## Phase-shifting interferometry corrupted by white and non-white additive noise |

Optics Express, Vol. 19, Issue 10, pp. 9529-9534 (2011)

http://dx.doi.org/10.1364/OE.19.009529

Acrobat PDF (886 KB)

### Abstract

The standard tool to estimate the phase of a sequence of phase-shifted interferograms is the Phase Shifting Algorithm (PSA). The performance of PSAs to a sequence of interferograms corrupted by *non-white* additive noise has *not* been reported before. In this paper we use the Frequency Transfer Function (FTF) of a PSA to generalize previous white additive noise analysis to non-white additive noisy interferograms. That is, we find the ensemble average and the variance of the estimated phase in a general PSA when interferograms corrupted by non-white additive noise are available. Moreover, for the special case of additive white-noise, and using the Parseval’s theorem, we show (for the first time in the PSA literature) a useful relationship of the PSA’s noise robustness; in terms of its FTF spectrum, and in terms of its coefficients. In other words, we find the PSA’s estimated phase variance, in the spectral space as well as in the PSA’s coefficients space.

© 2011 OSA

## 1. Introduction

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

3. M. Servin, J. C. Estrada, J. A. Quiroga, J. F. Mosiño, and M. Cywiak, “Noise in phase shifting interferometry,” Opt. Express **17**(11), 8789–8794 (2009). [CrossRef] [PubMed]

4. 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]

5. C. P. Brophy, “Effect of intensity error correlation on the computed phase of phase-shifting interferometry,” J. Opt. Soc. Am. A **7**(4), 537–541 (1990). [CrossRef]

5. C. P. Brophy, “Effect of intensity error correlation on the computed phase of phase-shifting interferometry,” J. Opt. Soc. Am. A **7**(4), 537–541 (1990). [CrossRef]

6. C. Rathjen, “Statistical properties of phase-shift algorithms,” J. Opt. Soc. Am. **12**(9), 1997–2008 (1995). [CrossRef]

*x*-characteristic polynomial [7

7. Y. Surrel, “Additive noise effect in digital phase detection,” Appl. Opt. **36**(1), 271–276 (1997). [CrossRef] [PubMed]

3. M. Servin, J. C. Estrada, J. A. Quiroga, J. F. Mosiño, and M. Cywiak, “Noise in phase shifting interferometry,” Opt. Express **17**(11), 8789–8794 (2009). [CrossRef] [PubMed]

4. 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. M. Servin, J. C. Estrada, J. A. Quiroga, J. F. Mosiño, and M. Cywiak, “Noise in phase shifting interferometry,” Opt. Express **17**(11), 8789–8794 (2009). [CrossRef] [PubMed]

*no-white*(pink) additive noise. Moreover using Parseval’s theorem we unify in a single theory our spectral approach [3

**17**(11), 8789–8794 (2009). [CrossRef] [PubMed]

4. 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]

5. C. P. Brophy, “Effect of intensity error correlation on the computed phase of phase-shifting interferometry,” J. Opt. Soc. Am. A **7**(4), 537–541 (1990). [CrossRef]

## 2. Analysis of the noise rejection of Phase-Shifting Algorithms

*F*[•]) of the system’s impulse response

*h*(

*t*) (

*H*(

*ω*) =

*F*[

*h*(

*t*)]), has been used as an efficient analyzing tool for their linear systems [9]. Here we use the PSA’s FTF to find the estimated phase variance due to a sequence of interferograms corrupted by

*non-white*additive noise. Moreover, for the special case of white additive noise, the equivalence between the noise analysis based on the PSA’s spectrum [3

**17**(11), 8789–8794 (2009). [CrossRef] [PubMed]

**17**(24), 21867–21881 (2009). [CrossRef] [PubMed]

**7**(4), 537–541 (1990). [CrossRef]

**17**(24), 21867–21881 (2009). [CrossRef] [PubMed]

*N*phase-shifted interferograms corrupted by additive noise with carrier

*ω*

_{0}(radians/interferogram) as,

*I*(

*x,y,k*) represents the noisy interferograms at site (

*x,y,k*);

*a*(

*x,y*) is the illumination background, and

*b*(

*x,y*) the fringe contrast. The interferogram’s additive noise

*n*(

*x,y,k*) is assumed to be

*non-white*, ergodic, zero-mean, and Gaussian. The

*non-flat*power spectral density of the non-white noise is

*F*[

*E*{

*n*(

*t*)

*n*(

*t*+ τ)}] =

*η*(

*ω*) [9]. Finally

*φ*(

*x,y*) is the phase that we want to estimate.

