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Triple correlation as a phase closure technique
Volume 60, number 3
OPTICS COMMUNICATIONS
1 November 1986
TRIPLE CORRELATION AS A PHASE CLOSURE TECHNIQUE Francois RODDIER National Optical Astronomy Observatories ~,Advanced Development Program, P.O. Box 26732, Tucson, AR, USA
Received 16 June 1986;revisedmanuscript received4 August 1986
A simple heuristic model is used to discuss the relationshipbetween triple correlationand phase closure in interferometric image reconstruction.The main propertiesof triple correlationare easilyrederivedgivingdeeper insight into the method.
1. Introduction N
Since Labeyrie's discovery of stellar speckle interferometry [ 1 ], many techniques have been proposed in order to fully reconstruct an image through turbulence. One of these, speckle masking [2], has recently been given strong theoretical support in terms of triple correlation theory or bispectral analysis [ 3-6 ]. Here, we describe this technique in terms of aperture synthesis theory. A simple heuristic model is used to describe the effect of atmospheric turbulence on image formation. Using this model, the main properties of the technique are easily rederived giving deeper insight into the method and its relationship to phase closure.
2. Image formation as an interference process
Let us consider a telescope aperture as made of N elementary sub-apertures. Each sub-aperture is taken small enough for the field to be coherent over its extent. According to diffraction theory, the complex amplitude ~ of the field in the telescope focal plane, assumed to be monochromatic, is obtained by adding the contribution ~k of each subaperture k and the associated illumination is
2
N
~'/k 2
The first term on the right hand side of eq. (1) is the sum of the irradiances produced by each sub-aperture. It contains no high angular resolution information. The second term is a sum of cross products describing interference. Eq. (1) shows that an image can be reconstructed from independent measurements of all the cross products using pairs of subapertures. This is called aperture synthesis. Since a pair of sub-apertures produces a fringe pattern, the image produced by the whole aperture can be viewed as a sum of fringe patterns. Through diffraction limited optics, pairs of subapertures with identical spacing and orientation produce identical fringe patterns or identical sinusoidal terms which add in phase to produce the image Fourier component at the fringe spatial frequency f The number N(J) of such pairs is called pupil redundancy. It is proportional to the overlap area of two pupil images shifted apart by that particular spacing. In the image, each object Fourier component is weighted by the pupil redundancy so that the image Fourier transform f(f) is related to the object Fourier transform O(f) by [(f) = O(f). T(./),
(2)
where the transfer function T(]) is given by Operated by the Association of Universities for Research in Astronomy, Inc., under contract with the National Science Foundation. 0 030-401/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
T ( f ) = N ( f ) / N( O) ,
(3) 145
Volume 60. number 3
OPTICS COMMUNICATIONS
the well known two-pupil overlap area normalized to unity at the origin.
Although it is not necessary [7], for the sake of simplicity we assume that atmospheric turbulence produces only phase perturbations on the telescope aperture. We assume that the random phase delays are roughly uniform over areas of linear size ro and we divide the telescope pupil into sub-apertures of that size. Let 0k be the phase delay introduced by turbulence on sub-aperture k. The complex amplitude ~k produced in the image plane by sub-aperture k is multiplied by exp(i0k). We further assume that the random phasors exp(i0D have zero mean and are statistically independent. The interference term ~k~u* is multiplied by exp(i0) where 0 = 0 k - 0 / describes a random shift of the fringe pattern produced by the two sub-apertures. Pairs of sub-apertures with the same spacing and orientation now produce fringe patterns with the same amplitude, spacing and orientation but randomly shifted. Adding these patterns still gives fringes with the same spacing and orientation but with a lower and random amplitude. The amplitude is obtained by adding the random phasors associated with each sub-aperture pair. In other words, the image Fourier transform [(f) is now related to the object Fourier transform O(f) by (4)
where the transfer function ~(f) is now random and given by ~¢(y)_
1
N(0)
NOr)
p~'~=lexp(i0p).
