Abstract

A phase shifting interferometer using a tunable laser as a light source is proposed for measuring shapes of both surfaces of a glass plate and the distribution of refractive index. To separate the superimposed interferograms generated with many wavefronts reflected from the plate, the phase shift associated with the wavelength shift is applied in the phase shifting interferometer with unequal optical paths in testing and reference beams. A laser diode is used for the tunable light source, and the data processing for obtaining phase distribution is based on the least-squares fitting in interferograms. The rms errors of the measurements are <1/50 wavelength for the surface shape, and 10−5 of the refractive index for a 5-mm thick optical glass plate.

© 1990 Optical Society of America

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References

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  1. Y. Ishii, J. Chen, K. Murata, “Digital Phase-Measureing Interferometry with a Tunable Laser Diode,” Opt. Lett. 12, 233–235 (1987).
    [CrossRef] [PubMed]
  2. A. Valentin, C. Nicolas, L. Henry, A. W. Mantz, “Tunable Diode Laser Control by a Stepping Michelson Interferometer,” Appl. Opt. 26, 41–46 (1987).
    [CrossRef] [PubMed]
  3. K. Tatsuno, Y. Tsunoda, “Diode Laser Direct Modulation Heterodyne Interferometer,” Appl. Opt. 26, 37–40 (1987).
    [CrossRef] [PubMed]
  4. J. Chen, Y. Ishii, M. Murata, “Heterodyne Interferometry with a Frequency-Modulated Laser Diode,” Appl. Opt. 27, 124–128 (1988).
    [CrossRef] [PubMed]
  5. J. E. Greivenkamp, “Generalized Data Reduction for Heterodyne Interferometry,” Opt. Eng. 23, 350–352 (1984).
    [CrossRef]
  6. K. Okada, J. Tujiuchi, “Wavelength Scanning Interferometry for the Measurement of Both Surface Shapes and Refractive Index Inhomogeneity,” Proc. Soc. Photo-Opt. Instrum. Eng. 1162, 395–401 (1989).
  7. K. Okada, J. Tsujiuchi, “Error Analysis on the Phase Calculation from Superposed Interferograms Generated by a Wavelength Scanning Interferometer,” Opt. Commun.76, (1990), in press.

1989

K. Okada, J. Tujiuchi, “Wavelength Scanning Interferometry for the Measurement of Both Surface Shapes and Refractive Index Inhomogeneity,” Proc. Soc. Photo-Opt. Instrum. Eng. 1162, 395–401 (1989).

1988

1987

1984

J. E. Greivenkamp, “Generalized Data Reduction for Heterodyne Interferometry,” Opt. Eng. 23, 350–352 (1984).
[CrossRef]

Chen, J.

Greivenkamp, J. E.

J. E. Greivenkamp, “Generalized Data Reduction for Heterodyne Interferometry,” Opt. Eng. 23, 350–352 (1984).
[CrossRef]

Henry, L.

Ishii, Y.

Mantz, A. W.

Murata, K.

Murata, M.

Nicolas, C.

Okada, K.

K. Okada, J. Tujiuchi, “Wavelength Scanning Interferometry for the Measurement of Both Surface Shapes and Refractive Index Inhomogeneity,” Proc. Soc. Photo-Opt. Instrum. Eng. 1162, 395–401 (1989).

K. Okada, J. Tsujiuchi, “Error Analysis on the Phase Calculation from Superposed Interferograms Generated by a Wavelength Scanning Interferometer,” Opt. Commun.76, (1990), in press.

Tatsuno, K.

Tsujiuchi, J.

K. Okada, J. Tsujiuchi, “Error Analysis on the Phase Calculation from Superposed Interferograms Generated by a Wavelength Scanning Interferometer,” Opt. Commun.76, (1990), in press.

Tsunoda, Y.

Tujiuchi, J.

K. Okada, J. Tujiuchi, “Wavelength Scanning Interferometry for the Measurement of Both Surface Shapes and Refractive Index Inhomogeneity,” Proc. Soc. Photo-Opt. Instrum. Eng. 1162, 395–401 (1989).

Valentin, A.

Appl. Opt.

Opt. Eng.

J. E. Greivenkamp, “Generalized Data Reduction for Heterodyne Interferometry,” Opt. Eng. 23, 350–352 (1984).
[CrossRef]

Opt. Lett.

Proc. Soc. Photo-Opt. Instrum. Eng.

K. Okada, J. Tujiuchi, “Wavelength Scanning Interferometry for the Measurement of Both Surface Shapes and Refractive Index Inhomogeneity,” Proc. Soc. Photo-Opt. Instrum. Eng. 1162, 395–401 (1989).

Other

K. Okada, J. Tsujiuchi, “Error Analysis on the Phase Calculation from Superposed Interferograms Generated by a Wavelength Scanning Interferometer,” Opt. Commun.76, (1990), in press.

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of a Twyman-Green interferometer. The optical path difference between the two arms is δL/2: L1, L2, lenses; M1 M2, mirrors; BS, a beam splitter; M 1 , the mirror image of M1 by BS; D, the CCD TV camera.

