Abstract

In phase-shifting interferometry spatial nonuniformity of the phase shift gives a significant error in the evaluated phase when the phase shift is nonlinear. However, current error-compensating algorithms can counteract the spatial nonuniformity only in linear miscalibrations of the phase shift. We describe an error-expansion method to construct phase-shifting algorithms that can compensate for nonlinear and spatially nonuniform phase shifts. The condition for eliminating the effect of nonlinear and spatially nonuniform phase shifts is given as a set of linear equations of the sampling amplitudes. As examples, three new algorithms (six-sample, eight-sample, and nine-sample algorithms) are given to show the method of compensation for a quadratic and spatially nonuniform phase shift.

© 1997 Optical Society of America

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  1. 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, 2693–2703 (1974).
    [Crossref] [PubMed]
  2. K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland Elsevier, Amsterdam, 1988), Vol. 26, pp. 349–393.
  3. K. G. Larkin, “Efficient nonlinear algorithm for envelope detection in white light interferometry,” J. Opt. Soc. Am. A 13, 832–843 (1996).
    [Crossref]
  4. P. Carre, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
    [Crossref]
  5. J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wavefront measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
    [Crossref]
  6. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [Crossref] [PubMed]
  7. J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
    [Crossref]
  8. K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
    [Crossref]
  9. Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32, 3598–3600 (1993).
    [Crossref] [PubMed]
  10. J. Schmit, K. Creath, “Extended averaging technique for derivation of error compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34, 3610–3619 (1995).
    [Crossref] [PubMed]
  11. P. de Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34, 4723–4730 (1995).
    [Crossref]
  12. K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting for nonsinusoidal waveforms with phase-shift errors,” J. Opt. Soc. Am. A 12, 761–768 (1995).
    [Crossref]
  13. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35, 51–60 (1996).
    [Crossref] [PubMed]
  14. P. J. de Groot, “Vibration in phase-shifting interferometry,” J. Opt. Soc. Am. A 12, 354–365 (1995).
    [Crossref]
  15. J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (North-Holland Elsevier, Amsterdam, 1990), Vol. 28, pp. 271–359.
  16. K. Freischlad, C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7, 542–551 (1990).
    [Crossref]
  17. K. Hibino, “Susceptibility of error-compensating algorithms to random noise in phase-shifting interferometry,” Appl. Opt. 36 (1997).
  18. C. P. Brophy, “Effect of intensity error correlation on the computed phase of phase-shifting interferometry,” J. Opt. Soc. Am. A 7, 537–541 (1990).
    [Crossref]
  19. K. A. Stetson, W. R. Brohinsky, “Electro-optic holography and its application to hologram interferometry,” Appl. Opt. 24, 3631–3637 (1985).
    [Crossref]
  20. I. S. Gradshteyn, I. M. Ryzhik, “Trigonometric and hyperbolic functions,” in Table of Integrals, Series, and Products, A. Jeffrey, ed. (Academic, London, 1980), p. 31.

1997 (1)

K. Hibino, “Susceptibility of error-compensating algorithms to random noise in phase-shifting interferometry,” Appl. Opt. 36 (1997).

1996 (2)

1995 (4)

1993 (2)

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[Crossref]

Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32, 3598–3600 (1993).
[Crossref] [PubMed]

1992 (1)

1990 (2)

1987 (1)

1985 (1)

1983 (1)

1974 (1)

1966 (1)

P. Carre, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[Crossref]

Brangaccio, D. J.

Brohinsky, W. R.

Brophy, C. P.

Bruning, J. H.

Burow, R.

Carre, P.

P. Carre, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[Crossref]

Creath, K.

J. Schmit, K. Creath, “Extended averaging technique for derivation of error compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34, 3610–3619 (1995).
[Crossref] [PubMed]

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland Elsevier, Amsterdam, 1988), Vol. 26, pp. 349–393.

de Groot, P.

de Groot, P. J.

Eiju, T.

Elssner, K. E.

Falkenstorfer, O.

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[Crossref]

Farrant, D. I.

Freischlad, K.

Gallagher, J. E.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, “Trigonometric and hyperbolic functions,” in Table of Integrals, Series, and Products, A. Jeffrey, ed. (Academic, London, 1980), p. 31.

Grzanna, J.

Hariharan, P.

Herriott, D. R.

Hibino, K.

K. Hibino, “Susceptibility of error-compensating algorithms to random noise in phase-shifting interferometry,” Appl. Opt. 36 (1997).

