Abstract

In optical aperture-synthesis imaging of stellar objects, different beam combination strategies are used and proposed. Coaxial Michelson interferometers are very common and a homothetic multiaxial interferometer is recently realized in the Large Binocular Telescope. Laboratory experiments have demonstrated the working principles of two new approaches: densified pupil imaging and wide field-of-view (FOV) coaxial imaging using a staircase-shaped mirror. We develop a common mathematical formulation for direct comparison of the resolution and noise sensitivity of these four telescope configurations for combining beams from multiple apertures for interferometric synthetic aperture, wide-FOV imaging. Singular value decomposition techniques are used to compare the techniques and observe their distinct signal-to-noise ratio behaviors. We conclude that for a certain chosen stellar object, clear differences in performance of the imagers are identifiable.

© 2007 Optical Society of America

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  1. A. Quirrenbach, "Optical interferometry," Annu. Rev. Astron. Astrophys. 39, 353-401 (2001).
    [CrossRef]
  2. S. K. Saha, "Modern optical astronomy: technology and impact of interferometry," Rev. Mod. Phys. 74, 551-600 (2002).
    [CrossRef]
  3. P. M. Hinz, J. R. P. Angel, D. W. McCarthy, Jr., W. F. Hoffman, and C. Y. Peng, "The Large Binocular Telescope Interferometer," Proc. SPIE 4838, 108-112 (2003).
    [CrossRef]
  4. J. E. Baldwin and C. A. Haniff, "The application of interferometry to optical astronomical imaging," Philos. Trans. R. Soc. London, Ser. A 360, 969-986 (2002).
    [CrossRef]
  5. A. Labeyrie, "Resolved imaging of extra-solar planets with future 10-100km optical interferometric arrays," Astron. Astrophys., Suppl. Ser. 118, 517-524 (1996).
    [CrossRef]
  6. I. Montilla, S. F. Pereira, and J. J. M. Braat, "Michelson wide-field stellar interferometry: principles and experimental verification," Appl. Opt. 44, 328-336 (2005).
    [CrossRef] [PubMed]
  7. DARWIN The Infrared Space Interferometer: Redbook (ESA-SCI, 2000), Vol. 12.
  8. A. A. Michelson, "On the application of interference methods to astronomical measurements," London, Edinburgh Dublin Philos. Mag. J. Sci. 30, 1-21 (1890).
  9. A. A. Michelson and F. G. Pease, "Measurement of the diameter of Alpha Orionis with the interferometer," Astrophys. J. 53, 249 (1921).
    [CrossRef]
  10. M. A. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), pp. 508-510.
  11. M. Young and P. D. Hale, "Off-axis illumination and its relation to partial coherence," Am. J. Phys. 63, 1136-1141 (1995).
    [CrossRef]
  12. A. N. Tikhonov and V. Y. Goncharsky, Solutions of Ill-Posed Problems (Winston & Sons, 1977).
  13. P. C. Hansen, "Regularization tools: a Matlab package for analysis and solution of discrete ill-posed problems," Numer. Algorithms 6, 1-35 (1994).
    [CrossRef]
  14. P. C. Hansen, "The discrete Picard condition for discrete ill-posed problems," BIT 30, 658-672 (1990).
    [CrossRef]
  15. P. C. Hansen, "Analysis of discrete ill-posed problems by means of the L-curve," SIAM Rev. 34, 561-580 (1992).
    [CrossRef]
  16. R. Visser, "Regularization in nearfield acoustic source identification," in Proceedings of the Eighth International Congress on Sound and Vibration (2001), pp. 1637-1644.

2005

2003

P. M. Hinz, J. R. P. Angel, D. W. McCarthy, Jr., W. F. Hoffman, and C. Y. Peng, "The Large Binocular Telescope Interferometer," Proc. SPIE 4838, 108-112 (2003).
[CrossRef]

2002

J. E. Baldwin and C. A. Haniff, "The application of interferometry to optical astronomical imaging," Philos. Trans. R. Soc. London, Ser. A 360, 969-986 (2002).
[CrossRef]

S. K. Saha, "Modern optical astronomy: technology and impact of interferometry," Rev. Mod. Phys. 74, 551-600 (2002).
[CrossRef]

2001

R. Visser, "Regularization in nearfield acoustic source identification," in Proceedings of the Eighth International Congress on Sound and Vibration (2001), pp. 1637-1644.

