Abstract

The fourth-order index coefficients of GRIN-rod lenses are estimated using the magnitudes of the third-order wave-front spherical aberration reported by Cline and Jander and with the relationship between the third-order wave-front spherical aberration and the fourth-order index coefficient derived for the model of a near one-quarter pitch GRIN-rod lens in a double-pass configuration using the optical path length which is found analytically by algebraic manipulation of the asymptotic solution for the meridional ray equation derived by Streifer and Paxton.

© 1986 Optical Society of America

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References

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  1. W. J. Tomlinson, “Applications of GRIN-Rod Lenses in Optical Fiber Communication Systems,” Appl. Opt. 19, 1127 (1980).
    [CrossRef] [PubMed]
  2. K. Thyragarajan, A. Rohra, A. K. Ghatak, “Aberration Losses of the Microoptic Directional Coupler,” Appl. Opt. 19, 1061 (1980).
    [CrossRef]
  3. J. C. Palais, “Fiber Coupling using Graded-Index Lenses,” Appl. Opt. 19, 2011 (1980).
    [CrossRef] [PubMed]
  4. A. Nicia, “Lens Coupling in Fiber-Optic Devices: Efficiency Limits,” Appl. Opt. 20, 3136 (1981).
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  5. B. D. Metcalf, L. Jou, “Dual GRIN Lens Wavelength Multiplexer,” Appl. Opt. 22, 455 (1983).
    [CrossRef] [PubMed]
  6. E. G. Rawson, R. G. Murray, “Interferometric Measurement of Selfoc Dielectric Constant Coefficients to Sixth Order,” IEEE J. Quantum Electron. QE-9, 1114 (1973).
    [CrossRef]
  7. M. K. Maeda, J. Hamasaki, “Determining the Refractive Index Profile of a Lenslike Medium,” J. Opt. Soc. Am. 67, 1672 (1977).
    [CrossRef]
  8. I. Kitano, M. Toyama, H. Nishi, “Spherical Aberration of Gradient-Index Rod Lenses,” Appl. Opt. 22, 396 (1983).
    [CrossRef] [PubMed]
  9. T. Sakamoto, “GRIN Lens Profile Measurement by Ray Trace Analysis,” Appl. Opt. 22, 3064 (1983).
    [CrossRef] [PubMed]
  10. T. W. Cline, R. B. Jander, “Wave-Front Aberration Measurements on GRIN-Rod Lenses,” Appl. Opt. 21, 1035 (1982).
    [CrossRef] [PubMed]
  11. W. Streifer, K. B. Paxton, “Analytic Solution of Ray Equations in Cylindrically Inhomogeneous Guiding Media. 1: Meridional Rays,” Appl. Opt. 10, 769 (1971).
    [CrossRef] [PubMed]
  12. N. Yamamoto, K. Iga, “Evaluation of Gradient-Index Rod Lenses by Imaging,” Appl. Opt. 19, 1101 (1980).
    [CrossRef] [PubMed]
  13. T. Sakamoto, “Wave-Front Spherical Aberration Model of a GRIN-Rod Lens with Quadratic Index Profile,” Trans. IECE Jpn. J68-C, 87 (1985).
  14. T. Sakamoto, “Accuracy of Optical Path Lengths Computed with an Asymptotic Solution for a Meridional Ray Equation in GRIN-Rod Lenses,” Trans. IECE Jpn. J68-C, 519 (1985).

1985 (2)

T. Sakamoto, “Wave-Front Spherical Aberration Model of a GRIN-Rod Lens with Quadratic Index Profile,” Trans. IECE Jpn. J68-C, 87 (1985).

T. Sakamoto, “Accuracy of Optical Path Lengths Computed with an Asymptotic Solution for a Meridional Ray Equation in GRIN-Rod Lenses,” Trans. IECE Jpn. J68-C, 519 (1985).

1983 (3)

1982 (1)

1981 (1)

1980 (4)

1977 (1)

1973 (1)

E. G. Rawson, R. G. Murray, “Interferometric Measurement of Selfoc Dielectric Constant Coefficients to Sixth Order,” IEEE J. Quantum Electron. QE-9, 1114 (1973).
[CrossRef]

1971 (1)

Cline, T. W.

