Abstract

The design of phase-shifting algorithms (PSA’s) has been carried out with diverse strategies by different authors. A generalized algebraic approach is employed to obtain a family of detuning-insensitive PSA’s; their behavior against a linear phase error is analyzed from a geometric point of view. The obtained results are compared with the conditions provided by the Fourier representation of the corresponding sampling reference functions. In our case, new equations as criteria for determining whether a PSA is detuning insensitive, new analytic expressions for the phase error, and new algorithms with interesting properties are achieved.

© 2000 Optical Society of America

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Errata

Daniel Malacara-Doblado and Benito V. Dorrı́o, "Family of detuning-insensitive phase-shifting algorithms: erratum," J. Opt. Soc. Am. A 18, 721-721 (2001)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-18-3-721

References

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  13. D. Malacara, M. Servı́n, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998), Chap. 5, pp. 113–167.
  14. 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, 3421–3432 (1983).
    [CrossRef] [PubMed]

1999

B. V. Dorrı́o, J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10, R33–R55 (1999).
[CrossRef]

1996

1995

1994

1991

1990

1987

1983

1966

P. Carré, “Installation et utilisation du comparateur photo-électrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Bruning, J. H.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley Interscience, New York, 1992), Chap. 14, pp. 501–598.

Burow, R.

Carré, P.

P. Carré, “Installation et utilisation du comparateur photo-électrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Creath, K.

Dorri´o, B. V.

B. V. Dorrı́o, J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10, R33–R55 (1999).
[CrossRef]

Eiju, T.

Elssner, K.-E.

Fernández, J. L.

B. V. Dorrı́o, J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10, R33–R55 (1999).
[CrossRef]

Frankena, H. J.

Freischlad, K.

Greivenkamp, J. E.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley Interscience, New York, 1992), Chap. 14, pp. 501–598.

Groot, P.

Grzanna, J.

Hariharan, P.

Joenathan, C.

Koliopoulos, Ch. L.

Malacara, D.

D. Malacara, M. Servı́n, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998), Chap. 5, pp. 113–167.

Malacara, Z.

D. Malacara, M. Servı́n, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998), Chap. 5, pp. 113–167.

Merkel, K.

Oreb, B. F.

Schmit, J.

Schwider, J.

Servi´n, M.

D. Malacara, M. Servı́n, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998), Chap. 5, pp. 113–167.

Smorenburg, C.

Spolaczyk, R.

Surrel, Y.

Wingerden, J.

Appl. Opt.

J. Opt. Soc. Am. A

Meas. Sci. Technol.

B. V. Dorrı́o, J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10, R33–R55 (1999).
[CrossRef]

Metrologia

P. Carré, “Installation et utilisation du comparateur photo-électrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Other

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley Interscience, New York, 1992), Chap. 14, pp. 501–598.

D. Malacara, M. Servı́n, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998), Chap. 5, pp. 113–167.

K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Reading, UK, 1993), Chap. 4, pp. 94–104.

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Figures (5)

Fig. 1
Fig. 1

Schematic representation of sampling points located symmetrically with respect to the origin and uniformly spaced.

Fig. 2
Fig. 2

Amplitudes of the Fourier transforms for sampling reference functions in six-point first-order detuning-insensitive PSA’s with a relative phase-shift increment β0 of 90°.

Fig. 3
Fig. 3

Amplitudes of the Fourier transforms for sampling reference functions in four-point first-order detuning-insensitive PSA’s.

Fig. 4
Fig. 4

Peak-to-valley phase error, as a function of normalized frequency, for the Schwider–Hariharan PSA, compared with the PSA’s of Fig. 2: (a) A1=-12, A2=-12, A3=1, B1=0, B2=1, B3=1; (b) A1=1, A2=0, A3=-1, B1=0, B2=1, B3=0; (c) A1=-12, A2=1, A3=-12, B1=1, B2=0, B3=0; and (d) A1=1, A2=-12, A3=-12, B1=1, B2=1, B3=1.

Fig. 5
Fig. 5

Peak-to-valley phase error, as a function of normalized frequency, for the Schwider–Hariharan PSA, compared with the PSA’s of Fig. 3: (a) A1=1, A2=-1, A3=0, B1=0, B2=1, B3=0; (b) A1=1, A2=0, A3=-1, B1=0, B2=0, B3=1; and (c) A1=0, A2=1, A3=-1, B1=0, B2=0, B3=1.

Tables (3)

Tables Icon

Table 1 Relative Phase-Shift Increment β0 in Degrees for Different Detuning Insensitive PSA’s

Tables Icon

Table 2 Proportionality Factor C(β0) for the Considered First-Order Detuning Insensitive PSA’s

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Table 3 C(β0) for the Considered First-Order Detuning-Insensitive PSA’s

Equations (41)

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sk=a+b cos(ϕ+αk),
αk=(2k-n-1)β/2,
αk=-αn-k+1,
sk-sn-k+1=-2b sin ϕ sin αk,
sk+sn-k+1=2a+2b cos ϕ cos αk.
n,βϕ=tan-1C(β) N(β)D(β),
N(β)=k=1KBk(sk-sn-k+1),
D(β)=k=1KAk(sk+sn-k+1),
k=1KAk=0
C(β)=k=1KAkcos αkk=1KBksin αk
n,βϕ=tan-1-s(x)gN(x)dx-s(x)gD(x)dx,
gN(x)=C(β)k=1KBk[δ(x-xk)-δ(x-xn-k+1)],
gD(x)=k=1KAk[δ(x-xk)+δ(x-xn-k+1)],
GN(f )=C(β)k=1KBk[exp(-i2πfxk)-exp(-i2πfxn-k+1)],
GD(f )=k=1KAk[exp(-i2πfxk)+exp(-i2πfn-k+1)].
2πfxk=αk(f/fr);
GN(f )=-2iC(β)k=1KBksin[αk(f/fr)],
GD(f )=2k=1KAkcos[αk(f/fr)].
GN(0)=GD(0)=0.
Am[GN(fr)]=Am[GD(fr)],
GN(fr)=±iGD(fr).
GN(mfr)=GD(mfr)=0,
β=β0+δβ,
β=β0(f/fr).
d[N(β)/D(β)]dββ=β0
=-dC(β)dββ=β0/[C2(β0)]tan ϕ=0,
dC(β)dββ=β0=C(β0)=0.
C(β)=k=1KAkαksin αkk=1KBkαkcos αk.
C(β)=-k=1KAkαk2cos αkk=1KBkαk2sin αk.
δϕ=12Am[GN(f )]Am[GD(f )]-1sin 2ϕ.
δϕ=12 ϕ(β0)(δβ)2;
δϕ=sin 2ϕ4C(β0)C(β0) (δβ)2.
P-Vphaseerror
=k=1KBkαk2sin αkk=1KBksin αk-k=1KAkαk2cos αkk=1KAkcos αk(δβ)22β02
 
C(β)=2 sin β
90°,6ϕ
=tan-1C(β0) s2+s3-s4-s5-12s1-12s2+s3+s4-12s5-12s6.
 
90°,6ϕ=tan-1C(β0) s2-s5s1-s3-s4+s6.
90°,4ϕ=tan-1C(β0) -s2+s3s2-s3-s4+s5.

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