Abstract

In phase sampling interferometry, existing temporal analysis methods are sensitive to border effects and cannot deal with missing data. In this work we propose a quadrature filter that allows a reliable dynamic phase measurement for every sample, even in the cases involving few samples or missing data. The method is based on the use of a regularized least squares cost function that enforces the quadrature character of the filter. A comparison with existing techniques shows the effectiveness of the proposed method.

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References

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  1. D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis For Optical Testing, 2nd. ed. (CRC Press, 2005.
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    [CrossRef]
  10. A. Papoulis, Signal analysis, McGraw-Hill (1977)
  11. J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 4, 350–352 (1984).
  12. http://www.mathworks.com/help/techdoc/ref/peaks.html

2009 (2)

2003 (2)

2000 (1)

1999 (1)

1997 (1)

1994 (1)

1984 (1)

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 4, 350–352 (1984).

Estrada, J. C.

Figueroa, J. E.

Greivenkamp, J. E.

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 4, 350–352 (1984).

Haible, P.

Huntley, J. M.

Kaufmann, G. H.

Kerr, D.

Kothiyal, M. P.

Marroquin, J. L.

Quiroga, J. A.

Ruiz, P. D.

Servin, M.

Takeda, M.

Tiziani, H. J.

Yamamoto, H.

Appl. Opt. (3)

J. Opt. Soc. Am. A (3)

Opt. Eng. (1)

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 4, 350–352 (1984).

Opt. Express (1)

Opt. Lett. (1)

Other (3)

http://www.mathworks.com/help/techdoc/ref/peaks.html

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis For Optical Testing, 2nd. ed. (CRC Press, 2005.

A. Papoulis, Signal analysis, McGraw-Hill (1977)

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Figures (5)

Fig. 1
Fig. 1

Infinite impulse responses for the RQF (red) and the RLSQF (blue) filters with a tuning frequency ω0=π/2 and regularization parameterλ=1.

Fig. 2
Fig. 2

Frequency responses for a N=15 samples DC suppressing RLSQF filter (see text for details). In this plot, we show the RLSQF frequency responses for samples k=1 (blue) and k=8 (green) together with the infinite frequency response (red).

Fig. 3
Fig. 3

Temporal PSI example for few samples. A) the signal spectrum (dashed line) is plotted together with the frequency response of the FT (blue), RQF (green) and RLSQF (red) filters. In the case of the RQF and LSRQF method, we are plotting the response for the sample k=8. B) Demodulated phase for the FT (blue), RQF (green) and LSRQF (red) methods compared with the actual phase (dashed line).

Fig. 4
Fig. 4

Temporal PSI example for few samples and low carrier frequency and missing data. A) plot of the masked signal B) Demodulated phase for the FT (blue), RQF (green) and LSRQF (red) methods compared with the actual phase (dashed black line)

Fig. 5
Fig. 5

Temporal demodulation of a time-varying saturated interferogram. From the 150 samples we depict the results for t=6 and t=75 (see text for details). In each row, we depict the instantaneous interferogram, the processing mask and the demodulated phase.

Equations (43)

