Abstract

Phase error analysis in Temporal Phase Shifting (TPS) algorithms due to frequency detuning has been to date only performed numerically. In this paper, we show an exact analytical expression to obtain this phase error due to detuning using the spectral TPS response. The new proposed method is based on the phasorial representation of the output of the TPS quadrature filter. Doing this, the detuning problem is reduced to a ratio of two symmetrical spectral responses of the quadrature filter at the detuned frequency. Finally, some popular cases of TPS algorithms are analyzed to show the usefulness of the proposed method.

© 2009 Optical Society of America

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References

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  1. J Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf ed., (North Holland, Amsterdam, Oxford, New York, Tokyo, 1990).
  2. J. H. Bruning, D. R. Herriot, J. E. Gallagher, D. P. Rosenfel, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometry for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [Crossref] [PubMed]
  3. H. Schreiber and J. H. Brunning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara ed., (John Wiley & Sons, Inc., Hoboken, New Jersey2007).
  4. M. Servin and M. Kujawinska, “Modern fringe pattern analysis in Interferometry,” in Handbook of Optical Engineering, D. Malacara and B. J. Thompson eds., (Marcel Dekker, 2001).
  5. J. Schwider, R. Burrow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
    [Crossref] [PubMed]
  6. J P. Hariharan, B. Oreb, and T. Eiju, “Digital phase shifting interferometry: a simple error compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2505 (1987).
    [Crossref] [PubMed]
  7. K. Freischland and C. L. Koliopoulos, “Fourier description of digital phase measuring interferometry,” J. Opt. Soc. Am. A 7, 542–551 (1990).
    [Crossref]
  8. M. Servin, D. Malacara, J. L. Marroquin, and F. J. Cuevas, “Complex linear filters for phase shifting with very low detuning sensitivity,” J. Mod. Opt. 44, 1269–1278 (1997).
    [Crossref]

1997 (1)

M. Servin, D. Malacara, J. L. Marroquin, and F. J. Cuevas, “Complex linear filters for phase shifting with very low detuning sensitivity,” J. Mod. Opt. 44, 1269–1278 (1997).
[Crossref]

1990 (1)

1987 (1)

1983 (1)

1974 (1)

Brangaccio, D. J.

Bruning, J. H.

Brunning, J. H.

H. Schreiber and J. H. Brunning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara ed., (John Wiley & Sons, Inc., Hoboken, New Jersey2007).

Burrow, R.

Cuevas, F. J.

M. Servin, D. Malacara, J. L. Marroquin, and F. J. Cuevas, “Complex linear filters for phase shifting with very low detuning sensitivity,” J. Mod. Opt. 44, 1269–1278 (1997).
[Crossref]

Eiju, T.

Elssner, K. E.

Freischland, K.

Gallagher, J. E.

Grzanna, J.

Hariharan, J P.

Herriot, D. R.

Koliopoulos, C. L.

Kujawinska, M.

M. Servin and M. Kujawinska, “Modern fringe pattern analysis in Interferometry,” in Handbook of Optical Engineering, D. Malacara and B. J. Thompson eds., (Marcel Dekker, 2001).

Malacara, D.

M. Servin, D. Malacara, J. L. Marroquin, and F. J. Cuevas, “Complex linear filters for phase shifting with very low detuning sensitivity,” J. Mod. Opt. 44, 1269–1278 (1997).
[Crossref]

Marroquin, J. L.

M. Servin, D. Malacara, J. L. Marroquin, and F. J. Cuevas, “Complex linear filters for phase shifting with very low detuning sensitivity,” J. Mod. Opt. 44, 1269–1278 (1997).
[Crossref]

Merkel, K.

Oreb, B.

Rosenfel, D. P.

Schreiber, H.

H. Schreiber and J. H. Brunning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara ed., (John Wiley & Sons, Inc., Hoboken, New Jersey2007).

Schwider, J

J Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf ed., (North Holland, Amsterdam, Oxford, New York, Tokyo, 1990).

Schwider, J.

Servin, M.

M. Servin, D. Malacara, J. L. Marroquin, and F. J. Cuevas, “Complex linear filters for phase shifting with very low detuning sensitivity,” J. Mod. Opt. 44, 1269–1278 (1997).
[Crossref]

M. Servin and M. Kujawinska, “Modern fringe pattern analysis in Interferometry,” in Handbook of Optical Engineering, D. Malacara and B. J. Thompson eds., (Marcel Dekker, 2001).

Spolaczyk, R.

White, A. D.

Appl. Opt. (3)

J. Mod. Opt. (1)

M. Servin, D. Malacara, J. L. Marroquin, and F. J. Cuevas, “Complex linear filters for phase shifting with very low detuning sensitivity,” J. Mod. Opt. 44, 1269–1278 (1997).
[Crossref]

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

Other (3)

H. Schreiber and J. H. Brunning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara ed., (John Wiley & Sons, Inc., Hoboken, New Jersey2007).

M. Servin and M. Kujawinska, “Modern fringe pattern analysis in Interferometry,” in Handbook of Optical Engineering, D. Malacara and B. J. Thompson eds., (Marcel Dekker, 2001).

J Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf ed., (North Holland, Amsterdam, Oxford, New York, Tokyo, 1990).

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Figures (3)

Fig. 1.
Fig. 1.

Graphical representation of the detuning components c and ε.

