Abstract

The main purpose of this paper is to present a method to design tunable quadrature filters in phase shifting interferometry. From a general tunable two-frame algorithm introduced, a set of individual filters corresponding to each quadrature conditions of the filter is obtained. Then, through a convolution algorithm of this set of filters the desired symmetric quadrature filter is recovered. Finally, the method is applied to obtain several tunable filters, like four and five-frame algorithms.

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References

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    [CrossRef] [PubMed]
  5. H. Schreiber, J. H. Brunning, and J. E. Greivenkamp, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara ed., (John Wiley & Sons, Inc., Hoboken, New Jersey 2007).
  6. M. Afifi, K. Nassim, and S. Rachafi, “Five-frame phase-shifting algorithm insensitive to diode laser power variation,” Opt. Commun. 197(1-3), 37–42 (2001).
    [CrossRef]
  7. 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).
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  8. J. F. Mosiño, D. M. Doblado, and D. M. Hernández, “Calculus of exact detuning phase shift error in temporal phase shifting algorithms,” Opt. Express 17(18), 15766–15771 (2009).
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2009

2001

M. Afifi, K. Nassim, and S. Rachafi, “Five-frame phase-shifting algorithm insensitive to diode laser power variation,” Opt. Commun. 197(1-3), 37–42 (2001).
[CrossRef]

1997

1996

1995

Afifi, M.

M. Afifi, K. Nassim, and S. Rachafi, “Five-frame phase-shifting algorithm insensitive to diode laser power variation,” Opt. Commun. 197(1-3), 37–42 (2001).
[CrossRef]

Creath, K.

Doblado, D. M.

Estrada, J. C.

Hernández, D. M.

Mosiño, J. F.

Nassim, K.

M. Afifi, K. Nassim, and S. Rachafi, “Five-frame phase-shifting algorithm insensitive to diode laser power variation,” Opt. Commun. 197(1-3), 37–42 (2001).
[CrossRef]

Phillion, D. W.

Quiroga, J. A.

Rachafi, S.

M. Afifi, K. Nassim, and S. Rachafi, “Five-frame phase-shifting algorithm insensitive to diode laser power variation,” Opt. Commun. 197(1-3), 37–42 (2001).
[CrossRef]

Schmit, J.

Servin, M.

Surrel, Y.

Appl. Opt.

Opt. Commun.

M. Afifi, K. Nassim, and S. Rachafi, “Five-frame phase-shifting algorithm insensitive to diode laser power variation,” Opt. Commun. 197(1-3), 37–42 (2001).
[CrossRef]

Opt. Express

Other

H. Schreiber, J. H. Brunning, and J. E. Greivenkamp, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara ed., (John Wiley & Sons, Inc., Hoboken, New Jersey 2007).

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Tables (1)

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Table 1 Several particular five-frame temporal phase shifting algorithms

Equations (39)

