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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 24 — Nov. 22, 2010
  • pp: 24405–24411
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Two-frame algorithm to design quadrature filters in phase shifting interferometry

J. F. Mosiño, J. C. Gutiérrez-García, T. A. Gutiérrez-García, and J. M. Macías-Preza  »View Author Affiliations


Optics Express, Vol. 18, Issue 24, pp. 24405-24411 (2010)
http://dx.doi.org/10.1364/OE.18.024405


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Abstract

The main purpose of this paper is to present a method to design tunable quadrature filters in phase shifting interferometry. From a general tunable two-frame algorithm introduced, a set of individual filters corresponding to each quadrature conditions of the filter is obtained. Then, through a convolution algorithm of this set of filters the desired symmetric quadrature filter is recovered. Finally, the method is applied to obtain several tunable filters, like four and five-frame algorithms.

© 2010 OSA

1. Introduction

The estimated phase of a quadrature filter order M is given by [1

1. J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34(19), 3610–3619 (1995). [CrossRef] [PubMed]

9

9. J. F. Mosiño, D. M. Doblado, and D. M. Hernández, “A method to design tunable quadrature filters in phase shifting interferometry,” Opt. Express 17(18), 15772–15777 (2009). [CrossRef] [PubMed]

],

tan(φ)=k=1MbkIkk=1MakIk=[b1b2...bM] I[a1a2...aM] I=N ID I.
(1)

H(α)=0;H(α)=0;H(α)=0;...Hm(α)=0.
(3)

In other words, m gives the order of insensitivity to the phase shift error, and Hm(ω) is the mth derivate of H(ω) with regard to ω. In the same way, the condition to be satisfied by a filter which is insensitive to the mth order bias variation error H(ω) is [9

9. J. F. Mosiño, D. M. Doblado, and D. M. Hernández, “A method to design tunable quadrature filters in phase shifting interferometry,” Opt. Express 17(18), 15772–15777 (2009). [CrossRef] [PubMed]

],

H(0)=0;H(0)=0;H(0)=0...Hm(0)=0.
(4)

2. Convolution algorithm

Assuming that the Fourier transform of a filter can be factorized in two functions such as H(ω)=H1(ω)  H2(ω), where H1(ω) and H2(ω) are the Fourier transforms of h1(t)=D1δ+i N1δ and h2(t)=D2δ+i N2δ which are an n and m order filters respectively, the individual estimated phases φ1 and φ2 are given by

tan(φ1)=N1 ID1 I,  and  tan(φ2)=N2 ID2 I.
(5)

Hence, the desired filter h(t) is obtained from the expression h(t)=h1(t)h2(t), where ∗ denotes the temporal discrete convolution, and h(t) becomes,
h(t)=[D1δ+i N1δ][D2δ+i N2δ]=[D1D2N1N2]δ+i[N1D2+D1N2]δ.
(6)
and this expression corresponds with the estimated phase ϕ and given by
tan(φ)=N ID I=[N1D2+D1N2] I[D1D2N1N2] I,
(7)
that is, the convolution algorithm can simply be represented as

ND=(N1D1)(N2D2)=[N1D2+D1N2][D1D2N1N2].
(8)

In other words, as mentioned before, a new (n + m-1) frame filter from two individual filters is obtained. Likewise, the design of a tunable quadrature filter is seen as an algebraic problem without the use of Fourier formalisms. The convolution properties allow this case to be extended for three or more filters.

3. Design of tunable filters

The design of a quadrature filter order M implies that only M-1 parameters are free, two of which are the quadrature conditions and the other M-3 are used to compensate some errors.

