Abstract

A method for measuring profiles along a circle on a flat surface with no standard is described. For the measurement, two unknown surfaces are placed almost parallel, and the distance between them is measured many times along a circle by rotation of one of the surfaces. Profiles of the two surfaces can be determined from the distance data. In this study the measuring method is explained: The space between two surfaces measured with a Fizeau interferometer. Four measuring experiments are carried out for determining the profile of a precision-grade half-mirror; in each experiment a different ordinary mirror with unknown profile is used as the second mirror. Profiles of the precise mirrors obtained by these experiments agree closely, with deviations of ∼2 nm. A similar experiment with many concentric circles was carried out with a precise half-mirror and another precise mirror. Although the profiles of many concentric circles were independent of one another, the result shows that the high-frequency component of a whole plane can be estimated.

© 2003 Optical Society of America

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References

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  1. J. Grzanna, G. Schulz, “Absolute testing of flatness standards at square-grid points,” Opt. Commun. 77, 107–112 (1990).
    [CrossRef]
  2. G. Schulz, J. Grzanna, “Absolute flatness testing by the rotation method with optimal measuring-error compensation,” Appl. Opt. 31, 3767–3780 (1992).
    [CrossRef] [PubMed]
  3. K. E. Elssner, J. Grzanna, G. Schulz, “Interferentielle Absolutprüfung von Sphärizitätsnormalen,” Opt. Acta 27, 563–580 (1980).
    [CrossRef]
  4. S. Sonozaki, K. Iwata, Y. Iwahashi, “Measurement of profile along a circle on flat surfaces with no standard,” in Proceedings of IMEKO-14 World Congress, (IMEKO, Budapest, 1997), Vol. 8, pp. 147–152.
  5. S. Sonozaki, K. Iwata, Y. Iwahashi, “Measurement of profile along a circle on a flat surface using a Fizeau-interferometer with no standard,” in Proceedings of IMEKO-15 World Congress, (IMEKO, Budapest, 1999), Vol. 9, pp. 49–54.
  6. M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe pattern analysis for computer based topography and interferometory,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]

1992

1990

J. Grzanna, G. Schulz, “Absolute testing of flatness standards at square-grid points,” Opt. Commun. 77, 107–112 (1990).
[CrossRef]

1982

1980

K. E. Elssner, J. Grzanna, G. Schulz, “Interferentielle Absolutprüfung von Sphärizitätsnormalen,” Opt. Acta 27, 563–580 (1980).
[CrossRef]

Elssner, K. E.

K. E. Elssner, J. Grzanna, G. Schulz, “Interferentielle Absolutprüfung von Sphärizitätsnormalen,” Opt. Acta 27, 563–580 (1980).
[CrossRef]

Grzanna, J.

G. Schulz, J. Grzanna, “Absolute flatness testing by the rotation method with optimal measuring-error compensation,” Appl. Opt. 31, 3767–3780 (1992).
[CrossRef] [PubMed]

J. Grzanna, G. Schulz, “Absolute testing of flatness standards at square-grid points,” Opt. Commun. 77, 107–112 (1990).
[CrossRef]

K. E. Elssner, J. Grzanna, G. Schulz, “Interferentielle Absolutprüfung von Sphärizitätsnormalen,” Opt. Acta 27, 563–580 (1980).
[CrossRef]

Ina, H.

Iwahashi, Y.

S. Sonozaki, K. Iwata, Y. Iwahashi, “Measurement of profile along a circle on a flat surface using a Fizeau-interferometer with no standard,” in Proceedings of IMEKO-15 World Congress, (IMEKO, Budapest, 1999), Vol. 9, pp. 49–54.

S. Sonozaki, K. Iwata, Y. Iwahashi, “Measurement of profile along a circle on flat surfaces with no standard,” in Proceedings of IMEKO-14 World Congress, (IMEKO, Budapest, 1997), Vol. 8, pp. 147–152.

Iwata, K.

S. Sonozaki, K. Iwata, Y. Iwahashi, “Measurement of profile along a circle on flat surfaces with no standard,” in Proceedings of IMEKO-14 World Congress, (IMEKO, Budapest, 1997), Vol. 8, pp. 147–152.

S. Sonozaki, K. Iwata, Y. Iwahashi, “Measurement of profile along a circle on a flat surface using a Fizeau-interferometer with no standard,” in Proceedings of IMEKO-15 World Congress, (IMEKO, Budapest, 1999), Vol. 9, pp. 49–54.

Kobayashi, S.

Schulz, G.

G. Schulz, J. Grzanna, “Absolute flatness testing by the rotation method with optimal measuring-error compensation,” Appl. Opt. 31, 3767–3780 (1992).
[CrossRef] [PubMed]

J. Grzanna, G. Schulz, “Absolute testing of flatness standards at square-grid points,” Opt. Commun. 77, 107–112 (1990).
[CrossRef]

K. E. Elssner, J. Grzanna, G. Schulz, “Interferentielle Absolutprüfung von Sphärizitätsnormalen,” Opt. Acta 27, 563–580 (1980).
[CrossRef]

Sonozaki, S.

S. Sonozaki, K. Iwata, Y. Iwahashi, “Measurement of profile along a circle on flat surfaces with no standard,” in Proceedings of IMEKO-14 World Congress, (IMEKO, Budapest, 1997), Vol. 8, pp. 147–152.

