfer function. It is shown both theoretically and experimentally that, in the case of circular fibers of diameter d, the modulation of the image of an object varies with the position of the fiber bundle and lies between two extreme values which become closer together as the spatial frequency decreases. Thus, for sufficiently low frequencies (of the order 1/8d), an optical transfer function can be used. The experimental results are in good agreement with the theory. A randomly vibrating bundle is found to have a spatially invariant point spread function and can therefore be characterized by an optical transfer function.

© 1964 Optical Society of America

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References

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  1. R. J. Potter, Ph.D. thesis, “A Theoretical and Experimental Study of Optical Fibers”, University of Rochester, Rochester, New York, 1960.
  2. R. J. Potter, J. Opt. Soc. Am. 51, 1079 (1961).
    [Crossref]
  3. N. S. KapanyConcept of Classical Optics, edited by J. Strong (W. H. Freeman and Company, San Francisco, 1958).
  4. E. Snitzer, J. Opt. Soc. Am. 51, 491 (1961).
    [Crossref]
  5. N. S. Kapany, J. A. Eyer, and R. E. Keim, J. Opt. Soc. Am. 47, 423 (1957).
    [Crossref]
  6. N. S. Kapany, J. Opt. Soc. Am. 49, 770 (1959).
    [Crossref]
  7. N. S. Kapany, J. Opt. Soc. Am. 49, 779 (1959).
    [Crossref]
  8. P. C. Roetling and W. P. Ganlev, J. Opt. Soc. Am. 52, 99 (1962).
    [Crossref]

1962 (1)

1961 (2)

1959 (2)

1957 (1)

J. Opt. Soc. Am. (6)

Other (2)

R. J. Potter, Ph.D. thesis, “A Theoretical and Experimental Study of Optical Fibers”, University of Rochester, Rochester, New York, 1960.

N. S. KapanyConcept of Classical Optics, edited by J. Strong (W. H. Freeman and Company, San Francisco, 1958).

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Figures (9)

F. 1
F. 1

The function of f′(x′) representing the image of a sine wave object through a static fiber bundle, shows the periodicity 1/ν of the object and the periodicity p of the fiber structure. Any series of points f′(x0+kp), k integer, falls on a same sine curve (dotted line).

F. 2
F. 2

Series of points of the image f′(x′) used to define the modulation Ci of the image.

F. 3
F. 3

Particular directions of the fiber structure.

F. 4
F. 4

Static fiber bundle, modulation transfer Ci/C0 vs spatial frequency ν. Curve J represents T(ν) = 2J1(πνd)/πνd, curve C represents T ( ν ) cos ( π ν R 3 ), curve C represents T ( ν ) × cos 2 ( π ν R 3 ). d = 2R, diameter of fibers. Almost all the values of Ci/C0 given by all series of points of f′(x′) for all positions of the fiber bundle are expected to be included between curves J and C, a very small part of them being between C and C′.

F. 5
F. 5

Setup for modulation measurements, viewed from top: L1, condenser; T, sine wave target; L2, cylindrical lens; G, frosted glass located in the plane of the vertical focal lines of L2; A, microscope objective imaging G on the entrance face of the fiber bundle F; B, microscope objective imaging the exit face of the fiber bundle F on the plane of the vertical slit S; PMT, photomultiplier; R, recorder.

F. 6
F. 6

Static fiber bundle, experimental results. Modulation transfer Ci/C0 vs spatial frequency ν. Diameter of fibers d = 2R ≃100 μ. J, C, and C′ are the curves of Fig. 5. Triangles correspond to the case No. 1 when the sine wave T alone is horizontally moving. Points and crosses are given by different series of points of the curve f′(x′) obtained by horizontal scanning of the exit face of the fiber bundle (case No. 2).

F. 7
F. 7

Image of bar charts given by a fiber bundle. Dimension of fibers d≃10 μ. Spatial period of object, 1/ν≃3d.

