Abstract

Astrometric measurements using stellar interferometry rely on the precise measurement of the central white-light fringe to accurately obtain the optical path-length difference of incoming starlight to the two arms of the interferometer. Because of dispersion in the optical system the optical path-length difference is a function of the wavelength of the light and extracting the proper astrometric signatures requires accommodating these effects. One standard approach to stellar interferometry uses a channeled spectrum to determine phases at a number of different wavelengths that are then converted to the path-length delay. Because of throughput considerations these channels are made sufficiently broad so that monochromatic models are inadequate for retrieving the phase∕delay information. The presence of dispersion makes the polychromatic modeling problem for phase estimation even more difficult because of its effect on the complex visibility function. We introduce a class of models that rely on just a few spectral and dispersion parameters. A phase-shifting interferometry algorithm is derived that exploits the model structure. Numerical examples are given to illustrate the robustness and precision of the approach.

© 2007 Optical Society of America

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References

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  1. R. Danner and S. Unwin, eds., Space Interferometry Mission: Taking the Measure of the Universe (JPL, 1999), pp. 400-811.
  2. K. Creath, "Phase-measurement interferometry techniques," in Progress in Optics XXVI (Elsevier Science, 1988), pp. 350-393.
  3. M. Milman, "Accurately computing the optical pathlength difference for a Michelson interferometer with minimal knowledge of the source spectrum," J. Opt. Soc. Am. A 22, 2774-2785 (2005).
    [CrossRef]
  4. P. Hariharan and M. Roy, "White-light phase-stepping interferometry: measurement of the fractional interference order," J. Mod. Opt. 42, 2357-2360 (1995).
    [CrossRef]
  5. A. Harasaki, J. Schmit, and J. C. Wyant, "Improved vertical-scanning interferometry," Appl. Opt. 39, 2107-2115 (2000).
    [CrossRef]
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    [CrossRef]
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  8. J. Davis, W. J. Tango, and E. D. Thorvaldson, "Dispersion in stellar interferometry: simultaneous optimization for delay tracking and visibility measurements," Appl. Opt. 37, 5132-5136 (1998).
    [CrossRef]
  9. T. A. ten Brummelaar, "Differential path considerations in optical stellar interferometry," Appl. Opt. 34, 2214-2219 (1995).
    [CrossRef] [PubMed]
  10. A. Harasaki, J. Schmit, and J. C. Wyant, "Offset of coherent envelope position due to phase change on reflection," Appl. Opt. 40, 2102-2106 (2001).
    [CrossRef]
  11. A. Pfortner and J. Schwider, "Dispersion error in white-light Linnik interferometers and its implications for evaluation procedures," Appl. Opt. 40, 6223-6228 (2001).
  12. P. Pavlicek and J. Soubusta, "Measurement of the influence of dispersion on white-light interferometry," Appl. Opt. 43, 766-770 (2004).
    [CrossRef] [PubMed]
  13. M. Milman, J. Catanzarite, and S. G. Turyshev, "The effect of wavenumber error on the computation of path-length delay in white-light interferometry," Appl. Opt. 41, 4884-4890 (2002).
    [CrossRef] [PubMed]
  14. A. F. Boden, "Elementary theory of interferometry," in Principles of Long Baseline Stellar Interferometry, P. B. Lawson, ed., Course Notes from the 1999 Michelson Interferometry Summer School (JPL, 1999).
  15. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).
  16. E. Hewitt and K. Stromberg, Real and Abstract Analysis (Springer-Verlag, 1965).
  17. D. G. Leuenberger, Optimization by Vector Space Methods (Wiley, 1969).
  18. V. Makarov and M. Milman, "Accuracy and covariance analysis for global astrometry with the Space Interferometry Mission," Publ. Astron. Soc. Pac. 117, 757-771 (2005).
    [CrossRef]
  19. R. Linfeld, "Effects of stellar spectral types and nongeometric phase on delay estimation," JPL memo (JPL, 2000).
  20. M. Milman, M. Regehr, and T. J. Shen, "White light modeling, algorithm development, and validation on the microarcsecond metrology testbed," Proc. SPIE 5491, 1813-1822 (2004).
    [CrossRef]
  21. R. Goullioud and T. J. Shen, "MAM testbed detailed description and alignment," in IEEE Aerospace Conference (IEEE, 2004).