(5)
The sum extends over all sub-aperture pairs contributing to the image Fourier component f Since this is a random walk problem (sum of independent random phasor-= with zero mean), on the average the resulting magnitude I~(f) I grows only as the square r o o t Nl/2(f) of the number N(f) of terms. Hence the average energy spectrum ( I ~ ( f ) [ 2 ) is proportional to N(f) instead ofN2(f). 146
4. Speckle interferometry Speckle interferometry consists in estimating the image average energy spectrum ( I / ( f ) 12) from a sequence of short exposures (short enough to freeze the effect of turbulence). Taking the average squared modulus of eq. (4) gives
3. The effect of atmospheric turbulence on image formation
[¢f) = 0(f).,~(f),
1 November 1986
< lift)12 ) = 10(f)I e'(IStf) 12>,
(6)
and taking (5) into account, the so-called speckle transfer function ( I S ( f ) 12 ) can be written ( I S ( f ) [ 2 > - N 2 ( 0 ) x),,=, N(I) -
N2(0)
1
-
- N(0)
(7)
Tq)
This relation holds only when sub-aperture separations are larger than ro i.e. at frequencies higher than the so-called seeing cut-off. It was first derived by Korff et al. [ 8 ] using this stochastic model. It shows that the speckle transfer function extends up to the telescope diffraction cut-off frequency. The number N(0) of areas of size ro in the telescope pupil is also the number of speckle in a point source image. Eq. (7) also shows that the speckle transfer function is seeing dependent and must be calibrated on a reference source under identical seeing conditions. Its magnitude varies as the inverse of the number of speckles in the image. Speckle interferometry allows to recover the magnitude of the object Fourier transform but the phase is lost.
5. Triple correlation Triple correlation is a method to recover the phase information. It consists in computing the third order moment ( [ ( f l ) [(I"2)/*0r~ +f2) ) called bi-spectrum. Using eq. (4), the image bi-spectrum can be written as a product of the object bi-spectrum times a transfer function we shall call the triple correlation transfer function
= 0 ~ e, ) ¢ ~ 2 ) 0 " 0 e, +f2)
x (~¢¢J'1) g(/:) ~*~ +f2)>.
(8)
Volume60, number 3
OPTICS COMMUNICATIONS
From eq. (5) we obtain an expression for the transfer function (~(.fl) ~(f2 ) ,~*(fl +f2 ) )
1
E r=l
(9)
where 0p, 0q, 0r are the phase shifts of the fringes at frequency f~, f2, and fl +f2 produced by three different pairs of sub-apertures. The sum is extended over all possible triplets of such pairs. Most of the terms in the sum involve six different sub-apertures. In some cases one sub-aperture may belong to two different pairs. In most cases at least one sub-aperture belongs to only one pair. If this is the case, the exp(i0) factor associated with this particular sub-aperture will average to zero independently of the others and the resulting contribution of the pair triplet will be zero. The only case where this does not occur is when each sub-aperture belongs to two pairs, i.e. when the three sub-aperture pairs form a close loop of three sub-apertures. Let k, l, m denote these three sub-apertures and let Ok, 01, Ore, denote the phase delays introduced by turbulence on each of them. We have
Op=Ol-Ok ,
(10a)
Oq =0,, - 0 l ,
(10b) (10c)
,
so that
~p + 0 q - 0 r = 0 .
(ll)
This is the usual phase closure relation in which turbulence terms cancel out. The contribution of such a triplet in eq. (9) is (exp(iOp) exp(iOq) exp(-iOr) ) = ( exp[i(Op+Oq-Or)]
) ~*~ +A) > - N(f,,f~)
(exp(i0~)
×exp(i0q) e x p ( - i 0 r ) ) ,
--Or=Ok --Om
to the overlap area of three pupil images shifted apart by the same spacings f~, f2, and f~ +f2. The transfer function for triple correlation is therefore equal to
<~(f, ) ~
N(fl)N(.f2) N(fl-t-f2)
-N~(0) pE E =l q=l
1 November 1986
) =
1.
