Abstract

Any linear phase sampling algorithm can be described as a linear filter characterized by its frequency response. In traditional phase sampling interferometry the phase of the frequency response has been ignored because the impulse responses can be made real selecting the correct sample offset. However least squares methods and recursive filters can have a complex frequency response. In this paper, we derive the quadrature equations for a general phase sampling algorithm and describe the role of the filter phase.

© 2011 OSA

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References

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  1. D. Malacara, M. Servín, and Z. Malacara Interferogram Analysis For Optical Testing, Second ed. CRC Press 2005.
  2. 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. J. L. Marroquin, J. E. Figueroa, and M. Servin, “Robust quadrature filters,” J. Opt. Soc. Am. A 14(4), 779–791 (1997).
    [CrossRef]
  4. J. A. Quiroga, J. C. Estrada, M. Servín, and J. Vargas, “Regularized least squares phase sampling interferometry,” Opt. Express 19(6), 5002–5013 (2011).
    [CrossRef] [PubMed]
  5. A. V. Oppenheim, and R. W. Schafer, Discrete-Time Signal processing, Pearson (2010)
  6. 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]
  7. http:\\goo.gl/BGjJ9

2011 (1)

2009 (1)

1997 (1)

1987 (1)

Eiju, T.

Estrada, J. C.

Figueroa, J. E.

Hariharan, P.

Marroquin, J. L.

Oreb, B. F.

Quiroga, J. A.

Servin, M.

Servín, M.

Vargas, J.

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Figures (3)

Fig. 1
Fig. 1

Amplitude and phase delay for the recursive filter of Eq. (18) for η=0.75 and ω 0 =π/2 rad/sample.

Fig. 2
Fig. 2

Demodulation results using the recursive filter of Eq. (18). a) input signal, b) demodulation results (red) and actual phase (blue). In this figure there is a marker every four samples to make easier the observation of the 4 samples delay between both signals.

Fig. 3
Fig. 3

Recursive filter demodulation results with a discontinuous rectangular phase. a) temporal interferogram, b) demodulated phase in the forward-time pass, c) demodulated phase after the bidirectional filtering.

Equations (22)

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g( t )=b+mcos( φ( t )+ ω 0 t )t=1...N,
H( ω )=FT( h )= t= h( t ) e itω   and   h( t )=F T 1 ( H )= π π H( ω ) e itω dω.
G( ω )=FT( g( t ) )=bδ( ω )+ 1 2 m[ C( ω ω 0 )+C ( ω+ ω 0 ) ],
Q( ω )=H( ω )G( ω )= 1 2 mH( ω )C( ω ω 0 ).
q( t )=h( t )g( t )= 1 2 mH( ω 0 ) e i( φ( t )+ ω 0 t ) .
H( Δ ω 0 )=H( 0 )=0, H( Δ ω 0 )0,
s( t )=p( t )cos( ω 0 t ).
Φ( ω )= Φ 0 τ( ω 0 )( ω ω 0 ),
τ( ω )= dΦ dω |.
F( ω )=H( ω )S( ω )= 1 2 e i Φ 0 A( ω )P( ω ω 0 ) e iτ( ω ω 0 ) ,
F( ω )= 1 2 e i Φ 0 A( ω )( P( ω ) e iτω δ( ω ω 0 ) ),
f( t )= 1 2 e i Φ 0 A( ω 0 )p( tτ ) e it ω 0 .
g( t )=b+mcosφ( t )cos ω 0 tmsinφ( t )sin ω 0 t.
q( t )= 1 2 m e i Φ 0 A( ω 0 ) e i( φ( tτ )+ ω 0 t ) .
h 1 ( t )=[ 1,2i,2,2i,1 ].
H 1 ( ω )= t=1 5 h( t ) e itω =( 2+4sinω2cos2ω ) e i3ω .
tanφ( 3 )= Im( h 1 g ) Re( h 1 g ) | t=0 = 2( g( 2 )g( 4 ) ) 2g( 3 )g( 1 )g( 5 ) .
q( t )=ηq( t1 ) e i ω 0 +( g( t1 ) e i ω 0 2g( t )+g( t+1 ) e i ω 0 )*( δ( t1 )δ( t+1 ) ).
H 2 ( ω )= 4isin( ω )( cos( ω+ ω 0 )1 ) 1ηexp( i( ω ω 0 ) ) .
tan φ r ( t4 )= Im( H 2 { g( t ) } ) Re( H 2 { g( t ) } ) .
h T ( t )= h 2 ( t )* h 2 * ( t ),
H T ( t )=H( ω ) H * ( ω )= | A( ω ) | 2 .

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