Abstract

A simple method to obtain the fourth-order coefficient of the series expansion for the radial distribution of a lenslike medium is described. The coefficient is obtained by measuring the position of a discontinuity of the far-field intensity pattern. The relation between the coefficient and this position is calculated by using either the ray theory or the wave theory. By using a ray model, the position of the discontinuity and the macroscopic intensity distribution is easily obtained.

© 1977 Optical Society of America

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References

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  1. T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and H. Matsumura, “Optical characteristics of a light-focusing fiber guide and its applications,” IEEE J. Quantum Electron. QE-6, 606–612 (1970).
    [CrossRef]
  2. M. Ikeda, “Propagation of multimode fibers with graded core index, ”IEEE J. Quantum Electron. QE-10, 362–371 (1974).
    [CrossRef]
  3. J. Hamasaki and K. Maeda, “Correction of image through lenslike medium by means of holography,” 1974National Conv. Rec. of IECE, Japan 1182.
  4. R. Olshansky and D. B. Keck, “Pulse broadening in graded-index optical fibers, ”Appl. Opt. 15, 483–491 (1976).
    [CrossRef] [PubMed]
  5. T. Kitano, H. Matsumura, M. Furukawa, and I. Kitano, “Measurement of fourth-order aberration in a lens-like medium, ”IEEE J. Quantum Electron. QE-9, 967–971 (1973).
    [CrossRef]
  6. Y. Suematsu and K. Furuya, “Measuring methods of characteristics in a lenslike medium stressed on spotsize divergence and group delay, ” IECE of Japan, Rep. Tech. Group on Quantum Electron. QE-70, 39 (1970).
  7. E. G. Rawson and R. G. Murray, “Interferometric measurement of SELFOC dielectric constant coefficients to sixth order,” IEEE J. Quantum Electron. QE-9, 1114–1118 (1973).
    [CrossRef]
  8. M. Ikeda, M. Tateda, and H. Yoshikiyo, “Refractive index profile of a graded index fiber; Measurement by a reflection method. ”Appl. Opt. 14, 814–815 (1975).
    [CrossRef] [PubMed]
  9. K. Terazawa, Sugaku Gairon (Iwanami, Tokyo, 1959).
  10. H. Matsumura and T. Kitano, “Geometrical analysis in a lens-medium with the fourth order aberration,” IECE of Japan, Rep. Tech. Group on Quantum Electron. QE70, 37 (1970).
  11. S. Kawakami and J. Nishizawa, “An optical waveguide with the optimum distribution of refractive index with reference to waveform distortion,” IEEE Trans. Microwave Theory Tech. MTT-16, 814–818 (1968).
    [CrossRef]
  12. J. Hamasald, “Gaussian wave in spatial frequency domain,” Seisan Kenkyu 24, 12–18 (1972).
  13. K. Maeda, “Research on image transmission characteristics of lens-like medium,” Ph. D. thesis, University of Tokyo (1976).

1976 (1)

1975 (1)

1974 (1)

M. Ikeda, “Propagation of multimode fibers with graded core index, ”IEEE J. Quantum Electron. QE-10, 362–371 (1974).
[CrossRef]

1973 (2)

T. Kitano, H. Matsumura, M. Furukawa, and I. Kitano, “Measurement of fourth-order aberration in a lens-like medium, ”IEEE J. Quantum Electron. QE-9, 967–971 (1973).
[CrossRef]

E. G. Rawson and R. G. Murray, “Interferometric measurement of SELFOC dielectric constant coefficients to sixth order,” IEEE J. Quantum Electron. QE-9, 1114–1118 (1973).
[CrossRef]

1972 (1)

J. Hamasald, “Gaussian wave in spatial frequency domain,” Seisan Kenkyu 24, 12–18 (1972).

1970 (3)

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and H. Matsumura, “Optical characteristics of a light-focusing fiber guide and its applications,” IEEE J. Quantum Electron. QE-6, 606–612 (1970).
[CrossRef]

Y. Suematsu and K. Furuya, “Measuring methods of characteristics in a lenslike medium stressed on spotsize divergence and group delay, ” IECE of Japan, Rep. Tech. Group on Quantum Electron. QE-70, 39 (1970).