A. Quirrenbach, "Optical interferometry," Annu. Rev. Astron. Astrophys. 39, 353-401 (2001).
[CrossRef]

2000

DARWIN The Infrared Space Interferometer: Redbook (ESA-SCI, 2000), Vol. 12.

1996

A. Labeyrie, "Resolved imaging of extra-solar planets with future 10-100km optical interferometric arrays," Astron. Astrophys., Suppl. Ser. 118, 517-524 (1996).
[CrossRef]

1995

M. Young and P. D. Hale, "Off-axis illumination and its relation to partial coherence," Am. J. Phys. 63, 1136-1141 (1995).
[CrossRef]

1994

P. C. Hansen, "Regularization tools: a Matlab package for analysis and solution of discrete ill-posed problems," Numer. Algorithms 6, 1-35 (1994).
[CrossRef]

1992

P. C. Hansen, "Analysis of discrete ill-posed problems by means of the L-curve," SIAM Rev. 34, 561-580 (1992).
[CrossRef]

1990

P. C. Hansen, "The discrete Picard condition for discrete ill-posed problems," BIT 30, 658-672 (1990).
[CrossRef]

1980

M. A. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), pp. 508-510.

1977

A. N. Tikhonov and V. Y. Goncharsky, Solutions of Ill-Posed Problems (Winston & Sons, 1977).

1921

A. A. Michelson and F. G. Pease, "Measurement of the diameter of Alpha Orionis with the interferometer," Astrophys. J. 53, 249 (1921).
[CrossRef]

1890

A. A. Michelson, "On the application of interference methods to astronomical measurements," London, Edinburgh Dublin Philos. Mag. J. Sci. 30, 1-21 (1890).

Angel, J. R. P.

P. M. Hinz, J. R. P. Angel, D. W. McCarthy, Jr., W. F. Hoffman, and C. Y. Peng, "The Large Binocular Telescope Interferometer," Proc. SPIE 4838, 108-112 (2003).
[CrossRef]

Baldwin, J. E.

J. E. Baldwin and C. A. Haniff, "The application of interferometry to optical astronomical imaging," Philos. Trans. R. Soc. London, Ser. A 360, 969-986 (2002).
[CrossRef]

Born, M. A.

M. A. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), pp. 508-510.

Braat, J. J. M.

Goncharsky, V. Y.

A. N. Tikhonov and V. Y. Goncharsky, Solutions of Ill-Posed Problems (Winston & Sons, 1977).

Hale, P. D.

M. Young and P. D. Hale, "Off-axis illumination and its relation to partial coherence," Am. J. Phys. 63, 1136-1141 (1995).
[CrossRef]

Haniff, C. A.

J. E. Baldwin and C. A. Haniff, "The application of interferometry to optical astronomical imaging," Philos. Trans. R. Soc. London, Ser. A 360, 969-986 (2002).
[CrossRef]

Hansen, P. C.

P. C. Hansen, "Regularization tools: a Matlab package for analysis and solution of discrete ill-posed problems," Numer. Algorithms 6, 1-35 (1994).
[CrossRef]

P. C. Hansen, "Analysis of discrete ill-posed problems by means of the L-curve," SIAM Rev. 34, 561-580 (1992).
[CrossRef]

P. C. Hansen, "The discrete Picard condition for discrete ill-posed problems," BIT 30, 658-672 (1990).
[CrossRef]

Hinz, P. M.

P. M. Hinz, J. R. P. Angel, D. W. McCarthy, Jr., W. F. Hoffman, and C. Y. Peng, "The Large Binocular Telescope Interferometer," Proc. SPIE 4838, 108-112 (2003).
[CrossRef]

Hoffman, W. F.

P. M. Hinz, J. R. P. Angel, D. W. McCarthy, Jr., W. F. Hoffman, and C. Y. Peng, "The Large Binocular Telescope Interferometer," Proc. SPIE 4838, 108-112 (2003).
[CrossRef]

Labeyrie, A.

A. Labeyrie, "Resolved imaging of extra-solar planets with future 10-100km optical interferometric arrays," Astron. Astrophys., Suppl. Ser. 118, 517-524 (1996).
[CrossRef]

McCarthy, D. W.

