Abstract

This paper describes a technique that combines ideas of phase shifting interferometry (PSI) and two-wavelength interferometry (TWLI) to extend the phase measurement range of conventional single-wavelength PSI. To verify theoretical predictions, experiments have been performed using a solid-state linear detector array to measure 1-D surface heights. Problems associated with TWLPSI and the experimental setup are discussed. To test the capability of the TWLPSI, a very fine fringe pattern was used to illuminate a 1024 element detector array. Without temporal averaging, the repeatability of measuring a surface having a sag of ~100 μm is better than 25-Å (0.0025%) rms.

© 1984 Optical Society of America

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References

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  1. J. H. Bruning, “Fringe Scanning Interferometers,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978).
  2. C. Koliopoulos, “Interferometric Optical Phase Measurement Techniques,” Ph.D. Dissertation, Optical Sciences Center, U. Arizona (1981).
  3. B. P. Hildebrand, K. A. Haines, “Multiple-Source Holography,” J. Opt. Soc. Am. 57, 155 (1967).
    [CrossRef]
  4. C. Polhemus, “Two-Wavelength Interferometry,” Appl. Opt. 12, 2071 (1973).
    [CrossRef] [PubMed]
  5. J. C. Wyant, “Testing Aspherics Using Two-Wavelength Holography,” Appl. Opt. 10, 2113 (1971).
    [CrossRef] [PubMed]
  6. J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital Wave-Front Measuring Interferometry: Some Systematic Error Sources,” Appl. Opt. 22, 3421 (1983).
    [CrossRef] [PubMed]

1983 (1)

1973 (1)

1971 (1)

1967 (1)

Bruning, J. H.

J. H. Bruning, “Fringe Scanning Interferometers,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978).

Burow, R.

Elssner, K.-E.

Grzanna, J.

Haines, K. A.

Hildebrand, B. P.

Koliopoulos, C.

C. Koliopoulos, “Interferometric Optical Phase Measurement Techniques,” Ph.D. Dissertation, Optical Sciences Center, U. Arizona (1981).

Merkel, K.

Polhemus, C.

Schwider, J.

Spolaczyk, R.

Wyant, J. C.

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Figures (7)

Fig. 1
Fig. 1

Phase error vs phase for a phase step of 88° rather than 90°

Fig. 2
Fig. 2

Arbitrary optical path difference (OPD) distribution across the detector array.

Fig. 3
Fig. 3

Computer simulation results for TWLPSI, longer λeq is needed for a steeper wave front. For comparison, the dotted line shows results for single-wavelength PSI.

Fig. 4
Fig. 4

Experimental setup for the TWLPSI.

Fig. 5
Fig. 5

(a) Single-wavelength data obtained at λ = 6328 Å showing breakdown of phase measurement when the wave front slope is too steep. (b) Similar data as in (a) but λ is changed to 5145 Å. (c) The equivalent wavelength (λeq = 2.75 μm) data. (d) Same data as in (c) but with both tilt and focus removed.

Fig. 6
Fig. 6

(a) Single-wavelength data obtained at λ = 6328 Å with 2π ambiguities corrected. (b) Same data as in (a) but with both tilt and focus removed. In (c) and (d) the wavelength was changed to 5145 Å.

Fig. 7
Fig. 7

(a) Repeatability of the surface height measurement without temporal averaging by the second method of TWLPSI (obtained at λ = 6328 Å). (b) Difference between the data sets in Fig. 6(b) (λ = 6328 Å) and Fig. 6(d) (λ = 5145 Å).

Tables (1)

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Table I Error (e.g., High Frequency Noise) Amplification Effect of the Equivalent-Wavelength Phase Calculations for Different λeq

Equations (13)

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OPD n = [ ϕ n a 2 π + m ] λ a ,
OPD n = [ ϕ n b 2 π + p ] λ b ,
OPD n + 1 = [ ϕ ( n + 1 ) a 2 π + m ] λ a ,
OPD n + 1 = [ ϕ ( n + 1 ) b 2 π + p ] λ b .
2 π [ OPD n + 1 - OPD n ] = [ ( ϕ ( n + 1 ) a - ϕ n a ) + 2 π ( m - m ) ] λ a .
2 π [ OPD n + 1 - OPD n ] = [ ( ϕ ( n + 1 ) b - ϕ n b ) + 2 π ( p - p ) ] λ b .
Δ OPD n + 1 = OPD n + 1 - OPD n , Δ ϕ ( n + 1 ) a = ϕ ( n + 1 ) a - ϕ n a , Δ ϕ ( n + 1 ) b = ϕ ( n + 1 ) b - ϕ n b .
Δ M n + 1 = Δ ϕ ( n + 1 ) b λ b - Δ ϕ ( n + 1 ) a λ a 2 π [ λ a - λ b ] .
Δ OPD n + 1 = [ ( Δ ϕ ( n + 1 ) b - Δ ϕ ( n + 1 ) a ) / 2 π ] ( λ a λ b ) / ( λ a - λ b ) .
Δ OPD n + 1 = { 1 2 π [ Δ ϕ ( n + 1 ) b - Δ ϕ ( n + 1 ) a ] λ eq if λ a > λ b , 1 2 π [ Δ ϕ ( n + 1 ) a - Δ ϕ ( n + 1 ) b ] λ eq if λ b > λ a .
V ( f x ) = K [ I ( f x ) sinc ( W d f x ) ] * [ comb ( S f x ) ] ,
Δ OPD n + 1 = 1 2 π [ Δ ϕ ( n + 1 ) b - Δ ϕ ( n + 1 ) a ] [ λ eq / ( λ a + λ b 2 ) ] ( λ a + λ b 2 ) = 1 2 π [ Δ ϕ ( n + 1 ) b - Δ ϕ ( n + 1 ) a ] M a b ( λ a + λ b 2 ) ,
Δ OPD n + 1 = 1 2 π { [ Δ ϕ ( n + 1 ) b - Δ ϕ ( n + 1 ) a ] S + ( Δ ϕ ( n + 1 ) b - Δ ϕ ( n + 1 ) a ) E } × M a b ( λ a + λ b 2 ) .

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