Abstract

The paper proposes a novel approach for estimating multiple phases in holographic moiré. The need to design such an algorithm is necessitated by the development of optical configurations containing two phase stepping devices, e.g. PZTs, with a view to measure simultaneously two phase distributions. The approach consists of first applying minimum-norm algorithm to extract phase steps imparted to the PZTs. Salient feature of the algorithm lies in its ability to handle nonsinusoidal waveforms and noise. This approach also provides the flexibility of using arbitrary phase steps, a feature most commonly attributed to generalized phase shifting interferometry. Once the phase steps are estimated for each PZT, the Vandermonde system of equations is designed to estimate the phase distributions.

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References

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  37. G. Bienvenu, �??Influence of the spatial coherence of the background noise on high resolution passive methods,�?? in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing, Washington, DC, 306-309 (1979).
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App. Opt. (8)

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, App. Opt. 13, 2693-2703 (1974).
[CrossRef]

Y. Zhu and T. Gemma, �??Method for designing error-compensating phase-calculation algorithms for phase shifting interferometry,�?? App. Opt. 40, 4540-4546 (2001).
[CrossRef]

P. Hariharan, B. F. Oreb, and T. Eiju, �??Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,�?? App. Opt. 26, 2504-2506 (1987).
[CrossRef]

Y. �??Y. Cheng and J. C. Wyant, �??Phase-shifter calibration in phase-shifting interferometry,�?? App. Opt. 24, 3049-3052 (1985).
[CrossRef]

P. de Groot and L. L. deck, �??Numerical simulations of vibration in phase-shifting interferometry,�?? App. Opt. 35, 2172-2178 (1996).
[CrossRef]

P. K. Rastogi, �??Phase shifting applied to four-wave holographic interferometers,�?? App. Opt. 31, 1680-1681 (1992).
[CrossRef]

P. K. Rastogi, �??Phase-shifting holographic moiré: phase-shifter error-insensitive algorithms for the extraction of the difference and sum of phases in holographic moiré,�?? App. Opt. 32, 3669-3675 (1993).
[CrossRef]

P. K. Rastogi and E. Denarié, �??Visualization of in-plane displacement fields using phase shifting holographic moiré: application to crack detection and propagation,�?? App. Opt. 31, 2402-2404 (1992).
[CrossRef]

Appl. Opt. (8)

IEEE Proceedings (1)

T. Söderström and P. Stoica, �??Accuracy of high-order Yule-Walker methods for frequency estimation of complex sine waves,�?? IEEE Proceedings-F 140, 71-80 (1993).

IEEE Trans. Acoustics (1)

R. Roy and T. Kailath, �??ESPRIT-Estimation of signal parameters via rotational invariance techniques,�?? IEEE Trans. Acoustics, Speech, and Signal Processing 37, 984-995 (1989).
[CrossRef]

IEEE Trans. on Acoustics, Speech, and Si (1)

M. Kaveh and A. J, Barabell, �??The statistical performance of the MUSIC and the Minimum-Norm algorithms in resolving plane waves in noise,�?? IEEE Transactions on Acoustics, Speech, and Signal Processing ASSP-34, 331-341 (1986).
[CrossRef]

IEEE Trans. on Aerospace and Electr. Sys (1)

R. Kumaresan and D. W. Tufts, �??Estimating the angles of arrival of multiple plane waves,�?? IEEE Transactions on Aerospace and Electronic Systems AES-19, 134-139 (1983).
[CrossRef]

IEEE Trans. Signal Processing (1)

B. D. Rao and K. V. S. Hari, �??Weighted subspace methods and spatial smoothing: analysis and comparison�??, IEEE Trans. Signal Processing 41, 788-803 (1993).
[CrossRef]

IEEE Transactions on Acoustics (1)

J. J. Fuchs, �??Estimating the number of sinusoids in additive white noise,�?? IEEE Transactions on Acoustics, Speech, and Signal Processing 36, 1846-1853 (1988).
[CrossRef]

Intl Conf. Acoustics, Speech and SP 1983 (1)

A. J. Barabell, �??Improving the resolution performance of eigenstructure-based direction-finding algorithms,�?? in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing, Boston, MA, 336-339 (1983).

Intl Conf. Acoustics, Speech, SP 1979 (1)

G. Bienvenu, �??Influence of the spatial coherence of the background noise on high resolution passive methods,�?? in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing, Washington, DC, 306-309 (1979).

