## Spectral analysis of phase shifting algorithms

Optics Express, Vol. 17, Issue 19, pp. 16423-16428 (2009)

http://dx.doi.org/10.1364/OE.17.016423

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### Abstract

Systematic spectral analysis of Phase Shifting Interferometry (PSI) algorithms was first proposed in 1990 by Freischlad and Koliopoulos (F&K). This analysis was proposed with the intention that “in a glance” the main properties of the PSI algorithms would be highlighted. However a major drawback of the F&K spectral analysis is that it changes when the PSI algorithm is rotated or its reference signal is time-shifted. In other words, the F&K spectral plot is different when the PSI algorithm is rotated or its reference is time-shifted. However, it is well known that these simple operations do not alter the basic phase demodulation properties of PSI algorithms, except for an unimportant piston. Here we propose a new way to analyze the spectra of PSI algorithms which is invariant to rotation and/or reference time-shift among other advantages over the nowadays standard PSI spectral analysis by F&K.

© 2009 OSA

## 1. Introduction

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

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

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

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

5. 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]

6. J. G. Proakis, and D. G. Manolakis, *Digital Signal Processing*, 4th-ed., (Prentice Hall, 2007). [PubMed]

*invariant to rotation and/or reference time-shift*. Also this new spectral representation has zero response at frequencies rejected by the PSI quadrature filter.

## 2. The aim of a quadrature filter in phase shifting interferometry

*a*(

*x,y*),

*b*(

*x,y*), and

*φ*(

*x,y*) are the background, contrast and the searched phase. Finally

*ω*

_{0}is the temporal carrier (in radians) of the phase shifted interferograms. Rewriting it as,The spatial dependence of the functions

*a*(

*x,y*),

*b*(

*x,y*) and

*φ*(

*x,y*) were omitted for clarity. The aim of a PSI algorithm is to filter out

*a*(

*x,y*), and

*one*of the two complex exponentials. Therefore the output signal

*Ic*(

*t*) is,Where the symbol * denotes a one-dimensional convolution, and

*h*(

*t*) is the PSI filter’s complex (quadrature) impulse response,Denoting by

*H*(

*ω*) =

*F*[

*h*(

*t*)] its Fourier transform. The minimal conditions on the frequency transfer function

*H*(

*ω*) to obtain the complex signal

*Ic*(

*t*) in Eq. (3) from the interferogram are:These conditions on

*H*(

*ω*) are not normally used in the field of few-steps PSI interferometry. Instead due to [1

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

## 3. Rotation and reference time-shift of PSI algorithms

*et. al*. [4], page 239, and Schmit

*et. al*. [5

5. 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]

**7**(4), 542–551 (1990). [CrossRef]

*hr*(

*t*) and imaginary

*hi*(

*t*) parts of a complex impulse response filter

*h*(

*t*) =

*hr*(

*t*) +

*i hi*(

*t*) as,The rotation (by Δ

_{0}) and the reference time-shift (

*t-t*

_{0}) of this complex linear filter are formally represented by,The most general impulse response of a linear-quadrature digital-filter (PSI algorithm) is,

*iw*

_{0}

*t*] is the reference signal (local oscillator),

*T*is the sampling rate, and the coefficients

*a*are weighting (possible complex) constants. Time shifting the reference one obtains,Therefore, time-shifting the reference signal also implies filter’s rotation, with Δ

_{n}_{0}= -

*ω*

_{0}

*t*

_{0}. This was first observed in Malacara

*et. al.*[4], page 239. Rotating

*h*(

*t*) by Δ

_{0}one obtains,As Eq. (10) shows, rotation and/or reference time-shift linearly mix up the real and imaginary parts of the original PSI filter

*h*(

*t*). This give apparently “new” PSI algorithms that looks quite different albeit being the same one [4,5

5. 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]

_{0}= -

*ω*

_{0}

*t*

_{0}) quadrature filter according to Eq. (10) looks now as,

## 4. The Freischlad and Koliopoulos spectral analysis

*et. al.*[1

**7**(4), 542–551 (1990). [CrossRef]

*h*(

*t*),

*separately*, that isHaving these Fourier transforms, F&K recommend to analyze these spectra

*without constant or common phase factors*[1

**7**(4), 542–551 (1990). [CrossRef]

*the F&K spectral representation would give different spectral plots for each rotation of the same PSI algorithm*. In our herein presented spectral representation, this drawback does not exist.

## 5. Our proposed spectral analysis

**7**(4), 542–551 (1990). [CrossRef]

*hr*(

*t*) and the imaginary

*hi*(

*t*) components of

*h*(

*t*)

*separately*; Eq. (12). This is followed by a rule to obtain two real functions to plot and analyze. On the other hand, we propose to analyze the spectra of the complex sum

*h*(

*t*) =

*hr*(

*t*) +

*i hi*(

*t*). So we only take the Fourier transform of

*h*(

*t*),

*i.e. H*(

*ω*) =

*F*[

*h*(

*t*)]. This new way of analyzing the spectra of PSI algorithms have (as shown here) interesting and useful consequences.

*H*(

*ω*) [6

6. J. G. Proakis, and D. G. Manolakis, *Digital Signal Processing*, 4th-ed., (Prentice Hall, 2007). [PubMed]

*Ic*(

*ω*) to the input’s signal spectra

*I*(

*ω*),Where the function |

*H*(

*ω*)| is the magnitude of

*H*(

*ω*), and Δ(

*ω*) is the phase introduced by the linear filter. Moreover, the filter’s phase Δ(

*ω*) is constant in many popular PSI algorithms [4]. The output of any PSI filter is the complex analytical signal

*Ic*(

*t*) in Eq. (3) associated with the interferogram’s real signal

*I*(

*t*). Therefore, as it is shown in Fig. 1 we may have an infinite number of transfer functions

*H*(

*ω*) that comply with Eq. (5). In Fig. 1 we show two block diagrams of linear quadrature PSI filters that may be used to obtain our searched analytical signal

*Ic*(

*t*) at –

*ω*

_{0}from the real interferometric signal

*I*(

*t*).

