Abstract

A new method for phase-shifting interferometry based on wave amplitude modulation is proposed and discussed. This proposal is based on the interference of three waves, where two waves attend as two reference waves and the other wave attends as a probe wave. Thereby, three interference terms are obtained, but because a phase difference of π/2 between the two references is kept constant, one of the three terms will be dropped, while the two remaining will be put in quadrature. Under these conditions, the resulting pattern is mathematically modeled by an interferogram of two waves, where an additional phase is given by the amplitude variations of the reference waves. In this Letter, both a theoretical model and some numerical simulations are presented.

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References

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2009 (1)

2008 (2)

2007 (1)

A. Martínez, M. del Mar Sánchez-López, and I. Moreno, Eur. J. Phys. 28, 805 (2007).
[CrossRef]

2006 (1)

J. F. James, A Student’s Guide to Fourier Transform(Cambridge U. Press, 2006).

1992 (1)

D. Malacara, Optical Shop Testing (Wiley, 1992).

1990 (1)

J. Schwider, in Progress in Optics, E.Wolf, ed. (Elsevier, 1990), Vol.  28, pp. 296–303.
[CrossRef]

1988 (1)

K. Creath, in Progress in Optics, E.Wolf, ed. (Elsevier, 1988), Vol.  26, pp. 358–365.
[CrossRef]

1987 (1)

1984 (2)

1978 (2)

B. J. Doerband and M. Stockmann, Dtsch. Ges. Angew. Optik (1978), A52.

J. C. Wyant and R. N. Shagam, in Optics Present and Future: Proceedings of the Eleventh Congress of the International Commission for Optics, ICO 11 (1978), pp. 659–662.

1974 (1)

1972 (1)

1970 (1)

1969 (2)

1967 (1)

Ai, C.

Brangaccio, D. J.

Bruning, J. H.

Cheng, Y.-Y.

Crane, R.

R. Crane, Appl. Opt. 8, 538 (1969).

Creath, K.

K. Creath, in Progress in Optics, E.Wolf, ed. (Elsevier, 1988), Vol.  26, pp. 358–365.
[CrossRef]

Deck, L.

del Mar Sánchez-López, M.

A. Martínez, M. del Mar Sánchez-López, and I. Moreno, Eur. J. Phys. 28, 805 (2007).
[CrossRef]

Delisle, C.

Doerband, B. J.

B. J. Doerband and M. Stockmann, Dtsch. Ges. Angew. Optik (1978), A52.

Gallagher, J. E.

Herriott, D. R.

Hioki, R.

James, J. F.

J. F. James, A Student’s Guide to Fourier Transform(Cambridge U. Press, 2006).

Kiire, T.

Kothiyal, M. P.

Malacara, D.

Martínez, A.

A. Martínez, M. del Mar Sánchez-López, and I. Moreno, Eur. J. Phys. 28, 805 (2007).
[CrossRef]

Morales, A.

Moreno, I.

A. Martínez, M. del Mar Sánchez-López, and I. Moreno, Eur. J. Phys. 28, 805 (2007).
[CrossRef]

Nakadate, S.

Olof, B.

Rizo, I.

Rosenfield, D. P.

Schwider, J.

J. Schwider, in Progress in Optics, E.Wolf, ed. (Elsevier, 1990), Vol.  28, pp. 296–303.
[CrossRef]

Shagam, R. N.

J. C. Wyant and R. N. Shagam, in Optics Present and Future: Proceedings of the Eleventh Congress of the International Commission for Optics, ICO 11 (1978), pp. 659–662.

Shibuya, M.

Stevenson, W. H.

Stockmann, M.

B. J. Doerband and M. Stockmann, Dtsch. Ges. Angew. Optik (1978), A52.

Susuki, T.

White, A. D.

Wyant, J. C.

C. Ai and J. C. Wyant, Appl. Opt. 26, 1112 (1987).
[CrossRef] [PubMed]

Y.-Y. Cheng and J. C. Wyant, Appl. Opt. 23, 4539 (1984).
[CrossRef] [PubMed]

J. C. Wyant and R. N. Shagam, in Optics Present and Future: Proceedings of the Eleventh Congress of the International Commission for Optics, ICO 11 (1978), pp. 659–662.

Yang, Q.

Zhao, B.

Zhou, R.

Appl. Opt. (9)

Dtsch. Ges. Angew. Optik (1)

B. J. Doerband and M. Stockmann, Dtsch. Ges. Angew. Optik (1978), A52.

Eur. J. Phys. (1)

A. Martínez, M. del Mar Sánchez-López, and I. Moreno, Eur. J. Phys. 28, 805 (2007).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Lett. (1)

Other (5)

J. F. James, A Student’s Guide to Fourier Transform(Cambridge U. Press, 2006).

J. C. Wyant and R. N. Shagam, in Optics Present and Future: Proceedings of the Eleventh Congress of the International Commission for Optics, ICO 11 (1978), pp. 659–662.

D. Malacara, Optical Shop Testing (Wiley, 1992).

J. Schwider, in Progress in Optics, E.Wolf, ed. (Elsevier, 1990), Vol.  28, pp. 296–303.
[CrossRef]

K. Creath, in Progress in Optics, E.Wolf, ed. (Elsevier, 1988), Vol.  26, pp. 358–365.
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Amplitudes of waves 1 and 3 put in a reference system. Any point on the arc keeps constant A r , and ψ is varied as required for PSI.

Fig. 2
Fig. 2

Amplitude variations for PSI of three steps: (a) a point on the arc yields a phase shift while the reference amplitude is kept constant; (b) interferograms shifted in phase by the amplitude variations indicated in (a).

Fig. 3
Fig. 3

Amplitude variations for PSI of four steps: (a) a point on the arc yields phase shifting while the reference amplitude is kept constant; (b) interferograms shifted in phase by the amplitude variations indicated in (a).

Equations (12)

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I k ( x , y ) = a ( x , y ) + b ( x , y ) cos [ ϕ ( x , y ) + α k ] ,
E n ( x , y ) = A n ( x , y ) exp [ i ϕ n ( x , y ) ] ,
I ( x , y ) = | E 1 ( x , y ) + E 2 ( x , y ) + E 3 ( x , y ) | 2 .
I = A 1 2 + A 2 2 + A 3 2 + 2 A 1 A 2 cos ( ϕ 1 ϕ 2 ) + 2 A 1 A 3 cos ( ϕ 1 ϕ 3 ) + 2 A 2 A 3 cos ( ϕ 2 ϕ 3 ) ,
ϕ 1 = 0 , ϕ 2 = ϕ , ϕ 3 = π / 2 ,
I = A 1 2 + A 2 2 + A 3 2 + 2 A 1 A 2 cos ϕ + 2 A 2 A 3 sin ϕ ,
I = A r 2 + A 2 2 + 2 A r A 2 cos ( ϕ ψ ) .
A r 2 = A 1 2 + A 3 2 , tan ψ = A 3 A 1 ,
A 1 = A r cos ψ , A 3 = A r sin ψ ,
A 2 = A r = 1 , A 1 , A 3 [ 0 , 1 ] , ϕ = x 2 + y 2 ,
E 1 = A 1 , E 2 = exp ( i ϕ ) , E 3 = i A 3 ,
I = 2 + 2 cos ( ϕ ψ ) ,

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