Abstract

A new type of phase shifting interferometer with a common-path arrangement using a polarization technique is proposed and discussed. In the interferometer, the dc (specular) component of an object beam is separated in the Fourier transform plane and used as a reference beam for its ac component. The phase of the dc component used as the reference beam is shifted by using a polarization technique for phase shifting interferometry. The present interferometer is different from a shearing type in that the phase distribution of an object beam is directly analyzed from the acquired intensity variations obtained by a 2-D detector such as a TV camera. Some experiments were conducted to verify the validity of the present phase shifting interferometer. They showed that high stability of the phase measurements is achieved up to λ/200 with an accuracy of λ/40 for wavelength λ light. The interferometer is suitable for obtaining 2-D phase information about the surface structure of small objects.

© 1987 Optical Society of America

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References

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  1. R. Dändliker, B. Ineichen, F. M. Mottier, “High Resolution Hologram Interferometry by Electronic Phase Measurement,” Opt. Commun. 9, 412 (1973).
    [Crossref]
  2. R. Dändliker, R. Thalmann, “Heterodyne and Quasi-Heterodyne Holographic Interferometry,” Opt. Eng. 24, 824 (1985).
  3. P. Carre, “Installation et utilisation du comparateur photoelectrique et interferential du bureau in ternational des poids et mesures,” Metrologia 2, 13 (1966).
    [Crossref]
  4. J. H. Bruning, D. R. Herriot, 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]
  5. P. Hariharan, B. F. Oreb, N. Brown, “A Digital Phase-Measurement System for Real-Time Holographic Interferometry,” Opt. Commun. 41, 393 (1982).
    [Crossref]
  6. G. E. Sommargren, “Up-Down Frequency Shifter for Optical Heterodyne Interferometry,” J. Opt. Soc. Am. 65, 960 (1975).
    [Crossref]
  7. R. N. Shagam, J. C. Wyant, “Optical Frequency Shifter for Heterodyne Interferometers Using Multiple Rotating Polarization Retarders,” Appl. Opt. 17, 3034 (1978).
    [Crossref] [PubMed]
  8. M. P. Kothiyal, C. Delisle, “Shearing Interferometer for Phase Shifting Interferometry with Polarization Phase Shifter,” Appl. Opt. 24, 4439 (1985).
    [Crossref] [PubMed]
  9. D. Malacara, Ed., Optical Shop Testing (Wiley, New York, 1978), Chaps. 4 and 5.
  10. R. C. Jones, “A New Calculus for the Treatment of Optical Systems: I. Description and Discussion of the Calculus,” J. Opt. Soc. Am. 31, 488 (1941).
    [Crossref]

1985 (2)

R. Dändliker, R. Thalmann, “Heterodyne and Quasi-Heterodyne Holographic Interferometry,” Opt. Eng. 24, 824 (1985).

M. P. Kothiyal, C. Delisle, “Shearing Interferometer for Phase Shifting Interferometry with Polarization Phase Shifter,” Appl. Opt. 24, 4439 (1985).
[Crossref] [PubMed]

1982 (1)

P. Hariharan, B. F. Oreb, N. Brown, “A Digital Phase-Measurement System for Real-Time Holographic Interferometry,” Opt. Commun. 41, 393 (1982).
[Crossref]

1978 (1)

1975 (1)

1974 (1)

1973 (1)

R. Dändliker, B. Ineichen, F. M. Mottier, “High Resolution Hologram Interferometry by Electronic Phase Measurement,” Opt. Commun. 9, 412 (1973).
[Crossref]

1966 (1)

P. Carre, “Installation et utilisation du comparateur photoelectrique et interferential du bureau in ternational des poids et mesures,” Metrologia 2, 13 (1966).
[Crossref]

1941 (1)

Brangaccio, D. J.

Brown, N.

