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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 18 — Aug. 31, 2009
  • pp: 15766–15771
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Calculus of exact detuning phase shift error in temporal phase shifting algorithms

J. F. Mosiño, D. Malacara Doblado, and D. Malacara Hernández  »View Author Affiliations


Optics Express, Vol. 17, Issue 18, pp. 15766-15771 (2009)
http://dx.doi.org/10.1364/OE.17.015766


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Abstract

The detuning phase shift error is a common systematic error observed in temporal phase shifting (TPS) algorithms. This error, generally due to miscalibration of the phase shifter, is solved by using a quadrature filter insensitive to this detuning error. To compare algorithms, this error is frequently analyzed numerically. However, in this work we present an exact and analytical expression to calculate such error which is applicable to any kind of filters with real or complex frequency response. Finally, a table with the detuning error for several algorithms is reported.

© 2009 OSA

1. Introduction

2. Error Detuning in Phase-Shifting Interferometry (PSI)

To solve Eq. (13), we take the tangent for both terms, and we have,
tan(Δϕ)=tan(ϕ)σtan(ϕ)1+σtan2(ϕ).
(14)
This expression can be rearranged to obtain,
tan(Δϕ)=   (1σ1+σ)2tan(ϕ)1+tan2(ϕ)1+(1σ1+σ)(1tan2(ϕ)1+tan2(ϕ)).
(15)
Then, by using Eq. (11) and after some trigonometric substitutions, Eq. (14) becomes,
tan(Δϕ)=rsin(2ϕ)1+rcos(2ϕ).
(16)
Although this analytical expression is similar to the equation previously reported [8

8. J. E. Hernández and D. Malacara, “Exact linear detuning error in phase shifting algorithms,” Opt. comm. 180, 9–14 (2000).

], the ratio reported in this work is more general than the previously reported ratio, in spite of both being referred to as r. That is, from Eq. (15) and the ratio r we can obtain the exact detuning error for any TPS algorithm. Then, from Eq. (15) we can observe that if |ε|0, then r0and σ1, no detuning error is present, and the erroneous phase ϕ becomes the desired phase ϕϕ. On the other hand, assuming a small detuning error, we have that tan(Δϕ)Δϕ and rcos(2ϕ)<<1. Hence, Eq. (16) is reduced to,
Δϕ=rsin(2ϕ)=|H(ω0Δ)||H(ω0+Δ)|sin(2ϕ)=(σ1σ+1)sin(2ϕ).
(17)
This expression may be further simplified by using σ1.0. Doing this, we recover the expression Δϕ0.5(σ1)sin(2ϕ), which was reported in literature [7

7. K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase measuring interferometry,” J. Opt. Soc. Am . A 7(4), 542–551 (1990). [CrossRef]

,8

8. J. E. Hernández and D. Malacara, “Exact linear detuning error in phase shifting algorithms,” Opt. comm. 180, 9–14 (2000).

]. Notice that this result is almost the same result presented here; however, we consider the expression here reported to be more practical. To compare against the results reported in literature, we maximize our exact result in Eq. (17) with respect to φ , obtaining the following expression,
|Δϕmax|=sin1|H(ω0Δ)/H(ω0+Δ)|.
(18)
We emphasize that this Eq. (18) for the maximum detuning error is exact when tuned onto the left side; in this fashion, it coincides exactly with the detuning error that was evaluated numerically [1

1. J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf ed., (North Holland, Amsterdam, Oxford, New York, Tokyo, 1990).

7

7. K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase measuring interferometry,” J. Opt. Soc. Am . A 7(4), 542–551 (1990). [CrossRef]

]. We can repeat all the steps described above for a quadrature filter tuned onto the right side of the frequency axis, or for sign minus in Eq. (8) and we obtain the following result, which is equivalent to having changed the sign of ω0+Δ, then, we have
|Δϕmax|=sin1|H(ω0+Δ)/H(ω0Δ)|.
(19)
In consequence, it can be said that the user must take whether the quadrature filter is tuned onto the left or onto the right before applying the formula. Finally, we must notice that for a symmetrical filter, the frequency response becomes a real function and the result coincides with what has been previously reported [9

9. J. F. Mosiño, M. Servin, J. C. Estrada, and J. A. Quiroga, “Phasorial analysis of detuning error in temporal phase shifting algorithms,” Opt. Express 17(7), 5618–5623 (2009). [CrossRef] [PubMed]

]. This expression is a very versatile way to evaluate the detuning phase shift error analytically or numerically, instead of the approximation reported in literature [7

7. K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase measuring interferometry,” J. Opt. Soc. Am . A 7(4), 542–551 (1990). [CrossRef]

,8

8. J. E. Hernández and D. Malacara, “Exact linear detuning error in phase shifting algorithms,” Opt. comm. 180, 9–14 (2000).

].