J. Mod. Opt. (1)

C. Joenathan and B. M. Khorana, �??Phase measurement by differentiating interferometric fringes, �??J. Mod. Opt. 39, 2075-2087 (1992).
[CrossRef]

J. Opt. Soc. Am. A (4)

Meas. Sci. Technol. (1)

B. Zhao, �??A statistical method for fringe intensity-correlated error in phase-shifting measurement: the effect of quantization error on the N-bucket algorithm,�?? Meas. Sci. Technol. 8, 147-153 (1997).
[CrossRef]

Metrologia (1)

P. Carré, �??Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,�?? Metrologia 2, 13-23 (1966).
[CrossRef]

Opt. Eng. (2)

J. Schwider, O. Falkenstorfer, H. Schreiber, and A. Zoller, �??New compensating four-phase algorithm for phase-shift interferometry,�?? Opt. Eng. 32, 1883-1885 (1993).
[CrossRef]

J. E. Grievenkamp, �??Generalized data reduction for heterodyne interferometry,�?? Opt. Eng. 23, 350-352 (1984).

Opt. Engg. (1)

R. Józwicki, M. Kujawinska, and M. Salbut, �??New contra old wavefront measurement concepts for interferometric optical testing, Opt. Engg. 31, 422-433 (1992).
[CrossRef]

Opt. Express (1)

Opt. Lasers Eng. (1)

P. K. Rastogi, M. Spajer, and J. Monneret, �??In-plane deformation measurement using holographic moiré,�?? Opt. Lasers Eng. 2, 79-103 (1981).
[CrossRef]

Opt. Lett. (1)

Proc. RADC, Spectral Estimation Wksp '79 (1)

R. O. Schmidt, �??Multiple emitter location and signal parameter estimation,�?? in Proceedings RADC, Spectral Estimation Workshop, Rome, NY, (243-258) 1979.

Other (3)

J. E. Greivenkamp and J. H. Bruning, "Phase shifting interferometry Optical Shop Testing ed D. Malacara (New York: Wiley ) 501-598 (1992).

K. Creath, �??Phase-shifting holographic interferometry,�?? Holographic Interferometry, P. K. Rastogi, ed. (Springer Series in Optical Sciences, Berlin 1994), Vol.68, pp. 109-150.

T. Kreis, Holographic interferometry Principles and Methods (Akademie Verlag, 1996) pp. 101-170.

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Figures (6)

Fig. 1.
Fig. 1.

Schematic of the optical setup in holographic moiré.

Fig. 2.
Fig. 2.

Fringe map corresponding to a) κ=1 and pure signal, a) κ=1 and 10 SNR, a) κ=2 and pure signal, a) κ=2 and 10 SNR.

Fig. 3.
Fig. 3.

Plot for phase step values α and β (in degrees) obtained at an arbitrary pixel location on a data frame for different values of N and m using the forward approach. During the simulation the phase steps are assumed to be α=45° and β=70°.

Fig. 4.
Fig. 4.

Plot for phase step values α and β (in degrees) obtained at an arbitrary pixel location on a data frame for different values of N and m using the forward-backward approach. During the simulation the phase steps are assumed to be α=45° and β=70°.

Fig. 5.
Fig. 5.

Plots show typical error in computation of phase distribution a) φ 1 (in radians), and b) φ 2 (in radians), for phase step obtained from Fig. 4(c) for 30dB noise.

Fig. 6.
Fig. 6.

Plot shows wrapped phase for a) φ 1 and b) φ 2 for phase step obtained from Fig. 4(c) for 30dB noise.

Equations (39)