H. Matsumura and T. Kitano, “Geometrical analysis in a lens-medium with the fourth order aberration,” IECE of Japan, Rep. Tech. Group on Quantum Electron. QE70, 37 (1970).

1968 (1)

S. Kawakami and J. Nishizawa, “An optical waveguide with the optimum distribution of refractive index with reference to waveform distortion,” IEEE Trans. Microwave Theory Tech. MTT-16, 814–818 (1968).
[CrossRef]

Furukawa, M.

T. Kitano, H. Matsumura, M. Furukawa, and I. Kitano, “Measurement of fourth-order aberration in a lens-like medium, ”IEEE J. Quantum Electron. QE-9, 967–971 (1973).
[CrossRef]

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and H. Matsumura, “Optical characteristics of a light-focusing fiber guide and its applications,” IEEE J. Quantum Electron. QE-6, 606–612 (1970).
[CrossRef]

Furuya, K.

Y. Suematsu and K. Furuya, “Measuring methods of characteristics in a lenslike medium stressed on spotsize divergence and group delay, ” IECE of Japan, Rep. Tech. Group on Quantum Electron. QE-70, 39 (1970).

Hamasaki, J.

J. Hamasaki and K. Maeda, “Correction of image through lenslike medium by means of holography,” 1974National Conv. Rec. of IECE, Japan 1182.

Hamasald, J.

J. Hamasald, “Gaussian wave in spatial frequency domain,” Seisan Kenkyu 24, 12–18 (1972).

Ikeda, M.

Kawakami, S.

S. Kawakami and J. Nishizawa, “An optical waveguide with the optimum distribution of refractive index with reference to waveform distortion,” IEEE Trans. Microwave Theory Tech. MTT-16, 814–818 (1968).
[CrossRef]

Keck, D. B.

Kitano, I.

T. Kitano, H. Matsumura, M. Furukawa, and I. Kitano, “Measurement of fourth-order aberration in a lens-like medium, ”IEEE J. Quantum Electron. QE-9, 967–971 (1973).
[CrossRef]

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and H. Matsumura, “Optical characteristics of a light-focusing fiber guide and its applications,” IEEE J. Quantum Electron. QE-6, 606–612 (1970).
[CrossRef]

Kitano, T.

T. Kitano, H. Matsumura, M. Furukawa, and I. Kitano, “Measurement of fourth-order aberration in a lens-like medium, ”IEEE J. Quantum Electron. QE-9, 967–971 (1973).
[CrossRef]

H. Matsumura and T. Kitano, “Geometrical analysis in a lens-medium with the fourth order aberration,” IECE of Japan, Rep. Tech. Group on Quantum Electron. QE70, 37 (1970).

Koizumi, K.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and H. Matsumura, “Optical characteristics of a light-focusing fiber guide and its applications,” IEEE J. Quantum Electron. QE-6, 606–612 (1970).
[CrossRef]

Maeda, K.

J. Hamasaki and K. Maeda, “Correction of image through lenslike medium by means of holography,” 1974National Conv. Rec. of IECE, Japan 1182.

K. Maeda, “Research on image transmission characteristics of lens-like medium,” Ph. D. thesis, University of Tokyo (1976).

Matsumura, H.

T. Kitano, H. Matsumura, M. Furukawa, and I. Kitano, “Measurement of fourth-order aberration in a lens-like medium, ”IEEE J. Quantum Electron. QE-9, 967–971 (1973).
[CrossRef]

H. Matsumura and T. Kitano, “Geometrical analysis in a lens-medium with the fourth order aberration,” IECE of Japan, Rep. Tech. Group on Quantum Electron. QE70, 37 (1970).

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and H. Matsumura, “Optical characteristics of a light-focusing fiber guide and its applications,” IEEE J. Quantum Electron. QE-6, 606–612 (1970).
[CrossRef]

Murray, R. G.

E. G. Rawson and R. G. Murray, “Interferometric measurement of SELFOC dielectric constant coefficients to sixth order,” IEEE J. Quantum Electron. QE-9, 1114–1118 (1973).
[CrossRef]

Nishizawa, J.

S. Kawakami and J. Nishizawa, “An optical waveguide with the optimum distribution of refractive index with reference to waveform distortion,” IEEE Trans. Microwave Theory Tech. MTT-16, 814–818 (1968).
[CrossRef]

Olshansky, R.