P. M. Hinz, J. R. P. Angel, D. W. McCarthy, Jr., W. F. Hoffman, and C. Y. Peng, "The Large Binocular Telescope Interferometer," Proc. SPIE 4838, 108-112 (2003).
[CrossRef]

Michelson, A. A.

A. A. Michelson and F. G. Pease, "Measurement of the diameter of Alpha Orionis with the interferometer," Astrophys. J. 53, 249 (1921).
[CrossRef]

A. A. Michelson, "On the application of interference methods to astronomical measurements," London, Edinburgh Dublin Philos. Mag. J. Sci. 30, 1-21 (1890).

Montilla, I.

Pease, F. G.

A. A. Michelson and F. G. Pease, "Measurement of the diameter of Alpha Orionis with the interferometer," Astrophys. J. 53, 249 (1921).
[CrossRef]

Peng, C. Y.

P. M. Hinz, J. R. P. Angel, D. W. McCarthy, Jr., W. F. Hoffman, and C. Y. Peng, "The Large Binocular Telescope Interferometer," Proc. SPIE 4838, 108-112 (2003).
[CrossRef]

Pereira, S. F.

Quirrenbach, A.

A. Quirrenbach, "Optical interferometry," Annu. Rev. Astron. Astrophys. 39, 353-401 (2001).
[CrossRef]

Saha, S. K.

S. K. Saha, "Modern optical astronomy: technology and impact of interferometry," Rev. Mod. Phys. 74, 551-600 (2002).
[CrossRef]

Tikhonov, A. N.

A. N. Tikhonov and V. Y. Goncharsky, Solutions of Ill-Posed Problems (Winston & Sons, 1977).

Visser, R.

R. Visser, "Regularization in nearfield acoustic source identification," in Proceedings of the Eighth International Congress on Sound and Vibration (2001), pp. 1637-1644.

Wolf, E.

M. A. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), pp. 508-510.

Young, M.

M. Young and P. D. Hale, "Off-axis illumination and its relation to partial coherence," Am. J. Phys. 63, 1136-1141 (1995).
[CrossRef]

Am. J. Phys.

M. Young and P. D. Hale, "Off-axis illumination and its relation to partial coherence," Am. J. Phys. 63, 1136-1141 (1995).
[CrossRef]

Annu. Rev. Astron. Astrophys.

A. Quirrenbach, "Optical interferometry," Annu. Rev. Astron. Astrophys. 39, 353-401 (2001).
[CrossRef]

Appl. Opt.

Astron. Astrophys., Suppl. Ser.

A. Labeyrie, "Resolved imaging of extra-solar planets with future 10-100km optical interferometric arrays," Astron. Astrophys., Suppl. Ser. 118, 517-524 (1996).
[CrossRef]

Astrophys. J.

A. A. Michelson and F. G. Pease, "Measurement of the diameter of Alpha Orionis with the interferometer," Astrophys. J. 53, 249 (1921).
[CrossRef]

BIT

P. C. Hansen, "The discrete Picard condition for discrete ill-posed problems," BIT 30, 658-672 (1990).
[CrossRef]

London, Edinburgh Dublin Philos. Mag. J. Sci.

A. A. Michelson, "On the application of interference methods to astronomical measurements," London, Edinburgh Dublin Philos. Mag. J. Sci. 30, 1-21 (1890).

Numer. Algorithms

P. C. Hansen, "Regularization tools: a Matlab package for analysis and solution of discrete ill-posed problems," Numer. Algorithms 6, 1-35 (1994).
[CrossRef]

Philos. Trans. R. Soc. London, Ser. A

J. E. Baldwin and C. A. Haniff, "The application of interferometry to optical astronomical imaging," Philos. Trans. R. Soc. London, Ser. A 360, 969-986 (2002).
[CrossRef]

Proc. SPIE

P. M. Hinz, J. R. P. Angel, D. W. McCarthy, Jr., W. F. Hoffman, and C. Y. Peng, "The Large Binocular Telescope Interferometer," Proc. SPIE 4838, 108-112 (2003).
[CrossRef]

Rev. Mod. Phys.

S. K. Saha, "Modern optical astronomy: technology and impact of interferometry," Rev. Mod. Phys. 74, 551-600 (2002).
[CrossRef]

SIAM Rev.

P. C. Hansen, "Analysis of discrete ill-posed problems by means of the L-curve," SIAM Rev. 34, 561-580 (1992).
[CrossRef]

Other

R. Visser, "Regularization in nearfield acoustic source identification," in Proceedings of the Eighth International Congress on Sound and Vibration (2001), pp. 1637-1644.