K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting for nonsinusoidal waveforms with phase-shift errors,” J. Opt. Soc. Am. A 12, 761–768 (1995).
[Crossref]

Koliopoulos, C. L.

Larkin, K. G.

Merkel, K.

Oreb, B. F.

Rosenfeld, D. P.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, “Trigonometric and hyperbolic functions,” in Table of Integrals, Series, and Products, A. Jeffrey, ed. (Academic, London, 1980), p. 31.

Schmit, J.

Schreiber, H.

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[Crossref]

Schwider, J.

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[Crossref]

J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wavefront measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
[Crossref]

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (North-Holland Elsevier, Amsterdam, 1990), Vol. 28, pp. 271–359.

Spolaczyk, R.

Stetson, K. A.

Streibl, N.

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[Crossref]

Surrel, Y.

White, A. D.

Zoller, A.

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[Crossref]

Appl. Opt. (9)

K. Hibino, “Susceptibility of error-compensating algorithms to random noise in phase-shifting interferometry,” Appl. Opt. 36 (1997).

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, 2693–2703 (1974).
[Crossref] [PubMed]

J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wavefront measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
[Crossref]

K. A. Stetson, W. R. Brohinsky, “Electro-optic holography and its application to hologram interferometry,” Appl. Opt. 24, 3631–3637 (1985).
[Crossref]

Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32, 3598–3600 (1993).
[Crossref] [PubMed]

J. Schmit, K. Creath, “Extended averaging technique for derivation of error compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34, 3610–3619 (1995).
[Crossref] [PubMed]

P. de Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34, 4723–4730 (1995).
[Crossref]

Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35, 51–60 (1996).
[Crossref] [PubMed]

P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
[Crossref] [PubMed]

J. Opt. Soc. Am. A (6)

Metrologia (1)

P. Carre, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[Crossref]

Opt. Eng. (1)

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[Crossref]

Other (3)

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (North-Holland Elsevier, Amsterdam, 1990), Vol. 28, pp. 271–359.

I. S. Gradshteyn, I. M. Ryzhik, “Trigonometric and hyperbolic functions,” in Table of Integrals, Series, and Products, A. Jeffrey, ed. (Academic, London, 1980), p. 31.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland Elsevier, Amsterdam, 1988), Vol. 26, pp. 349–393.

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Figures (3)

Fig. 1
Fig. 1

(a) Functions F01(ν) and iF02(ν) and (b). Real and imaginary parts of their derivatives for the new six-sample algorithm, where F01(ν) and iF02(ν) are defined by F01(ν)=(i/12)×[5 sin(5πν/6)-6 sin(πν/2)-17 sin(πν/6)] and iF02(ν)=(3/36)i[cos(5πν/6)-26 cos(πν/2)+25 cos(πν/6)].

Fig. 2
Fig. 2

(a) Functions F01(ν) and iF02(ν) and (b) imaginary parts of their second-order derivatives for the new eight-sample algorithm, where F01(ν) and iF02(ν) are defined by F01(ν)=(2/32)i[-4 sin(7πν/4)+2 sin(5πν/4)-14 sin(3πν/4)-20×sin(πν/4)] and iF02(ν)=(2/32)i[-3 cos(7πν/4)+cos(5πν/4)-17 cos(3πν/4)+19 cos(πν/4)].

Fig. 3
Fig. 3

(a) Functions F01(ν) and iF02(ν) and (b) imaginary parts of their second-order derivatives for the new nine-sample algorithm, where F01(ν) and iF02(ν) are defined by F01(ν)=(i/16)×[sin(2πν)-2 sin(3πν/2)-14 sin(πν)-18 sin(πν/2)] and iF02(ν)=(i/8)[-cos(2πν)-4 cos(3πν/2)-4 cos(πν)+4 cos ×(πν/2)+5].

