## Phasorial analysis of detuning error in temporal phase shifting algorithms

Optics Express, Vol. 17, Issue 7, pp. 5618-5623 (2009)

http://dx.doi.org/10.1364/OE.17.005618

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### Abstract

Phase error analysis in Temporal Phase Shifting (TPS) algorithms due to frequency detuning has been to date only performed numerically. In this paper, we show an exact analytical expression to obtain this phase error due to detuning using the spectral TPS response. The new proposed method is based on the phasorial representation of the output of the TPS quadrature filter. Doing this, the detuning problem is reduced to a ratio of two symmetrical spectral responses of the quadrature filter at the detuned frequency. Finally, some popular cases of TPS algorithms are analyzed to show the usefulness of the proposed method.

© 2009 Optical Society of America

## 1. Introduction

*ω*

_{0}has exactly the carrier frequency used in the TPS algorithm. When this is not the case, the actual temporal carrier used to obtain our interferograms differs from

*ω*

_{0}and an erroneous phase is estimated. This erroneous phase estimation is called detuning error. To evaluate the detuning error from any quadrature filter used in TPS the common procedure has been numeric evaluation, then, it would be good to obtain an exact-analytical expression for the detuning to better understand this phase estimation error [1-8]. The analytical expression herein derived for the detuning error in TPS algorithms allows us to minimize some TPS quadrature filters for detuning error. Most TPS algorithms used nowadays have no free parameters that permit us this kind of detuning optimization. However, some new techniques may be used to generate more general TPS algorithms including free parameters that allow us to improve the robustness of these new TPS filters to detuning. Even if no optimization is possible, one may always compare among the myriad of TPS algorithms available in the literature using the analytical expression given in this paper. The input of any TPS quadrature filter is a set of real cosine signals which are the intensity of a series of TPS interferograms captured by a CCD camera. The output of these TPS quadrature filters is the complex analytical signal at the temporal carrier frequency -

*ω*

_{0}associated with the real intensity of our interferograms. As mentioned before, when the carrier frequency of the interferograms is detuned, an additional undesired complex signal with carrier frequency +

*ω*

_{0}is added to our desired signal at -

*ω*

_{0}. Hence the complex-vectorial (or phasor) sum of both signals give rise to an erroneously (detuned) estimated phase from this set of temporal interferograms.

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

*ϕ*(

*x*,

*y*) denotes the unknown phase,

*a*(

*x*,

*y*) is the background illumination, and

*b*(

*x*,

*y*) is the contrast of interference fringes; these two signals are low frequency. The temporal carrier

*ω*

_{0}is a linear phase shift among the set of interferograms which is introduced in the data gathering process. Meanwhile,

*t*corresponds with the temporal sampling which is taken as a natural number in this paper. Taking the Fourier transform of

*I*(

*x*,

*y*,

*t*), we have

*I*(

*x*,

*y*,

*ω*) as,

*g*(

*t*) is obtained by convolving a discrete temporal quadrature filter with several temporal phase shifted interferograms expressed as,

*g*(

*t*) =

*h*(

*t*) ∗

*I*(

*x*,

*y*,

*t*) where (

^{*}) denotes one dimensional temporal convolution.

*G*(

*ω*) =

*H*(

*ω*)

*I*(

*ω*). Notice that, the quadrature filter

*h*(

*t*) is a one-sided (complex) convolution filter that is tuned at frequency (

*ω*

_{0}= ω). Quadrature filters are complex linear systems having a symmetric real component

*hr*(

*n*) and an antisymmetric imaginary component

*hi*(

*n*) or

*h*(

*t*) =

*hr*(

*t*) +

*i hi*(

*t*). As a consequence, we have the very important fact that the resulting Fourier transform of

*h*(

*t*) being

*H*(

*ω*) always becomes a real function of the frequency

*ω*. Then, the output signal

*G*(

*ω*) =

*I*(

*ω*)

*H*(

*ω*) is expressed as,

*x*,

*y*) of the functions

*I*,

*a*,

*b*and

*ϕ*has been dropped. Now, we can observe in Fig. 1, that at (exactly) the carrier frequency

*ω*=

*ω*

_{0}, we have

*H*(

*ω*

_{0}) = 0,

*H*(0) = 0 and

*H*(−

*ω*

_{0}) ≠ 0.