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g ( t ) = b + m cos ( φ ( t ) + ω 0 t ) t = 1... N ,
G ( ω ) = b δ ( ω ) + 1 2 m [ C ( ω ω 0 ) + C ( ω + ω 0 ) ] ,
H ( Δ ω 0 ) = H ( 0 ) = 0 ,
H ( Δ ω 0 ) 0
Q ( ω ) = H ( ω ) G ( ω ) = 1 2 m H ( ω ) C ( ω ω 0 ) ,
q ( t ) = h ( t ) g ( t ) 1 2 m H ( u ) e i ( φ ( t ) + ω 0 t ) ,
C ( ω ω 0 ) = e i φ 0 δ ( ω ω 0 )
q ( t ) = h ( t ) g ( t ) = 1 2 m H ( ω 0 ) W ( ω 0 ) e i ( φ 0 + ω 0 t ) ,
w ( t ) = { 1 t = 1... N 0 o t h e r w i s e .
W ( ω ) = sin ( 0.5 ω N ) sin ( 0.5 ω ) e i ω ( N + 1 ) / 2 .
Δ ω g ω 0 2 π / N ,
P = ω 0 2 π / N Δ ω g
f ( t ) = m e i ( φ ( t ) + ω 0 t )
t = 1 N
U ( f , f ) = R ( f , f * , g ) + λ V ( f ) ,
R ( f , f * ) = t = 1 N | f ( t ) + f ( t ) g ( t ) | 2 M ( t ) ,
R ( f , f * ) = t = 2 N | f ( t ) f ( t 1 ) + f ( t ) f ( t 1 ) g ( t ) + g ( t 1 ) | 2 M ( t ) .
V M ( f ) = t = 2 N | f ( t ) f ( t 1 ) e i ω 0 | 2
V T ( f ) = t = 2 N 1 | 2 f ( t ) f ( t 1 ) e i ω 0 f ( t + 1 ) e i ω 0 | 2 .
K ( f , f , g ) + λ ( f ( t ) h M ) = 0 , K ( f , f , g ) + λ ( f ( t ) h M ) = 0 ,
K ( f , f * , g ) = M ( t ) ( f ( t ) + f ( t ) 2 g ( t ) ) h D M ( t + 1 ) ( f ( t + 1 ) + f ( t + 1 ) 2 g ( t + 1 ) ) h D ,
h D = δ ( t ) δ ( t 1 )
h M = 2 δ ( t ) δ ( t 1 ) e i ω 0 δ ( t + 1 ) e i ω 0 .
( F ( ω ) + F ˜ ( ω ) 2 G ( ω ) ) H D C ( ω ) + λ F ˜ ( ω ) H V M ( ω + ω 0 ) = 0 , ( F ( ω ) + F ˜ ( ω ) 2 G ( ω ) ) H D C ( ω ) + λ F ( ω ) H V M ( ω ω 0 ) = 0 ,
H D C ( ω ) = 2 ( 1 cos ω )
H V M ( ω ) = H D C ( ω ) + 2 λ ( 1 c o s ω ) .
H M ( ω ) = F ( ω ) G ( ω ) = 2 H D C ( ω ) ( H D C ( ω ) H V M ( ω + ω 0 ) ) H D C ( ω ) H V M ( ω + ω 0 ) H V M ( ω ω 0 ) .
h T = 6 δ ( t ) 4 [ δ ( t 1 ) e i ω 0 + δ ( t + 1 ) e i ω 0 ] + δ ( t 2 ) e 2 i ω 0 + δ ( t + 2 ) e 2 i ω 0 .
H V T ( ω ) = H D C ( ω ) + 2 λ ( 6 + 2 c o s 2 ω 8 cos ω ) .
H R Q F ( ω ) = F ( ω ) G ( ω ) = 2 H D C ( ω ) H V T ( ω ω 0 ) .
f ( t ) = f ( t 1 ) e i ω 0
t < 1
t > N .
A f ˜ = 2 L g ˜ ,
L = ( M ( 1 ) + M ( 2 ) 0 M ( 2 ) 0 ... 0 M ( 1 ) + M ( 2 ) 0 M ( 2 ) 0 ... M ( 2 ) 0 M ( 2 ) + M ( 3 ) 0 M ( 3 ) ... 0 M ( 2 ) 0 M ( 2 ) + M ( 3 ) 0 M ( 3 ) ... ... ... ... ... 0 M ( N 1 ) 0 M ( N 1 ) + M ( N ) 0 M ( N ) 0 ... 0 M ( N 1 ) 0 M ( N 1 ) + M ( N ) 0 M ( N ) ... 0 M ( N ) 0 M ( N ) + M ( N + 1 ) 0 ... 0 M ( N ) 0 M ( N ) + M ( N + 1 ) ) .
f = Q T 2 g .
f ( k ) = m e i ( φ ( k ) + ω 0 k ) = n = 1 N h k ( n ) g ( n ) .
f ( t ) + f ( t ) = m cos φ cos α ( t ) m sin φ sin α ( t ) ,
U ( a 2 , a 3 ) = t = 1 N | a 2 ( t ) cos α ( t ) + a 3 ( t ) sin α ( t ) g ( t ) | 2 M ( t ) + ...         λ t = 2 N | a 2 ( t ) a 2 ( t 1 ) | 2 + | a 3 ( t ) a 3 ( t 1 ) | 2 .
φ ( t ) = 0.25 cos ( 2 π t / N ) .
H F T ( ω ) = { sin ( π ω 2 ω 0 ) 4 ω > 0 0 ω 0 , ω > 2 ω 0 .
φ ( t ) = 1.25 cos ( 2 π t / N ) .
g n ( x , y , t ) = b ( x , y , t ) + m ( x , y , t ) cos ( φ ( x , y , t ) + ω 0 t ) + n ( x , y , t ) ,

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