*ω*) = Δ

_{0}[4], therefore in these cases the plot of |

*H*(

*ω*)| may be used as its spectrum, which is given byWhere

*H*(

*ω*) =

*Re*(

*ω*) +

*i Im*(

*ω*), being

*Re*(

*ω*) and

*Im*(

*ω*) real functions of

*ω*. As seen in the next section, using this new way of plotting the PSI spectra, one can easily visualize the main properties of a PSI algorithm. Moreover, it is well known that [6

6. J. G. Proakis, and D. G. Manolakis, *Digital Signal Processing*, 4th-ed., (Prentice Hall, 2007). [PubMed]

*h*(

*t*)exp[

*i*Δ

_{0}]; the time-shifted

*h*(

*t-t*

_{0}), and the original PSI filter

*h*(

*t*) have all the same magnitude |

*H*(

*ω*)|,

## 6. Examples

*H*(

*ω*)| avoids them.

### 6.1 Two four-step PSI algorithms

*π*/4, -

*π*/4,

*π*/4, and 3

*π*/4 radians; while in the second case: -

*π*/2, 0,

*π*/2, and

*π*radians. Denoting by

*h*1(

*t*) =

*hr*1(

*t*) +

*i hi*1(

*t*) and

*h*2(

*t*) =

*hr*2(

*t*) +

*i hi*2(

*t*), the two impulse responses associated with these PSI algorithms are:

**7**(4), 542–551 (1990). [CrossRef]

*h*1(

*t*) and

*h*2(

*t*) and realize that both are proportional,The plot of |

*H*1(ω)| and |

*H*2(ω)| are shown in Fig. 2 . We can see that in this representation both filters have proportional spectra and therefore (except for a piston phase equals to

*π*/4 introduced by

*H*2(

*ω*) [5

**34**(19), 3610–3619 (1995). [CrossRef] [PubMed]

*have identical phase demodulating properties*.

### 6.2 Five-step Schwider-Hariharan PSI algorithm

*π*, -

*π*/2, 0,

*π*/2, and

*π*. This PSI algorithm has the following associated impulse response

*h*(

*t*),According to [1

**7**(4), 542–551 (1990). [CrossRef]

*h*(

*t*) by Δ

_{0}radians we obtain,If the rotation angle is (for example) Δ

_{0}=

*π*/4 then the rotated impulse response becomes,

**34**(19), 3610–3619 (1995). [CrossRef] [PubMed]

**7**(4), 542–551 (1990). [CrossRef]

*H*(

*ω*) =

*F*[

*h*(

*t*)] of both algorithms are (of course) identicalThis is shown in Fig. 3 , where we can see that this 5-step PSI filter will eliminate the complex exponential at

*ω*= 1.0 and the background of the interferogram

*I*(

*t*) at

*ω*= 0.

### 6.3 Discusion

- 1. The magnitude of the frequency response of the filter is invariant to the PSI filter rotations and/or constant time-shift of the reference signal (local oscillator).
- 2. The signals at frequencies
*ω*_{1},…,*ω*that the PSI filter rejects are clearly seen as zeroes over the frequency axis_{n}*i.e.*|*H*(*ω*_{1})| = … = |*H*(*ω*)| = 0._{n} - 3. The properties of the PSI algorithms in the neighborhood of the rejected frequencies are also clearly shown. For example the detuning robustness of the Schwider-Hariharan 5-step algorithm is shown as a zero for the first derivative of
*H*(*ω*) at*ω*= 1.0. - 4. In [7] we show that the phase noise in a PSI algorithm is proportional to the integral of |
7. 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]*H*(*ω*)|^{2}. So at a glance one may estimate the noise rejection of two “competing” PSI algorithms by their area under |*H*(*ω*)|^{2}for the same output signal’s energy.

## 7. Conclusions

**34**(19), 3610–3619 (1995). [CrossRef] [PubMed]

**7**(4), 542–551 (1990). [CrossRef]

*H*(

*ω*) the rejected frequencies, and no need for further interpretation is required. On top of this, the spectral analysis based on

*H*(

*ω*) is the most usual (and intuitive) manner to graphically represent the spectra of any optical and/or electrical filter in engineering.

## Acknowledgements

## References and links

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

2. | D. W. Phillion, “General methods for generating phase-shifting interferometry algorithms,” Appl. Opt. |

3. | Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. |

4. | D. Malacara, M. Servin, and Z. Malacara, |

5. | J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. |

6. | J. G. Proakis, and D. G. Manolakis, |

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

**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: May 26, 2009

Revised Manuscript: August 14, 2009

Manuscript Accepted: August 15, 2009

Published: August 31, 2009

**Citation**

M. Servin, J. C. Estrada, and J. A. Quiroga, "Spectral analysis of phase
shifting algorithms," Opt. Express **17**, 16423-16428 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-19-16423

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### References

- K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7(4), 542–551 (1990). [CrossRef]
- D. W. Phillion, “General methods for generating phase-shifting interferometry algorithms,” Appl. Opt. 36(31), 8098–8115 (1997). [CrossRef]
- Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996). [CrossRef] [PubMed]
- D. Malacara, M. Servin, and Z. Malacara, Interferogram analysis for Optical Testing, 2th ed., (Marcel Deker, 2003).
- 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]
- J. G. Proakis, and D. G. Manolakis, Digital Signal Processing, 4th-ed., (Prentice Hall, 2007). [PubMed]
- 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]

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