Fig. 2
Fig. 2

Schematic diagram of the interferometer to measure parallel plate S of thickness d. The optical path difference between M1 and the front surface of S is δL1/2.

Fig. 3
Fig. 3

Schematic diagram of the interferometer to measure the shape of both the front and rear surfaces and the inhomogeneity of refractive index. Mirror M2 is added behind parallel plate S at a distance of δL2/2.

Fig. 4
Fig. 4

Example of an interferogram obtained by the interferometer shown in Fig. 2.

Fig. 5
Fig. 5

Experimental results obtained by the interferometer shown in Fig. 2: (a) phase ϕ1 defined by Eq. (6) along the vertical line in Fig. 4; (b) ϕ1 + ϕ2; and (c) ϕ2.

Fig. 6
Fig. 6

Experimental results obtained by the interferometer shown in Fig. 3: (a) phase ϕ1 defined by Eq. (16); (b) ϕ3; (c) change of the thickness of the sample; and (d) change of the refractive index.

Equations (28)

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I ( x ) = I 0 ( x ) + I 1 ( x ) cos { k [ δ L + 2 a ( x ) ] } ,
I = I 0 + I 1 cos { k [ δ L + 2 a ( x ) ] k / k } = I 0 + I 1 cos { k [ δ L + 2 a ( x ) ] - k [ δ L + 2 a ( x ) ] Δ λ / ( λ + Δ λ ) } ,
I = I 0 + I 1 cos [ k δ L + ϕ ( x ) - k ( Δ λ / λ ) δ L ] ,
I = I 0 + I 1 cos { k [ 2 a ( x ) + δ L 1 ] } + I 2 cos [ k ( { 2 n ( x ) [ b ( x ) - a ( x ) ] + 2 a ( x ) } + δ L 1 + 2 n ( x ) d 0 ) ] + I 3 cos ( k { 2 n ( x ) [ b ( x ) - a ( x ) ] + 2 n ( x ) d 0 } ) ,
I = I 0 + I 1 cos [ ϕ 1 ( x ) - k ( Δ λ / λ ) δ L 1 ] + I 2 cos { [ ϕ 1 ( x ) + ϕ 2 ( x ) ] - k ( Δ λ / λ ) ( δ L 1 + 2 n 0 d 0 ) } + I 3 cos [ ϕ 2 ( x ) - k ( Δ λ / λ ) 2 n 0 d 0 ] ,
ϕ 1 ( x ) = 2 k a ( x ) , ϕ 2 ( x ) = 2 k n ( x ) · [ b ( x ) - a ( x ) ] .
I m = I 0 + I 1 cos [ ϕ 1 ( x ) - k ( m δ λ / λ ) δ L 1 ] + I 2 cos { [ ϕ 1 ( x ) + ϕ 2 ( x ) ] - k ( m δ λ / λ ) ( δ L 1 + 2 n 0 d 0 ) } + I 3 cos [ ϕ 2 ( x ) - k ( m δ λ / λ ) 2 n 0 d 0 ] ,
I m ( x ) = I 0 + I 1 cos [ ϕ 1 ( x ) ] cos [ k ( m δ λ / λ ) δ L 1 ] + I 1 sin [ ϕ 1 ( x ) ] × sin [ k ( m δ λ / λ ) δ L 1 ] + I 2 cos [ ϕ 1 ( x ) + ϕ 2 ( x ) ] cos [ k ( m δ λ / λ ) ( δ L 1 + 2 n 0 d 0 ) ] + I 2 sin [ ϕ 1 ( x ) + ϕ 2 ( x ) ] sin [ k ( m δ λ / λ ) ( δ L 1 + 2 n 0 d 0 ) ] + I 3 cos [ ϕ 2 ( x ) ] cos [ k ( m δ λ / λ ) 2 n 0 d 0 ] + I 3 sin [ ϕ 2 ( x ) ] sin [ k ( m δ λ / λ ) 2 n 0 d 0 ] ,
A · X = Y ,