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Fig. 2.
Fig. 2.

Phasor representation of G(x,y,ω).

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Fig. 3.
Fig. 3.

Error detuning for TPS algorithms for five, seven and eleven steps.

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Equations (22)

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I ( x , y , t ) = a ( x , y ) + b ( x , y ) cos [ ϕ ( x , y ) + ω 0 t ] .
I ( x , y , ω ) = a ( x , y ) δ ( ω ) + b ( x , y ) 2 exp [ − i ϕ ( x , y ) ] δ ( ω − ω 0 ) + b ( x , y ) 2 exp [ i ϕ ( x , y ) ] δ ( ω + ω 0 ) .
G ( ω ) = a H ( ω ) δ ( ω ) + b 2 H ( ω ) exp ( − i ϕ ) δ ( ω − ω 0 ) + b 2 H ( ω ) exp ( i ϕ ) δ ( ω + ω 0 ) .
G ( x , y , ω ) = b 2 exp ( i ϕ ) H ( − ω 0 − Δ ) δ ( ω + ω 0 + Δ ) + b 2 exp ( − i ϕ ) H ( ω 0 + Δ ) δ ( ω − ω 0 − Δ ) .
c = ( b / 2 ) H ( − ω 0 − Δ ) , ε = ( b / 2 ) H ( + ω 0 + Δ ) .
G ( x , y , ω ) = c exp ( i ϕ ) δ ( ω + ω 0 + Δ ) + ε exp ( − i ϕ ) δ ( ω − ω 0 − Δ ) .
H ( ω 0 + Δ ) H ( − ω 0 − Δ ) = ε c = sin ( ϕ − ϕ ' ) sin ( ϕ + ϕ ' ) = tan ( ϕ ) − tan ( ϕ ' ) tan ( ϕ ) + tan ( ϕ ' ) .
Δ ϕ = tan − 1 [ ( c − ε c + ε ) tan ( ϕ ) ] − ϕ .
tan ( Δ ϕ ) = − ( ε / c ) sin ( 2 ϕ ) 1 + ( ε / c ) cos ( 2 ϕ ) .
Δϕ ≈ − ε c sin ( 2 ϕ ) = − H ( ω 0 + Δ ) H ( − ω 0 − Δ ) sin ( 2 ϕ ) .
Δ ϕ max = sin − 1 ∣ H ( ω 0 + Δ ) H ( − ω 0 − Δ ) ∣ ≈ ∣ H ( ω 0 + Δ ) H ( − ω 0 − Δ ) ∣ .
ϕ ( x , y , α ) = tan − 1 { 2 [ I ( α ) − I ( − α ) ] [ I ( − 2 α ) + I ( 2 α ) − 2 I ( 0 ) ] } , α = π / 2
h ( t ) = [ δ ( t + 2 α ) + δ ( t − 2 α ) − 2 δ ( 0 ) ] + i 2 [ δ ( t − α ) − δ ( t + α ) ] , α = π / 2
H ( ω , α = π / 2 ) = 4 sin ( ω π / 2 ) − 2 [ 1 − cos ( ω π ) ] .
ε c = H ( ω = 1 , α = π / 2 + Δ ) H ( ω = 1 , α = − π / 2 − Δ ) = − tan 2 ( Δ 2 ) .
Δ α = tan − 1 [ tan 2 ( Δ / 2 ) 1 + tan 2 ( Δ / 2 ) cos ( 2 ϕ ) sin ( 2 ϕ ) ] .
tan [ ϕ ( x , y , α ) ] = I ( − 3 α ) + 4.3 I ( − 2 α ) − 14 I ( − α ) + 14 I ( α ) − 4.3 I ( 2 α ) − I ( 3 α ) 1.5 I ( − 3 α ) − 6 I ( − 2 α ) − 4.5 I ( − α ) + 18 I ( 0 ) − 4.5 I ( α ) − 6 I ( 2 α ) + 1.5 I ( 3 α ) .
H ( ω ) = 2 [ sin ( 3 ω α ) + 1.5 cos ( 3 ω α ) + 4.3 sin ( 2 ω α ) − 6 cos ( 2 ω α ) − 14 sin ( ω α ) − 4.5 cos ( ω α ) + 9 ]
Δ ϕ max = sin − 1 ∣ cos ( 3 Δ ) + 4.3 sin ( 2 Δ ) + 14 cos ( Δ ) − 1.5 sin ( 3 Δ ) − 6 cos ( 2 Δ ) − 4.5 sin ( Δ ) + 9 cos ( 3 Δ ) + 4.3 sin ( 2 Δ ) + 14 cos ( Δ ) + 1.5 sin ( 3 Δ ) + 6 cos ( 2 Δ ) + 4.5 sin ( Δ ) + 9 ∣ .
Δ ϕ max = ∣ Δ 75 + 44 Δ ∣ .
ϕ ( x , y , α = π / 2 ) = tan − 1 { [ I ( 5 α ) − I ( 5 α ) ] − 8 [ I ( − 3 α ) − I ( 3 α ) ] + 15 [ I ( − α ) − I ( α ) ] 4 [ I ( − 4 α ) + I ( 4 α ) ] − 12 [ I ( − 2 α ) + I ( 2 α ) ] + 16 I ( 0 ) } .
Δ ϕ max = sin − 1 ∣ tan 4 ( Δ / 2 ) ∣ .

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