2005 (2)

M. Milman, "Accurately computing the optical pathlength difference for a Michelson interferometer with minimal knowledge of the source spectrum," J. Opt. Soc. Am. A 22, 2774-2785 (2005).
[CrossRef]

V. Makarov and M. Milman, "Accuracy and covariance analysis for global astrometry with the Space Interferometry Mission," Publ. Astron. Soc. Pac. 117, 757-771 (2005).
[CrossRef]

2004 (2)

M. Milman, M. Regehr, and T. J. Shen, "White light modeling, algorithm development, and validation on the microarcsecond metrology testbed," Proc. SPIE 5491, 1813-1822 (2004).
[CrossRef]

P. Pavlicek and J. Soubusta, "Measurement of the influence of dispersion on white-light interferometry," Appl. Opt. 43, 766-770 (2004).
[CrossRef] [PubMed]

2002 (1)

2001 (2)

2000 (1)

1998 (1)

1996 (1)

1995 (2)

P. Hariharan and M. Roy, "White-light phase-stepping interferometry: measurement of the fractional interference order," J. Mod. Opt. 42, 2357-2360 (1995).
[CrossRef]

T. A. ten Brummelaar, "Differential path considerations in optical stellar interferometry," Appl. Opt. 34, 2214-2219 (1995).
[CrossRef] [PubMed]

1990 (1)

Appl. Opt. (8)

J. Mod. Opt. (1)

P. Hariharan and M. Roy, "White-light phase-stepping interferometry: measurement of the fractional interference order," J. Mod. Opt. 42, 2357-2360 (1995).
[CrossRef]

J. Opt. Soc. Am. A (2)

Proc. SPIE (1)

M. Milman, M. Regehr, and T. J. Shen, "White light modeling, algorithm development, and validation on the microarcsecond metrology testbed," Proc. SPIE 5491, 1813-1822 (2004).
[CrossRef]

Publ. Astron. Soc. Pac. (1)

V. Makarov and M. Milman, "Accuracy and covariance analysis for global astrometry with the Space Interferometry Mission," Publ. Astron. Soc. Pac. 117, 757-771 (2005).
[CrossRef]

Other (8)

R. Linfeld, "Effects of stellar spectral types and nongeometric phase on delay estimation," JPL memo (JPL, 2000).

R. Danner and S. Unwin, eds., Space Interferometry Mission: Taking the Measure of the Universe (JPL, 1999), pp. 400-811.

K. Creath, "Phase-measurement interferometry techniques," in Progress in Optics XXVI (Elsevier Science, 1988), pp. 350-393.

A. F. Boden, "Elementary theory of interferometry," in Principles of Long Baseline Stellar Interferometry, P. B. Lawson, ed., Course Notes from the 1999 Michelson Interferometry Summer School (JPL, 1999).

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

E. Hewitt and K. Stromberg, Real and Abstract Analysis (Springer-Verlag, 1965).

D. G. Leuenberger, Optimization by Vector Space Methods (Wiley, 1969).

R. Goullioud and T. J. Shen, "MAM testbed detailed description and alignment," in IEEE Aerospace Conference (IEEE, 2004).

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Figures (8)

Fig. 1
Fig. 1

Schematic of a stellar interferometer.

Fig. 2
Fig. 2

Dispersion function for 5 µm misbalance of BK7 glass at red end of spectrum (left); nonlinear residual after affine model is removed (right).

Fig. 3
Fig. 3

Dispersion function for 5 µm misbalance of BK7 glass at blue end of spectrum (left); nonlinear residual after affine model is removed (right).

Fig. 4
Fig. 4

Graph of ψ over entire passband.

Fig. 5
Fig. 5

(Color online) Spectrum (top left), dispersion function (top right), and interferograms (bottom).

Fig. 6
Fig. 6

(Color online) Model approximation error for monochromatic and second moment models.

Fig. 7
Fig. 7

(Color online) Group delay estimation error.

Fig. 8
Fig. 8

(Color online) Group delay estimation error.

