Abstract

We have been reporting several new techniques of analysis and synthesis applied to Phase Shifting Interferometry (PSI). These works are based upon the Frequency Transfer Function (FTF) and how this new tool of analysis and synthesis in PSI may be applied to obtain very general results, among them; rotational invariant spectrum; complex PSI algorithms synthesis based on simpler first and second order quadrature filters; more accurate formulae for estimating the detuning error; output-power phase noise estimation. We have made our cases exposing these aspects of PSI separately. Now in the light of a better understanding provided by our past works we present and expand in a more coherent and holistic way the general theory of PSI algorithms. We are also providing herein new material not reported before. These new results are on; a well defined way to combine PSI algorithms and recursive linear PSI algorithms to obtain resonant quadrature filters.

© 2009 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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] [PubMed]
  3. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996).
    [CrossRef] [PubMed]
  4. D. Malacara, M. Servin, and Z. Malacara, Interferogram analysis for Optical Testing, 2th ed., (Marcel Deker, 2003).
  5. M. Servin, J. C. Estrada, and J. A. Quiroga, “Spectral analysis of phase shifting algorithms,” Opt. Express 17(19), 16423–16428 (2009).
    [CrossRef] [PubMed]
  6. J. G. Proakis, and D. G. Manolakis, Digital Signal Processing, 4th-ed., (Prentice Hall, 2007).
    [PubMed]
  7. 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]
  8. 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]
  9. 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]
  10. J. C. Estrada, M. Servin, and J. A. Quiroga, “Easy and straightforward construction of wideband phase-shifting algorithms for interferometry,” Opt. Lett. 34(4), 413–415 (2009).
    [CrossRef] [PubMed]
  11. K. G. Larkin, and B. F. Oreb, “Propagation of errors in different phase-shifting algorithms: a special property of the arctangent function,” presented at the SPIE International Symposium on Optical Applied Science and Engineering, San Diego, California, SPIE, 1755, 219–227 (1992).
  12. F. G. Stremler, Introduction to Communications Systems, 3rd ed., (Addison-Wesley, 1990).
  13. 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]
  14. K. Hibino, “Susceptibility of systematic error-compensating algorithms to random noise in phase-shifting interferometry,” Appl. Opt. 36(10), 2084–2093 (1997).
    [CrossRef] [PubMed]
  15. C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. 7(8), 368–370 (1982).
    [CrossRef] [PubMed]
  16. V. K. Madisetti, and D. B. Williams, eds., Digital Signal Processing Handbook, (CRC Press, IEEE Press, 1998).

2009 (4)

1997 (2)

1996 (1)

1995 (1)

1990 (1)

1982 (1)

1974 (1)

Brangaccio, D. J.

Bruning, J. H.

Creath, K.

Cywiak, M.

Estrada, J. C.

Freischlad, K.

Gallagher, J. E.

Herriott, D. R.

Hibino, K.

Koliopoulos, C. L.

Morgan, C. J.

Mosiño, J. F.

Phillion, D. W.

Quiroga, J. A.

Rosenfeld, D. P.

Schmit, J.

Servin, M.

Surrel, Y.

White, A. D.

Appl. Opt. (5)

J. Opt. Soc. Am. A (1)

Opt. Express (3)

Opt. Lett. (2)

Other (5)

D. Malacara, M. Servin, and Z. Malacara, Interferogram analysis for Optical Testing, 2th ed., (Marcel Deker, 2003).

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

K. G. Larkin, and B. F. Oreb, “Propagation of errors in different phase-shifting algorithms: a special property of the arctangent function,” presented at the SPIE International Symposium on Optical Applied Science and Engineering, San Diego, California, SPIE, 1755, 219–227 (1992).

F. G. Stremler, Introduction to Communications Systems, 3rd ed., (Addison-Wesley, 1990).

V. K. Madisetti, and D. B. Williams, eds., Digital Signal Processing Handbook, (CRC Press, IEEE Press, 1998).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Two different spectral magnitudes for |H(ω)| that comply with Eq. (5). The input to these filters is the interferogram I(t) and the output the searched analytical signal Ic(t). These two FTFs are valid PSI algorithms with temporal carrier ω 0. Both have the minimum required zeroes for H(ω), these are H(0) = 0, and H(ω 0) = 0.

