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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 21 — Oct. 10, 2011
  • pp: 19987–19992
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Role of the filter phase in phase sampling interferometry

Juan Antonio Quiroga, Manuel Servín, Julio Cesar Estrada, Javier Vargas, and Francisco Javier Torre-Belizon  »View Author Affiliations


Optics Express, Vol. 19, Issue 21, pp. 19987-19992 (2011)
http://dx.doi.org/10.1364/OE.19.019987


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Abstract

Any linear phase sampling algorithm can be described as a linear filter characterized by its frequency response. In traditional phase sampling interferometry the phase of the frequency response has been ignored because the impulse responses can be made real selecting the correct sample offset. However least squares methods and recursive filters can have a complex frequency response. In this paper, we derive the quadrature equations for a general phase sampling algorithm and describe the role of the filter phase.

© 2011 OSA

1 Quadrature phase detection

Phase sampling interferometry (PSI) [1

1. D. Malacara, M. Servín, and Z. Malacara Interferogram Analysis For Optical Testing, Second ed. CRC Press 2005.

] is an experimental technique for phase measurement based on the introduction of a linear phase shift between a set of interferograms. In the temporal case, if we describe the interferogram as a rectangular 2D matrix, the measured intensity at every pixel can be described as a set of temporal samples given by
g(t)=b+mcos(φ(t)+ω0t)t=1...N,
(1)
where b is the background or DC term, m the modulation or AC term, and ω0 the carrier frequency. In PSI, the main objective is the computation of the modulating phase φ(t) for every sample.

H(ω)=FT(h)=t=h(t)eitω  and   h(t)=FT1(H)=ππH(ω)eitωdω.
(2)

The interferogram spectrum is given by
G(ω)=FT(g(t))=bδ(ω)+12m[C(ωω0)+C(ω+ω0)],
(3)
where C(ω)=FT(eiφ(t)). If C(ω) is narrowband, C(ωω0) will be a side-lobe contained in an interval Δω0 with ω0Δω0. The objective of any PSA is the filtering of one of the lobes of G(ω) so that the output signal spectrum is given by

Q(ω)=H(ω)G(ω)=12mH(ω)C(ωω0).
(4)

Transforming back Eq. (4) in the temporal domain, we obtain the analytical signal associated with g

q(t)=h(t)g(t)=12mH(ω0)ei(φ(t)+ω0t).
(5)

2 The role of the filter phase in quadrature phase detection: the group delay

s(t)=p(t)cos(ω0t).
(7)

In (7) we assume that the envelope p(t) is narrowband. If we factorize the filter frequency response in terms of amplitude and phase, H(ω)=A(ω)eiΦ(ω), near ω0 we can always approximate the filter phase as
Φ(ω)=Φ0τ(ω0)(ωω0),
(8)
with

τ(ω)=dΦdω|.
(9)

When we apply the PSA to g(t), taking into account the quadrature conditions (6) and that the envelope p(t) is narrow-band, we will obtain
F(ω)=H(ω)S(ω)=12eiΦ0A(ω)P(ωω0)eiτ(ωω0),
(10)
where P(ω)=FT(p(t)). Expression (10) can be rewritten as
F(ω)=12eiΦ0A(ω)(P(ω)eiτωδ(ωω0)),
(11)
where * denotes the convolution product. And, using the convolution and modulation properties of the FT we obtain

f(t)=12eiΦ0A(ω0)p(tτ)eitω0.
(12)

That is, the filter phase introduces a time delay τ in the signal envelope. In signal processing, the time delay τ as given by (9) is denominated group delay [5

5. A. V. Oppenheim, and R. W. Schafer, Discrete-Time Signal processing, Pearson (2010)

]. The physical interpretation of τ is that the different packets of a signal will have different delays after passing through a linear system. For this reason, a desirable property for any digital filter is to have a linear phase for all ω so that the group delay is constant. In the case of quadrature phase detection, it is enough to have a linear phase around the tuning frequency ω0 given that the interferogram is quasi-monochromatic.