3. Some Examples of Error Detuning in Phase-Shifting Interferometry

3.1 Three frame algorithm case

One three frame non symmetric TPS algorithm is given by
tan[ϕ(x,y,α=π/2)]=I(α)I(0)I(0)I(α).
(20)
The time response of this quadrature filter with α=π/2 is,
h(t,α=π/2)=[δ(t)δ(tα)]+i[δ(t+α)δ(t)].
(21)
The frequency response becomes non-real; then, H(ω,α) is
H(ω,α)=iexp[ωαi]exp[ωαi]+1i;α=π/2.
(22)
This filter is tuned at frequency ω=1 with α=π/2. That is, the filter satisfies H(ω=1,α=π/2)=H(ω=0,α=π/2)=0, meaning that it is tuned onto the right side. Now, from Eq. (18), the exact detuning error for α=π/2+Δ is
Δϕmax=sin1|H(ω=1,α=π/2+Δ)H(ω=1,α=π/2Δ)|.
(23)
Then, calculating this ratio we have

Δϕmax=sin1|2i(i+1)sin(Δ/2)[sin(Δ/2)+cos(Δ/2)]2i(i1)cos(Δ/2)[sin(Δ/2)+cos(Δ/2)]|=sin1|tan(Δ2)||Δ|/2.
(24)

3.2 Four frame algorithm

The four non symmetric frame TPS in cross algorithm is given by
ϕ(x,y,α=π/2)=tan1(I(α)I(α)I(0)I(2α)).
(25)
Then, the frequency response of this TPS algorithm is a complex function given by,
H(ω,α)=12sin(ωα)exp(2ωαi).
(26)
Notice that this filter is also tuned onto the right side at ω=1 with a phase step α=π/2, and we have that H(ω=1,α=π/2)=0 and H(ω=0,α=π/2)=0. Now, from Eq. (19) we obtain,

Δϕmax=sin1|iexp(Δi)tan(Δ/2)|=sin1|tan(Δ/2)||Δ|/2.
(27)

3.2 Other algorithms

In Table 1

Table 1. Detuning Phase Shift error for several TPS algorithms

table-icon
View This Table
, the value r, for some detuning phase shift errors for several TPS algorithms are presented. We notice that many of them have the form tann(Δ/2) for n integer.

4. Conclusions

An exact and analytical algorithm to evaluate the detuning error in phase shifting algorithms was obtained from algebraic methods. The expression is applicable to any kind of (PSI) algorithms, symmetrical or not. The derived expression was compared with other well known approximations. Finally, this expression was successfully applied to evaluate and obtain the detuning error for some well known quadrature filters.

Acknowledgments

This work was partially supported by CONACyT under grant No. 42771.

References and links

1.

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf ed., (North Holland, Amsterdam, Oxford, New York, Tokyo, 1990).

2.

J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometry for testing optical surfaces and lenses,” Appl. Opt. 13(11), 2693–2703 (1974). [CrossRef] [PubMed]

3.

H. Schreiber, J. H. Brunning, and J. E. Greivenkamp, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara ed., (John Wiley & Sons, Inc., Hoboken, New Jersey 2007).

4.

M. Servin, and M. Kujawinska, “Modern fringe pattern analysis in Interferometry,” in Handbook of Optical Engineering, D. Malacara and B. J. Thompson eds., (Marcel Dekker, 2001).

5.

D. Malacara, M. Servin, and Z. Malacara, “Phase Detection Algorithms,” in Interferogram Analysis for Optical Testing, D. Malacara ed., (Taylor & Francis Group, 2005).

6.

J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22(21), 3421–3432 (1983). [CrossRef] [PubMed]

7.

K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase measuring interferometry,” J. Opt. Soc. Am . A 7(4), 542–551 (1990). [CrossRef]

8.

J. E. Hernández and D. Malacara, “Exact linear detuning error in phase shifting algorithms,” Opt. comm. 180, 9–14 (2000).

9.

J. F. Mosiño, M. Servin, J. C. Estrada, and J. A. Quiroga, “Phasorial analysis of detuning error in temporal phase shifting algorithms,” Opt. Express 17(7), 5618–5623 (2009). [CrossRef] [PubMed]

OCIS Codes
(120.2650) Instrumentation, measurement, and metrology : Fringe analysis
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(120.5050) Instrumentation, measurement, and metrology : Phase measurement

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: June 17, 2009
Revised Manuscript: July 19, 2009
Manuscript Accepted: July 21, 2009
Published: August 20, 2009

Citation
J. F. Mosiño, D. Malacara Doblado, and D. Malacara Hernández, "Calculus of exact detuning phase shift error in temporal phase shifting algorithms," Opt. Express 17, 15766-15771 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-18-15766


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References

  1. J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf ed., (North Holland, Amsterdam, Oxford, New York, Tokyo, 1990).
  2. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometry for testing optical surfaces and lenses,” Appl. Opt. 13(11), 2693–2703 (1974). [CrossRef] [PubMed]
  3. H. Schreiber, J. H. Brunning, and J. E. Greivenkamp, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara ed., (John Wiley & Sons, Inc., Hoboken, New Jersey 2007).
  4. M. Servin, and M. Kujawinska, “Modern fringe pattern analysis in Interferometry,” in Handbook of Optical Engineering, D. Malacara and B. J. Thompson eds., (Marcel Dekker, 2001).
  5. D. Malacara, M. Servin, and Z. Malacara, “Phase Detection Algorithms,” in Interferogram Analysis for Optical Testing, D. Malacara ed., (Taylor & Francis Group, 2005).
  6. J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22(21), 3421–3432 (1983). [CrossRef] [PubMed]
  7. K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase measuring interferometry,” J. Opt. Soc. Am . A 7(4), 542–551 (1990). [CrossRef]
  8. J. E. Hernández and D. Malacara, “Exact linear detuning error in phase shifting algorithms,” Opt. comm. 180, 9–14 (2000).
  9. J. F. Mosiño, M. Servin, J. C. Estrada, and J. A. Quiroga, “Phasorial analysis of detuning error in temporal phase shifting algorithms,” Opt. Express 17(7), 5618–5623 (2009). [CrossRef] [PubMed]

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