Abstract

We describe a sinusoidal phase modulating interferometer in which a CCD image sensor detects four values by integrating the time-varying intensity in an interference pattern for intervals of one-quarter period of the phase modulation. The optimum amplitude and phase of the sinusoidal phase modulation are determined. The measurement error caused by the additive noise and the deviation from the optimum phase modulation is analyzed. The experimental results for surface profiles of magnetic sliders show that the sinusoidal phase modulating interferometer proposed here yields a measurement accuracy of the order of 1 nm.

© 1987 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. Crane, “Interference Phase Measurement,” Appl. Opt. 8, 538 (1969).
  2. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital Wavefront Measuring Interferometer for Testing Optical Surfaces and Lenses,” Appl. Opt. 13, 2693 (1974).
    [CrossRef] [PubMed]
  3. K. D. Stumpf, “Real-Time Interferometer,” Opt. Eng. 18, 648 (1979).
    [CrossRef]
  4. N. A. Massie, “Real-Time Digital Heterodyne Interferometry: a System,” Appl. Opt. 19, 154 (1980).
    [CrossRef] [PubMed]
  5. G. E. Sommargren, “Optical Heterodyne Profilometry,” Appl. Opt. 20, 610 (1981).
    [CrossRef] [PubMed]
  6. T. Yatagai, T. Kanou, “Aspherical Surface Testing with Shearing Interferometer Using Fringe Scanning Detection Method,” Opt. Eng. 23, 357 (1984).
    [CrossRef]
  7. B. Bhushan, J. C. Wyant, C. L. Koliopoulos, “Measurement of Surface Topography of Magnetic Tapes by Mirau Interferometry,” Appl. Opt. 24, 1489 (1985).
    [CrossRef] [PubMed]
  8. K. Creath, Y.-Y. Cheng, J. C. Wyant, “Contouring Aspheric Surfaces Using Two-Wavelength Phase-Shifting Interferometry,” Opt. Acta 32, 1455 (1985).
    [CrossRef]
  9. O. Sasaki, H. Okazaki, “Sinusoidal Phase Modulating Interferometry for Surface Profile Measurement,” Appl. Opt. 25, 3137 (1986).
    [CrossRef] [PubMed]
  10. O. Sasaki, H. Okazaki, “Analysis of Measurement Accuracy in Sinusoidal Phase Modulating Interferometry,” Appl. Opt. 25, 3152 (1986).
    [CrossRef] [PubMed]

1986 (2)

1985 (2)

K. Creath, Y.-Y. Cheng, J. C. Wyant, “Contouring Aspheric Surfaces Using Two-Wavelength Phase-Shifting Interferometry,” Opt. Acta 32, 1455 (1985).
[CrossRef]

B. Bhushan, J. C. Wyant, C. L. Koliopoulos, “Measurement of Surface Topography of Magnetic Tapes by Mirau Interferometry,” Appl. Opt. 24, 1489 (1985).
[CrossRef] [PubMed]

1984 (1)

T. Yatagai, T. Kanou, “Aspherical Surface Testing with Shearing Interferometer Using Fringe Scanning Detection Method,” Opt. Eng. 23, 357 (1984).
[CrossRef]

1981 (1)

1980 (1)

1979 (1)

K. D. Stumpf, “Real-Time Interferometer,” Opt. Eng. 18, 648 (1979).
[CrossRef]

1974 (1)

1969 (1)

R. Crane, “Interference Phase Measurement,” Appl. Opt. 8, 538 (1969).

Bhushan, B.

Brangaccio, D. J.

Bruning, J. H.

Cheng, Y.-Y.

K. Creath, Y.-Y. Cheng, J. C. Wyant, “Contouring Aspheric Surfaces Using Two-Wavelength Phase-Shifting Interferometry,” Opt. Acta 32, 1455 (1985).
[CrossRef]

Crane, R.

R. Crane, “Interference Phase Measurement,” Appl. Opt. 8, 538 (1969).

Creath, K.

K. Creath, Y.-Y. Cheng, J. C. Wyant, “Contouring Aspheric Surfaces Using Two-Wavelength Phase-Shifting Interferometry,” Opt. Acta 32, 1455 (1985).
[CrossRef]

Gallagher, J. E.

Herriott, D. R.

Kanou, T.