(12)
The number of such terms in eq. (9) is equal to the number N ~ , f2) of such triplets of sub-apertures in the telescope pupil. For each pair of frequency vectors (f,, f2) there are two possible triplet configurations (with 180 ° rotation symmetry) and for each configuration the number of triplets is proportional
N3(0)
1 - N2(0 ) [A,(f,,f2) + A 2 ( f , , f 2 ) ] ,
(13)
where A, ~ , f 2 ) and A2(f,,f2) are the overlap area of the three pupil images in each configuration normalized to unity at the origin (they are equal if the pupil has a center of symmetry).
6. Properties of triple correlation Eq. (l 3) generalizes eq. (7) to triple correlation analysis. It holds only when sub-aperture separations are larger than ro, i.e. when fl,f2, andf~ +f2 are all larger than the seeing cut-off frequency. It shows that the triple correlation transfer function extends in the (fl, f2) space up to the limits imposed by fl, f2 to be all below the telescope diffraction cut-off frequency. Its magnitude is seeing dependent and equal to zero. From eq. (8) we conclude that the phase of the image bi-spectrum is equal to the phase of the object bi-spectrum. Since the phase of the object can be recovered, but for a linear term, from the phase of its bi-spectrum by solving the phase closure equations [5 ], triple correlation does solve the phase problem encountered in classical speckle interferometry. The transfer function for triple correlation has zero phase independently of seeing conditions. Therefore recovery of the object phase does not require calibration on a reference source. Since use is made of the phase closure relations (11 ), this is true even,if the telescope produces permanent wavefront aberrations. Indeed let 0~, represent a permanent phase delay on sub-aperture k. The phasor exp[i(0k+0~)] still has zero mean (exp [i(O~ + 0~,)] ) = (exp(iOk)) exp(iO~) = 0
(14)
and all the terms in (9) will still cancel out except the closure terms where the permanent phase terms disappear. We therefore conclude that, at least beyond 147
Volume 60, number 3
OPTICS COMMUNICATIONS
the seeing cut-off frequency, the phase o f the image bi-spectrum is insensitive to telescope aberrations. The above relations hold whatever the shape o f the telescope pupil is. They apply to a diluted aperture as well. W h e n the aperture reduces to a non-redundant set o f small apertures o f linear size ro, the speckle transfer function ( 9 ) m a y contain only phase closure terms in which case it is no longer r a n d o m and the closure phases can be o b t a i n e d from a single outcome o f the process ( p r o v i d e d that the S N R is good enough). This is the case when a non-redundant mask with three holes is used as recently done by Baldwin et al. [9]. The two techniques are then strictly equivalent. Triple correlation can therefore be considered as a generalization o f phase closure in optics. It extends the technique to any k i n d o f aperture r e d u n d a n t or not.
7. Conclusion We have been able to rederive the m a i n properties o f triple correlation using a simple m o d e l to describe
148
1 November 1986
the effect o f a t m o s p h e r i c turbulence on image formation. Triple correlation appears to be a generalisation o f phase closure in optics. It should allow recovery o f the object phases without reference sources even through a b e r r a t e d optics.
References [ 1] A. Labeyrie, Astron. Astrophys 6 (1970) 85. [2] G.P. Weigelt, Optics Comm. 21 (1977) 55. [3] G. Weigelt and B. Wirnitzer, Optics Lett. 8 (1983) 389. [4] A.W. Lohmann, G. Weigelt and B. Wirnitzer, Appl. Optics 22 (1983) 4028. [ 5 ] H. Bartelt, A.W. Lohmann and B. Wirnitzer, Appl. Optics 23 (1984) 3121. [6] B. Wirnitzer, J. Opt. Soc. Am. A 2 (1984) 14. [ 7 ] F. Roddier, Progress in Optics 19 ( 1981 ) 315. [8] D. Korff, G. Dryden and M.G. Miller, Optics Comm. 5 (1972) 187. [9] J.E. Baldwin, C.A. Haniff, C.D. Mackay and P.J. Warner, Nature 320 (1986) 595.
