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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References

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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 (1)

2003 (1)

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 (2)

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 (2)

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

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

2000 (1)

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

1996 (1)

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 (1)

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

1994 (1)

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 (1)

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

1990 (1)

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

1980 (1)

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

1977 (1)

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

1921 (1)

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

1890 (1)

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. (1)

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. (1)

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

Appl. Opt. (1)

Astron. Astrophys., Suppl. Ser. (1)

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. (1)

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

BIT (1)

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. (1)

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 (1)

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 (1)

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 (1)

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. (1)

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

SIAM Rev. (1)

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

Other (4)

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

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

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.

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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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