*G*(

*ω*

_{0}) becomes

*G*(

*x*,

*y*,

*ω*) = (

*b*/2)

*H*(−

*ω*

_{0})exp(

*iϕ*). Then the searched phase

*ϕ*is recovered from the complex analytical signal

*g*(

*x*,

*y*,

*t*) which is its inverse Fourier transform. Now let us consider that our phase stepping interferograms are not sampled at the expected frequency rate

*ω*

_{0}, but at an erroneous temporal frequency given by

*ω*=

*ω*

_{0}+ Δ, then we have that the spectral response

*G*(

*x*,

*y*,

*ω*) becomes,

*c*and the undesired or spurious signal as

*ε*,

*c*with an angle

*ϕ*(

*x*,

*y*) and the erroneous spurious signal

*ε*with angle −

*ϕ*(

*x*,

*y*).

*c*,

*ε*and

*R*which stand for,

*ϕ*=

*ϕ*'−

*ϕ*may be defined as the difference between the desired phase

*ϕ*and the undesired phase

*ϕ*'. This expression for the detuning error Δ

*ϕ*is widely used to evaluate the robustness to detuning in TPS and it is has been evaluated only numerically [1-8]. From of Eq. (8) we obtain

*ϕ*', then by using this into Δ

*ϕ*=

*ϕ*'-

*ϕ*, we find that the detuning error is expressed as,

*ε*→0 no detuning error is present and the erroneous phase

*ϕ*' becomes the desired phase

*ϕ*' →

*ϕ*. The main objective in this paper is to find an easy, useful and analytical detuning robustness equation in terms of the frequency response

*H*(

*ω*) of any TPS algorithm. To this end, we must substitute Eq. (8) into Eq. (9) and after using some trigonometric relations, the searched expression for the detuning error Δ

*ϕ*in terms of the frequency response of the TPS algorithm is,

*ε*/

*c*) << 1.0 , as a consequence of this tan(Δ

*ϕ*) ≈ Δ

*ϕ*the finally Eq. (10) becomes,

*ϕ*). The Eq. (11) is the most important result in this paper. A short-cut way to obtain the last result given by Eq. (11) from Eq. (8), is to consider that the detuning error is small. Therefore the following approximations apply

*ϕ*≈

*ϕ*', sin(

*ϕ*−

*ϕ*') ≈ − Δ

*ϕ*and sin(

*ϕ*' +

*ϕ*) ≈ sin(2

*ϕ*), as a consequence the detuning error becomes Δ

*ϕ*= −(

*ε*/

*c*)sin(2

*ϕ*) which is equal to Eq. (11).

*ϕ*obtaining the following analytical result,

*H*(

*ω*) real or complex.

## 3. Some examples of error detuning in Phase-Shifting Interferometry.

6. J P. Hariharan, B. Oreb, and T. Eiju, “Digital phase shifting interferometry: a simple error compensating phase calculation algorithm,” Appl. Opt. **26**, 2504–2505 (1987). [CrossRef] [PubMed]

### 3.1 The five frames algorithm:

6. J P. Hariharan, B. Oreb, and T. Eiju, “Digital phase shifting interferometry: a simple error compensating phase calculation algorithm,” Appl. Opt. **26**, 2504–2505 (1987). [CrossRef] [PubMed]

*ϕ*(

*x*,

*y*,

*α*) is the interferogram’s phase for phase step

*α*=

*π*/2. We first obtain the time response of the quadrature filter from the TPS formula Eq. (13) with as,

*i*= √−1; then, we obtain the frequency response

*H*(

*ω*,

*α*=

*π*/2) of this quadrature filter as,

*ω*= 1. Now assuming a detuning error in the phase step of

*α*=

*π*/2 + Δ and according to Eq. (8) we obtain,

*α*=

*π*/2 + Δ. This result may be approximated for small Δ (as reported in [3,4,6

6. J P. Hariharan, B. Oreb, and T. Eiju, “Digital phase shifting interferometry: a simple error compensating phase calculation algorithm,” Appl. Opt. **26**, 2504–2505 (1987). [CrossRef] [PubMed]