S. Sonozaki, K. Iwata, Y. Iwahashi, “Measurement of profile along a circle on a flat surface using a Fizeau-interferometer with no standard,” in Proceedings of IMEKO-15 World Congress, (IMEKO, Budapest, 1999), Vol. 9, pp. 49–54.

Takeda, M.

Appl. Opt.

J. Opt. Soc. Am.

Opt. Acta

K. E. Elssner, J. Grzanna, G. Schulz, “Interferentielle Absolutprüfung von Sphärizitätsnormalen,” Opt. Acta 27, 563–580 (1980).
[CrossRef]

Opt. Commun.

J. Grzanna, G. Schulz, “Absolute testing of flatness standards at square-grid points,” Opt. Commun. 77, 107–112 (1990).
[CrossRef]

Other

S. Sonozaki, K. Iwata, Y. Iwahashi, “Measurement of profile along a circle on flat surfaces with no standard,” in Proceedings of IMEKO-14 World Congress, (IMEKO, Budapest, 1997), Vol. 8, pp. 147–152.

S. Sonozaki, K. Iwata, Y. Iwahashi, “Measurement of profile along a circle on a flat surface using a Fizeau-interferometer with no standard,” in Proceedings of IMEKO-15 World Congress, (IMEKO, Budapest, 1999), Vol. 9, pp. 49–54.

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Figures (9)

Fig. 1
Fig. 1

Two unknown profiles along a circle on two surfaces, A and B, placed face to face.

Fig. 2
Fig. 2

Schematic diagram of a slight inclination of surface B relative to surface A, showing the relation of space W to profiles η and ξ and the inclination.

Fig. 3
Fig. 3

Schematic of a Fizeau interferometer for measuring space between upper surface B and lower surface A.

Fig. 4
Fig. 4

Example of an interference fringe pattern.

Fig. 5
Fig. 5

Three examples of space between two surfaces along a circle with 10-mm radius in the first experiment. These three curves were obtained with the same upper plate and the same lower plate but with a different rotation angle of the lower plate. Filled circle, rotation angle 0 (deg); filled square, rotation angle 240 (deg); open circle, rotation angle 120 (deg).

Fig. 6
Fig. 6

Profiles of four ordinary mirrors obtained by four experiments. Filled circle, ordinary mirror 1 (1st exp.); open circle, ordinary mirror 2 (2nd exp.); filled square, ordinary mirror 3 (3rd exp.); open square, ordinary mirror 4 (4th exp.).

Fig. 7
Fig. 7

Profiles of the precise half-mirror obtained by the four experiments. These profiles were obtained with the same precise half-mirror used as the upper plate but with the different ordinary mirrors shown in Fig. 6 as the lower plate. Filled circle, 1st exp.; open circle, 2nd exp.; filled square, 3rd exp.; open square, 4th exp.

Fig. 8
Fig. 8

Profiles of the precise half-mirror calculated by the spacing at 72 measuring points on a circle with 10-mm radius. The position of the measuring circle is slightly different from that of the circle in the previous experiments. Linear algebraic equations method (N = 72). Filled circle, three rotations: ϕ of 0, 5, and 10 deg; open circle, four rotations: ϕ of 0, 5, 10, and 15 deg; filled triangle, five rotations: ϕ of 0, 5, 10, 15, and 20 deg; open triangle, six rotations: ϕ of 0, 5, 10, 15, 20, and 25 deg. Fourier transformation method (N = 72) (-4 nm shifted).

Fig. 9
Fig. 9

Profiles of the precise half-mirror and of the other precise mirror along ten concentric circles.

Equations (24)

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Wr, θ, ϕ=-ξ-θ+ϕ-ηθ+αϕr cos θ+βϕr sin θ+εϕ.
z=ar cos θ+br sin θ+c.
S=02π ξ2dθ=02πξ-ar cos θ+br sin θ+c2dθ,
δSδa=0, δSδb=0, δSδc=0.
02π ξ cos θdθ=0,02π ξ sin θdθ=0,02π ξdθ=0.
02π η cos θdθ=0,02π η sin θdθ=0,02π ηdθ=0.
θn=2πnN, n=1, 2, , N,
ϕm=2πmN, m=1, 2, , M,
Wn,m=-ξ-n+m-ηn+αmr cos θn+βm sin θn+εm,
n=1N ξn cos θn=0, n=1N ξn sin θn=0, n=1N ξn=0,
n=1N ηn cos θn=0, n=1N ηn sin θn=0, n=1N ηn=0,
P=MN+6,
Q=2N+3M.
PQ
εϕ=12π02π Wr, θ, ϕdθ,
αϕ=1π02π Wr, θ, ϕcos θdθ,
βϕ=1π02π Wr, θ, ϕsin θdθ.
sθ, ϕ-ξ-θ+ϕ-ηθ
ηθ=-12π02π sθ, ϕdϕ,
gx, y=ax, y+bx, ycos2πμ0x+ν0y+ϕx, y.
ϕ=4πW/λ.
gx, y=ax, y+cx, yexp2πiμ0x+ν0y+c*x, yexp-2πiμ0x+ν0y,
Gμ, ν=Aμ, ν+Cμ-μ0, ν-ν0+C*-μ+μ0, -ν+ν0,
logcx, y=log1/2bx, y+iϕx, y.

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