Rawson, E. G.

E. G. Rawson and R. G. Murray, “Interferometric measurement of SELFOC dielectric constant coefficients to sixth order,” IEEE J. Quantum Electron. QE-9, 1114–1118 (1973).
[CrossRef]

Suematsu, Y.

Y. Suematsu and K. Furuya, “Measuring methods of characteristics in a lenslike medium stressed on spotsize divergence and group delay, ” IECE of Japan, Rep. Tech. Group on Quantum Electron. QE-70, 39 (1970).

Tateda, M.

Terazawa, K.

K. Terazawa, Sugaku Gairon (Iwanami, Tokyo, 1959).

Uchida, T.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and H. Matsumura, “Optical characteristics of a light-focusing fiber guide and its applications,” IEEE J. Quantum Electron. QE-6, 606–612 (1970).
[CrossRef]

Yoshikiyo, H.

Appl. Opt. (2)

IECE of Japan, Rep. Tech. Group on Quantum Electron. (2)

Y. Suematsu and K. Furuya, “Measuring methods of characteristics in a lenslike medium stressed on spotsize divergence and group delay, ” IECE of Japan, Rep. Tech. Group on Quantum Electron. QE-70, 39 (1970).

H. Matsumura and T. Kitano, “Geometrical analysis in a lens-medium with the fourth order aberration,” IECE of Japan, Rep. Tech. Group on Quantum Electron. QE70, 37 (1970).

IEEE J. Quantum Electron. (4)

E. G. Rawson and R. G. Murray, “Interferometric measurement of SELFOC dielectric constant coefficients to sixth order,” IEEE J. Quantum Electron. QE-9, 1114–1118 (1973).
[CrossRef]

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and H. Matsumura, “Optical characteristics of a light-focusing fiber guide and its applications,” IEEE J. Quantum Electron. QE-6, 606–612 (1970).
[CrossRef]

M. Ikeda, “Propagation of multimode fibers with graded core index, ”IEEE J. Quantum Electron. QE-10, 362–371 (1974).
[CrossRef]

T. Kitano, H. Matsumura, M. Furukawa, and I. Kitano, “Measurement of fourth-order aberration in a lens-like medium, ”IEEE J. Quantum Electron. QE-9, 967–971 (1973).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

S. Kawakami and J. Nishizawa, “An optical waveguide with the optimum distribution of refractive index with reference to waveform distortion,” IEEE Trans. Microwave Theory Tech. MTT-16, 814–818 (1968).
[CrossRef]

Seisan Kenkyu (1)

J. Hamasald, “Gaussian wave in spatial frequency domain,” Seisan Kenkyu 24, 12–18 (1972).

Other (3)

K. Maeda, “Research on image transmission characteristics of lens-like medium,” Ph. D. thesis, University of Tokyo (1976).

J. Hamasaki and K. Maeda, “Correction of image through lenslike medium by means of holography,” 1974National Conv. Rec. of IECE, Japan 1182.

K. Terazawa, Sugaku Gairon (Iwanami, Tokyo, 1959).

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Figures (15)

FIG. 1
FIG. 1

Schematic diagram of the measurement of the angle of the cliff.

FIG. 2
FIG. 2

Bundle of rays representing an incident Gaussian beam.

FIG. 3
FIG. 3

Output angle of the exit ray versus the displacement x0 of the incident ray. a = 0. 25 mm. (a) h = 1. 1, (b) h = 2.

FIG. 4
FIG. 4

Ray model of the far-field pattern.

FIG. 5
FIG. 5

Excitation efficiency |Cl0|2 versus L1.

FIG. 6
FIG. 6

Calculated intensity distributions of the far-field pattern for different values of h. (a) Far-field pattern for h = 2 3, (b) far-field pattern for h = 1. 1, (c) far-field pattern for h = 2.

FIG. 7
FIG. 7

Theoretical curves of cliff angle θp versus h.

FIG. 8
FIG. 8

Experimental arrangement for measuring the intensity distribution of the far-field pattern.

FIG. 9
FIG. 9

Pictures of the far-field pattern for a sample, (z = 292 mm, L1 15 mm, W 0 = W 0.)