OPTICS COMMUNICATIONS
1 November 1986
TRIPLE CORRELATION AS A PHASE CLOSURE TECHNIQUE Francois RODDIER National Optical Astronomy Observatories ~,Advanced Development Program, P.O. Box 26732, Tucson, AR, USA
Received 16 June 1986;revisedmanuscript received4 August 1986
A simple heuristic model is used to discuss the relationshipbetween triple correlationand phase closure in interferometric image reconstruction.The main propertiesof triple correlationare easilyrederivedgivingdeeper insight into the method.
1. Introduction N
Since Labeyrie's discovery of stellar speckle interferometry [ 1 ], many techniques have been proposed in order to fully reconstruct an image through turbulence. One of these, speckle masking [2], has recently been given strong theoretical support in terms of triple correlation theory or bispectral analysis [ 3-6 ]. Here, we describe this technique in terms of aperture synthesis theory. A simple heuristic model is used to describe the effect of atmospheric turbulence on image formation. Using this model, the main properties of the technique are easily rederived giving deeper insight into the method and its relationship to phase closure.
2. Image formation as an interference process
Let us consider a telescope aperture as made of N elementary sub-apertures. Each sub-aperture is taken small enough for the field to be coherent over its extent. According to diffraction theory, the complex amplitude ~ of the field in the telescope focal plane, assumed to be monochromatic, is obtained by adding the contribution ~k of each subaperture k and the associated illumination is
2
N
~'/k 2
The first term on the right hand side of eq. (1) is the sum of the irradiances produced by each sub-aperture. It contains no high angular resolution information. The second term is a sum of cross products describing interference. Eq. (1) shows that an image can be reconstructed from independent measurements of all the cross products using pairs of subapertures. This is called aperture synthesis. Since a pair of sub-apertures produces a fringe pattern, the image produced by the whole aperture can be viewed as a sum of fringe patterns. Through diffraction limited optics, pairs of subapertures with identical spacing and orientation produce identical fringe patterns or identical sinusoidal terms which add in phase to produce the image Fourier component at the fringe spatial frequency f The number N(J) of such pairs is called pupil redundancy. It is proportional to the overlap area of two pupil images shifted apart by that particular spacing. In the image, each object Fourier component is weighted by the pupil redundancy so that the image Fourier transform f(f) is related to the object Fourier transform O(f) by [(f) = O(f). T(./),
(2)
where the transfer function T(]) is given by Operated by the Association of Universities for Research in Astronomy, Inc., under contract with the National Science Foundation. 0 030-401/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
T ( f ) = N ( f ) / N( O) ,
(3) 145
Volume 60. number 3
OPTICS COMMUNICATIONS
the well known two-pupil overlap area normalized to unity at the origin.
Although it is not necessary [7], for the sake of simplicity we assume that atmospheric turbulence produces only phase perturbations on the telescope aperture. We assume that the random phase delays are roughly uniform over areas of linear size ro and we divide the telescope pupil into sub-apertures of that size. Let 0k be the phase delay introduced by turbulence on sub-aperture k. The complex amplitude ~k produced in the image plane by sub-aperture k is multiplied by exp(i0k). We further assume that the random phasors exp(i0D have zero mean and are statistically independent. The interference term ~k~u* is multiplied by exp(i0) where 0 = 0 k - 0 / describes a random shift of the fringe pattern produced by the two sub-apertures. Pairs of sub-apertures with the same spacing and orientation now produce fringe patterns with the same amplitude, spacing and orientation but randomly shifted. Adding these patterns still gives fringes with the same spacing and orientation but with a lower and random amplitude. The amplitude is obtained by adding the random phasors associated with each sub-aperture pair. In other words, the image Fourier transform [(f) is now related to the object Fourier transform O(f) by (4)
where the transfer function ~(f) is now random and given by ~¢(y)_
1
N(0)
NOr)
p~'~=lexp(i0p).