F. 8
F. 8

Same as Fig. 7. Spatial period of the object, 1/ν≃10 or 12d.

F. 9
F. 9

Fiber bundle with a rotation of π/3 around the center O of any fiber.

Equations (32)

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T ( ν ) = 2 J 1 ( π ν d ) / π ν d ,
τ ( ν ) = [ 2 J 1 ( π ν d ) / π ν d ] 2 ,
f ( x , y ) = 1 π R 2 f ( x , y ) d x d y ,
f ( x , y ) = 1 π R 2 + f ( x , y ) e ( x x , y y ) d x d y ,
{ e ( x x , y y ) = 1 for ( x x ) 2 + ( y y ) 2 < R 2 , = 0 elsewhere .
T ( ν ) = 2 J 1 ( π ν d ) / π ν d ,
f ( x , y ) e ( x x , y y ) ,
f ( x , y ) = f ( x , y ) e ( x x , y y ) d x d y .
f ( x , y ) = 1 π R 2 f ( x , y ) e ( x x , y y ) × e ( x x , y y ) d x d y d x d y ,
+ e ( x x , y y ) e ( x x , y y ) d x d y ,
E ( α , β ) = + e ( s , t ) e ( α s , β s ) d s d t .
f ( x , y ) = 1 π R 2 + E ( x x , y y ) f ( x , y ) d x d y .
τ ( ν ) = [ 2 J 1 ( π ν d ) / π ν d ] 2
f ( x , y ) = 1 π R 2 + i = 0 N f ( x , y ) e ( x i x , y i y ) × e ( x x i , y y i ) d x d y ,
E = i e ( x x , y i y ) e ( x x i , y y i ) .
f ( x , y ) = a + b M ( ν ) cos [ 2 π ν x + ϕ ( ν ) ] ,
f ( x + k p ) = f ( x ) ,
a + b T ( ν ) cos 2 π ν ( x + h n ) , ( h n < R ) ,
f ( x ) = n = 1 N ( R 2 h n 2 ) 1 2 [ a + b T ( ν ) cos 2 π ν ( x + h n ) ] .
f ( x ) = A + B cos 2 π ν ( x + H ) ,
A = a n = 1 N ( R 2 h n 2 ) 1 2 , and B b T ( ν ) n = 1 N ( R 2 h n 2 ) 1 2 .
C i = ( f M f m ) / f M + f m ) .
B = b T ( ν ) n = 1 N ( R 2 h n 2 ) 1 2 , and B A = b a T ( ν ) = C 0 T ( ν ) .
B / A = { N b T ( ν ) cos ( π ν R 3 ) } / ( N a ) = C 0 T ( ν ) cos ( π ν R 3 )
T ( ν ) , T ( ν ) cos ( π ν R 3 )
T ( ν ) cos 2 ( π ν R 3 )
2 π ν p / 2 = π ν d 3 / 2 = 0.44 π ν d ,
f ( x ) = R + R ( R 2 h 2 ) 1 2 [ a + b 2 J 1 ( π ν d ) π ν d cos 2 π ν ( x + h ) ] d h .
1 + 1 ( 1 u 2 ) 1 2 cos m u d u = π 2 J 1 ( m ) m , f ( x ) = π 2 R 2 { a + b [ 2 J 1 ( π ν d ) π ν d ] 2 cos 2 π ν x } .
O O = m O T 1 + n O T 2 ,
d sin α 1 m = d sin α 2 n = p , α 2 = α 1 π 3 , sin α 1 m = sin α 1 sin α 2 m + n = sin [ α 1 + ( π / 3 ) ] m + n = p d . ( m + n ) sin α 1 = 1 2 m sin α 1 + ( m 3 / 2 ) cos α 1 , tan α 1 = m 3 m + 2 n .
p = d x sin α 1 m = d 3 / ( m + 2 n ) { 1 + [ 3 m 2 / ( m + 2 n ) 2 ] } 1 2 = d 3 2 ( m 2 + n 2 + m n ) 1 2 ,