Abstract

An improved approach to interferometry using sinusoidal phase shifting balances several harmonic components in the interference signal against each other. The resulting computationally efficient phase- estimation algorithms have low sensitivity to errors such as spurious intensity noise, vibration, and errors in the phase shift pattern. Specific example algorithms employing 8 and 12 camera frames illustrate design principles that are extendable to algorithms of any length for applications that would benefit from a simplified, sinusoidal phase shift.

© 2009 Optical Society of America

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References

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  1. D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, Vol. 61 of Optical Engineering Series (Marcel Dekker, 1998), pp. 169-245.
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  6. X. Zhongbao and N. Zhang, “Sinusoidal phase modulation interferometer based on integration method,” Proc. SPIE 6357, 635725 (2006).
    [CrossRef]
  7. P. J. de Groot and L. L. Deck, “New algorithms and error analysis for sinusoidal phase shifting interferometry,” Proc. SPIE 7063, 706301 (2008).
  8. P. de Groot, “Sinusoidal phase shifting interferometry,” U.S. patent application 2008/0180679 Al (2008).
  9. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980), p. 973.
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    [CrossRef] [PubMed]
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    [CrossRef]
  12. P. de Groot, “Vibration in phase shifting interferometry,” J. Opt. Soc. Am. A 12, 354-365 (1995).
    [CrossRef]
  13. K. Falaggis, D. P. Towers, and C. E. Towers, “Phase measurement through sinusoidal excitation with application to multi-wavelength interferometry.” J. Opt. A Pure Appl. Opt. 11, 054008 (2009).
    [CrossRef]

2009 (1)

K. Falaggis, D. P. Towers, and C. E. Towers, “Phase measurement through sinusoidal excitation with application to multi-wavelength interferometry.” J. Opt. A Pure Appl. Opt. 11, 054008 (2009).
[CrossRef]

2008 (1)

P. J. de Groot and L. L. Deck, “New algorithms and error analysis for sinusoidal phase shifting interferometry,” Proc. SPIE 7063, 706301 (2008).

2006 (1)

X. Zhongbao and N. Zhang, “Sinusoidal phase modulation interferometer based on integration method,” Proc. SPIE 6357, 635725 (2006).
[CrossRef]

2001 (1)

1995 (1)

1990 (2)

1987 (2)

1986 (1)

de Groot, P.

P. de Groot, “Vibration in phase shifting interferometry,” J. Opt. Soc. Am. A 12, 354-365 (1995).
[CrossRef]

P. de Groot, “Sinusoidal phase shifting interferometry,” U.S. patent application 2008/0180679 Al (2008).

de Groot, P. J.

P. J. de Groot and L. L. Deck, “New algorithms and error analysis for sinusoidal phase shifting interferometry,” Proc. SPIE 7063, 706301 (2008).

Deck, L. L.

P. J. de Groot and L. L. Deck, “New algorithms and error analysis for sinusoidal phase shifting interferometry,” Proc. SPIE 7063, 706301 (2008).

Dubois, A.

Eiju, T.

Falaggis, K.

K. Falaggis, D. P. Towers, and C. E. Towers, “Phase measurement through sinusoidal excitation with application to multi-wavelength interferometry.” J. Opt. A Pure Appl. Opt. 11, 054008 (2009).
[CrossRef]

Freischlad, K.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980), p. 973.

Hariharan, P.

Koliopoulos, C. L.

Malacara, D.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, Vol. 61 of Optical Engineering Series (Marcel Dekker, 1998), pp. 169-245.

Malacara, Z.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, Vol. 61 of Optical Engineering Series (Marcel Dekker, 1998), pp. 169-245.

Nakamura, T.

Okamura, T.

Okazaki, H.

Oreb, B. F.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980), p. 973.

Sakai, M.

Sasaki, O.

Servin, M.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, Vol. 61 of Optical Engineering Series (Marcel Dekker, 1998), pp. 169-245.