(5)
The sum extends over all sub-aperture pairs contributing to the image Fourier component f Since this is a random walk problem (sum of independent random phasor-= with zero mean), on the average the resulting magnitude I~(f) I grows only as the square r o o t Nl/2(f) of the number N(f) of terms. Hence the average energy spectrum ( I ~ ( f ) [ 2 ) is proportional to N(f) instead ofN2(f). 146
4. Speckle interferometry Speckle interferometry consists in estimating the image average energy spectrum ( I / ( f ) 12) from a sequence of short exposures (short enough to freeze the effect of turbulence). Taking the average squared modulus of eq. (4) gives
3. The effect of atmospheric turbulence on image formation
[¢f) = 0(f).,~(f),
1 November 1986
< lift)12 ) = 10(f)I e'(IStf) 12>,
(6)
and taking (5) into account, the so-called speckle transfer function ( I S ( f ) 12 ) can be written ( I S ( f ) [ 2 > - N 2 ( 0 ) x),,=, N(I) -
N2(0)
1
-
- N(0)
(7)
Tq)
This relation holds only when sub-aperture separations are larger than ro i.e. at frequencies higher than the so-called seeing cut-off. It was first derived by Korff et al. [ 8 ] using this stochastic model. It shows that the speckle transfer function extends up to the telescope diffraction cut-off frequency. The number N(0) of areas of size ro in the telescope pupil is also the number of speckle in a point source image. Eq. (7) also shows that the speckle transfer function is seeing dependent and must be calibrated on a reference source under identical seeing conditions. Its magnitude varies as the inverse of the number of speckles in the image. Speckle interferometry allows to recover the magnitude of the object Fourier transform but the phase is lost.
5. Triple correlation Triple correlation is a method to recover the phase information. It consists in computing the third order moment ( [ ( f l ) [(I"2)/*0r~ +f2) ) called bi-spectrum. Using eq. (4), the image bi-spectrum can be written as a product of the object bi-spectrum times a transfer function we shall call the triple correlation transfer function
x (~¢¢J'1) g(/:) ~*~ +f2)>.
(8)
Volume60, number 3
OPTICS COMMUNICATIONS
From eq. (5) we obtain an expression for the transfer function (~(.fl) ~(f2 ) ,~*(fl +f2 ) )
1
E r=l
(9)
where 0p, 0q, 0r are the phase shifts of the fringes at frequency f~, f2, and fl +f2 produced by three different pairs of sub-apertures. The sum is extended over all possible triplets of such pairs. Most of the terms in the sum involve six different sub-apertures. In some cases one sub-aperture may belong to two different pairs. In most cases at least one sub-aperture belongs to only one pair. If this is the case, the exp(i0) factor associated with this particular sub-aperture will average to zero independently of the others and the resulting contribution of the pair triplet will be zero. The only case where this does not occur is when each sub-aperture belongs to two pairs, i.e. when the three sub-aperture pairs form a close loop of three sub-apertures. Let k, l, m denote these three sub-apertures and let Ok, 01, Ore, denote the phase delays introduced by turbulence on each of them. We have
Op=Ol-Ok ,
(10a)
Oq =0,, - 0 l ,
(10b) (10c)
,
so that
~p + 0 q - 0 r = 0 .
(ll)
This is the usual phase closure relation in which turbulence terms cancel out. The contribution of such a triplet in eq. (9) is (exp(iOp) exp(iOq) exp(-iOr) ) = ( exp[i(Op+Oq-Or)]
) ~*~ +A) > - N(f,,f~)
(exp(i0~)
×exp(i0q) e x p ( - i 0 r ) ) ,
--Or=Ok --Om
to the overlap area of three pupil images shifted apart by the same spacings f~, f2, and f~ +f2. The transfer function for triple correlation is therefore equal to
<~(f, ) ~
N(fl)N(.f2) N(fl-t-f2)
-N~(0) pE E =l q=l
1 November 1986
) =
1.