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

I ( t ) = I dc + k = 1 κ a k exp [ ik ( φ 1 + t α ) ] + k = 1 κ a k exp [ ik ( φ 1 + t α ) ] +
k = 1 κ b k exp [ ik ( φ 2 + t β ) ] + k = 1 κ b k exp [ ik ( φ 2 + t β ) ] ;
for t = 0 , 1 , 2 , ... , m , ... , N 1
I ( t ) = I dc + k = 1 κ k u k t + k = 1 κ k * ( u k * ) t + k = 1 κ k v k t + k = 1 κ k * ( v k * ) t + η ( t ) ;
for t = 0 , 1 , ... , m , ... , N 1
r ( p ) = E [ I ( t ) I * ( t p ) ]
I ( t ) = I dc + a 1 e i φ 1 e i α t + a 1 e i φ 1 e e α t + b 1 e i φ 2 e i β t + b 1 e i φ 2 e i β t + η ( t )
I * ( t p ) = I dc + a 1 e i φ 1 e i α ( t p ) + a 1 e φ 1 e i α ( t p ) +
b 1 e i φ 2 e i β ( t p ) + b 1 e i φ 2 e i β ( t p ) + η * ( t p )
r ( p ) = E { I dc 2 + c 1 + e i α p ( a 1 2 + c 2 ) + e i α p ( a 1 2 + c 3 ) + e i β p ( b 1 2 + c 4 ) + e i β p ( b 1 2 + c 5 ) + η ( t ) η * ( t p ) }
r ( p ) = A 0 2 + A 1 2 e i α p + A 2 2 e i α p + A 3 2 e i β p + A 4 2 e i β p + σ 2 δ p , 0
E [ η ( k ) η * ( j ) ] = σ 2 δ k , j
r ( p ) = E [ I ( t ) I * ( t p ) ] = n = 0 4 κ A n 2 e i ω n p + σ 2 δ p , 0
R I = E [ I * ( t ) I ( t ) ] = [ r ( 0 ) r * ( 1 ) r * ( 2 ) . r * ( m 1 ) r ( 1 ) r ( 0 ) . . . r ( 2 ) . . . . . . . . r * ( 1 ) r ( m 1 ) . . . r ( 0 ) ]
R I = A P A c R s + σ 2 I R ε
P = [ A 0 2 0 . 0 0 A 1 2 . . . . . . 0 . . A m 2 ]
R I G = G [ A 4 κ + 1 2 0 . . 0 0 A 4 κ + 2 2 . . . . . . . . . . . . 0 0 0 . . A m 2 ] = σ 2 G = AP A c G + σ 2 G
A c G = 0
R ( G ) = N ( A c )
S c G = 0
R ( S ) = R ( A )
a T ( z 1 ) G ̂ G ̂ c a ( z ) = 0
a T ( z 1 ) [ 1 g ̂ ] = 0
S ̂ = [ χ c S ¯ ] } m 1 } 1
S ̂ c [ 1 g ̂ ] = 0
S ¯ c g ̂ = χ
g ̂ = S ¯ ( S ¯ c S ¯ ) 1 χ
I = S ̂ c S ̂ = χ χ c + S ¯ c S ¯
χ 2 1
rank ( S ¯ ) = n
I ( x , y ; t ) = I dc + a 1 exp [ i ( φ 1 + t α ) ] + a 1 exp [ i ( φ 1 + t α ) ] +
a 2 exp [ 2 i ( φ 1 + t α ) ] + a 2 exp [ 2 i ( φ 1 + t α ) ] +
b 1 exp [ i ( φ 2 + t β ) ] + b 1 exp [ i ( φ 2 + t β ) ] +
b 2 exp [ 2 i ( φ 2 + t β ) ] + b 2 exp [ 2 i ( φ 2 + t β ) ] + η ( t )
φ 1 ( x , y ) = 2 π λ ( p x ) 2 + ( p y ) 2 + φ R 1
φ 2 ( x , y ) = 2 π λ ( p x ) 2 + ( p y ) 2 + φ R 2 .
R ̂ I = 1 N t = m N [ I * ( t 1 ) I * ( t 2 ) . . I * ( t m ) ] [ I ( t 1 ) I ( t 2 ) . . I ( t m ) ]
R ̂ I = 1 2 N t = m N { [ I * ( t 1 ) I * ( t 2 ) . I * ( t m ) ] [ I ( t 1 ) I ( t 2 ) . . I ( t m ) ] + [ I * ( t m ) . I * ( t 2 ) I * ( t 1 ) ] [ I ( t m ) . . I ( t 2 ) I ( t 1 ) ] }
[ e i κ α 0 e i κ α 0 e i κ β 0 . e i ( κ 1 ) α 0 . . 1 e i κ α 1 e i κ α 1 e i κ β 1 . e i ( κ 1 ) α 1 . . 1 . . . . . . . . . . . . . . . . e i κ α ( N 1 ) e i κ α ( N 1 ) e i κ β ( N 1 ) . e i ( κ 1 ) α ( N 1 ) . . 1 ] [ κ κ * κ . I dc ] = [ I 0 I 1 I 2 . I N 1 ]

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