3.1 The tunable two-frame algorithm

The general form of a symmetric two-frame algorithm is [9

9. J. F. Mosiño, D. M. Doblado, and D. M. Hernández, “A method to design tunable quadrature filters in phase shifting interferometry,” Opt. Express 17(18), 15772–15777 (2009). [CrossRef] [PubMed]

],

tan(φ)=b1I1b1I2a1I1+a1I2=b1[11] Ia1[11] I.
(9)

H(ω,α)=2[a1cos(ω/2)b1sin(ω/2)].
(10)

Tuning this filter to cut off the frequency ω=α, the condition H(ω=α,α)=0 is satisfied. Hence, the solution of Eq. (10) is given by the ratio b1/a1=cos(α/2)/sin(α/2). And from Eq. (9) the corresponding tunable two-frame filter and the ratio N/D are respectively,

tan(φ)=NIDI=cos(α/2)[11]Isin(α/2)[1  1] I=cos(α/2)sin(α/2)[I1I2I1+I2].
(11)
ND=cos(α/2)[11]sin(α/2)[1  1] .
(12)

Furthermore, this two-frame filter has the following Fourier transform

H(ω,α)=2sin[(ωα)/2].
(13)

Therefore, from Eq. (3) the Fourier transform and the corresponding algorithm for a filter insensitive to the 0, 1, 2 … and m orders phase shift detuning error are respectively
Hm+1(ω,α)=(2)m+1sinm+1[(ωα)/2].
(14)
ND={cos(α/2)sin(α/2)[11][11]}m+1.
(15)
where m + 1 denotes the times that the convolution is applied. From Eq. (13), for α = 0, the other quadrature condition H(ω=0,α)=0 is recovered and its Fourier transform becomes
H(ω,α)=H(ω,0)=2sin(ω/2).
(16)
and from Eq. (11) the corresponding two-frame algorithm that cuts off the frequency ω=0 is

tan(φ)=[11]I[00]I=[I1I20I1+0I2].
(17)

Then, from Eqs. (4) and (16) the filter insensitive to the m order bias modulation error is,
Hm+1(ω,0)=(2)m+1sinm+1(ω/2).
(18)
and from Eq. (15) for α = 0, the corresponding algorithm is

ND={[11][00]}m+1.
(19)

In the same way, other possible conditions working over the frequency ω=π are,
H(π)=0H(π)=0;H(π)=0;H(π)=0...Hm(π)=0.
(20)
where the Fourier transform for the filter insensitive to the 0, 1, 2 and m order of the derivative function at frequency ω = π is
H(ω,π)=(2)m+1cosm+1(ω/2).
(21)
and from Eqs. (12) and (20) the respective algorithm becomes

ND={[00][11]}m+1.
(22)

3.2 The tunable three-frame algorithm

The Fourier transform that meets the two quadrature conditions H(0) = 0 and H(α) = 0 is

H(ω)=(2)2sin(ω/2) sin[(ωα)/2]  .
(23)

This function is the product of the two filters that cuts off individually both frequencies ω = 0 and ω = α. Then, in terms of the corresponding two-frame filters, and through the convolution algorithm Eq. (8), the desired N/D becomes
ND={[11][00]}{cos(α/2)[11]sin(α/2)[1  1]}=sin(α/2)[1  1][11]cos(α/2)[11][11].
(24)
and from Eq. (7) the estimated phase is expressed as [1

1. J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34(19), 3610–3619 (1995). [CrossRef] [PubMed]

5

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

],

tan(φ)=NIDI=sin(α/2)[101]Icos(α/2)[121]I=tan(α/2)I1I3I1+2I2I3.
(25)

3.3 The tunable four-frame algorithm

Like Eq. (1), the general form for the estimated phase of a four-frame algorithm is

tan(φ)=[b2b1b1b2] I[a2a1a1a2] I.
(26)

From Eq. (13), the Fourier transform that cuts off the frequencies ω=0, ω=α and ω=β is

H(ω)=H(ω,0)H(ω,α)H(ω,β)=8sin(ω/2) sin[(ωα)/2] sin[(ωβ)/2].
(27)

That is, two conditions correspond to the necessary quadrature conditions, and H(ω,β)=0 is the particular condition that cuts off the frequency β. Then, by applying the corresponding two-frame filters, the result is

ND={[11][00]}{cos(α/2)[11]sin(α/2)[1  1]}{cos(β/2)[11]sin(β/2)[1  1]}.
(28)

