Abstract

A nondestructive measurement method that makes possible the measurement of a three-dimensional refractive-index distribution of any shape plastic lens is presented. In this method, a Mach–Zehnder interferometer and shearing interferometer are combined into a single optical system and are used selectively. Interference fringes of a test object that is immersed in matching liquid are detected at various rotation angles. And transmitted wave fronts are calculated with these interference fringes. Finally, the refractive-index distribution is obtained by computed tomography analysis. In addition, accurate control of the matching liquid temperature makes it possible to measure the absolute refractive index of the test object. This system has good performance with a measurement accuracy of 10-4 or better peak to valley.

© 2002 Optical Society of America

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References

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  1. Y. Otani, T. Shimada, T. Yoshizawa, N. Umeda, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33, 1604–1609 (1994).
    [CrossRef]
  2. L. A. Gerasimova, “Interferometric measurement of the refractive-index gradient distribution in gradient-index optical blanks,” Appl. Opt. 35, 2997–3000 (1996).
    [CrossRef] [PubMed]
  3. J. Radon, “Über die bestimmung von funktionen durch ihre integralwerte längs gewisser mannigfaltigkeiten,” Ber. Verh. Saechs. Akad. Wiss. Leipzig Math. Phys. Kl. 69, 262–267 (1917).
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    [CrossRef]
  5. Y. Maruyama, K. Iwata, R. Nagata, “Measurement of refractive index distribution in the interior of a solid object from multi-directional interferograms,” Jpn. J. Appl. Phys. 16, 1171–1176 (1977).
    [CrossRef]
  6. H. Suhara, T. Nakase, “Refractive-index distributions interferometry in optical lenses using computed tomography,” in Japan Optics ’96 Annual Meeting (Optical Society of Japan, Tokyo, Japan), pp. 331–332 (in Japanese).
  7. H. Suhara, S. Itoh, T. Ueda, T. Ito, “Development of the refractive index measuring interferometer (RIMEI),” Ricoh Tech. Rep. 25 (Ricoh Co., Ltd., Tokyo, Japan, 1999), pp. 120–124, in Japanese.
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  11. G. W. R. Leibbrandt, G. Harbers, P. J. Kunst, “Wave-front analysis with high accuracy by use of a double-grating lateral shearing interferometer,” Appl. Opt. 35, 6151–6161 (1996).
    [CrossRef] [PubMed]
  12. K. Iwata, H. Kikuta, “Measurement of dynamic flow field by optical computed tomography with shearing interferometers,” Opt. Rev. 7, 415–419 (2000).
    [CrossRef]
  13. L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).
    [CrossRef]

2000 (1)

K. Iwata, H. Kikuta, “Measurement of dynamic flow field by optical computed tomography with shearing interferometers,” Opt. Rev. 7, 415–419 (2000).
[CrossRef]

1996 (2)

1995 (1)

1994 (1)

Y. Otani, T. Shimada, T. Yoshizawa, N. Umeda, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33, 1604–1609 (1994).
[CrossRef]

1991 (1)

1988 (1)

R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[CrossRef]

1977 (1)

Y. Maruyama, K. Iwata, R. Nagata, “Measurement of refractive index distribution in the interior of a solid object from multi-directional interferograms,” Jpn. J. Appl. Phys. 16, 1171–1176 (1977).
[CrossRef]

1974 (1)

L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).
[CrossRef]

1973 (1)

G. N. Hounsfield, “Computerized transverse axial scanning tomography: Part I. Description of system,” J. Radiol. 46, 1016–1022 (1973).
[CrossRef]

1917 (1)

J. Radon, “Über die bestimmung von funktionen durch ihre integralwerte längs gewisser mannigfaltigkeiten,” Ber. Verh. Saechs. Akad. Wiss. Leipzig Math. Phys. Kl. 69, 262–267 (1917).

Bone, D. J.

Farrant, D. I.

Gerasimova, L. A.

Goldstein, R. M.

R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[CrossRef]

Harbers, G.

Hibino, K.

Hounsfield, G. N.

G. N. Hounsfield, “Computerized transverse axial scanning tomography: Part I. Description of system,” J. Radiol. 46, 1016–1022 (1973).
[CrossRef]

Ito, T.

H. Suhara, S. Itoh, T. Ueda, T. Ito, “Development of the refractive index measuring interferometer (RIMEI),” Ricoh Tech. Rep. 25 (Ricoh Co., Ltd., Tokyo, Japan, 1999), pp. 120–124, in Japanese.

Itoh, S.