Fig. 2
Fig. 2

The magnitude of the spectrum of the complex quadrature filter associated with the “two” 5-step Schwieder-Hariharan algorithms in Eq. (17) is identical. The spectral components of the real-valued interferogram that are rejected are the background H(0) = 0 and the complex signal at ω 0, i.e. H(ω 0) = 0 This spectrum remains invariant to the reference (local oscillator) time-shift.

Fig. 3
Fig. 3

The interferogram data has a detuning error equal to Δ radians/interferogram. The spurious signal that contributes to the erroneous phase demodulation is H(ω 0 + Δ)exp[]. In a well tuned PSI algorithm (Δ = 0), this signal is zero as shown in Fig. 1 and Fig. 2.

Fig. 4
Fig. 4

This is a phasorial representation of the two complex signals in Eq. (19). The desired signal is H(ω 0 + Δ)exp[], while the undesired one is H(-ω 0-Δ)exp[-]. Their phasorial sum gives the erroneous demodulated phase ψ.

Fig. 5
Fig. 5

This is the phasorial representation of the complex sum in Eq. (25). Ic(0) is the vectorial (phasorial) sum of the noise free phasor (b/2)H(-ω 0)exp[-], and the complex band-pass noise nc(0)exp[-iΦ(0)]. The angle of Ic(0) gives the noisy demodulated phase ϕn .

Fig. 6
Fig. 6

Here we show the difference of noise rejection between the standard 3-step PSI algorithm [13] and the 5-step Schwider-Hariharan algorithm [4]. For the same output signal power the integral under |H(ω)|2 is greater for the case of a 3-step algorithm than for the 5-step one. As a consequence, the 5-step algorithm rejects more noise than the 3-step one.

Fig. 7
Fig. 7

Spectral plot of the FTF in Eq. (42). This is an ultra-wide-band filter that corresponds to a 9-step PSI algorithm. This algorithm would demodulate any 9-step sequence of interferograms no matter what carrier they may have. However to obtain the maximum signal to noise ratio, according to Eq. (27), the best carrier is π/2 radians/frame. The “flat zone” is strictly zero only at the tuning frequencies: ω 1 = π/4, ω 2 = = π/2 and ω 3 = 3π/4. But compared to the amount of measuring noise, the non-zero ripples in the right-hand “flat-zone” are small enough to be negligible..

Fig. 8
Fig. 8

Block diagram of the resonant PSI algorithm. This is a very simple quadrature filter which requires only a single memory frame to keep the last output complex signal Ic(t). If the η parameter equals 0.99, one would obtain a high Q resonator with an extremely narrow band-width roughly equivalent to a 200-step PSI algorithm.

Fig. 9
Fig. 9

This is the magnitude |H(ω)| and phase delay introduced by the recursive quadrature filter shown in Eq. (48). The magnitude response |H(ω)| is resonant at our temporal carrier ω 0 = -π/2. Here we have used η = 0.8 which produces a broad resonant peak. The angle introduced by this recursive filter is however “new” in PSI. This is because standard PSI algorithms have all a constant phase delay. This delay in this case is always less than π/2 radians.

Fig. 10
Fig. 10

This is the spectrum of the 5-step recursive resonator in Eq. (50). This PSI filter has the same pole at ω =π/2 as the basic resonator (with η = 0.8), but now we have added two zeroes at ω = 0, and at the unwanted conjugate signal at ω = + π/2. We have now obtained an almost flat-zero right hand side frequency response.

Equations (51)

Equations on this page are rendered with MathJax. Learn more.