Now we are ready to go forward and analyse the effect of the filter phase on a PSI temporal signal like (1). First, we must rewrite g(t) as

g(t)=b+mcosφ(t)cosω0tmsinφ(t)sinω0t.
(13)

q(t)=12meiΦ0A(ω0)ei(φ(tτ)+ω0t).
(14)

Again, we can compute the phase φ(tτ) from the angle of q(t). Equation (14) is the generalization of the classical result given by Eq. (5) and it is the main result of this paper: the PSA filter phase determines the sample where we are measuring the modulating phase, and the time-shift is the group delay given in Eq. (9). Another interesting result is that if around the tuning frequency the phase is not linear, τ=τ(ω), the recovered phase will be distorted by the variable group delay.

In conclusion, the filter phase determines the temporal origin of the PSA and adds an additional requirement to the usual quadrature conditions. That is, the group delay must be constant around ω0, τ(Δω0) τ0.

3 Two illustrative examples

We are going to clarify the former discussion by analysing two PSAs. The first example is a filter with impulse response given by

h1(t)=[1,2i,2,2i,1].
(15)

Its frequency response is

H1(ω)=t=15h(t)eitω=(2+4sinω2cos2ω)ei3ω.
(16)

In this case H1(π/2)=H1(0)=0 and τ=3. This means that h1 represents a 5 samples PSA method tuned at ω0=π/2 and constant group-delay of 3 samples. For this PSA the modulating phase will be obtained from

tanφ(3)=Im(h1g)Re(h1g)|t=0=2(g(2)g(4))2g(3)g(1)g(5).
(17)

The reader will recognise this equation as the standard form of the Hariharan 5 step PSA [6

6. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26(13), 2504–2506 (1987). [CrossRef] [PubMed]

]. In general, in classical linear PSAs, if the filter length is K and the first sample start in t=1 the group delay is τ=round((K+1)/2), where round(.) represents the rounding to the closest integer operation.

The second example we present is more interesting from the point of view of the group delay. We are speaking of the first order recursive PSI filter presented in [2

2. M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009). [CrossRef] [PubMed]

] given by

q(t)=ηq(t1)eiω0+(g(t1)eiω02g(t)+g(t+1)eiω0)*(δ(t1)δ(t+1)).
(18)

H2(ω)=4isin(ω)(cos(ω+ω0)1)1ηexp(i(ωω0)).
(19)

It is easy to verify that H2(ω) fulfills the quadrature conditions (6) and therefore represents a true PSA. As before, we measure the instantaneous phase determining the angle of the analytical signal q(t). However, in this case the question is: at which sample are we measuring the phase? As we have shown, the answer depends on the group delay. In Fig. 1
Fig. 1 Amplitude and phase delay for the recursive filter of Eq. (18) for η=0.75 and ω0=π/2 rad/sample.
we show the amplitude and the group delay of the PSA represented by H2(ω) with η=0.75 and ω0=π/2 rad/sample. As can be seen, due to the non-linear phase, the group delay has a strong dependence with the frequency. As τ(π/2)4 samples, if we apply the recursive PSI filter to a carrier interferogram with ω0π/2 we will obtain a demodulated phase with a delay of 4 samples with respect to the input signal, that is

tanφr(t4)=Im(H2{g(t)})Re(H2{g(t)}).
(20)

To show the effect of this delay on the phase, Fig. 2
Fig. 2 Demodulation results using the recursive filter of Eq. (18). a) input signal, b) demodulation results (red) and actual phase (blue). In this figure there is a marker every four samples to make easier the observation of the 4 samples delay between both signals.
shows the demodulation of a temporal interferogram given by g(t)=100+50cos{2cos(4πt/N)+0.5πt} with N=100 samples, using the PSA represented in Fig. 1. Figure 2a shows a plot of the input temporal signal and Fig. 2b shows the demodulated phase, φr(t) together with the actual input phase. As expected there is a 4 samples delay between both phases.