A. N. Tikhonov and V. Y. Goncharsky, Solutions of Ill-Posed Problems (Winston & Sons, 1977).

DARWIN The Infrared Space Interferometer: Redbook (ESA-SCI, 2000), Vol. 12.

M. A. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), pp. 508-510.

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Figures (11)

Fig. 1
Fig. 1

Sketches of four stellar interferometers as described in the literature. Note that Michelson’s experiments on stellar interferometry were performed with a configuration similar to sketch C. The label MI as used here and in the literature refers to the coaxial MI–Morley interferometer. Configuration D is commonly addressed as a Fizeau interferometer, but it should be noted that Fizeau experimented with a masked aperture; no beam relay optics were needed.

Fig. 2
Fig. 2

This illustration indicates the coordinates used in the derivation of the point-source response functions for all types of beam combination (any B out 0 ), when a general beam compression by a factor M is applied in each arm. The diffraction integral is constructed symmetrically, leading to a compact expression. The phase differences at the collection plane x are indicated with black arrows at the locations numbered 1, 2, and 3. After beam compression, the phase slope in an aperture in the plane x is proportional to x M θ in . The arrow at location 2 indicates that for proper exit pupil placement at a separation B in ( k ) M , the light paths of a masked aperture are followed, since the wavefronts in the apertures are part of a single wavefront. Generally, see indicator 3, the aperture separation after beam relay is B out , which is zero for coaxial beam combination. As a result, the wavefronts in the exit apertures, see indicator 3, are parallel to the dotted line x M θ in but have an offset. As a result, a diffraction integral can be constructed, in which the phases proportional to [ x M θ in ± 1 2 ( B in ( k ) M B out ) M θ in ] occur.

Fig. 3
Fig. 3

Staircase function h ( θ in ) , as described in the text.

Fig. 4
Fig. 4

Naïve inversion of interferometric data. Panel (a) shows a reorganized measurement vector b ¯ , resembling intensity interferograms recorded at four different baseline lengths, obtained as b ¯ = A x , where x is a positive source function L ( θ in ) with maximum amplitude 1 in arbitrary units (arb. units). The results are then photon quantized, resulting in b = b ¯ + e b . Panel (b) then shows the naive reconstruction x ̂ = ( A T A ) 1 A T b , where the pseudoinverse or Moore–Penrose inverse is used since A is not square. The result is an estimate L ̂ ( θ in ) of very high magnitude ( 10 4 instead of 1 for the source), with a large number of sign changes. Note the scale of the vertical axis causing the source to appear as a straight line at zero.

Fig. 5
Fig. 5

Singular value spectrum of a 1592 × 151 image transfer matrix A, as in Eq. (16). The condition number is cond ( A ) = 2.40 × 10 16 .

Fig. 6
Fig. 6

Singular vectors show more oscillations for higher index i. Vectors u i can be interpreted as measurement modes whereas vectors v i represent source modes. The vectors u i are plotted with offsets in the vertical direction, and only for three baselines B in ( k ) , where k = 1 , 3 , 5 .

Fig. 7
Fig. 7

L curve is a method to blindly find the optimum regularization parameter Λ. The norms A x Λ b and x Λ are plotted for a range of Λ values. The corner is found as the minimum radius of curvature in a spline fit through the calculated points. The standard two norms are taken.

Fig. 8
Fig. 8

Plot of the mode participation, expressed as u i T b σ i , for a mode with index i. The thin curve (no regularization, Λ = 0 ) shows that indeed the participation of noise-sensitive, higher index modes is strong. The thick curves show the regularized mode participation for Λ = Λ opt ( 1 10 , 1 , 10 ) .

Fig. 9
Fig. 9

Reconstructions or estimates L ̂ ( θ in ) for three regularization parameters Λ = Λ opt ( 1 10 , 1 , 10 ) , where Λ opt was automatically found using the L curve. In the figure, the size of the detector pixels in angular measure is indicated (short horizontal line), as well as the approximate size of the single-dish-diffraction envelope (long horizontal line). The coaxial MI interferometer that was simulated here, is able to recover spatial details in L ( θ in ) that are much smaller than the incoherent diffraction limit. A complete set of baselines B in ( k ) was taken.