Towers, C. E.

K. Falaggis, D. P. Towers, and C. E. Towers, “Phase measurement through sinusoidal excitation with application to multi-wavelength interferometry.” J. Opt. A Pure Appl. Opt. 11, 054008 (2009).
[CrossRef]

Towers, D. P.

K. Falaggis, D. P. Towers, and C. E. Towers, “Phase measurement through sinusoidal excitation with application to multi-wavelength interferometry.” J. Opt. A Pure Appl. Opt. 11, 054008 (2009).
[CrossRef]

Zhang, N.

X. Zhongbao and N. Zhang, “Sinusoidal phase modulation interferometer based on integration method,” Proc. SPIE 6357, 635725 (2006).
[CrossRef]

Zhongbao, X.

X. Zhongbao and N. Zhang, “Sinusoidal phase modulation interferometer based on integration method,” Proc. SPIE 6357, 635725 (2006).
[CrossRef]

Appl. Opt. (4)

J. Opt. A Pure Appl. Opt. (1)

K. Falaggis, D. P. Towers, and C. E. Towers, “Phase measurement through sinusoidal excitation with application to multi-wavelength interferometry.” J. Opt. A Pure Appl. Opt. 11, 054008 (2009).
[CrossRef]

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

Proc. SPIE (2)

X. Zhongbao and N. Zhang, “Sinusoidal phase modulation interferometer based on integration method,” Proc. SPIE 6357, 635725 (2006).
[CrossRef]

P. J. de Groot and L. L. Deck, “New algorithms and error analysis for sinusoidal phase shifting interferometry,” Proc. SPIE 7063, 706301 (2008).

Other (3)

P. de Groot, “Sinusoidal phase shifting interferometry,” U.S. patent application 2008/0180679 Al (2008).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980), p. 973.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, Vol. 61 of Optical Engineering Series (Marcel Dekker, 1998), pp. 169-245.

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Figures (10)

Fig. 1
Fig. 1

Laser Fizeau interferometer for optical testing. Mechanical motion of approximately 1 micron excursion introduces controlled phase shifts while the camera captures a sequence of interference patterns.

Fig. 2
Fig. 2

Sinusoidal phase shift ϕ ( α ) for a phase shift amplitude u = π and φ = 0 .

Fig. 3
Fig. 3

Interference signal I ( α ) resulting from the sinusoidal phase shift ϕ ( α ) shown in Fig. 2 for an interference phase θ = π / 2 .

Fig. 4
Fig. 4

Strength of even (solid lines) and odd (dashed lines) harmonics in the SinPSI signal as a function of the sinusoidal phase shift amplitude u.

Fig. 5
Fig. 5

Variation in the normalized filter functions F odd / Γ odd and F even / Γ even as a function of the sinusoidal phase shift amplitude u for the 4-frame sinusoidal PSI algorithm in Eq. (43). The design amplitude u 0 is shown by the vertical line at 2.35 rad.

Fig. 6
Fig. 6

Data acquisition pattern for the 8-frame algorithm.

Fig. 7
Fig. 7

Variation in the normalized filter functions F odd / Γ odd and F even / Γ even as a function of the sinusoidal phase shift amplitude u for the 8-frame sinusoidal PSI algorithm in Eq. (43). The design amplitude u 0 is shown by the vertical line at 2.95 rad.

Fig. 8
Fig. 8

Variation in the normalized filter functions F odd / Γ odd and F even / Γ even as a function of the sinusoidal phase shift amplitude u for the 12-frame sinusoidal PSI algorithm in Eq. (60). The design amplitude u 0 shown by the vertical line at 3 . 384 rad is positioned so that the algorithm sensitivity is zero at u = 2 u 0 = 6 . 768 .

Fig. 9
Fig. 9

Measurement error in nm rms over a full cycle of phase at 633 nm wavelength as a function of calibration error, for algorithms based on 4, 8, and 12 camera frames per sinusoidal phase shift cycle.