An undesired effect of the nonlinear filter phase is the distortion induced by the different group delays of the spectral components of the interferogram. Practically, one consequence is that the filter response to an edge is not symmetrical, and depends on the edge direction. Figure 3
Fig. 3 Recursive filter demodulation results with a discontinuous rectangular phase. a) temporal interferogram, b) demodulated phase in the forward-time pass, c) demodulated phase after the bidirectional filtering.
shows the demodulation results for a 100 samples temporal interferogram with a rectangular pulse of 2 rad. Figure 3a shows the interferogram and Fig. 3b shows the demodulation results using the same filter represented in Fig. 1. As can be seen, there is clear asymmetry in the recovered phase. This is a well-known effect and can be suppressed if the interferogram can be stored. In this case, the solution is to process back the output of the recursive filter in reverse order: qT(t)=H2{q*(t)}. In digital signal processing this method is referred as bidirectional filtering. Mathematically, this is equivalent to demodulating back the signal with the same impulse response but time-inverted and complex conjugated. Therefore, if h2(t) is the impulse response of the recursive PSA, the total impulse response of the bidirectional filtering will be
hT(t)=h2(t)*h2*(t),
(21)
from which it is easy to see that total frequency response is real and given by

HT(t)=H(ω)H*(ω)=|A(ω)|2.
(22)

The resulting PSA has zero-phase as desired, and all the flexibility of a recursive filter. In Fig. 3c we show the demodulation results for the bidirectional filtering version of the recursive filter represented in Fig. 1. In Figs. 2 and 3 we have used η=0.75. For smaller values, typically η0.2 we have practically a linear PSA with some noise rejection thanks to the recursive averaging. On the other side, for η0.9 we have a recursive filter with a highly non-linear phase. To test this behaviour, the MATLAB code to run all the examples presented in this paper can be downloaded from [7

7. http:\\goo.gl/BGjJ9

]

4 Conclusions

In this work, we have demonstrated the importance of the filter phase for the correct interpretation of the output of a general PSA. We have shown how the group delay determines the sample at which we are actually measuring the phase. We have analyzed under this new perspective a classical PSA and a more sophisticated recursive filter. Finally, in the case of a non-constant group delay we have shown how the bidirectional filtering can be used to compensate the filter phase effects.

References

1.

D. Malacara, M. Servín, and Z. Malacara Interferogram Analysis For Optical Testing, Second ed. CRC Press 2005.

2.

M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009). [CrossRef] [PubMed]

3.

J. L. Marroquin, J. E. Figueroa, and M. Servin, “Robust quadrature filters,” J. Opt. Soc. Am. A 14(4), 779–791 (1997). [CrossRef]

4.

J. A. Quiroga, J. C. Estrada, M. Servín, and J. Vargas, “Regularized least squares phase sampling interferometry,” Opt. Express 19(6), 5002–5013 (2011). [CrossRef] [PubMed]

5.

A. V. Oppenheim, and R. W. Schafer, Discrete-Time Signal processing, Pearson (2010)

6.

P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26(13), 2504–2506 (1987). [CrossRef] [PubMed]

7.

http:\\goo.gl/BGjJ9

OCIS Codes
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(120.3940) Instrumentation, measurement, and metrology : Metrology
(120.5050) Instrumentation, measurement, and metrology : Phase measurement

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: March 7, 2011
Revised Manuscript: April 6, 2011
Manuscript Accepted: April 21, 2011
Published: September 28, 2011

Citation
Juan Antonio Quiroga, Manuel Servín, Julio Cesar Estrada, Javier Vargas, and Francisco Javier Torre-Belizon, "Role of the filter phase in phase sampling interferometry," Opt. Express 19, 19987-19992 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-21-19987


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