Equations (95)

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[ K , K + ] = [ k i , k + i ] ,
Y ( x ) k k + I ( k ) { 1 + V 0 ( k ) cos ( k x + ψ ( k ) ) } d k .
ψ ( k ) = a 0 + a 1 k + r ( k ) .
Y ( x ) = I ¯ + k k + I ( k ) V 0 ( k ) cos ( k ( x + a 1 ) + a 0 + r ( k ) ) d k ,
  I ¯ = k k + I ( k ) d k .
  V ¯ = k k + I ( k ) V 0 ( k ) d k / I ¯ , p ( k ) = I ( k ) I ¯ V ¯ ,
μ j = k k + ( k k ¯ ) j p ( k ) d k ,
  k ¯ = k k + k p ( k ) d k .
Y ( x ) = I ¯ + k k + I ( k ) V 0 ( k ) cos ( [ k k ¯ ] u + r ( k ) + a 0 + k ¯ u ) d k = I ¯ + cos ( a 0 + k ¯ u ) k k + I ( k ) V 0 ( k ) cos ( [ k k ¯ ] u + r ( k ) ) d k sin ( a 0 + k ¯ u ) k k + I ( k ) V 0 ( k ) sin ( [ k k ¯ ] u + r ( k ) ) d k = I ¯ + I ¯ V ¯ cos ( a 0 + k ¯ u ) C ( u ) I ¯ V ¯ sin ( a 0 + k ¯ u ) S ( u ) ,
C ( u ) = k k + cos ( [ k k ¯ ] u + r ( k ) ) p ( k ) d k ,
S ( u ) = k k + sin ( [ k k ¯ ] u + r ( k ) ) p ( k ) d k .
δ ( u ) = tan 1 [ S ( u ) C ( u ) ] ,
C ( u ) = cos ( δ ( u ) ) S 2 ( u ) + C 2 ( u ) ,
S ( u ) = sin ( δ ( u ) ) S 2 ( u ) + C 2 ( u ) .
Y ( x ) = I ¯ + cos ( k ¯ u + a 0 ) cos ( δ ( u ) ) I ¯ V ¯ S 2 ( u ) + C 2 ( u ) sin ( k ¯ u + a 0 ) sin ( δ ( u ) ) I ¯ V ¯ S 2 ( u ) + C 2 ( u ) = I ¯ [ 1 + V ¯ S 2 ( u ) + C 2 ( u ) cos ( k ¯ u + a 0 + δ ( u ) ) ] .
Y ( x ) = I ¯ [ 1 + V ( u ) cos ( k ¯ u + a 0 + δ ( u ) ) ] ,
V ( u ) V ¯ S 2 ( u ) + C 2 ( u ) .
0 V ( u ) V ¯ , lim | u | V ( u ) = 0 ,
δ ( 0 ) = S ( 0 ) C ( 0 ) C ( 0 ) S ( 0 ) S 2 ( 0 ) + C 2 ( 0 ) = 0 ,
S ( 0 ) = ( k k ¯ ) p ( k ) d k = 0 ,
k * + δ ( 0 ) = k ¯ .
V ( 0 ) = V ¯ , V ( 0 ) = 0 , V ( 0 ) = μ 2 V ¯ ,
V ( 3 ) ( 0 ) = 0 , V ( 4 ) ( 0 ) = μ 4 V ¯ ,
δ ( 0 ) = δ ( 0 ) = δ ( 0 ) = 0 , δ ( 3 ) ( 0 ) = μ 3 .
Y ( x ) = I ¯ [ 1 + V ¯ [ 1 μ 2 2 u 2 + μ 4 24 u 4 ] cos ( k ¯ u + a 0 μ 3 6 u 3 ) ] .
S ( u ) S ( u ) + C ( u ) C ( u ) = 0.
Ω ( t , u ) = S ( t , u ) S ( t , u ) + C ( t , u ) C ( t , u ) ,
C ( t , u ) = k k + cos ( [ k k ¯ ] u + tr ( k ) ) p ( k ) d k ,
S ( t , u ) = k k + sin ( [ k k ¯ ] u + tr ( k ) ) p ( k ) d k ,