P. Hariharan, B. F. Oreb, N. Brown, “A Digital Phase-Measurement System for Real-Time Holographic Interferometry,” Opt. Commun. 41, 393 (1982).
[Crossref]

Bruning, J. H.

Carre, P.

P. Carre, “Installation et utilisation du comparateur photoelectrique et interferential du bureau in ternational des poids et mesures,” Metrologia 2, 13 (1966).
[Crossref]

Dändliker, R.

R. Dändliker, R. Thalmann, “Heterodyne and Quasi-Heterodyne Holographic Interferometry,” Opt. Eng. 24, 824 (1985).

R. Dändliker, B. Ineichen, F. M. Mottier, “High Resolution Hologram Interferometry by Electronic Phase Measurement,” Opt. Commun. 9, 412 (1973).
[Crossref]

Delisle, C.

Gallagher, J. E.

Hariharan, P.

P. Hariharan, B. F. Oreb, N. Brown, “A Digital Phase-Measurement System for Real-Time Holographic Interferometry,” Opt. Commun. 41, 393 (1982).
[Crossref]

Herriot, D. R.

Ineichen, B.

R. Dändliker, B. Ineichen, F. M. Mottier, “High Resolution Hologram Interferometry by Electronic Phase Measurement,” Opt. Commun. 9, 412 (1973).
[Crossref]

Jones, R. C.

Kothiyal, M. P.

Mottier, F. M.

R. Dändliker, B. Ineichen, F. M. Mottier, “High Resolution Hologram Interferometry by Electronic Phase Measurement,” Opt. Commun. 9, 412 (1973).
[Crossref]

Oreb, B. F.

P. Hariharan, B. F. Oreb, N. Brown, “A Digital Phase-Measurement System for Real-Time Holographic Interferometry,” Opt. Commun. 41, 393 (1982).
[Crossref]

Rosenfeld, D. P.

Shagam, R. N.

Sommargren, G. E.

Thalmann, R.

R. Dändliker, R. Thalmann, “Heterodyne and Quasi-Heterodyne Holographic Interferometry,” Opt. Eng. 24, 824 (1985).

White, A. D.

Wyant, J. C.

Appl. Opt. (3)

J. Opt. Soc. Am. (2)

Metrologia (1)

P. Carre, “Installation et utilisation du comparateur photoelectrique et interferential du bureau in ternational des poids et mesures,” Metrologia 2, 13 (1966).
[Crossref]

Opt. Commun. (2)

P. Hariharan, B. F. Oreb, N. Brown, “A Digital Phase-Measurement System for Real-Time Holographic Interferometry,” Opt. Commun. 41, 393 (1982).
[Crossref]

R. Dändliker, B. Ineichen, F. M. Mottier, “High Resolution Hologram Interferometry by Electronic Phase Measurement,” Opt. Commun. 9, 412 (1973).
[Crossref]

Opt. Eng. (1)

R. Dändliker, R. Thalmann, “Heterodyne and Quasi-Heterodyne Holographic Interferometry,” Opt. Eng. 24, 824 (1985).

Other (1)

D. Malacara, Ed., Optical Shop Testing (Wiley, New York, 1978), Chaps. 4 and 5.

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Figures (8)

Fig. 1
Fig. 1

Optical arrangement of a common-path phase shifting interferometer using a polarization technique. In the figure, PL1 and PL2 are polarizers, Q is a quarterwave plate, and H is a halfwave plate.

Fig. 2
Fig. 2

Phase shifter to introduce the phase difference between the ac and dc components of the object wave. This consists of the quarterwave plate Q, the halfwave plate H, and the linear polarizer PL2.

Fig. 3
Fig. 3

(a) Variation of the phase difference Δψ between the ac and dc components of the object introduced by the phase shifter for ϕ = −π/8. (b) Normalized amplitudes of t dc and t ac ( x ) as a function of the rotation angle θh of the halfwave plate.

Fig. 4
Fig. 4

Vector diagram in the complex plane for explaining the relation between the complex amplitude to(x) of the object and the other related complex amplitudes and quantities.