H. Suhara, S. Itoh, T. Ueda, T. Ito, “Development of the refractive index measuring interferometer (RIMEI),” Ricoh Tech. Rep. 25 (Ricoh Co., Ltd., Tokyo, Japan, 1999), pp. 120–124, in Japanese.

Iwata, K.

K. Iwata, H. Kikuta, “Measurement of dynamic flow field by optical computed tomography with shearing interferometers,” Opt. Rev. 7, 415–419 (2000).
[CrossRef]

Y. Maruyama, K. Iwata, R. Nagata, “Measurement of refractive index distribution in the interior of a solid object from multi-directional interferograms,” Jpn. J. Appl. Phys. 16, 1171–1176 (1977).
[CrossRef]

Kikuta, H.

K. Iwata, H. Kikuta, “Measurement of dynamic flow field by optical computed tomography with shearing interferometers,” Opt. Rev. 7, 415–419 (2000).
[CrossRef]

Kunst, P. J.

Larkin, K. G.

Leibbrandt, G. W. R.

Logan, B. F.

L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).
[CrossRef]

Maruyama, Y.

Y. Maruyama, K. Iwata, R. Nagata, “Measurement of refractive index distribution in the interior of a solid object from multi-directional interferograms,” Jpn. J. Appl. Phys. 16, 1171–1176 (1977).
[CrossRef]

Nagata, R.

Y. Maruyama, K. Iwata, R. Nagata, “Measurement of refractive index distribution in the interior of a solid object from multi-directional interferograms,” Jpn. J. Appl. Phys. 16, 1171–1176 (1977).
[CrossRef]

Nakase, T.

H. Suhara, T. Nakase, “Refractive-index distributions interferometry in optical lenses using computed tomography,” in Japan Optics ’96 Annual Meeting (Optical Society of Japan, Tokyo, Japan), pp. 331–332 (in Japanese).

Oreb, B. F.

Otani, Y.

Y. Otani, T. Shimada, T. Yoshizawa, N. Umeda, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33, 1604–1609 (1994).
[CrossRef]

Radon, J.

J. Radon, “Über die bestimmung von funktionen durch ihre integralwerte längs gewisser mannigfaltigkeiten,” Ber. Verh. Saechs. Akad. Wiss. Leipzig Math. Phys. Kl. 69, 262–267 (1917).

Shepp, L. A.

L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).
[CrossRef]

Shimada, T.

Y. Otani, T. Shimada, T. Yoshizawa, N. Umeda, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33, 1604–1609 (1994).
[CrossRef]

Suhara, H.

H. Suhara, S. Itoh, T. Ueda, T. Ito, “Development of the refractive index measuring interferometer (RIMEI),” Ricoh Tech. Rep. 25 (Ricoh Co., Ltd., Tokyo, Japan, 1999), pp. 120–124, in Japanese.

H. Suhara, T. Nakase, “Refractive-index distributions interferometry in optical lenses using computed tomography,” in Japan Optics ’96 Annual Meeting (Optical Society of Japan, Tokyo, Japan), pp. 331–332 (in Japanese).

Ueda, T.

H. Suhara, S. Itoh, T. Ueda, T. Ito, “Development of the refractive index measuring interferometer (RIMEI),” Ricoh Tech. Rep. 25 (Ricoh Co., Ltd., Tokyo, Japan, 1999), pp. 120–124, in Japanese.

Umeda, N.

Y. Otani, T. Shimada, T. Yoshizawa, N. Umeda, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33, 1604–1609 (1994).
[CrossRef]

Werner, C. L.

R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[CrossRef]

Yoshizawa, T.

Y. Otani, T. Shimada, T. Yoshizawa, N. Umeda, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33, 1604–1609 (1994).
[CrossRef]

Zebker, H. A.

R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[CrossRef]

Appl. Opt. (3)

Ber. Verh. Saechs. Akad. Wiss. Leipzig Math. Phys. Kl. (1)

J. Radon, “Über die bestimmung von funktionen durch ihre integralwerte längs gewisser mannigfaltigkeiten,” Ber. Verh. Saechs. Akad. Wiss. Leipzig Math. Phys. Kl. 69, 262–267 (1917).