C ( t , u ) = k k + ( k k ¯ ) sin ( [ k k ¯ ] u + tr ( k ) ) p ( k ) d k ,
S ( t , u ) = k k + ( k k ¯ ) cos ( [ k k ¯ ] u + tr ( k ) ) p ( k ) d k .
Ω ( t , u ) = 0
Ω u ( 0 , 0 ) = S ( 0 ) 2 + C ( 0 ) 2 + S ( 0 ) S ( 0 ) + C ( 0 ) C ( 0 ) = C ( 0 ) C ( 0 ) = μ 2 ,
  u ( t ) = Ω t [ Ω u ] 1 , u ( 0 ) = 0.
u ( 0 ) = 1 μ 2 ( k k ¯ ) r ( k ) p ( k ) d k .
a 1 = 1 μ 2 ( k k ¯ ) ψ ( k ) p ( k ) d k ,
a 0 = ( ψ ( k ) a 1 k ) p ( k ) d k = ψ ( k ) p ( k ) d k a 1 k ¯ .
r ( k ) p ( k ) d k = 0 , ( k k ¯ ) r ( k ) p ( k ) d k = 0 .
min a 0 , a 1 [ ψ ( k ) a 1 k a 0 ] 2 p ( k ) d k ,
f , g = f ( k ) g ( k ) p ( k ) d k ,
V ( 0 ) = V ¯ [ 1 1 2 r 2 ( k ) p ( k ) d k ] , V ( 0 ) = 0 ,
V ( 0 ) = μ 2 V ¯ [ 1 1 2 r 2 ( k ) p ( k ) d k ] ,
δ ( 0 ) = 0 , δ ( 0 ) = 1 2 ( k k ¯ ) r 2 ( k ) p ( k ) d k ,
δ ( 0 ) = ( k k ¯ ) 2 r ( k ) p ( k ) d k .
Y ( x ) = I ¯ { 1 + V ¯ [ 1 μ 2 2 u 2 ] cos ( k ¯ u + a 0 ) } ,
Y ( x ) = I ¯ { 1 + V ¯ [ 1 μ 2 2 u 2 + μ 4 24 u 4 ] × cos ( k ¯ u + a 0 1 2 δ ( 0 ) u 2 μ 3 6 u 3 ) } .
k k + ( k k ¯ ) 2 r ( k ) p ( k ) d k μ 2 | r | ,
X = [ I 0 I 0 V 0 cos ( k ¯ u 0 + a 0 ) I 0 V 0 sin ( k ¯ u 0 + a 0 ) ] ,
Y = A X μ 2 2 B ( X ) , B ( X ) [ u 0 2 B 0 + 2 u 0 B 1 + B 2 ] X ,
A = [ 1 cos ( k ¯ u 1 ) sin ( k ¯ u 1 ) 1 cos ( k ¯ u n ) sin ( k ¯ u n ) ] ,
B i = [ 0 u 1 i cos ( k ¯ u 1 ) u 1 i sin ( k ¯ u 1 ) 0 u n i cos ( k ¯ u n ) u n i sin ( k ¯ u n ) ] , i = 0 , 1 , 2 .
K Y = X μ 2 2 K B ( X ) ,
Ψ ( X ) = K Y + μ 2 2 K B ( X ) .
X i + 1 = Ψ ( X i ) , X 0 = A Y
u 0 = [ tan 1 ( X 3 / X 2 ) a 0 ] / k ¯ .
δ d = δ ϕ λ 2 π + ϕ δ λ 2 π ( λ = 2 π / k ) .
Y = A ( k ) X ( k ) ,
X ^ ( k ) = K A ( k ) X ( k ) .
d ^ ( k ) = ϕ ( X ^ ( k ) ) / k 0 ,   where   ϕ ( X ) = tan 1 ( X 3 / X 2 ) a 0 .
d = ϕ ( X ( k ) ) / k ,
ϵ ( k ) = ϕ ( X ^ ( k ) ) k 0 ϕ ( X ( k ) ) k ,
ϕ ( X ) = [ 0 sin ( ϕ 0 ) / ( I 0 V 0 ) cos ( ϕ 0 ) / ( I 0 V 0 ) ] ,
ϕ 0 = tan 1 ( X 3 / X 2 ) ,
X ^ ( k 0 ) = K A ( k 0 ) X ( k 0 ) + K A ( k 0 ) X ( k 0 ) = K A ( k 0 ) X ( k 0 ) + X ( k 0 ) ,   since   K A ( k 0 ) = I .