Ghatak, A. K.

Hamasaki, J.

Iga, K.

Jander, R. B.

Jou, L.

Kitano, I.

Maeda, M. K.

Metcalf, B. D.

Murray, R. G.

E. G. Rawson, R. G. Murray, “Interferometric Measurement of Selfoc Dielectric Constant Coefficients to Sixth Order,” IEEE J. Quantum Electron. QE-9, 1114 (1973).
[CrossRef]

Nicia, A.

Nishi, H.

Palais, J. C.

Paxton, K. B.

Rawson, E. G.

E. G. Rawson, R. G. Murray, “Interferometric Measurement of Selfoc Dielectric Constant Coefficients to Sixth Order,” IEEE J. Quantum Electron. QE-9, 1114 (1973).
[CrossRef]

Rohra, A.

Sakamoto, T.

T. Sakamoto, “Wave-Front Spherical Aberration Model of a GRIN-Rod Lens with Quadratic Index Profile,” Trans. IECE Jpn. J68-C, 87 (1985).

T. Sakamoto, “Accuracy of Optical Path Lengths Computed with an Asymptotic Solution for a Meridional Ray Equation in GRIN-Rod Lenses,” Trans. IECE Jpn. J68-C, 519 (1985).

T. Sakamoto, “GRIN Lens Profile Measurement by Ray Trace Analysis,” Appl. Opt. 22, 3064 (1983).
[CrossRef] [PubMed]

Streifer, W.

Thyragarajan, K.

Tomlinson, W. J.

Toyama, M.

Yamamoto, N.

Appl. Opt. (10)

IEEE J. Quantum Electron. (1)

E. G. Rawson, R. G. Murray, “Interferometric Measurement of Selfoc Dielectric Constant Coefficients to Sixth Order,” IEEE J. Quantum Electron. QE-9, 1114 (1973).
[CrossRef]

J. Opt. Soc. Am. (1)

Trans. IECE Jpn. (2)

T. Sakamoto, “Wave-Front Spherical Aberration Model of a GRIN-Rod Lens with Quadratic Index Profile,” Trans. IECE Jpn. J68-C, 87 (1985).

T. Sakamoto, “Accuracy of Optical Path Lengths Computed with an Asymptotic Solution for a Meridional Ray Equation in GRIN-Rod Lenses,” Trans. IECE Jpn. J68-C, 519 (1985).

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Figures (4)

Fig. 1
Fig. 1

Model for the wave-front spherical aberration analysis of a near one-quarter pitch GRIN-rod lens in a double-pass configuration.

Fig. 2
Fig. 2

Accuracy of the derived optical path length. Solid and dashed curves denote the normalized maximum errors gΔl/n0 for |gxi| ≤ 0.2 and γi = 0 and for gxi = 0 and |sinγi| ≤ 0.2, respectively.

Fig. 3
Fig. 3

Fitting of the derived optical path length to the measured phase shifts: ●, measured data by Rawson and Murray.6 The solid curve is the theoretical phase shifts calculated by Eq. (5) with g = 0.2135 mm−1, h4 = 1.36, h6 = −3.0, Z = 14.7 mm, and λ = 0.6328 μm. The theoretical interference pattern in a double-pass configuration is also shown in the lower part of the figure.

Fig. 4
Fig. 4

Polynomial fitting for the measured data: ● is as in Fig. 3. (a), (b), (c), and (d) are the results of the least-squares analysis using polynomials of degrees two, four, six, and eight, respectively. The corresponding rms deviations σN are 0.968, 0.045, 0.037, and 0.045 rad. The best fitting polynomial is shown as a solid curve. The magnitudes of the terms composing the polynomial are also shown as the dashed curves.