Fig. 5
Fig. 5

Four intensity patterns obtained for a phase grating object by changing the phase shifts as Δψ = (a) π/4, (b) 3π/4, (c) −3π/4, and (d) −π/4.

Fig. 6
Fig. 6

Phase distribution evaluated experimentally from the four intensity patterns in Fig. 5. The phase of the object is distributed approximately within a range of 2.48 rad.

Fig. 7
Fig. 7

Four intensity patterns obtained for a sinusoidal phase grating by changing the phase shifts as Δψ = (a) π/4, (b) 3π/4, (c) −3π/4, and (d) −π/4.

Fig. 8
Fig. 8

Phase distribution analyzed experimentally from the four intensity patterns in Fig. 7. The measured phases are distributed approximately within a range of 2.63 rad.

Equations (45)

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I ( x ) = I o ( x ) + I r ( x ) + 2 I o ( x ) I r ( x ) cos [ ψ + θ ( x ) ] ,
I 1 ( x ) = I o ( x ) + I r ( x ) + 2 I o ( x ) I r ( x ) cos θ ( x )             for ψ = 0 ,
I 2 ( x ) = I o ( x ) + I r ( x ) - 2 I o ( x ) I r ( x ) sin θ ( x )             for ψ = π / 2 ,
I 3 ( x ) = I o ( x ) + I r ( x ) - 2 I o ( x ) I r ( x ) cos θ ( x )             for ψ = π ,
I 4 ( x ) = I o ( x ) + I r ( x ) + 2 I o ( x ) I r ( x ) sin θ ( x )             for ψ = - π / 2.
θ ( x ) = tan - 1 { [ I 4 ( x ) - I 2 ( x ) ] / [ I 1 ( x ) - I 3 ( x ) ] } ,
I 1 ( x ) - I 3 ( x ) = 4 I o ( x ) I r ( x ) cos θ ( x ) ,
I 4 ( x ) - I 2 ( x ) = 4 I o ( x ) I r ( x ) sin θ ( x ) .
t o ( x ) = t o ( x ) exp [ i θ o ( x ) ]
A o ( x ) = t o ( x ) ( cos ϕ sin ϕ ) .
A o ( x ) = [ t dc + t ac ( x ) ] ( cos ϕ sin ϕ ) ,
t ac ( x ) = t ac ( x ) exp [ i θ ac ( x ) ] .
A ¯ o ( ξ ) = F [ A o ( x ) ] = [ t ¯ dc ( ξ ) + t ¯ ac ( ξ ) ] ( cos ϕ sin ϕ ) ,
t ¯ dc ( ξ ) = F [ t dc ] ,
t ¯ ac ( ξ ) = F [ t ac ( x ) ] .
A ¯ 1 ( ξ ) = P 1 A ¯ o ( ξ ) , = A ¯ dc ( ξ ) + A ¯ ac ( ξ ) ,
P 1 = { β ( cos 2 θ p 1 sin θ p 1 cos θ p 1 sin θ p 1 cos θ p 1 sin 2 θ p 1 ) for ξ 0 , ( 1 0 0 1 ) for ξ = 0 ,
A ¯ dc ( ξ ) = t ¯ dc ( ξ ) ( cos ϕ sin ϕ ) ,
A ¯ ac ( ξ ) = β cos ( θ p 1 - ϕ ) t ¯ ac ( ξ ) ( cos θ p 1 sin θ p 1 ) .
Q = ( 1 0 0 i ) ,             for θ q = 0 ,