Fig. 10
Fig. 10

Sensitivity of the 4-, 8-, and 12-frame sinusoidal PSI algorithms to a 1 nm vibrational amplitude. The measurement error in nm rms over a full cycle of phase at 633 nm wavelength is plotted as a function of vibrational frequency normalized to the camera frame rate.

Tables (3)

Tables Icon

Table 1 Error in nm rms Over a Full Phase Cycle at 633 nm for 1% Random Noise

Tables Icon

Table 2 Error in nm rms Over a Full Phase Cycle at 633 nm for a 1% Detector Nonlinearity of Order n

Tables Icon

Table 3 Error in nm rms Over a Full Phase Cycle at 633 nm for a 1% Phase Shift Nonlinearity of Order n

Equations (71)

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tan ( θ ) = I 0 + I 1 I 2 I 3 I 0 I 1 + I 2 I 3 ,
θ = 4 π h / λ .
I ( θ , t ) = q { 1 + V cos [ θ + ϕ ( t ) ] } ,
ϕ ( t ) = u cos [ α ( t ) + φ ] ,
α ( t ) = 2 π f t .
I ( θ , α ) = q + q V cos ( θ ) cos [ ϕ ( α ) ] q V sin ( θ ) sin [ ϕ ( α ) ] .
exp [ i u cos ( α ) ] = J 0 ( u ) + 2 ν = 1 i ν J ν ( u ) cos ( ν α ) ,
I ( θ , α ) = q D ( θ ) + q V cos ( θ ) C ( α ) + q V sin ( θ ) S ( α ) ,
D ( θ ) = 1 + V J 0 ( u ) cos ( θ ) ,
S ( α ) = 2 ν = 1 , 3 , 5 ( 1 ) ( ν + 1 ) / 2 J ν ( u ) cos [ ν ( α + φ ) ] ,
C ( α ) = 2 v = 2 , 4 , 6 ( 1 ) ν / 2 J ν ( u ) cos [ ν ( α + φ ) ] ,
I ¯ ( θ , α ) = α β / 2 α + β / 2 I ( θ , α ) d α ,
B ( ν ) = sin ( ν β / 2 ) ν β / 2 .
I ¯ ( θ , α ) = q D ( θ ) + q V cos ( θ ) C ¯ ( α ) + q V sin ( θ ) S ¯ ( α ) ,
S ¯ ( α ) = 2 ν = 1 , 3 , 5 ( 1 ) ( ν + 1 ) / 2 B ( ν ) J ν ( u ) cos [ ν ( α + φ ) ] ,
C ¯ ( α ) = 2 v = 2 , 4 , 6 ( 1 ) ν / 2 B ( ν ) J ν ( u ) cos [ ν ( α + φ ) ] .
ϕ j = u cos ( α j ) ,
α j = j Δ α + α 0 ,
j = 0 , 1 , 2 P 1.
H ( θ , ν ) = j cos ( ν α j ) I ¯ j ( θ ) j cos ( ν α j ) 2
h ν , j = cos ( ν α j ) j cos ( ν α j ) 2
H ν ( θ ) = j h ν , j I ¯ j ( θ ) .
j h ν , j = 0 ,
h odd , j = ν = odd γ ν h ν , j ,
h even , j = ν = even γ ν h ν , j .
H odd ( θ ) = j h odd , j I ¯ j ( θ ) ,
H even ( θ ) = j h even , j I ¯ j ( θ ) .