T. Yatagai, T. Kanou, “Aspherical Surface Testing with Shearing Interferometer Using Fringe Scanning Detection Method,” Opt. Eng. 23, 357 (1984).
[CrossRef]

Koliopoulos, C. L.

Massie, N. A.

Okazaki, H.

Rosenfeld, D. P.

Sasaki, O.

Sommargren, G. E.

Stumpf, K. D.

K. D. Stumpf, “Real-Time Interferometer,” Opt. Eng. 18, 648 (1979).
[CrossRef]

White, A. D.

Wyant, J. C.

B. Bhushan, J. C. Wyant, C. L. Koliopoulos, “Measurement of Surface Topography of Magnetic Tapes by Mirau Interferometry,” Appl. Opt. 24, 1489 (1985).
[CrossRef] [PubMed]

K. Creath, Y.-Y. Cheng, J. C. Wyant, “Contouring Aspheric Surfaces Using Two-Wavelength Phase-Shifting Interferometry,” Opt. Acta 32, 1455 (1985).
[CrossRef]

Yatagai, T.

T. Yatagai, T. Kanou, “Aspherical Surface Testing with Shearing Interferometer Using Fringe Scanning Detection Method,” Opt. Eng. 23, 357 (1984).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Sinusoidal phase modulating interferometer.

Fig. 2
Fig. 2

Four integration values yi(i = 1–4).

Fig. 3
Fig. 3

Values of B2 and phase θ as a function of z when B1 = 0. The optimum values of z and θ are 2.45 and 56°, respectively.

Fig. 4
Fig. 4

Values of γ as a function of Δz and Δθ at z0 = 2.45 and θ0 = 56°.

Fig. 5
Fig. 5

Error Δα for the various values of γ as a function of α.

Fig. 6
Fig. 6

Values of B1 as a function of Δz and Δθ at z0 = 2.45 and θ0 = 56°.

Fig. 7
Fig. 7

Values of B2 as a function of Δz and Δθ at z0 = 2.45 and θ0 = 56°.

Fig. 8
Fig. 8

Measured surface profile of two rails of a magnetic head. The P–V of this surface is 42 nm.

Fig. 9
Fig. 9

Measured surface profile of two rails of another magnetic head. The P–V of this surface is 17 nm.

Tables (1)

Tables Icon

Table I Measurement Accuracy at ρ = 1000

Equations (43)