I ( x , y , t ) = a ( x , y ) + b ( x , y ) cos [ φ ( x , y ) + ω 0 t ] .
I ( x , y , t ) = a + ( b / 2 ) exp [ i ( φ + ω 0 t ) ] + ( b / 2 ) exp [ i ( φ + ω 0 t ) ] .
I c ( x , y , t ) = H ( ω 0 ) [ b ( x , y ) / 2 ] exp { i [ φ ( x , y ) + ω 0 t ] } = h ( t ) * I ( x , y , t ) .
h ( t ) = h r ( t ) + i h i ( t ) .
H ( ω 0 ) 0 ,     a n d       H ( ω 0 ) = H ( 0 ) = 0 .
h ( t ) = exp [ i ω 0 t ] {     n a n δ ( t n T ) }     .
I s ( t ) * h ( t ) = { I ( t ) n δ ( t n T ) } * {     exp [ i ω 0 t ] n a n δ ( t n T ) }     .
I c ( 0 ) = I s ( t ) * h ( t ) | t = 0 = [ I s ( t ) * h r ( t ) + i I s ( t ) * h i ( t ) ] | t = 0 .
tan [ φ ( x , y ) ] = h i ( t ) * I s ( t ) h r ( t ) * I s ( t ) | t = 0 = n a n sin ( ω 0 n T ) I ( n T ) n a n cos ( ω 0 n T ) I ( n T ) .
H i ( ω ) = F [ h i ( t ) ] H r ( ω ) = F [ h r ( t ) ] .
H ( ω ) = F [ h ( t ) ] = F [ h r ( t ) + i     h i ( t ) ] .
h t 0 ( t ) = exp[ i ω 0 ( t t 0 ) ] { n a n δ ( t n T ) } = h ( t ) exp [ i ω 0 t 0 ] .
h ( t ) exp [ i Δ 0 ] = h r ( t ) cos ( Δ 0 ) h i ( t ) sin ( Δ 0 ) + i [ h r ( t ) sin ( Δ 0 ) + h i ( t ) cos ( Δ 0 ) .
tan [ φ ( x , y ) + Δ 0 ] = [ h r ( t ) sin ( Δ 0 ) + h i ( t ) cos ( Δ 0 ) ] * I ( t ) [ h r ( t ) cos ( Δ 0 ) h i ( t ) sin ( Δ 0 ) ] * I ( t ) | t = 0 .
H i ( ω ) = F [ h r ( t ) sin ( Δ 0 ) + h i ( t ) cos ( Δ 0 ) ] H r ( ω ) = F [ h r ( t ) cos ( Δ 0 ) h i ( t ) sin ( Δ 0 ) ] .
| H ( ω ) | = | F [ h ( t ) exp [ i Δ 0 ] | = | F [ h ( t ) ] | = R e ( ω ) 2 + I m ( ω ) 2 .
tan [ φ ( x , y ) ] = 2 ( I 2 I 4 ) I 1 2 I 3 + I 5 , a n d     tan [ φ ( x , y ) + π / 4 ] = I 1 + 2 I 2 2 I 3 2 I 4 + I 5 I 1 2 I 2 2 I 3 + 2 I 4 + I 5 .
F [ h i d e a l ( t ) ] = H i d e a l ( ω ) = { 1 , f o r ω < 0 0 , f o r ω 0 .
I c ( x , y , 0 ) = ( b / 2 ) { H ( ω 0 Δ ) exp [ i φ ] + H ( ω 0 + Δ ) exp [ i φ ] } .
ψ = tan 1 [ H ( ω 0 Δ ) H ( ω 0 + Δ ) H ( ω 0 Δ ) + H ( ω 0 + Δ ) tan ( φ ) ] .
ψ = φ + | H ( ω 0 + Δ ) | | H ( ω 0 Δ ) | sin ( 2 φ ) .
I n ( t ) = a + b cos ( φ + ω 0 t ) + n ( t ) .
I c ( t ) = I n ( t ) * h ( t ) = H ( ω 0 ) ( b / 2 ) exp [ i ( φ + ω 0 t ) ] + n c ( t ) exp [ i ( Φ ( t ) + ω 0 t ) ] .
σ n c 2 = E { | n c ( t ) | 2 } = η 2 ( π , π ) H ( ω ) H * ( ω )     d ω .
I c ( 0 ) = [ I n ( t ) * h ( t ) ] t = 0 = H ( ω 0 ) ( b / 2 ) exp [ i φ ] + n c ( 0 ) exp [ i Φ ( 0 ) ] .
E { φ n 2 } = tan 1 [ σ n c 2 ( b 2 / 4 ) | H ( ω 0 ) | 2 ] .