ϵ ( k 0 ) = 1 k 0 ϕ ( X ^ ( k 0 ) ) X ^ ( k 0 ) 1 k 0 2 [ k ϕ ( X ( k 0 ) ) X ( k 0 ) ϕ ( X ( k 0 ) ) ] = 1 k 0 ϕ ( X ( k 0 ) ) [ X ^ ( k 0 ) X ( k 0 ) ] + ϕ ( X ( k 0 ) ) k 0 2 = 1 k 0 ϕ ( X ( k 0 ) ) K A ( k 0 ) X ( k 0 ) + ϕ ( X ( k 0 ) ) k 0 2 .
k 0 ϕ ( X ( k 0 ) ) K A ( k 0 ) X ( k 0 ) + ϕ ( X ( k 0 ) ) = 0 .
sin 2 ( ϕ 0 ) k 2 , a 3 cos 2 ( ϕ 0 ) k 3 , a 2 + sin ( 2 ϕ 0 ) 2 { k 2 , a 2 k 3 , a 3 } = [ ϕ a 0 ] / k 0 .
F ( X , μ 2 ) = X μ 2 2 K B ( X ) K Y .
F X X ( μ 2 ) + F μ 2 = 0 ,
X ( μ 2 ) = [ F X ] 1 F μ 2 , X ( μ 2 * ) = X * .
X ( μ 2 * + h ) = X ( μ 2 * ) [ F X ] 1 F μ 2 h .
F X = I + μ 2 * K B ( X ) ,
[ F X ] 1 I μ 2 * K B ( X ) ,
F μ 2 h = h K B ( X ) .
X ( μ 2 * + h ) X ( μ 2 * ) h K B ( X ) .
δ ϕ = h sin ( 2 ϕ ) 4 { K 2 , u 0 2 C u 0 + 2 u 0 C u + C u 2 + K 3 , u 0 2 S u 0 + 2 u 0 S u + S u 2 } + h 2 { sin 2 ( ϕ ) K 2 , u 0 2 S u 0 + 2 u 0 S u + S u 2 + cos 2 ( ϕ ) K 3 , u 0 2 C u 0 + 2 u 0 C u + C u 2 } ,
C u k = ( u 1 k cos ( k u 1 ) ,  …  , u n k cos ( k u n ) ) T ,
S u k = ( u 1 k sin ( k u 1 ) ,  …  , u n k sin ( k u n ) ) T ,
k = 0 , 1 , 2 .
F ( X , a 0 ) = X K Y μ 2 K [ u 2 ( X , a 0 ) B 0 + 2 u ( X , a 0 ) B 1 + B 2 ] X ,
u ( X , u 0 ) = 1 k ¯ [ tan 1 ( X 3 , X 2 ) a 0 ] .
δ ϕ = δ a 0 μ 2 k ¯ [ u K B 0 X + K B 1 X ] .
Y ( x ) = I 0 { 1 + V 0 [ 1 μ 2 u 2 2 ] cos ( k ¯ u + a 0 ) } ,
x = ϕ k ¯ a 1 a 0 k ¯ .
a 1 + a 0 / k ¯ = 1 k ¯ ψ ( k ) p ( k ) d k = α 0 k ¯ + α 1 + 1 k ¯ r ( k ) p ( k ) d k .
x = ϕ k ¯ α 1 α 0 + r k ¯ ,
r = r ( k ) p ( k ) d k .
x j + α 1 i = ϕ i j α 0 i k ¯ i j r i k ¯ i j , i = 1 ,  …  , N , j = 1 , 2 .
y i j = x j + 1 k ¯ i j δ r i ( k ) p j ( k ) d k + c + η i j ,
y j = i w i j y i j ,
w i j = 1 , w i j 0 .
y 2 y 1 = x 2 x 1 + i [ w i 2 δ r i 2 w i 1 δ r i 1 ] ;
δ r i j = 1 k ¯ i j δ r i ( k ) p j ( k ) d k .
δ r i j = 1 k ¯ i j [ δ α 0 i + δ α 1 i k ] p i j ( k ) d k = δ α 0 i k ¯ i j + δ α 1 i .
i [ w i 2 δ r i 2 w i 1 δ r i 1 ] = i δ α 0 i [ w i 2 k ¯ i 2 w i 1 k ¯ i 1 ] .

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