Tables (3)

Tables Icon

Table 1 Phase-Shifting Algorithms and Their Immunity to Phase-Shift Errors

Tables Icon

Table 2 Number of Independent Equations Necessary for Constructing the Algorithms When the Sampling Amplitudes Have Symmetriesa

Tables Icon

Table 3 Peak-to-Valley Phase Errors That Are Due to a Quadratic and Spatially Nonuniform Phase Shift for the New Six-Sample and for the Previously Reported Seven-Sample and Five-Sample Algorithms

Equations (92)

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ϕ=arctan7(I2-I4)-(I0-I6)-4(I1+I5)+8I3,
αr=π2(r-3)1+1+22(r-3)
forr=0, , 6,
Δϕ=π22-π22264sin(2ϕ)+π414256sin(2ϕ)+.
I(x, y, α)=s0(x, y)+k=1j sk(x, y)×cos[kα-ϕk(x, y)],
ϕ=arctanr=1m brIrr=1m arIr,
αr=α0r[1+(α0r)]=α0r1+1+2α0rπ+3α0rπ2++pα0rπp-1forr=1, 2, , m,
Δϕ=ϕ-ϕ1=o(sk)+o(q)+o(qsk)+o(s2)+o(2),
r=1m ar sin(kα0r)=0fork=1, , j,
r=1m ar cos(kα0r)=δ(k, 1)fork=0, 1, , j,
r1m br sin(kα0r)=δ(k, 1)fork=1, , j,
r=1m br cos(kα0r)=0fork=0, 1, , j,
ϕ=arctanr=1mbrs0+k=1jsk coskα0r1+q=1pqα0rπq-1-ϕkr=1mars0+k=1jsk coskα0r1+q=1pqα0rπq-1-ϕk=arctansin ϕ1-πr=1mk=1jq=1psks1kqα0rπqbr sin(kα0r-ϕk)cos ϕ1-πr=1mk=1jq=1psks1kqα0rπqar sin(kα0r-ϕk)=arctansin ϕ1-q=1pqBq(ϕ1)-q=1pk=2jqsks1Dk,q(ϕk)cos ϕ1-q=1pqAq(ϕ1)-q=1pk=2jqsks1Ck,q(ϕk)arctan1+q=1p qAq(ϕ1)cos ϕ1-Bq(ϕ1)sin ϕ1+q=1pk=2j qsks1Ck,q(ϕk)cos ϕ1-Dk,q(ϕk)sin ϕ1tan ϕ1ϕ1+12q=1p qAq(ϕ1)cos ϕ1-Bq(ϕ1)sin ϕ1sin(2ϕ1)+12q=1pk=2j qsks1Ck,q(ϕk)cos ϕ1-Dk,q(ϕk)sin ϕ1sin(2ϕ1),
Aq(ϕ1)=π r=1m arα0rπq sin(α0r-ϕ1),
Bq(ϕ1)=π r=1m brα0rπq sin(α0r-ϕ1),
Ck,q(ϕk)=πk r=1m arα0rπq sin(kα0r-ϕk),
Dk,q(ϕk)=πk r=1m brα0rπq sin(kα0r-ϕk),
Aq(ϕ1)cos ϕ1-Bq(ϕ1)sin ϕ1sin(2ϕ1)=0forq=1, 2, , p.
r=1m α0rq(ar sin α0r+br cos α0r)sin(2ϕ1)+r=1m α0rq(ar cos α0r-br sin α0r)cos(2ϕ1)+r=1m α0rq(ar cos α0r+br sin α0r)=0.
r=1m α0rq(ar cos α0r+br sin α0r)=0
forq=1, 2, , p,
r=1m α0rq(ar cos α0r-br sin α0r)=0
forq=1, 2, , p,
r=1m α0rq(ar sin α0r+br cos α0r)=0
forq=1, 2, .., p.
Aq(ϕ1)cos ϕ1-Bq(ϕ1)sin ϕ1sin(2ϕ1)=const.
forq=1, 2, , p
Ck,q(ϕk)cos ϕ1-Dk,q(ϕk)sin ϕ1sin(2ϕ1)=0
fork=2, , jandq=1, .., p
r=1m α0rqar sin(kα0r)=0,
r=1m α0rqar cos(kα0r)=0,
r=1m α0rqbr sin(kα0r)=0,
r=1m α0rqbr cos(kα0r)=0,
fork=2, , jandq=1, 2, , p.
Ck,q(ϕk)cos ϕ1-Dk,q(ϕk)sin ϕ1sin(2ϕ1)=const.
fork=2, , jandq=1, , p
{ar}=(a1, a2, , a2, a1),
{br}=(b1, b2, , -b2, -b1).
[x]=nforx=integernorx=n+1/2.