*N*-steps linear PSA may be written as,

*c*are the PSA’s coefficients. The estimated phase

_{k}*φ*which is the true modulating phase. This PSA is tuned at

*ω*=

*ω*

_{0}(radians/sample). The estimated phase

*N*-step digital-filter

*h*(

*t*) (the PSI algorithm) and the data

*I*(

*t*) evaluated at the middle-point

*t*=

_{m}*N*-1. At this middle-point both

*I*(

*t*) and

*h*(

*t*) fully overlap (see Fig. 1 ). The full convolution product

*S*(

*t*) =

*I*(

*t*)*

*h*(

*t*) is,

*t*.

*a*(

*x,y*) and one of the two complex

*b*(

*x,y*)/2 terms. Arbitrarily choosing to keep the term at +

*ω*

_{0},

*H*(

*ω*) must have at least two zeroes: one at

*H*(0) = 0, and the other at

*H*(-

*ω*

_{0}) = 0. After convolution we have,

*ω*

_{0}

*t*which may be set to zero. Note that the amplitude of our input signal (

_{m}*b*/2)exp(

*iφ*) is now multiplied by the PSA filter’s response

*H*(

*ω*

_{0}) at +

*ω*

_{0}. The PSA’s response

*H*(

*ω*

_{0}) at

*ω*

_{0}is a very important piece of information that has not been taken explicitly into account before [5

**7**(4), 537–541 (1990). [CrossRef]

*n*(

_{H}*t*) is the output noise

_{m}*after filtering*, which is now

*complex and non-white*with variance [9],

*σ*is equal to the variance of the noise

_{S}^{2}*after being filtered*by the PSA. The ensemble average of the output

*E*{

*S*} and its variance

*E*{

*S*

^{2}} (Eq. (6)) are,

*E*{

*S*} is the searched noiseless phase,

*i.e. angle*[

*E*{

*S*}] =

*ϕ*. A phasorial representation of Eq. (6), is shown in Fig. 2 . From the triangle having as “base”

*σ*and as “height”

_{S}*H*(

*ω*

_{0})

*b*/2, the standard deviation of the estimated phase is,where the approximation for small additive noise, tan(

*x*)≈

*x*was made. On the other hand, combining Eq. (8) and Eq. (9) we find its variance as,

*non-white*(pink) noise having a

*non-flat*spectrum

*η*(

*ω*);

*and this is our first main result*.

*white-noise*, having a flat spectral density

*η*(

*ω*) =

*η*/2 (Watts/Hertz). Additionally, using Parseval’s theorem [9], and the response of

*h*(

*t*) at

*ω*

_{0}, or

*H*(

*ω*

_{0}) [4

**17**(24), 21867–21881 (2009). [CrossRef] [PubMed]

*H*(

*ω*) [3

**17**(11), 8789–8794 (2009). [CrossRef] [PubMed]

**17**(24), 21867–21881 (2009). [CrossRef] [PubMed]

*c*[5

_{k}**7**(4), 537–541 (1990). [CrossRef]

*σ*/

_{n}^{2}*b*

^{2}depends only on the interferogram’ data; the rest depends, on the spectrum, or on the coefficients of the PSA. The compact and useful formula using the PSA’s coefficients in Eq. (11) has not been stated in this handy form before [5

**7**(4), 537–541 (1990). [CrossRef]

*and it is our second main result.*

## 3. Blindness of Phase Shifting Algorithms to its filter’s output amplitude

*S*| = (

*b*/2)

*H*(

*ω*

_{0}), is canceled-out by the PSA’s ratio. In other words, the PSA’ ratio is blind to the quadrature filter’s output. Therefore, one may (erroneously) assume that the filter’s output is (among other possibilities),

## 4. Two examples

### 4.1 The Least Squares Phase Shifting Algorithm

*LS*)

*N*-steps PSA is [2,10

10. C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. **7**(8), 368–370 (1982). [CrossRef] [PubMed]

*LS*-PSA are one (

*c*= 1). The impulse response of this PSA is,and its FTF is given by,

_{k}*c*in

_{k}*h*(

_{LS}*t*), and the spectrum |

*H*(

_{LS}*ω*)| = |

*F*[

*h*(

_{LS}*t*)] of this PSA. Figure 3 also shows that, the 6-steps

*LS*-PSA is insensitive to at least the 2nd, 3rd, and 4th real-valued harmonics of the interferogram signal.

*N*observations corrupted by white-additive noise is reduced by a factor of 1/

*N*[9]. For the case shown in Fig. 3, this factor is 1/6. This sampled-mean (the

*LS*-PSA), is the Maximum Likelihood (ML) estimator of a parameter corrupted by white-additive Gaussian noise [9].