Tables (2)

Tables Icon

Table I Measured Parameters (λ = 0.6328 μm)a

Equations (18)

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n 2 ( r ) = n 0 2 [ 1 - ( g r ) 2 + h 4 ( g r ) 4 + h 6 ( g r ) 6 + ] ,
g x = { g x i + h 4 2 4 [ ( g x i ) 3 - 3 g x i sin 2 γ i + 3 ( g x i ) 3 sin 2 γ i ] + h 4 2 2 8 [ 23 ( g x i ) 5 - 170 ( g x i ) 3 sin 2 γ i - 113 g x i sin 4 γ i ] + h 6 2 8 [ 32 ( g x i ) 5 - 80 ( g x i ) 3 sin 2 γ i - 80 g x i sin 4 γ i ] } cos Ω z + { sin γ i - ( g x i ) 2 2 sin γ i + ( h 4 2 - 1 8 ) ( g x i ) 4 sin γ i + h 4 2 4 [ 9 sin 3 γ i + 21 ( g x i ) 2 sin γ i - 27 2 ( g x i ) 2 sin 3 γ i - 21 2 ( g x i ) 4 sin γ i ] + h 4 2 2 8 [ 271 sin 5 γ i + 854 ( g x i ) 2 sin 3 γ i + 599 ( g x i ) 4 sin γ i ] + h 6 2 8 [ 160 sin 5 γ i + 560 ( g x i ) 2 sin 3 γ i + 560 ( g x i ) 4 sin γ i ] } sin Ω z - { h 4 2 4 [ ( g x i ) 3 - 3 g x i sin 2 γ i + 3 ( g x i ) 3 sin 2 γ i ] + h 4 2 2 8 [ 24 ( g x i ) 5 - 180 ( g x i ) 3 sin 2 γ i - 108 g x i sin 4 γ i ] + h 6 2 8 [ 30 ( g x i ) 5 - 60 ( g x i ) 3 sin 2 γ i - 90 g x i sin 4 γ i ] } cos 3 Ω z + { h 4 2 4 [ sin 3 γ i - 3 ( g x i ) 2 sin γ i - 3 2 ( g x i ) 2 sin 3 γ i + 3 2 ( g x i ) 4 sin γ i ] + h 4 2 2 8 [ 48 sin 5 γ i + 12 ( g x i ) 2 sin 3 γ i - 132 ( g x i ) 4 sin γ i ] + h 6 2 8 [ 30 sin 5 γ i - 60 ( g x i ) 2 sin 3 γ i - 90 ( g x i ) 4 sin γ i ] } sin 3 Ω z + 1 2 8 ( h 4 2 - 2 h 6 ) [ ( g x i ) 5 - 10 ( g x i ) 3 sin 2 γ i + 5 g x i sin 4 γ i ] cos 5 Ω z + 1 2 8 ( h 4 2 - 2 h 6 ) [ sin 5 γ i - 10 ( g x i ) 2 sin 3 γ i + 5 ( g x i ) 4 sin γ i ] sin 5 Ω z ,
Ω g = 1 - 3 4 ( h 4 - 2 3 ) [ ( g x i ) 2 + sin 2 γ i ] - [ 21 64 ( h 4 - 2 3 ) 2 + 21 16 ( h 4 - 2 3 ) ] ( g x i ) 4 - [ 69 32 ( h 4 - 2 3 ) 2 + 23 8 ( h 4 - 2 3 ) ] ( g x i ) 2 sin 2 γ i - [ 69 64 ( h 4 - 2 3 ) 2 + 29 16 ( h 4 - 2 3 ) ] sin 4 γ i - 15 16 ( h 6 + 17 15 ) [ ( g x i ) 2 + sin 2 γ i ] 2 .
l = 0 z n 2 ( r ) n i cos γ i d z .