Fig. 10
Fig. 10

(a) Comparison of the first 20 solution modes v i for the HM and the densified methods. Note how the FOV narrows down in the densified case, as the spatial frequency increases. (b) Comparison of the first 20 solution modes v i for the MI and the staircase methods. Generally, the modes are a sum of several harmonics. Note the occurrence of symmetrical and antisymmetrical modes, e.g., in the staircase modes 5 and 6.

Fig. 11
Fig. 11

Normalized SVs σ i σ 1 for the four interferometry methods. A flat line up to the cutoff induced by the maximum baseline B in , max ( i = 100 ) indicates a proper transfer of all spatial frequencies. See text for details.

Tables (2)

Tables Icon

Table 1 Characterization of the Beam Combiner Optics in the Interferometers under Consideration a

Tables Icon

Table 2 Discretization of the ( θ out , d ) Detection Space for the Simulations a

Equations (28)

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A ( k ) ( θ out , d ) = ( 1 2 ) B out ( 1 2 ) D M ( 1 2 ) B out + ( 1 2 ) D M e i ( 2 π x θ out λ ) M exp { i ( 2 π [ x M θ in + 1 2 ( B in ( k ) M B out ) M θ in ] λ ) } d x + e i ( 2 π d λ ) ( 1 2 ) B out ( 1 2 ) D M ( 1 2 ) B out + ( 1 2 ) D M e i ( 2 π x θ out λ ) M exp { i ( 2 π [ x M θ in 1 2 ( B in ( k ) M B out ) M θ in ] λ ) } d x .
sinc ( x ) = { 1 for x = 0 sin ( x ) x , otherwise ,
I k ( θ out ) = 4 D 2 sinc 2 [ D π M λ ( θ out M θ in ) ] cos 2 [ B in ( k ) π M λ ( θ out M θ in ) ] ,
I k ( θ out , d ) = 4 D 2 sinc 2 [ D π M λ ( θ out M θ in ) ] cos 2 [ π λ ( d + B out θ out B in ( k ) θ in ) ] .
I k ( θ out , d ) = 4 D 2 sinc 2 [ D π M λ ( θ out M θ in ) ] cos 2 [ π λ ( d B in ( k ) θ in ) ] .
I k ( θ out , d ) = 4 D 2 sinc 2 [ D π M λ ( θ out M θ in ) ] cos 2 [ π λ ( d B in ( k ) θ in + h ( θ in ) ) ] ,
h ( θ in ) = n = + n h 0 rect [ θ in n θ s θ s ] = n = + n h 0 [ H { θ in ( n 1 2 ) θ s θ s } H { θ in ( n + 1 2 ) θ s θ s } ] ,
H ( x ) = { 0 , x < 0 1 2 , x = 0 1 , x > 0 .
θ in = ( θ in ( 1 ) , θ in ( 2 ) , , θ in ( n ) , , θ in ( N ) ) ,
d = ( d 1 , d 2 , , d p , , d P ) ,
θ out = ( θ out ( 1 ) , θ out ( 2 ) , , θ out ( j ) , , θ out ( J ) )
I j k ( d p ) = g [ L ( θ in ) , B in ( k ) , θ out ( j ) ] ,
L ̂ ( θ in ) = g 1 [ I j k ( d p ) ] ,
b = ( I j 1 ( d ) I j 2 ( d ) I j k ( d ) ) ,
B in ( k ) = { 2 D , k = 1 4 ( k 1 ) D , k = 2 , 3 , , 9 .
cos [ π B out λ ( θ out [ B in B out θ in ] + d B out ) ] = 1 for θ in = θ in ( max ) .
d = B out θ in ( 1 B in B out ) ,
b = A x ,
A = U S V T = i u i σ i v i T .
U = [ u 1 u 2 u min ( m , n ) ] , V = [ v 1 v 2 v min ( m , n ) ] .
cond ( A ) = σ 1 σ min ( m , n ) .
b = U S V T x , or b = i σ i ( v i T x ) u i .
x LS = i u i T b σ i v i ,
x k = i = 1 k u i T b σ i v i , with k min ( m , n ) .
min x { A x b 2 + Λ 2 L x 2 } ,
x Λ = i u i T b σ i σ i 2 + Λ 2 v i .
A x Λ b , x Λ ,
x Λ = i = 1 n u i T b σ i σ i 2 + Λ 2 v i ,

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