*ϕ*= (Δ

^{2}/4)sin(2

*ϕ*). It can be seen that the maximum detuning is when the sine function equals one so Δ

*ϕ*

_{max}= Δ

^{2}/4 renders the well known quadratic detuning error behavior of the five step Schwider-Hariharan TPS algorithm [5

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

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

### 3.2 Seven frames algorithm

8. M. Servin, D. Malacara, J. L. Marroquin, and F. J. Cuevas, “Complex linear filters for phase shifting with very low detuning sensitivity,” J. Mod. Opt. **44**, 1269–1278 (1997). [CrossRef]

*ω*= 1 for a phase step

*α*=

*π*/2, and we have that,

*H*(1) = 0 and

*H*(0) = 0. Now let us detune our phase step to

*α*=

*π*/2 + Δ then we obtain,

*n*Δ)=1.0 and sin(

*n*Δ)=

*n*Δ for any

*n*in Eq. (20) the maximum detuning error is

*only*numerically in [4,8

8. M. Servin, D. Malacara, J. L. Marroquin, and F. J. Cuevas, “Complex linear filters for phase shifting with very low detuning sensitivity,” J. Mod. Opt. **44**, 1269–1278 (1997). [CrossRef]

### 3.3 Eleven frames algorithm

*ϕ*

_{max}≈ Δ

^{4}/16.

## 4. Conclusions

## Acknowledgments

## References and links

1. | J Schwider, “Advanced evaluation techniques in interferometry,” in |

2. | J. H. Bruning, D. R. Herriot, J. E. Gallagher, D. P. Rosenfel, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometry for testing optical surfaces and lenses,” Appl. Opt. |

3. | H. Schreiber and J. H. Brunning, “Phase shifting interferometry,” in |

4. | M. Servin and M. Kujawinska, “Modern fringe pattern analysis in Interferometry,” in |

5. | J. Schwider, R. Burrow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. |

6. | J P. Hariharan, B. Oreb, and T. Eiju, “Digital phase shifting interferometry: a simple error compensating phase calculation algorithm,” Appl. Opt. |

7. | K. Freischland and C. L. Koliopoulos, “Fourier description of digital phase measuring interferometry,” J. Opt. Soc. Am. A |

8. | M. Servin, D. Malacara, J. L. Marroquin, and F. J. Cuevas, “Complex linear filters for phase shifting with very low detuning sensitivity,” J. Mod. Opt. |

**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: December 17, 2008

Revised Manuscript: February 23, 2009

Manuscript Accepted: February 26, 2009

Published: March 25, 2009

**Citation**

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**, 5618-5623 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-7-5618

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### References

- J. Schwider, "Advanced evaluation techniques in interferometry," in Progress in Optics, E. Wolf ed., (North Holland, Amsterdam, Oxford, New York, Tokyo, 1990).
- J. H. Bruning, D. R. Herriot, J. E. Gallagher, D. P. Rosenfel, A. D. White, and D. J. Brangaccio, "Digital wavefront measuring interferometry for testing optical surfaces and lenses," Appl. Opt. 13, 2693-2703 (1974). [CrossRef] [PubMed]
- H. Schreiber and J. H. Brunning, "Phase shifting interferometry," in Optical Shop Testing, D. Malacara ed., (John Wiley and Sons, Inc., Hoboken, New Jersey 2007).
- 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).
- J. Schwider, R. Burrow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, "Digital wave-front measuring interferometry: some systematic error sources," Appl. Opt. 22, 3421-3432 (1983). [CrossRef] [PubMed]
- J. P. Hariharan, B. Oreb, and T. Eiju, "Digital phase shifting interferometry: a simple error compensating phase calculation algorithm," Appl. Opt. 26, 2504-2505 (1987). [CrossRef] [PubMed]
- K. Freischland and C. L. Koliopoulos, "Fourier description of digital phase measuring interferometry," J. Opt. Soc. Am. A 7, 542-551 (1990). [CrossRef]
- M. Servin, D. Malacara, J. L. Marroquin, and F. J. Cuevas, "Complex linear filters for phase shifting with very low detuning sensitivity," J. Mod. Opt. 44, 1269-1278 (1997). [CrossRef]

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