n 2 ( z ) n 0 2 = 1 - 1 2 [ ( g x i ) 2 + sin 2 γ i ] + ( g x i ) 2 2 sin 2 γ i + h 4 2 4 [ 5 ( g x i ) 4 - 6 ( g x i ) 2 sin 2 γ i - 3 sin 4 γ i - 2 ( g x i ) 4 sin 2 γ i + 6 ( g x i ) 2 sin 4 γ i ] - h 4 2 2 8 [ 8 ( g x i ) 6 + 216 ( g x i ) 4 sin 2 γ i + 312 ( g x i ) 2 sin 4 γ i + 104 sin 6 γ i ] + h 6 2 8 [ 48 ( g x i ) 6 - 240 ( g x i ) 4 sin 2 γ i - 240 ( g x i ) 2 sin 4 γ i - 80 sin 6 γ i ] + { - 1 2 [ ( g x i ) 2 - sin 2 γ i ] - ( g x i ) 2 2 sin 2 γ i + h 4 2 4 [ 8 ( g x i ) 4 + 24 ( g x i ) 2 sin 2 γ i - 16 ( g x i ) 4 sin 2 γ i ] + h 4 2 2 9 [ 19 ( g x i ) 6 + 1763 ( g x i ) 4 sin 2 γ i + 797 ( g x i ) 2 sin 4 γ i - 19 sin 6 γ i ] + h 6 2 8 [ 118 ( g x i ) 6 + 598 ( g x i ) 4 sin 2 γ i + 682 ( g x i ) 2 sin 4 γ i + 10 sin 6 γ i ] } cos 2 Ω z + { - g x i sin γ i + ( g x i ) 3 2 sin γ i + ( g x i ) 5 8 sin γ i + h 4 2 4 [ - 4 ( g x i ) 3 sin γ i + 12 g x i sin 3 γ i - 6 ( g x i ) 5 sin γ i + 18 ( g x i ) 3 sin 3 γ i ] + h 4 2 2 8 [ - 217 ( g x i ) 5 sin γ i + 510 ( g x i ) 3 sin 3 γ i + 295 g x i sin 5 γ i ] + h 6 2 8 [ - 292 ( g x i ) 5 sin γ i - 264 ( g x i ) 3 sin 3 γ i + 220 g x i sin 5 γ i ] } sin 2 Ω z + { h 4 2 4 [ 3 ( g x i ) 4 - 18 ( g x i ) 2 sin 2 γ i + 3 sin 4 γ i + 18 ( g x i ) 4 sin 2 γ i - 6 ( g x i ) 2 sin 4 γ i ] + h 4 2 2 8 [ 8 ( g x i ) 6 - 808 ( g x i ) 4 sin 2 γ i + 56 ( g x i ) 2 sin 4 γ i + 104 sin 6 γ i ] + h 6 2 8 [ 80 ( g x i ) 6 - 400 ( g x i ) 4 sin 2 γ i - 400 ( g x i ) 2 sin 4 γ i + 80 sin 6 γ i ] } cos 4 Ω z + { h 4 2 4 [ 12 ( g x i ) 3 sin γ i - 12 g x i sin 3 γ i - 6 ( g x i ) 5 sin γ i + 18 ( g x i ) 3 sin 3 γ i ] + h 4 2 2 4 [ 272 ( g x i ) 5 sin γ i - 768 ( g x i ) 3 sin 3 γ i - 272 g x i sin 5 γ i ] + h 6 2 8 [ 320 ( g x i ) 5 sin γ i - 320 g x i sin 5 γ i ] } sin 4 Ω z + { - h 4 2 2 9 [ 19 ( g x i ) 6 - 285 ( g x i ) 4 sin 2 γ i + 285 ( g x i ) 2 sin 4 γ i - 19 sin 6 γ i ] + h 6 2 8 [ 10 ( g x i ) 6 - 150 ( g x i ) 4 sin 2 γ i + 150 ( g x i ) 2 sin 4 γ i - 10 sin 6 γ i ] } cos 6 Ω z + { - h 4 2 2 8 [ 57 ( g x i ) 5 sin γ i - 190 ( g x i ) 3 sin 3 γ i + 57 g x i sin 5 γ i ] + h 6 2 8 [ 60 ( g x i ) 5 sin γ i - 200 ( g x i ) 3 sin 3 γ i + 60 g x i sin 5 γ i ] } sin 6 Ω z .