H ( θ h ) = - i ( cos 2 θ h sin 2 θ h sin 2 θ h - cos 2 θ h ) ,
P 2 = ( 1 0 0 0 ) ,             for θ p 2 = 0.
A dc = F - 1 [ P 2 H ( θ h ) Q A ¯ dc ( ξ ) ] = t dc P 2 H ( θ h ) Q ( cos ϕ sin ϕ ) = t dc ( sin ϕ sin 2 θ h - i cos ϕ cos 2 θ h 0 ) = ( t dc exp [ i ψ dc ( θ h ) ] 0 ) ,
t dc = t dc f ( θ h , ϕ ) ,
ψ dc ( θ h ) = tan - 1 ( tan ϕ tan 2 θ h ) - π / 2 ,
f ( θ h , ϕ ) = ( sin 2 ϕ sin 2 2 θ h + cos 2 ϕ cos 2 2 θ h ) 1 / 2 .
A ac ( x ) = F - 1 [ P 2 H ( θ h ) Q A ¯ ac ( ξ ) ] = β cos ( θ p 1 - ϕ ) t ac ( x ) P 2 H ( θ h ) Q ( cos θ p 1 sin θ p 1 ) = β cos ( θ p 1 - ϕ ) t ac ( x ) ( sin θ p 1 sin 2 θ h - i cos θ p 1 cos 2 θ h 0 ) = ( t ac ( x ) exp [ i ψ ac ( θ h ) ] 0 ) ,
t ac ( x ) = β cos ( θ p 1 - ϕ ) t ac ( x ) f ( θ h , θ p 1 ) ,
ψ ac ( θ h ) = tan - 1 ( tan θ p 1 tan 2 θ h ) - π / 2 ,
Δ ψ = ψ ac ( θ h ) - ψ dc ( θ h ) = - 2 tan - 1 ( tan ϕ tan 2 θ h ) .
θ h = π / 8 for Δ ψ = π / 4 ,
θ h = 1 2 tan - 1 ( 1 / tan 2 π / 8 ) 0.7 for Δ ψ = 3 π / 4 ,
θ h = - 1 2 tan - 1 ( 1 / tan 2 π / 8 ) = - 0.7 for Δ ψ = - 3 π / 4 ,
θ h = - π / 8 for Δ ψ = - π / 4.
f 2 ( ± π / 8 , - π / 8 ) = 1 / 2             for Δ ψ = ± π / 4 ,
f 2 ( ± 0.7 , - π / 8 ) = 1 / 6             for Δ ψ = ± 3 π / 4.
I ( x ) = t dc exp [ i ψ dc ( θ h ) ] + t ac ( x ) exp [ i ψ ac ( θ h ) ] 2 = f 2 ( θ h , - π / 8 ) t dc + β t ac ( x ) exp ( i Δ ψ ) / 2 2 ,
I 1 ( x ) = 1 2 { t dc 2 + β 2 2 t ac ( x ) 2 + 2 β t dc t ac ( x ) × cos [ θ ac ( x ) + ɛ + π / 4 ] } ,
I 2 ( x ) = 1 6 { t dc 2 + β 2 2 t ac ( x ) 2 - 2 β t dc t ac ( x ) × sin [ θ ac ( x ) + ɛ + π / 4 ] } ,
I 3 ( x ) = 1 6 { t dc 2 + β 2 2 t ac ( x ) 2 + 2 β t dc t ac ( x ) × cos [ θ ac ( x ) + ɛ + π / 4 ] } ,
I 4 ( x ) = 1 2 { t dc 2 + β 2 2 t ac ( x ) 2 + 2 β t dc t ac ( x ) × sin [ θ ac ( x ) + ɛ + π / 4 ] } ,
θ ac ( x ) = tan - 1 { [ I 4 ( x ) - 3 I 2 ( x ) ] / [ I 1 ( x ) - 3 I 3 ( x ) ] } - ɛ - π / 4.
θ o ( x ) = tan - 1 { t ac ( x ) sin θ ac ( x ) / [ t dc + t ac ( x ) cos θ ac ( x ) ] } .
t ac ( x ) = 2 [ I 1 ( x ) + 3 I 3 ( x ) - t dc 2 ] 1 / 2 / β .
t dc = t dc / f ( θ h , - π / 8 ) .

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