From Eq. (8) the last two terms become,
ND=[11][00]sin[(α+β)/2] [   1,0,1] [cos[(α+β)/2] ,2cos[(αβ)/2] ,cos[(α+β)/2]].
(29)
therefore, by applying Eqs. (7) and (8) the estimated phase is expressed as

tan(φ)=NIDI=cos[(α+β)/2](I1I4){2sin[(αβ)/2]+cos[(α+β)/2]}(I2I3) sin[(α+β)/2] (I1I2I3I4).
(30)

This is the general four-frame filter that cuts off both frequencies, the α step and an arbitrary frequency β. Thus, by using the geometrical condition β=π, a new four-frame algorithm is obtained. Therefore, as reported in [5

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

] we named it “tunable four-frame algorithm in X”,
tan(φ)=sin(α/2)cos(α/2) [1 111]  I [1-1-11]  I=tan(α/2)I1I2+I3+I4I1I2I3+I4.
(31)
and for α=π/2, the well known four-frame algorithm in X is recovered [5

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

]. On the other hand, Eq. (30) for β=α gives a filter that cuts off twice the same frequency α, then the derivative at frequency α is zero. That is to say, that the filter becomes insensitive to the linear phase shift detuning error, and gives the reported four-frame algorithm class B [9

9. J. F. Mosiño, D. M. Doblado, and D. M. Hernández, “A method to design tunable quadrature filters in phase shifting interferometry,” Opt. Express 17(18), 15772–15777 (2009). [CrossRef] [PubMed]

]. Finally, Eq. (30) for β=0 recovers the reported four-frame algorithm class C [9

9. J. F. Mosiño, D. M. Doblado, and D. M. Hernández, “A method to design tunable quadrature filters in phase shifting interferometry,” Opt. Express 17(18), 15772–15777 (2009). [CrossRef] [PubMed]

], which is a four-frame algorithm insensitive to the linear bias modulation error.

3.4 The tunable five-frame algorithm

For the α step and the two arbitrary frequencies β and γ, the Fourier transform of the filter is,

H(ω)=16sin(ω/2)sin[(ωα)/2]sin[(ωβ)/2]sin[(ωγ)/2].
(32)

From Eqs. (7) and (8) the desired phase is obtained through the following expression

ND={[11][00]}{cos(α/2)[11]sin(α/2)[1  1]}{cos(β/2)[11]sin(β/2)[1  1]}{cos(γ/2)[11]sin(γ/2)[1  1]}.
(33)

In Table 1

Table 1. Several particular five-frame temporal phase shifting algorithms

table-icon
View This Table
, particular cases of several five-frame algorithms are shown. Case 1 meets the conditions H(0)=H(0)=H(0)=H(α)=0, therefore from Eq. (32) H(ω) becomes,

H(ω)=16sin3(ω/2)sin[(ωα)/2].
(34)

In case 2, H(0)=H(α)=H(0)=H(π)=0 and H(ω) becomes,

H(ω)=16sin2(ω/2)cos(ω/2)sin[(ωα)/2].
(35)

In case 3, H(0)=H(α)=H(α)=H(π)=0 and H(ω) gives

H(ω)=16sin(ω/2)cos(ω/2)sin2[(ωα)/2].
(36)

In case 4, H(0)=H(α)=H(π)=H(π)=0 and H(ω) results

H(ω)=16sin(ω/2)cos2(ω/2)sin[(ωα)/2].
(37)

In case 5, the filter meets the conditions H(0)=H(α)=H(0)=H(α)=0 and gives

H(ω)=16sin2(ω/2)sin2[(ωα)/2].
(38)

This is a tunable five-frame algorithm which is simultaneously insensitive to linear bias modulation and linear phase shift detuning errors. It comprises the results reported in [6

6. M. Afifi, K. Nassim, and S. Rachafi, “Five-frame phase-shifting algorithm insensitive to diode laser power variation,” Opt. Commun. 197(1-3), 37–42 (2001). [CrossRef]

]. To conclude, in case 6 the tunable filter insensitive to second order phase shift detuning error is,