mj+[3p/2]+2,
sin(j+2-k)α0r=(-1)m sin(kα0r),
cos(j+2-k)α0r=(-1)m+1 cos(kα0r),
m=j+[3p/2]+2+p(j-1)=(p+1)j+[p/2]+2,
α0r=2πnr-m2forevenm2πnr-m+12foroddm.
{ar}=(a1, a2, a3, a3, a2, a1),
{br}=(b1, b2, b3, -b3, -b2, -b1),
a1+a2+a3=0,-a1+a3=1/3,
b1+2b2+b3=-1,
25a1-a3+25b1+18b2+b3=0,
25a1-a3-25b1-18b2-b3=0,
5a1+6a2+a3+53b1-3b3=0.
{ar}=(3/72, -133/36, 253/72, 253/72, -133/36, 3/72),
{br}=(5/24, -1/4, -17/24, 17/24, 1/4, -5/24).
ϕ=arctan3(5I1-6I2-17I3+17I4+6I5-5I6)I1-26I2+25I3+25I4-26I5+I6.
αr=π3r-721+1+23r-72,
Δϕ=ϕ-ϕ1=-1975124,416(π2)2 sin(2ϕ1)-π212568643cos(2ϕ1)+2658643+o(3).
ϕ=arctanI1-4I2+4I4-I5-I1-2I2+6I3-2I4-I5
2a1+2a2+2a3+a4=0,-2a2+a4=1,
-2a1+2a2-2a3+a4=0,b1-b3=1/2,
-3a1+a3+2b2=0,8a2-9b1+b3=0,
8a2+9b1-b3=0.
{ar}=-3322, 1322, -17322, 19322, 19322, -17322, 1322, -3322,
{br}=-4322, 2322, -14322, -20322, 20322, 14322, -2322, 4322,
ϕ=arctan-4(I1-I8)+2(I2-I7)-14(I3-I6)-20(I4-I5)-3(I1+I8)+I2+I7-17(I3+I6)+19(I4+I5).
{ar}=-116, -14, -14, 14, 58, 14, -14, -14, -116,
{br}=132, -116, -716, -916, 0, 916, 716, 116, -132,
ϕ=arctan1/2(I1-I9)-(I2-I8)-7(I3-I7)-9(I4-I6)-(I1+I9)-4(I2+I8)-4(I3+I7)+4(I4+I6)+10I5.
{ar}=0, -12, 12, 0, 12, -12, 0,
{br}=133, -123, -123, 0, 123, 123, -133,
ϕ=arctan-I2-I3+I5+I6+(2/3)(I1-I7)3(-I2+I3+I5-I6).
{ar}=(0, a2, a3, , aj+2, aj+3, 0)+(e1, -e1, 0, 0, , 0, -e1, e1),
{br}=(0, b2, b3, , bj+2, bj+3, 0)+(e2, e2, 0, 0, , 0, -e2, -e2),
ar=2j+2cos α0r,br=2j+2sin α0r
forr=2, 3, , j+3,
α0r=2πj+2r-j+52.
2j+2r=2j+3 sin(kα0r)sin α0r=2j+2r=2j+3 cos(kα0r)cos α0r=δ(k, 1),
2j+2r=2j+3 sin(kα0r)cos α0r=2j+2r=2j+3 cos(kα0r)sin α0r=0.
e1(sin α01-sin α02)+e2(cos α01+cos α02)=1π(j+2)r=2j+3 α0r sin(2α0r),
e1(cos α01-cos α02)-e2(sin α01+sin α02)=1π(j+2)r=2j+3 α0r cos(2α0r),
e1=12(j+2)cos3πj+2sin22πj+2,
e2=12(j+2)sin3πj+2sin22πj+2,
ϕ=arctan14(I1+I2-Ij+3-Ij+4)sin3πj+2sin22πj+2+r=2j+3 Irsin2πj+2r-j+5214(I1-I2-Ij+3+Ij+4)cos3πj+2sin22πj+2+r=2j+3 Ircos2πj+2r-j+52.
ϕ=arctanI1-3I2-4I3+4I4+3I5-I6-I1-3I2+4I3+4I4-3I5-I6.
f1(α)=r=1m brδ(α-αr),
f2(α)=r=1m arδ(α-αr),
F1(ν)=r=1m br exp(-iαrν),
F2(ν)=r=1m ar exp(-iαrν),
F1(0)=F2(0)=0,
F1(ν0)+iF2(ν0)=0.
dqF01dνq+idqF02dνq=0atν=ν0forq=1, , p,
dqF01dνq-idqF02dνq=(-i)q r=1m {[-arα0rq sin(α0rν)+brα0rq cos(α0rν)]-i[arα0rq cos(α0rν)+brα0rq sin(α0rν)]}.
ImiqdqF01dνq-idqF02dνq=0atν=ν0,

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