j h odd , j h even , j = 0 ,
j h odd , j cos ( ν α j ) = 0 for     ν = 2 , 4 , 6 ,
j h even , j cos ( ν α j ) = 0 for     ν = 1 , 3 , 5 .
tan ( θ ) = Γ even Γ odd H odd ( θ ) H even ( θ )
Γ odd = 2 ν = 1 , 3 , 5 ( 1 ) ( ν + 1 ) / 2 J ν ( u 0 ) B ( ν ) j h odd , j cos ( ν α j ) ,
Γ even = 2 ν = 2 , 4 , 6 ( 1 ) ν / 2 J ν ( u 0 ) B ( ν ) j h even , j cos ( ν α j ) .
F odd ( u ) = 2 ν = 1 , 3 , 5 ( 1 ) ( ν + 1 ) / 2 J ν ( u ) B ( ν ) j h odd , j cos ( ν α j ) ,
F even ( u ) = 2 ν = 2 , 4 , 6 ( 1 ) ν / 2 J ν ( u ) B ( ν ) j h even , j cos ( ν α j ) ,
Γ odd = F odd ( u 0 ) ,
Γ even = F even ( u 0 ) .
α j = j π / 2
h odd = ( 1 0 1 0 ) ,
h even = ( 1 1 1 1 ) .
Γ odd = 1 . 5718 ,
Γ even = 2.2283 ,
tan ( θ ) = 1 . 4176 I ¯ 0 I ¯ 2 I ¯ 0 I ¯ 1 + I ¯ 2 I ¯ 3 .
α j = j π / 4 + π / 8
h 1 = 1 4 [ ς 1 ς 3 ς 3 ς 1 ς 1 ς 3 ς 3 ς 1 ] ,
h 2 = 1 4 2 ( 1 1 1 1 1 1 1 1 ) ,
h 3 = 1 4 [ ς 3 ς 1 ς 1 ς 3 ς 3 ς 1 ς 1 ς 3 ] ,
ς 1 = cos ( π / 8 ) ,
ς 3 = cos ( 3 π / 8 ) .
γ 1 = 4 ς 3 / ( ς 1 2 + ς 3 2 ) ,
γ 2 = 4 2 ,
γ 3 = 4 ς 1 / ( ς 1 2 + ς 3 2 ) ,
h odd = ( 0 1 1 0 0 1 1 0 ) ,
h even = ( 1 1 1 1 1 1 1 1 ) .
Γ odd = 2 . 9432 ,
Γ even = 4 . 8996 ,
Γ even / Γ odd = 1 . 6647.
tan ( θ ) = 1 . 6647 ( g 1 g 2 ) g 0 + g 1 + g 2 g 3 ,
g j = I ¯ j + I ¯ 7 j ; j = 0 , 1 , 2 , 3.
tan ( θ ) = 1.2461 ( g 0 g 5 ) 1.5525 ( g 1 g 4 ) 2.5746 ( g 2 g 3 ) 0.2707 ( g 0 + g 5 ) 2.6459 ( g 1 + g 4 ) + 2.3753 ( g 2 + g 3 ) ,
g j = I ¯ j + I ¯ 11 j ; j = 0 , 1 , 2 , 3.
ε st dv ( δ u ) = 1 2 2 | ρ ( δ u ) 1 | ,
ρ ( δ u ) = F even ( u 0 + δ u ) F odd ( u 0 + δ u ) Γ odd Γ even .
I ( θ , t ) = q { 1 + V cos [ θ + ϕ ( t ) ] } + n ( σ , t ) ,
ε rms = ( σ q V ) 1 2 ( p even Γ even ) 2 + ( p odd Γ odd ) 2 ,
p odd = j ( h odd ) j 2 ,
p even = j ( h even ) j 2 .
I ( θ , α , ν ) = q { 1 + V cos [ θ + ϕ ( α ) + n ( α , ν ) ] } ,
n ( α , ν ) = 2 σ cos [ ν α + ξ ] .
Δ I = q ζ [ ( I q q ) 2 1 2 ] ( I q q ) ( n 2 ) ,
Δ ϕ = u ζ [ ( ϕ ϕ u ) 2 1 2 ] ( ϕ ϕ u ) ( n 2 ) ,

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