H(ω)=16sin(ω/2)sin3[(ωα)/2].
(39)

Finally, from [7

7. 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]

,8

8. J. F. Mosiño, D. M. Doblado, and D. M. Hernández, “Calculus of exact detuning phase shift error in temporal phase shifting algorithms,” Opt. Express 17(18), 15766–15771 (2009). [CrossRef] [PubMed]

], the detuning error of the function sin[(ωα)/2] gives Δφ=tan(Δ/2) for the angle Δ. Therefore, for sinq[(ωα)/2] the detuning error is Δφ=(1)qtanq(Δ/2). Thus, cases 3 and 5 have q = 2, while case 6 has q = 3, and the others cases have the value q = 1.

4. Conclusions

A new method to design quadrature filters is presented. The design problem is reduced to an algebraic problem that through the convolution of a set of two-frame filters gives the desired phase without the use of Fourier’s formalisms. Each individual condition of the filter corresponds with a specific two-frame filter. Therefore, several new tunable four and five-frame symmetrical algorithms are reported. This method is easily evaluated numerically.

Acknowledgments

This work was partially supported by CONACyT México through scholarship granted #175434. The authors also acknowledge the help granted by MDD. J. J. Lozano in the revision of this paper.

References and links

1.

J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34(19), 3610–3619 (1995). [CrossRef] [PubMed]

2.

Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996). [CrossRef] [PubMed]

3.

D. W. Phillion, “General methods for generating phase-shifting interferometry algorithms,” Appl. Opt. 36(31), 8098–8115 (1997). [CrossRef]

4.

Y. Surrel, “Design of phase-detection algorithms insensitive to bias modulation,” Appl. Opt. 36(4), 805–807 (1997). [CrossRef] [PubMed]

5.

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

6.

M. Afifi, K. Nassim, and S. Rachafi, “Five-frame phase-shifting algorithm insensitive to diode laser power variation,” Opt. Commun. 197(1-3), 37–42 (2001). [CrossRef]

7.

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]

8.

J. F. Mosiño, D. M. Doblado, and D. M. Hernández, “Calculus of exact detuning phase shift error in temporal phase shifting algorithms,” Opt. Express 17(18), 15766–15771 (2009). [CrossRef] [PubMed]

9.

J. F. Mosiño, D. M. Doblado, and D. M. Hernández, “A method to design tunable quadrature filters in phase shifting interferometry,” Opt. Express 17(18), 15772–15777 (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: September 7, 2010
Revised Manuscript: October 25, 2010
Manuscript Accepted: October 27, 2010
Published: November 8, 2010

Citation
J. F. Mosiño, J. C. Gutiérrez-García, T. A. Gutiérrez-García, and J. M. Macías-Preza, "Two-frame algorithm to design quadrature filters in phase shifting interferometry," Opt. Express 18, 24405-24411 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-24-24405


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References

  1. J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34(19), 3610–3619 (1995). [CrossRef] [PubMed]
  2. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996). [CrossRef] [PubMed]
  3. D. W. Phillion, “General methods for generating phase-shifting interferometry algorithms,” Appl. Opt. 36(31), 8098–8115 (1997). [CrossRef]
  4. Y. Surrel, “Design of phase-detection algorithms insensitive to bias modulation,” Appl. Opt. 36(4), 805–807 (1997). [CrossRef] [PubMed]
  5. 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).
  6. M. Afifi, K. Nassim, and S. Rachafi, “Five-frame phase-shifting algorithm insensitive to diode laser power variation,” Opt. Commun. 197(1-3), 37–42 (2001). [CrossRef]
  7. 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]
  8. J. F. Mosiño, D. M. Doblado, and D. M. Hernández, “Calculus of exact detuning phase shift error in temporal phase shifting algorithms,” Opt. Express 17(18), 15766–15771 (2009). [CrossRef] [PubMed]
  9. J. F. Mosiño, D. M. Doblado, and D. M. Hernández, “A method to design tunable quadrature filters in phase shifting interferometry,” Opt. Express 17(18), 15772–15777 (2009). [CrossRef] [PubMed]

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