FIG. 10
FIG. 10

An example of measurement of the intensity distribution of the far-field pattern.

FIG. 11
FIG. 11

Effects of the insertion of a knife edge on the far-field pattern.

FIG. 12
FIG. 12

Theoretical curves of cliff angle θ versus h for different values of pitch number.

FIG. 13
FIG. 13

Theoretical curves of cliff angle θ versus h for different values of L1.

FIG. 14
FIG. 14

L1 dependence of the cliff angle.

FIG. 15
FIG. 15

Diagram for Appendix E.

Equations (63)

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n 2 ( r ) = n 0 2 ( 1 ( g r ) 2 + h ( g r ) 4 ) ,
n 2 ( x ) = n 0 2 ( 1 ( g x ) 2 + h ( g x ) 4 ) .
x = ( 1 / u ) s n ( υ z + s n 1 ( u x 0 | m ) | m ) ,
0 z d z [ ( 1 z ) ( 1 m z ) ] 1 / 2 = s n 1 ( z | m ) .
u = g ( h / m ) 1 / 4 ( 1 c 2 ) 1 / 4 ,
υ = g ( h / m ) 1 / 4 ( 1 c 2 ) 1 / 4 c 1 ,
m = 4 h ( 1 c 2 ) / { 1 + [ 1 4 h ( 1 c 2 ) ] 1 / 2 } 2
c = n ( x 0 ) cos γ 0 / n 0 .
d x d z = υ u c n ( υ z + s n 1 ( u x 0 | m ) | m ) d n ( υ z + s n 1 ( u x 0 | m ) | m ) ,
θ sin θ = n 0 t [ sin Z + 3 4 ( sin Z sin 3 Z ) h t + ] ,
Z = g z [ 1 3 4 ( h 2 3 ) t + ( 69 64 h 2 9 24 h + 3 8 ) t 2 + ] ,
t = 1 c 2 .
θ 2 = n 0 2 sin 2 Z tan Z g z ( h 2 / 3 ) ,
Z = g z [ 1 3 4 ( h 2 3 ) g 2 x 0 2 + ] .
Δ Ω = 2 π sin θ d θ .
Δ N = A i 2 π x i exp ( 2 x i 2 / W 2 ) | d x i | ,
f ( θ ) = Δ N / Δ Ω .
E 1 ( r , θ ) = l m C l m ϕ l m ( r , θ ) ,
ϕ l m ( r , θ ) = ( 2 π ) 1 / 2 1 W 0 ( l ! ( l + | m | ) ! ) 1 / 2 ( 2 r W 0 ) | m | L l ( | m | ) × [ 2 ( 2 W 0 ) 2 ] exp [ ( r W 0 ) 2 ] exp ( j m θ ) ,
W 0 = ( 2 n 0 k 0 g ) 1 / 2 ,
E 2 ( r , θ ) = l m C l m ϕ l m ( r , θ ) exp ( j β l m z ) ,
β l m 2 = n 0 2 k 0 2 2 n 0 k 0 g ( 2 l + | m | + 1 ) + h g ( 6 l 2 + 6 l ( | m | + 1 ) + ( | m | + 1 ) ( | m | + 2 ) ) ,
E f ( r , θ ) = 1 j λ 0 f 0 0 2 π E 2 ( r , θ ) × exp [ j 2 π ( r r cos θ cos θ λ 0 f + r r sin θ sin θ λ 0 f ) ] r d r d θ ,
E f ( r , θ ) = 1 λ 0 f ( 2 π ) 1 / 2 1 W f l m C l m exp ( j β l m z ) j | m | ( 1 ) l × exp ( j m θ ) ( l ! ( l + | m | ) ! ) 1 / 2 × ( 2 r W f ) | m | L l ( | m | ) [ 2 ( r W f ) 2 ] exp [ ( r W f ) 2 ] ,
W f = λ 0 f / π W 0 .