g l n 0 = { 1 - 3 16 ( h 4 - 2 3 ) [ ( g x i ) 2 + sin 2 γ i ] 2 - [ 1 32 ( h 4 - 2 3 ) 2 + 1 96 ( h 4 - 2 3 ) + 5 16 ( h 6 + 17 45 ) ] ( g x i ) 6 - [ 27 32 ( h 4 - 2 3 ) 2 + 41 32 ( h 4 - 2 3 ) + 15 16 ( h 6 - 17 45 ) ] ( g x i ) 4 sin 2 γ i - [ 39 32 ( h 4 - 2 3 ) 2 + 49 32 ( h 4 - 2 3 ) + 15 16 ( h 6 - 17 45 ) ] ( g x i ) 2 sin 4 γ i - [ 13 32 ( h 4 - 2 3 ) 2 + 61 96 ( h 4 - 2 3 ) + 5 16 ( h 6 - 17 45 ) ] sin 6 γ i } g z + { - 1 4 [ ( g x i ) 2 - sin 2 γ i ] - ( g x i ) 2 4 sin 2 γ i + h 4 2 4 [ ( g x i ) 4 + 12 ( g x i ) 2 sin 2 γ i + 3 sin 4 γ i - 8 ( g x i ) 4 sin 2 γ i - 6 ( g x i ) 2 sin 4 γ i ] + h 4 2 2 10 [ - 17 ( g x i ) 6 + 1436 ( g x i ) 4 sin 2 γ i + 1793 ( g x i ) 2 sin 4 γ i + 401 sin 6 γ i ] + h 6 2 8 [ - ( g x i ) 6 + 239 ( g x i ) 4 sin 2 γ i + 401 ( g x i ) 2 sin 4 γ i + 65 sin 6 γ i ] } sin 2 Ω z + { g x i 2 sin γ i - ( g x i ) 3 4 sin γ i - ( g x i ) 5 16 sin γ i + h 4 2 4 [ 8 ( g x i ) 3 sin γ i - 18 ( g x i ) 3 sin 3 γ i ] + h 4 2 2 9 [ 517 ( g x i ) 5 sin γ i + 258 ( g x i ) 3 sin 3 γ i - 19 g x i sin 5 γ i ] + h 6 2 8 [ 532 ( g x i ) 5 sin γ i + 744 ( g x i ) 3 sin 3 γ i + 20 g x i sin 5 γ i } ( cos 2 Ω z - 1 ) + { h 4 2 6 [ 3 ( g x i ) 4 - 18 ( g x i ) 2 sin 2 γ i + 3 sin 4 γ i + 18 ( g x i ) 4 sin 2 γ i - 6 ( g x i ) 2 sin 4 γ i ] + h 4 2 2 8 [ 11 ( g x i ) 6 - 247 ( g x i ) 4 sin 2 γ i - 31 ( g x i ) 2 sin 4 γ i + 35 sin 6 γ i ] + h 6 2 8 [ 20 ( g x i ) 6 - 100 ( g x i ) 4 sin 2 γ i - 100 ( g x i ) 2 sin 4 γ i + 20 sin 6 γ i } sin 4 Ω z - { h 4 2 5 [ 6 ( g x i ) 3 sin γ i - 6 g x i sin 3 γ i - 3 ( g x i ) 5 sin γ i + 9 ( g x i ) 3 sin 3 γ i ] + h 4 2 2 8 [ 104 ( g x i ) 5 sin γ i - 192 ( g x i ) 3 sin 3 γ i - 104 g x i sin 5 γ i ] + h 6 2 8 [ 80 ( g x i ) 5 sin γ i - 80 g x i sin 5 γ i ] } ( cos 4 Ω z - 1 ) + { - h 4 2 3 · 2 10 [ 19 ( g x i ) 6 - 285 ( g x i ) 4 sin 2 γ i + 285 ( g x i ) 2 sin 4 γ i - 19 sin 6 γ i ] + h 6 3 · 2 8 [ 5 ( g x i ) 6 - 75 ( g x i ) 4 sin 2 γ i + 75 ( g x i ) 2 sin 4 γ i - 5 sin 6 γ i ] } sin 6 Ω z + { h 4 2 3 · 2 9 [ 57 ( g x i ) 5 sin γ i - 190 ( g x i ) 3 sin 3 γ i + 57 g x i sin 5 γ i ] - h 6 3 · 2 8 [ 30 ( g x i ) 5 sin γ i - 100 ( g x i ) 3 sin 3 γ i + 30 g x i sin 5 γ i ] } ( cos 6 Ω z - 1 ) .