### 4.2 The Schwider-Hariharan 5-steps PSA

*SH*) 5-steps PSA [11

11. 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]

12. 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]

*SH*5-steps PSA is,

*SH*-PSA are

*c*

_{0}= 1,

*c*

_{1}= 2,

*c*

_{2}= 2,

*c*

_{3}= 2, and

*c*

_{4}= 1. The variance of the estimated phase is therefore,

*SH*-PSA is just 9% below from the noise-rejection of a 5-step

*LS*one.

*H*(

_{SH}*ω*) touches tangentially the

*ω*axis at -

*ω*

_{0}. This second-order detuning robustness is well known and widely used in PSI [2,11

11. 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]

12. 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]

*SH*-PSA penalizes 9% of noise robustness to gain second-order detuning robustness at -

*ω*

_{0};

*i.e.*the

*SH*algorithm has two spectral zeroes at -

*ω*

_{0}. On the other hand (see Fig. 3), a

*N*-steps

*LS*-PSA has only single-order spectral zeroes.

## 5. Conclusions

*non-white*noise of a general PSA using the Frequency Transfer Function (FTF) or

*H*(

*ω*). Using the pink-noise spectrum of the interferometric data and the PSA’s FTF the first, and the second-order ensemble averages of the estimated phase

*white-noise*, and using Parseval’s theorem, we have shown the equivalence (in Eq. (11)) between the spectral-based noise-variance [3

**17**(11), 8789–8794 (2009). [CrossRef] [PubMed]

**17**(24), 21867–21881 (2009). [CrossRef] [PubMed]

*c*of the PSA [5

_{k}**7**(4), 537–541 (1990). [CrossRef]

*N*-steps Least-Squares PSAs, and the noise analysis for the 5-steps Schwider-Hariharan PSA were presented as examples.

## Acknowledgments

## References and links

1. | J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. |

2. | D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, Taylor & Francis CRC Press, 2th edition (2005). |

3. | M. Servin, J. C. Estrada, J. A. Quiroga, J. F. Mosiño, and M. Cywiak, “Noise in phase shifting interferometry,” Opt. Express |

4. | M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express |

5. | C. P. Brophy, “Effect of intensity error correlation on the computed phase of phase-shifting interferometry,” J. Opt. Soc. Am. A |

6. | C. Rathjen, “Statistical properties of phase-shift algorithms,” J. Opt. Soc. Am. |

7. | Y. Surrel, “Additive noise effect in digital phase detection,” Appl. Opt. |

8. | K. Hibino, “Susceptibility of systematic error-compensating algorithms to random noise in phase-shiftinginterferometry,” Appl. Opt. |

9. | A. Papoulis, |

10. | C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. |

11. | 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. |

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

**OCIS Codes**

(120.2650) Instrumentation, measurement, and metrology : Fringe analysis

(120.3180) Instrumentation, measurement, and metrology : Interferometry

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: February 16, 2011

Revised Manuscript: April 18, 2011

Manuscript Accepted: April 21, 2011

Published: May 2, 2011

**Citation**

M. Servin, J. A. Quiroga, and J. C. Estrada, "Phase-shifting interferometry corrupted by white and non-white additive noise," Opt. Express **19**, 9529-9534 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-10-9529

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### References

- J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13(11), 2693–2703 (1974). [CrossRef] [PubMed]
- D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, Taylor & Francis CRC Press, 2th edition (2005).
- M. Servin, J. C. Estrada, J. A. Quiroga, J. F. Mosiño, and M. Cywiak, “Noise in phase shifting interferometry,” Opt. Express 17(11), 8789–8794 (2009). [CrossRef] [PubMed]
- 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]
- C. P. Brophy, “Effect of intensity error correlation on the computed phase of phase-shifting interferometry,” J. Opt. Soc. Am. A 7(4), 537–541 (1990). [CrossRef]
- C. Rathjen, “Statistical properties of phase-shift algorithms,” J. Opt. Soc. Am. 12(9), 1997–2008 (1995). [CrossRef]
- Y. Surrel, “Additive noise effect in digital phase detection,” Appl. Opt. 36(1), 271–276 (1997). [CrossRef] [PubMed]
- K. Hibino, “Susceptibility of systematic error-compensating algorithms to random noise in phase-shiftinginterferometry,” Appl. Opt. 36, 2064–2093 (1997).
- A. Papoulis, Probability Random Variables and Stochastic Processes, 4th ed. (McGraw-Hill Series in Electrical Engineering, 2001).
- C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. 7(8), 368–370 (1982). [CrossRef] [PubMed]
- 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]
- 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]

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