E 1 ( r , θ ) = E 1 ( r ) = ( 2 π ) 1 / 2 1 W ( L 1 ) × exp [ j k 0 2 r 2 q ( L 1 ) + j tan 1 ( 2 L 1 k 0 W 0 2 ) ] ,
1 q ( L 1 ) = 1 R ( L 1 ) = j 2 k 0 W ( L 1 ) 2 ,
W ( L 1 ) 2 = W 0 2 [ 1 + ( 2 L 1 / k 0 W 0 2 ) 2 ] ,
R ( L 1 ) = L 1 [ 1 + ( k 0 W 0 2 / 2 L 1 ) 2 ] .
0 2 π 0 | E 1 ( r , θ ) | 2 r d r d θ = 1 .
C l 0 = 0 2 π 0 2 π E 1 ( r , θ ) ϕ l 0 * ( r , θ ) r d r d θ = 2 η exp [ j tan 1 ( 2 L 1 / k 0 W 0 2 ) ] ( η 2 + j k 0 W 2 / 2 R + 1 ) l ( η 2 + j k 0 W 2 / 2 R + 1 ) l + 1 ,
η = W / W 0 .
l max = 1 2 [ ( a / W ) 2 1 ] .
Δ θ θ 0 = π 2 ( 2 l max + 2 ) 1 / 2 .
s n 1 ( x 0 u | m ) = K ( m ) ,
u x 0 = 1 ,
K ( m ) = π 2 [ 1 + ( 1 2 ) 2 m + ( 1 3 2 4 ) 2 m 2 + ( 1 3 5 2 4 6 ) 2 m 3 + ] .
x = ( 1 / u ) s n ( υ z + K | m ) .
θ n ( x ) sin γ = n 0 c d x d y ,
d x d z = υ u s n ( υ z + K | m ) .
s n ( s | m ) = ( 1 + m 16 + 7 256 m 2 + ) sin q + ( m 16 + m 2 32 + ) sin 3 q + ( m 2 256 + ) sin 5 q + ,
q = π s / 2 K ,
s n ( υ z + K | m ) = π 2 K [ ( 1 + m 16 + 7 m 2 256 + ) sin Z + 3 ( m 16 + m 2 32 + ) sin 3 Z 5 ( m 2 256 + ) sin 5 Z + ] ,
Z = ( π / 2 K ) υ z .
θ sin θ = n 0 ( 1 c 2 ) 1 / 2 π 2 K [ ( 1 + m 16 + 7 m 2 256 + ) sin Z + 3 ( m 16 + m 2 32 + ) sin 3 Z 5 ( m 2 256 + ) sin 5 Z + ] .
1 n 0 2 d θ 2 d t = sin 2 Z + [ 3 ( sin 2 Z sin 4 Z ) h 3 2 ( h 2 3 ) g z sin Z cos Z ] t + .
t = 2 3 / [ ( h 2 3 ) g z sin Z cos Z ] .
E f ( r ) = exp [ ( r W f ) 2 ] f [ 2 ( r W f ) 2 ] ,
f [ 2 ( r W f ) 2 ] = l = 0 l max C l L l ( 0 ) [ 2 ( r W f ) 2 ] .
f ( x ) = k = 1 l max + 1 f ( α k ) L l max + 1 ( 0 ) ( x ) L l max + 1 ( 0 ) ( α k ) ( x α k ) ,
0 < α 1 < α 2 < α l max + 1 < ,
L l max + 1 ( 0 ) ( α k ) = d d x [ L l max + 1 ( 0 ) ( x ) ] x = α k .
E f ( r ) = k = 1 l max + 1 f ( α k ) S k ( r ) ,
S k ( r ) = L l max + 1 ( 0 ) [ 2 ( r W f ) 2 ] L l max + 1 ( 0 ) ( α k ) [ 2 ( r W f ) 2 α k ] exp [ ( r W f ) 2 ] ,
α k = 2 r k E f = 2 θ k θ 0 1 ( l max + 1 ) 1 / 2 ( π 2 ( k 1 4 ) 1 16 + O ( k 3 ) ) .
z p p [ 1 + 1 4 ( 3 h 2 ) ( g x 0 ) 2 ] , ( g x 0 1 )
p = 2 π / g .
g = 2 π / z p 0 .
z m 1 g tan 1 1 n 0 g ( L 1 2 + x 0 2 ) 1 / 2 ,
x m x 0 sec ( g z m ) ,
t = 1 n 2 ( x m ) / n 0 2
z = z z m ,
L 1 p / 4 .