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

A ( t ) = a cos ( ω c t + θ ) .
s ( t ) = x 1 + s 0 cos [ z cos ( ω c t + θ ) + α ] ,
s ( t ) = x 1 + x 2 cos [ z cos ( ω c t + θ ) ] + x 3 sin [ z cos ( ω c t + θ ) ] .
y i = ( 4 / T ) ( T / 4 ) ( i - 1 ) ( T / 4 ) i s ( t ) d t             i = 1 - 4 ,
y i = x 1 + g i x 2 + h i x 3             i = 1 - 4 ,
g 1 = g 3 = J 0 ( z ) + ( 4 / π ) n = 1 [ J 2 n ( z ) / 2 n ] [ 1 - ( - 1 ) n ] sin ( 2 n θ ) , g 2 = g 4 = J 0 ( z ) - ( 4 / π ) n = 1 [ J 2 n ( z ) / 2 n ] [ 1 - ( - 1 ) n ] sin ( 2 n θ ) , h 1 = h 3 = - ( 4 / π ) n = 1 [ J 2 n - 1 ( z ) / ( 2 n - 1 ) ] × { ( - 1 ) n sin [ ( 2 n - 1 ) θ ] + cos [ ( 2 n - 1 ) θ ] } , h 2 = - h 4 = - ( 4 / π ) n = 1 [ J 2 n - 1 ( z ) / ( 2 n - 1 ) ] × { ( - 1 ) n sin [ ( 2 n - 1 ) θ ] - cos [ ( 2 n - 1 ) θ ] } ,
x 2 = Y g / 2 B g ,             x 3 = Y h / 2 B h ,
Y g = y 1 - y 2 + y 3 - y 4 , B g = g 1 - g 2 , Y h = y 1 + y 2 - y 3 - y 4 , B h = h 1 + h 2 .
α = tan - 1 ( x 3 / x 2 ) = tan - 1 [ ( B g / B h ) ( Y h / Y g ) ] .
n i n j = { σ 2 i = j , 0 i j ,
Y g = Y g + N g ,             Y h = Y h + N h ,
N g = n 1 - n 2 + n 3 - n 4 , N h = n 1 + n 2 - n 2 - n 4 .
tan α = ( B g / B h ) ( Y h / Y g ) = tan ( α + ɛ α ) ,
η t = tan ( α + ɛ α ) = a 0 + a 1 ɛ α + a 2 ɛ α 2 + ,
η t = a 0 + a 1 ɛ α + a 2 ɛ α 2 ,
η t 2 = a 0 2 + 2 a 0 a 1 ɛ α + ( a 1 2 + 2 a 0 a 1 ) ɛ α 2 .
ɛ α = ( 1 / a 1 3 ) [ ( a 1 2 + 2 a 0 a 2 ) ( η t - a 0 ) - a 2 ( η t 2 - a 0 2 ) ] ,
ɛ α 2 = ( 1 / a 1 2 ) [ - 2 a 0 ( η t - a 0 ) + ( η t 2 - a 0 2 ) ] .
η t = tan α 1 + ( N h / Y h ) 1 + ( N g / Y g ) .
Y g = 2 s 0 B g cos α N g .
η t = ( tan α ) [ 1 + ( N h / Y h ) ] [ 1 - ( N g / Y g ) + ( N g / Y g ) 2 ] .
η t 2 = ( tan 2 α ) [ 1 + 2 ( N h / Y h ) + ( N h / Y h ) 2 ] × [ 1 - 2 ( N g / Y g ) + 3 ( N g / Y g ) 2 ] .
η t = ( tan α ) [ 1 + ( 4 σ 2 / Y g 2 ) ] ,
η t 2 = ( tan 2 α ) { 1 + 4 σ 2 [ ( 3 / Y g 2 ) + ( 1 / Y h 2 ) ] } ,
N g = N h = 0 ,             N g 2 = 4 σ 2 , N g N h = 0 ,             N g 2 N h .
ɛ α = ( sin α ) ( cos α ) [ ( 1 / B g 2 ) - ( 1 / B h 2 ) ] ( σ 2 / s 0 2 ) ,
ɛ α 2 = { [ ( sin 2 α ) / B g 2 ] + [ ( cos 2 α ) / B h 2 ] } ( σ 2 / s 0 2 ) .
η c = cot ( α + ɛ α ) = b 0 + b 1 ɛ α + b 2 ɛ α 2 + ,
ɛ α = ( 1 / b 1 3 ) [ ( b 1 2 + 2 b 0 b 2 ) ( η c - b 0 ) - b 2 ( η c 2 - b 0 2 ) ] ,
ɛ α 2 = ( 1 / b 1 2 ) [ ( - 2 b 0 ( η c - b 0 + ( η c 2 - b 0 2 ) ] .
η c = ( cot α ) [ 1 + ( 4 σ 2 / Y h 2 ) ] ,
η c 2 = ( cot 2 α ) { 1 + 4 σ 2 [ ( 3 / Y h 2 ) + ( 1 / Y g 2 ) ] } .
B 1 = ( 1 / 4 ) [ ( 1 / B g 2 ) - ( 1 / B h 2 ) ] , B 2 = ( 1 / 4 ) [ ( 1 / B g 2 ) + ( 1 / B h 2 ) ]
ρ = ( s 0 2 / 2 ) / σ 2 ,
α = ( 1 / ρ ) B 1 sin 2 α ,
α 2 = ( 1 / ρ ) ( B 2 - B 1 cos 2 α ) .
z ˜ = z 0 + Δ z ,             θ ˜ = θ 0 + Δ θ .
Y ˜ g = 2 B ˜ g s 0 cos α ,             Y ˜ h = 2 B ˜ h s 0 sin α ,
tan ( α + Δ α ) = ( B g / B h ) ( Y ˜ h / Y ˜ g ) = ( B g / B ˜ g ) ( B ˜ h / B h ) tan α .
γ = ( B g / B ˜ g ) ( B ˜ h / B h ) ,
Δ α = tan - 1 ( γ tan α ) - α .
e α = α + Δ α .
e α 2 1 / 2 [ B 2 / ρ + ( Δ α ) 2 ] 1 / 2 .

Metrics