E { φ n } = φ , a n d E { φ n 2 } = σ n c 2 ( b 2 / 4 ) | H ( ω 0 ) | 2 = 2 η b 2 | H ( ω 0 ) | 2 ( π , π ) H ( ω ) H * ( ω ) d ω .
tan [ φ ] = h i 1 ( t ) * I ( t ) h r 1 ( t ) * I ( t ) | t = 0 , tan [ φ ] = h i 2 ( t ) * I ( t ) h r 2 ( t ) * I ( t ) | t = 0 .
h 1 ( t ) = h r 1 ( t ) + i h i 1 ( t ) ,     h 2 ( t ) = h r 2 ( t ) + i h i 2 ( t ) .
h ( t ) = h 1 ( t ) * h 2 ( t ) = h r 1 * h r 2 h i 1 * h i 2 + i     [ h r 1 * h i 2 + h i 1 * h r 2 ] .
tan [ φ ] = [ h i 1 ( t ) * h r 2 ( t ) h r 1 ( t ) * h i 2 ( t ) ] * I ( t ) [ h r 1 ( t ) * h r 2 ( t ) + h i 1 ( t ) * h i 2 ( t ) ] * I ( t ) | t = 0 .
H ( ω ) = F [ h 1 ( t ) * h 2 ( t ) ] = H 1 ( ω ) H 2 ( ω ) .
h 1 ( t ) = exp [ i ω 0 t ] [ δ ( t + 1 ) δ ( t 1 ) ] .
H 1 ( ω ) = F [ h 1 ( t ) ] = 2 i sin ( ω ω 0 ) .
H ( ω ) = 4 sin ( ω ) sin ( ω ω 0 ) .
h ( t ) = F 1 [ H ( ω ) ] = exp ( i ω 0 ) δ ( t + 2 ) 2 cos ( ω 0 ) δ ( t ) + exp ( i ω 0 ) δ ( t 2 ) .
tan ( φ ) = h i ( t ) * I ( t ) | t = 0 h r ( t ) * I ( t ) | t = 0 = [ I ( 2 ) + I ( 2 ) ] sin ( ω 0 ) [ I ( 2 ) 2 I ( 0 ) + I ( 2 ) ] cos ( ω 0 ) .
h 2 ( t ) = exp [ i ω 0 t ] [ δ ( t + 1 ) 2 δ ( t ) + δ ( t 1 ) ] .
H 2 ( ω ) = F [ h 2 ( t ) ] = 2 + 2 cos ( ω ω 0 ) .
H ( ω ) = F [ h 1 ( t ) * h 2 ( t ) ] = 4 i sin ( ω ) [ 1 + cos ( ω ω 0 ) ] .
tan ( φ ) = 2 I ( 1 ) 2 I ( 1 ) I ( 2 ) 2 I ( 0 ) + I ( 2 ) .
H ( ω ) = 8 i sin ( ω ) [ 1 + cos ( ω ω 1 ) ] [ 1 + cos ( ω ω 2 ) ] [ 1 + cos ( ω ω 3 ) ] .
tan ( φ ) = ( 3 2 + 5 ) [ I ( 1 ) I ( 1 ) ] + ( 2 + 1 ) [ I ( 3 ) I ( 3 ) ] ( 4 2 + 5 ) I ( 0 ) ( 2 2 + 3 ) [ I ( 2 ) + I ( 2 ) ] + 0.5 [ I ( 4 ) + I ( 4 ) ] .
I c ( x , y , t ) = η     I c ( x , y , t 1 ) exp [ i ω 0 ] + I ( x , y , t ) .
φ ( x , y ) = tan 1 [ I m [ I c ( x , y , t ) ] R e [ I c ( x , y , t ) ] ] .
I c ( x , y , t ) = h ( t ) * I ( x , y , t ) = { n = 1 η n exp [ i n ω 0 ] δ [ t + n ] } * I ( x , y , t ) .
I c ( x , y , ω ) = η I c ( x , y , ω ) exp [ i ω ] exp [ i ω 0 ] + I ( x , y , ω ) .
H ( x , y , ω ) = F [ h ( x , y , t ) ] = I c ( x , y , ω ) I ( x , y , ω ) = 1 1 η exp [ i ( ω + ω 0 ) ] .
| H ( ω ) | exp [ i Δ ( ω ) ] = 1 1 + η 2 2 η cos ( ω + ω 0 ) exp { i η sin [ ω + ω 0 ] 1 η cos [ ω + ω 0 ] } .
H ( x , y , ω ) = 4 i sin ( ω ) ( 1 + cos ( ω π / 2 ) ) 1 0.8 exp [ i ( ω + π / 2 ) ] .
I c n = η I c n 1 exp [ i ω 0 ] + { I n 1 exp [ i ω 0 ] 2 I n + I n + 1 exp [ i ω 0 ] } * ( I n 1 I n + 1 ) .

Metrics