Abstract

Some radiometric properties of gradient-index fiber lenses and lens arrays are explored. Consideration is restricted to the paraxial region of fibers that produce erect images at unit magnification. In two instances the radiometric properties of these lenses and lens arrays are considerably different from the properties of conventional (nongradient) lenses. First, the off-axis image plane irradiance for a single gradient-index fiber falls off far more rapidly than the familiar cos4 law. Second, the exposure (integrated image irradiance) for a fiber array is independent of the object-to-image distance.

© 1980 Optical Society of America

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References

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  1. E. G. Rawson, D. R. Herriott, J. McKenna, Appl. Opt. 9, 753 (1970).
    [CrossRef] [PubMed]
  2. Selfoc is a registered trade name of the Nippon Sheet Glass Company, Ltd., of Osaka, Japan.
  3. K. Matsushita, K. Ikeda, Proc. Soc. Photo-Opt. Instrum. Eng. 31, 23 (1972).
  4. Optical Society of America Topical Meeting on Gradient Index Optical Imaging Systems, U. Rochester, 15–16 May 1979.
  5. F. P. Kapron, J. Opt. Soc. Am. 60, 1433 (1970).
    [CrossRef]
  6. K. Matsushita, M. Toyama, in Digest of the Topical Meeting on Gradient Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1979), paper TuDl.
  7. N. Sullo, W. Lama, unpublished Xerox report.

1972 (1)

K. Matsushita, K. Ikeda, Proc. Soc. Photo-Opt. Instrum. Eng. 31, 23 (1972).

1970 (2)

Herriott, D. R.

Ikeda, K.

K. Matsushita, K. Ikeda, Proc. Soc. Photo-Opt. Instrum. Eng. 31, 23 (1972).

Kapron, F. P.

Lama, W.

N. Sullo, W. Lama, unpublished Xerox report.

Matsushita, K.

K. Matsushita, K. Ikeda, Proc. Soc. Photo-Opt. Instrum. Eng. 31, 23 (1972).

K. Matsushita, M. Toyama, in Digest of the Topical Meeting on Gradient Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1979), paper TuDl.

McKenna, J.

Rawson, E. G.

Sullo, N.

N. Sullo, W. Lama, unpublished Xerox report.

Toyama, M.

K. Matsushita, M. Toyama, in Digest of the Topical Meeting on Gradient Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1979), paper TuDl.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

K. Matsushita, K. Ikeda, Proc. Soc. Photo-Opt. Instrum. Eng. 31, 23 (1972).

Other (4)

Optical Society of America Topical Meeting on Gradient Index Optical Imaging Systems, U. Rochester, 15–16 May 1979.

Selfoc is a registered trade name of the Nippon Sheet Glass Company, Ltd., of Osaka, Japan.

K. Matsushita, M. Toyama, in Digest of the Topical Meeting on Gradient Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1979), paper TuDl.

N. Sullo, W. Lama, unpublished Xerox report.

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Figures (11)

Fig. 1
Fig. 1

Side view of a cylindrical GRIN optical fiber. The index of refraction varies radially with distance r from the optical axis. The case of unit magnification is illustrated, where k is the maximum field height for image-forming rays. Note the sinusoidal ray path, with coordinates z,x inside the fiber. Also note the inverted image midway through the fiber.

Fig. 2
Fig. 2

Plot of calculated vertex distance l and total conjugate 2l + L vs fiber length L at unit magnification. Note that a factor of 2 change in fiber length causes l to vary from zero to infinity. In this figure, n0 = 1.53 and (A)1/2 = 0.127 mm−1.

Fig. 3
Fig. 3

Side view of a GRIN fiber. For an object point (Lambertian source Δa) on the fiber axis, all image-forming rays are within the cone defined by the angle α.

Fig. 4
Fig. 4

Relative irradiance distribution in the image plane for a single GRIN fiber. The right abscissa is the position in the image plane measured from the optical axis. The left abscissa is the corresponding field angle θ. Theoretically, the irradiance distribution is ellipsoidal [Eq. (12)]. For this fiber, R = 0.48 mm, L = 31.0 mm, and (A)1/2 = 0.122 mm−1. These values yield a maximum field height k = 1.52 mm, in good agreement with the measured value. For comparison, the cos4θ falloff of a conventional lens is also plotted.

Fig. 5
Fig. 5

End view of a GRIN fiber. The irradiance profile in the x–y image plane is given by the ellipsoidal distribution [Eq. (12)]. Point p in the image plane moves at constant velocity υ through the irradiance distribution. The exposure received by this point is given by Eq.(14).