n 2 ( r ) = n 0 2 sech 2 g r = n 0 2 [ 1 - ( g r ) 2 + 2 3 ( g r ) 4 - 17 45 ( g r ) 6 + ] , g l s n 0 = arctan { 1 2 cosh g x i cos γ i [ ( cosh 2 g x i + cos 2 γ i ) × tan ( g z + κ 2 ) + sinh 2 g x i + sin 2 γ i ] } - arctan { 1 2 cosh g x i cos γ i [ ( cosh 2 g x i + cos 2 γ i ) tan κ 2 + sinh 2 g x i + sin 2 γ i ] } ,
tan κ = sinh 2 g x i cos 2 γ i - cosh 2 g x i sin 2 γ i cosh g x i sin 2 γ i ,
n 2 ( r ) = n 0 2 [ 1 - ( g r ) 2 ] , g l q n 0 = 1 2 cos γ i [ 2 - ( g x i ) 2 1 - ( g x i ) 2 - 1 - ( g x i ) 2 sin 2 γ i ] g z - 1 4 { ( g x i ) 2 - [ 1 - ( g x i ) 2 ] sin 2 γ i } sin 2 Ω q z - g x i 2 1 - ( g x i ) 2 sin γ i ( cos 2 Ω q z - 1 ) ,
Ω q g = 1 1 - ( g x i ) 2 cos γ i .
g l i n n 0 = π 2 + g Δ Z + 3 π 32 ( h 4 - 2 3 ) sin 4 γ i - g Δ Z ( 1 2 sin 2 γ i + 1 8 sin 4 γ i ) .
g x r e = sin γ i + h 4 2 sin 3 γ i + 3 π 4 ( h 4 - 2 3 ) g d n 0 sin 3 γ i + g Δ Z [ - 2 g d n 0 sin γ i + 3 π 8 ( h 4 - 2 3 ) sin 3 γ i + ( 3 h 4 - 1 ) g d n 0 sin 3 γ i ] , sin γ r e = 3 π 8 ( h 4 - 2 3 ) sin 3 γ i + g Δ Z [ - sin γ i + 3 2 ( h 4 - 2 3 ) sin 3 γ i ] .
g l r e n 0 = π 2 + g Δ Z - 27 π 32 ( h 4 - 3 2 ) sin 4 γ i + g Δ Z [ 2 sin 2 γ i + ( - h 4 2 + 11 8 ) sin 4 γ i + 21 π 4 ( h 4 - 2 3 ) g d n 0 sin 4 γ i ] .
g W lens n 0 = - 3 π 4 ( h 4 - 2 3 ) sin 4 γ i + g Δ Z [ 3 2 sin 2 γ i + ( - h 4 2 + 5 4 ) sin 4 γ i + 21 π 4 ( h 4 - 2 3 ) g d n 0 sin 4 γ i ] .
g W ref n 0 = - g Δ Z 3 π 4 ( h 4 - 2 3 ) g d n 0 sin 4 γ i .
g W n 0 = - 3 π 4 ( h 4 - 2 3 ) sin 4 γ i + g Δ Z [ 3 2 sin 2 γ i + ( - h 4 2 + 5 4 ) sin 4 γ i + 9 π 2 ( h 4 - 2 3 ) g d n 0 sin 4 γ i ]
h 4 = g W 040 n 0 ( n 0 N . A ) 4 - π 2 - g Δ Z ( 5 4 - 3 π g d n 0 ) - 3 π 4 + g Δ Z ( - 1 2 + 9 π 2 g d n 0 ) .
Δ φ = 2 π n 0 g λ [ - 9 π 8 ( h 4 - 2 3 ) sin 4 γ i + g Δ Z ( sin 2 γ i + 3 4 sin 4 γ i ) ] ,

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