IEEE Trans. Nucl. Sci. (1)

L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Radiol. (1)

G. N. Hounsfield, “Computerized transverse axial scanning tomography: Part I. Description of system,” J. Radiol. 46, 1016–1022 (1973).
[CrossRef]

Jpn. J. Appl. Phys. (1)

Y. Maruyama, K. Iwata, R. Nagata, “Measurement of refractive index distribution in the interior of a solid object from multi-directional interferograms,” Jpn. J. Appl. Phys. 16, 1171–1176 (1977).
[CrossRef]

Opt. Eng. (1)

Y. Otani, T. Shimada, T. Yoshizawa, N. Umeda, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33, 1604–1609 (1994).
[CrossRef]

Opt. Rev. (1)

K. Iwata, H. Kikuta, “Measurement of dynamic flow field by optical computed tomography with shearing interferometers,” Opt. Rev. 7, 415–419 (2000).
[CrossRef]

Radio Sci. (1)

R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[CrossRef]

Other (2)

H. Suhara, T. Nakase, “Refractive-index distributions interferometry in optical lenses using computed tomography,” in Japan Optics ’96 Annual Meeting (Optical Society of Japan, Tokyo, Japan), pp. 331–332 (in Japanese).

H. Suhara, S. Itoh, T. Ueda, T. Ito, “Development of the refractive index measuring interferometer (RIMEI),” Ricoh Tech. Rep. 25 (Ricoh Co., Ltd., Tokyo, Japan, 1999), pp. 120–124, in Japanese.

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Figures (13)

Fig. 1
Fig. 1

Mach–Zehnder interferometer system for measurement of the refractive-index distribution: ND, neutral-density filter; M1–M5, mirrors; POL, polarizing plate; NPBS1–NPBS4, nonpolarizing beam splitters; FL1–FL3, image-forming lenses.

Fig. 2
Fig. 2

Projection slice theorem. G(ζ, θ) agrees with F(ζ cos θ, ζ sin θ). FT, Fourier transform.

Fig. 3
Fig. 3

Temperature control unit.

Fig. 4
Fig. 4

Schematic diagram of the dual-interferometer system: M3-1 and M3-2, shearing beam generating plates; M6, mirror; FLx_1 and FLx_2, image-forming lenses with the same optical performance.

Fig. 5
Fig. 5

Optical plates for generating shearing beams: inside surfaces of M3-1 and M3-2, reflectivity of 13% and 17%, respectively; B1 and B2, shearing beam in a light-intensity ratio of 1:1; B3, surface-reflection ghost; B4, multiple-reflection ghost; B5, rear surface-reflection ghost; S, distance between two surfaces.

Fig. 6
Fig. 6

RIMEI system.

Fig. 7
Fig. 7

Experimental results without a test object: (a) interference fringe pattern and (b) measured wave front of the cross section from the center in (a).

Fig. 8
Fig. 8

Temperature characteristics of the matching liquid. The horizontal axis represents the refractive index of reference glass objects. The accuracy is 10-5. Glass objects were semicircular rods of 15-mm radius. The vertical axis represents the measurement results of the matching liquid temperature at which transmitted wave-front distortion is at minimum. The accuracy is 0.1 °C.

Fig. 9
Fig. 9

Schematic illustration of sample A.

Fig. 10
Fig. 10

Experimental results of sample A: (a) interference fringe pattern and (b) CT image reconstruction of the refractive-index distribution of the cross section indicated by the broken line in Fig. 9. The peak to valley of the refractive-index distribution in effective area was 47.7 × 10-5.

Fig. 11
Fig. 11

Experimental results of the CT image reconstruction of the refractive-index distribution for sample B. The peak-to-valley value was 21.5 × 10-5.

Fig. 12
Fig. 12

Experimental results for sample C obtained with the lateral shearing interferometer: (a) interference fringe pattern and (b) CT image reconstruction of the refractive-index distribution. The peak-to-valley value was 277.3 × 10-5.

Fig. 13
Fig. 13

Interference fringe pattern for sample C obtained with the Mach–Zehnder interferometer.

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

gX, θ=-+ fx, ydY,
gX, θ=-+-+ fx, yδX-x cos θ-y sin θdxdy.
Gζ, θ=-+ gX, θexp-iXζdX.
Gζ, θ=-+-+ fx, yexp-iζx cos θ+y sin θdxdy.
Fξ, η=-+-+ fx, yexp-iξx+ηydxdy.
Fζ cos θ, ζ sin θ=-+-+ fx, yexp-iζx cos θ+y sin θdxdy.
Fζ cos θ, ζ sin θ=Gζ, θ.
fx, y=12π2-+-+ Fξ, ηexpiξx+ηydξdη.
fx, y=12π20π-+ Gζ, θexp-iXζ|ζ|dζdθ.
Hζ, θ gX, θXexp-iXζdX,
Hζ, θ=-iζGζ, θ.
fx, y=12π2 i 0π-+ Hζ, θexp-iXζ×sgnζdζdθ,
sgnζ|ζ|/ζ.
n=-0.000411T+1.58956.

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