Fig. 6
Fig. 6

End view of a two-row Selfoc array. The center-to-center distance between adjacent fibers is 2bR. The irradiance boundary of an individual fiber is illustrated. A point p in the image plane moves at speed υ through the irradiance distribution composed of overlapping individual profiles. The exposure of this point is given by Eq.(16).

Fig. 7
Fig. 7

Measured image plane relative irradiance distribution vs distance perpendicular to the long dimension of a two-row Selfoc array. For this array R = 0.48 mm, L = 31.0 mm, b = 1.03, (A)1/2 = 0.122 mm−1. Theoretically, the total width of this distribution is given by 2k + (3)1/2bR = 3.9 mm, which is in good agreement with the measured width shown in this figure.

Fig. 8
Fig. 8

Calculated normalized image plane exposure E/∊ vs distance y along the rows of a two-row Selfoc array. Parameters: R = 0.5 mm, b = 1.0, a = 1.22. Note the good spatial uniformity (modulation = ±3%) in this case.

Fig. 9
Fig. 9

Calculated normalized exposure E/∊ for a two-row Selfoc array vs the overlap parameter a. Note that the exposure variation EmaxEmin generally decreases with increasing a. The average exposure 〈E〉 is independent of a. For these calculations, b = 1.0. Also note that 〈E〉 ≠ (Emax + Emin)/2.

Fig. 10
Fig. 10

Calculated normalized exposure vs object-to-image distance for a two-row Selfoc array with n0 = 1.53, b = 1.0, (A)1/2 = 0.127 mm−1, L is variable. Note that the average exposure is independent of object-to-image distance.

Fig. 11
Fig. 11

Calculated exposure modulation vs overlap parameter a for a two-row Selfoc array. Note that nonuniformity is minimized at discrete values of a given by Eq. (18). In this figure b = 1.0. The first minimum at a = 1.22 corresponds to the case plotted in Fig. 8.

Equations (22)

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n ( r ) = n 0 ( 1 A r 2 / 2 ) , A r 2 1 ,
l = tan [ ( A ) 1 / 2 L / 2 ] / n 0 ( A ) 1 / 2 , π ( A ) 1 / 2 L 2 π .
x ( z ) = R cos [ ( A ) 1 / 2 z ( A ) 1 / 2 L / 2 ] ,
x ( 0 ) = R cos [ ( A ) 1 / 2 L / 2 ] ,
sin θ = n 0 ( A ) 1 / 2 R sin [ ( A ) 1 / 2 L / 2 ] .
k = x ( 0 ) + l tan θ .
k = R sec [ ( A ) 1 / 2 L / 2 ] .
P = π N Δ a sin 2 ( α ) .
h 0 = TP Δ a = π NT sin 2 ( α ) .
tan α = n 0 ( A ) 1 / 2 R cos [ ( A ) 1 / 2 L / 2 ] .
h 0 = π N T n 0 2 A R 2 cos 2 [ ( A ) 1 / 2 L / 2 ] .
a = k / R = sec [ ( A ) 1 / 2 L / 2 ] .
h 0 = π N T n 0 2 A R 2 / a 2 .
( h / h 0 ) 2 + ( x / k ) 2 + ( y / k ) 2 = 1 , x 2 + y 2 k 2 .
e ( y ) = hdt = 1 υ x 0 x 0 h ( x , y ) d x ,
e ( y ) = π h 0 ( k 2 y 2 ) / 2 k υ , | y | k .
e ( y ) = ( π 2 NT n 0 2 A R 3 ) 2 υ ( k 2 y 2 ) k 3 R = ( k 2 y 2 ) k 3 R .
E ( y ) = R k 3 { m = 1 M 1 [ k 2 ( mbR y ) 2 ] + m = 0 M 2 [ k 2 ( mbR + y ) 2 ] } ,
M 1 = greatest integer < k + y bR , M 2 = greatest integer < k y bR .
E ( y ) = R k 3 { ( k 2 y 2 ) ( 1 + M 1 + M 2 ) + bRy [ M 1 ( 1 + M 1 ) M 2 ( 1 + M 2 ) ] 1 6 b 2 R 2 [ M 1 ( 1 + M 1 ) ( 1 + 2 M 1 ) + M 2 ( 1 + M 2 ) ( 1 + 2 M 2 ) ] } .
E = ( 4 3 ) b = 2 π 2 NT n 0 2 A R 3 3 υ b .
a = ( b 2 ) [ s ( s + 1 ) ] 1 / 2 , s = 2 , 3 , 4 . . . .

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