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tan ( φ ) = k = 1 M b k I k k = 1 M a k I k = [ b 1 b 2 ... b M ]   I [ a 1 a 2 ... a M ]   I = N   I D   I .
h ( t ) = k = 1 M a k [ δ ( t p ) ] + i   k = 1 M b k [ δ ( t p ) ] = D δ   + i   N δ  ,
H ( α ) = 0 ; H ( α ) = 0 ; H ( α ) = 0 ; ... H m ( α ) = 0.
H ( 0 ) = 0 ; H ( 0 ) = 0 ; H ( 0 ) = 0 ... H m ( 0 ) = 0.
tan ( φ 1 ) = N 1   I D 1   I ,   and   tan ( φ 2 ) = N 2   I D 2   I .
h ( t ) = [ D 1 δ + i   N 1 δ ] [ D 2 δ + i   N 2 δ ] = [ D 1 D 2 N 1 N 2 ] δ + i [ N 1 D 2 + D 1 N 2 ] δ .
tan ( φ ) = N   I D   I = [ N 1 D 2 + D 1 N 2 ]   I [ D 1 D 2 N 1 N 2 ]   I ,
N D = ( N 1 D 1 ) ( N 2 D 2 ) = [ N 1 D 2 + D 1 N 2 ] [ D 1 D 2 N 1 N 2 ] .
tan ( φ ) = b 1 I 1 b 1 I 2 a 1 I 1 + a 1 I 2 = b 1 [ 1 1 ]   I a 1 [ 1 1 ]   I .
H ( ω , α ) = 2 [ a 1 cos ( ω / 2 ) b 1 sin ( ω / 2 ) ] .
tan ( φ ) = N I D I = cos ( α / 2 ) [ 1 1 ] I sin ( α / 2 ) [ 1    1 ]   I = cos ( α / 2 ) sin ( α / 2 ) [ I 1 I 2 I 1 + I 2 ] .
N D = cos ( α / 2 ) [ 1 1 ] sin ( α / 2 ) [ 1    1 ]   .
H ( ω , α ) = 2 sin [ ( ω α ) / 2 ] .
H m + 1 ( ω , α ) = ( 2 ) m + 1 sin m + 1 [ ( ω α ) / 2 ] .
N D = { cos ( α / 2 ) sin ( α / 2 ) [ 1 1 ] [ 1 1 ] } m + 1 .
H ( ω , α ) = H ( ω , 0 ) = 2 sin ( ω / 2 ) .
tan ( φ ) = [ 1 1 ] I [ 0 0 ] I = [ I 1 I 2 0 I 1 + 0 I 2 ] .
H m + 1 ( ω , 0 ) = ( 2 ) m + 1 sin m + 1 ( ω / 2 ) .
N D = { [ 1 1 ] [ 0 0 ] } m + 1 .
H ( π ) = 0 H ( π ) = 0 ; H ( π ) = 0 ; H ( π ) = 0 ... H m ( π ) = 0.
H ( ω , π ) = ( 2 ) m + 1 cos m + 1 ( ω / 2 ) .
N D = { [ 0 0 ] [ 1 1 ] } m + 1 .
H ( ω ) = ( 2 ) 2 sin ( ω / 2 )   sin [ ( ω α ) / 2 ]   .
N D = { [ 1 1 ] [ 0 0 ] } { cos ( α / 2 ) [ 1 1 ] sin ( α / 2 ) [ 1    1 ] } = sin ( α / 2 ) [ 1    1 ] [ 1 1 ] cos ( α / 2 ) [ 1 1 ] [ 1 1 ] .
tan ( φ ) = N I D I = sin ( α / 2 ) [ 1 0 1 ] I cos ( α / 2 ) [ 1 2 1 ] I = tan ( α / 2 ) I 1 I 3 I 1 + 2 I 2 I 3 .
tan ( φ ) = [ b 2 b 1 b 1 b 2 ]   I [ a 2 a 1 a 1 a 2 ]   I .
H ( ω ) = H ( ω , 0 ) H ( ω , α ) H ( ω , β ) = 8 sin ( ω / 2 )   sin [ ( ω α ) / 2 ]   sin [ ( ω β ) / 2 ] .
N D = { [ 1 1 ] [ 0 0 ] } { cos ( α / 2 ) [ 1 1 ] sin ( α / 2 ) [ 1    1 ] } { cos ( β / 2 ) [ 1 1 ] sin ( β / 2 ) [ 1    1 ] } .
N D = [ 1 1 ] [ 0 0 ] sin [ ( α + β ) / 2 ]   [     1 , 0 , 1 ]   [ cos [ ( α + β ) / 2 ]   , 2 cos [ ( α β ) / 2 ]   , cos [ ( α + β ) / 2 ] ] .
tan ( φ ) = N I D I = cos [ ( α + β ) / 2 ] ( I 1 I 4 ) { 2 sin [ ( α β ) / 2 ] + cos [ ( α + β ) / 2 ] } ( I 2 I 3 )   sin [ ( α + β ) / 2 ]   ( I 1 I 2 I 3 I 4 ) .
tan ( φ ) = sin ( α / 2 ) cos ( α / 2 )   [ 1   1 1 1 ]    I   [ 1 -1 -1 1 ]    I = tan ( α / 2 ) I 1 I 2 + I 3 + I 4 I 1 I 2 I 3 + I 4 .
H ( ω ) = 16 sin ( ω / 2 ) sin [ ( ω α ) / 2 ] sin [ ( ω β ) / 2 ] sin [ ( ω γ ) / 2 ] .
N D = { [ 1 1 ] [ 0 0 ] } { cos ( α / 2 ) [ 1 1 ] sin ( α / 2 ) [ 1    1 ] } { cos ( β / 2 ) [ 1 1 ] sin ( β / 2 ) [ 1    1 ] } { cos ( γ / 2 ) [ 1 1 ] sin ( γ / 2 ) [ 1    1 ] } .
H ( ω ) = 16 sin 3 ( ω / 2 ) sin [ ( ω α ) / 2 ] .
H ( ω ) = 16 sin 2 ( ω / 2 ) cos ( ω / 2 ) sin [ ( ω α ) / 2 ] .
H ( ω ) = 16 sin ( ω / 2 ) cos ( ω / 2 ) sin 2 [ ( ω α ) / 2 ] .
H ( ω ) = 16 sin ( ω / 2 ) cos 2 ( ω / 2 ) sin [ ( ω α ) / 2 ] .
H ( ω ) = 16 sin 2 ( ω / 2 ) sin 2 [ ( ω α ) / 2 ] .
H ( ω ) = 16 sin ( ω / 2 ) sin 3 [ ( ω α ) / 2 ] .

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