A = [ N , Σ c 1 , Σ s 1 , Σ c 2 , Σ s 2 , Σ c 3 , Σ s 3 Σ c 1 , Σ c 1 2 , Σ s 1 c 1 , Σ c 2 c 1 , Σ s 2 c 1 , Σ c 3 c 1 , Σ s 3 c 1 Σ s 1 , Σ c 1 s 1 , Σ s 1 2 , Σ c 2 s 1 , Σ s 2 s 1 , Σ c 3 s 1 , Σ s 3 s 1 Σ c 2 , Σ c 1 c 2 , Σ s 1 c 2 , Σ c 2 2 , Σ s 2 c 2 , Σ c 3 c 2 , Σ s 3 c 2 Σ s 2 , Σ c 1 s 2 , Σ s 1 s 2 , Σ c 2 s 2 , Σ s 2 2 , Σ c 3 s 2 , Σ s 3 s 2 Σ c 3 , Σ c 1 c 3 , Σ s 1 c 3 , Σ c 2 c 3 , Σ s 2 c 3 , Σ c 3 2 , Σ s 3 s 3 Σ s 3 , Σ c 1 s 3 , Σ s 1 s 3 , Σ c 2 s 3 , Σ s 2 s 3 , Σ c 3 s 3 , Σ s 3 2 ] ,
X = [ I 0 I 1 cos [ ϕ 1 ( x ) ] I 1 sin [ ϕ 1 ( x ) ] I 2 cos [ ϕ 1 ( x ) + ϕ 2 ( x ) ] I 2 sin [ ϕ 1 ( x ) + ϕ 2 ( x ) ] I 3 cos [ ϕ 2 ( x ) ] I 3 sin [ ϕ 2 ( x ) ] ] ,
Y = [ Σ I m Σ I m · c 1 Σ I m · s 1 Σ I m · c 2 Σ I m · s 2 Σ I m · c 3 Σ I m · s 3 ] ,
c 1 = cos [ k ( m δ λ / λ ) δ L 1 ] , s 1 = sin [ k ( m δ λ / λ ) δ L 1 ] , c 2 = cos [ k ( m δ λ / λ ) ( δ L 1 + 2 n 0 d 0 ) ] , s 2 = sin [ k ( m δ λ / λ ) ( δ L 1 + 2 n 0 d 0 ) ] , c 3 = cos [ k ( m δ λ / λ ) 2 n 0 d 0 ) ] , s 3 = sin [ k ( m δ λ / λ ) 2 n 0 d 0 ) ] ,
ϕ 1 = tan - 1 ( I 1 sin ϕ 1 / I 1 cos ϕ 1 ) , ϕ 1 + ϕ 2 = tan - 1 [ I 1 sin ( ϕ 1 + ϕ 2 ) / I 1 cos ( ϕ 1 - ϕ 2 ) ] , ϕ 2 = tan - 1 [ I 1 sin ϕ 2 / I 1 cos ϕ 2 ] .
I m = I 0 + I 1 cos [ ϕ 1 ( x ) + k ( m δ λ / λ ) δ L 1 ] + I 2 cos { [ ϕ 1 ( x ) + ϕ 2 ( x ) ] + k ( m δ λ / λ ) ( δ L 1 + 2 n 0 d 0 ) } + I 3 cos { [ ϕ 1 ( x ) + ϕ 2 ( x ) + ϕ 3 ( x ) ] + k ( m δ λ / λ ) ( δ L 1 + 2 n 0 d 0 + δ L 2 ) } + I 4 cos [ ϕ 2 ( x ) ] + k ( m δ λ / λ ) 2 n 0 d 0 ] + I 5 cos { [ ϕ 2 ( x ) + ϕ 3 ( x ) ] + k ( m δ λ / λ ) ( 2 n 0 d 0 + δ L 2 ) } + I 6 cos [ ϕ 3 ( x ) + k ( m δ λ / λ ) δ L 2 ] ,
ϕ 1 ( x ) = 2 k a ( x ) , ϕ 2 ( x ) = 2 k n ( x ) [ b ( x ) - a ( x ) ] , ϕ 3 ( x ) = 2 k [ c ( x ) - b ( x ) ] .
ϕ 4 ( x ) = 2 k c ( x ) .
b ( x ) = [ ϕ 3 ( x ) - ϕ 4 ( x ) ] / ( 2 k )
d ( x ) = d 0 + a ( x ) - b ( x ) = d 0 + [ ϕ 1 ( x ) - ϕ 3 ( x ) + ϕ 4 ( x ) ] / ( 2 k ) ,
n ( x ) = [ n 0 d 0 + ϕ 2 / ( 2 k ) ] / d ( x ) .
n ( λ ) = n ( λ 0 ) + n 1 δ λ + ,
n ( λ ) d 0 = n ( λ 0 ) d 0 + n 1 d 0 δ λ .
I = I 0 + I 1 cos [ k 2 n ( λ ) d 0 + k 2 ( δ λ / λ ) n ( λ ) d 0 + ϕ ( x ) n ( λ ) / n ( λ 0 ) ] = I 0 + I 1 cos { k 2 n ( λ 0 ) d 0 + k 2 ( δ λ / λ ) n ( λ 0 ) d 0 + k 2 n 1 d 0 δ λ + k 2 [ ( δ λ ) 2 / λ ] n 1 d 0 + ϕ ( x ) + ϕ ( x ) n 1 δ λ / n ( λ 0 ) } .
δ ϕ = 2 k [ ( 1 / λ ) n ( λ 0 ) + n 1 ] d 0 δ λ .
n 1 = - [ n ( λ d ) - 1 ] / [ ν d ( λ C - λ F ) ] ,
δ ϕ = 2 k [ ( 1 / λ ) n ( λ 0 ) + n 1 ] d 0 δ λ = 2 k ( 1 / λ ) n ( λ 0 ) d 0 δ λ ,
d 0 = d 0 [ n ( λ 0 + λ n 1 ) ] / n ( λ 0 ) .
ϕ = ϕ ( x ) + ϕ ( x ) n 1 δ λ / n ( λ 0 ) .

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