(12)
The number of such terms in eq. (9) is equal to the number N ~ , f2) of such triplets of sub-apertures in the telescope pupil. For each pair of frequency vectors (f,, f2) there are two possible triplet configurations (with 180 ° rotation symmetry) and for each configuration the number of triplets is proportional
N3(0)
1 - N2(0 ) [A,(f,,f2) + A 2 ( f , , f 2 ) ] ,
(13)
where A, ~ , f 2 ) and A2(f,,f2) are the overlap area of the three pupil images in each configuration normalized to unity at the origin (they are equal if the pupil has a center of symmetry).
6. Properties of triple correlation Eq. (l 3) generalizes eq. (7) to triple correlation analysis. It holds only when sub-aperture separations are larger than ro, i.e. when fl,f2, andf~ +f2 are all larger than the seeing cut-off frequency. It shows that the triple correlation transfer function extends in the (fl, f2) space up to the limits imposed by fl, f2 to be all below the telescope diffraction cut-off frequency. Its magnitude is seeing dependent and equal to zero. From eq. (8) we conclude that the phase of the image bi-spectrum is equal to the phase of the object bi-spectrum. Since the phase of the object can be recovered, but for a linear term, from the phase of its bi-spectrum by solving the phase closure equations [5 ], triple correlation does solve the phase problem encountered in classical speckle interferometry. The transfer function for triple correlation has zero phase independently of seeing conditions. Therefore recovery of the object phase does not require calibration on a reference source. Since use is made of the phase closure relations (11 ), this is true even,if the telescope produces permanent wavefront aberrations. Indeed let 0~, represent a permanent phase delay on sub-aperture k. The phasor exp[i(0k+0~)] still has zero mean (exp [i(O~ + 0~,)] ) = (exp(iOk)) exp(iO~) = 0
(14)
and all the terms in (9) will still cancel out except the closure terms where the permanent phase terms disappear. We therefore conclude that, at least beyond 147
Volume 60, number 3
OPTICS COMMUNICATIONS
the seeing cut-off frequency, the phase o f the image bi-spectrum is insensitive to telescope aberrations. The above relations hold whatever the shape o f the telescope pupil is. They apply to a diluted aperture as well. W h e n the aperture reduces to a non-redundant set o f small apertures o f linear size ro, the speckle transfer function ( 9 ) m a y contain only phase closure terms in which case it is no longer r a n d o m and the closure phases can be o b t a i n e d from a single outcome o f the process ( p r o v i d e d that the S N R is good enough). This is the case when a non-redundant mask with three holes is used as recently done by Baldwin et al. [9]. The two techniques are then strictly equivalent. Triple correlation can therefore be considered as a generalization o f phase closure in optics. It extends the technique to any k i n d o f aperture r e d u n d a n t or not.
7. Conclusion We have been able to rederive the m a i n properties o f triple correlation using a simple m o d e l to describe
148
1 November 1986
the effect o f a t m o s p h e r i c turbulence on image formation. Triple correlation appears to be a generalisation o f phase closure in optics. It should allow recovery o f the object phases without reference sources even through a b e r r a t e d optics.
References [ 1] A. Labeyrie, Astron. Astrophys 6 (1970) 85. [2] G.P. Weigelt, Optics Comm. 21 (1977) 55. [3] G. Weigelt and B. Wirnitzer, Optics Lett. 8 (1983) 389. [4] A.W. Lohmann, G. Weigelt and B. Wirnitzer, Appl. Optics 22 (1983) 4028. [ 5 ] H. Bartelt, A.W. Lohmann and B. Wirnitzer, Appl. Optics 23 (1984) 3121. [6] B. Wirnitzer, J. Opt. Soc. Am. A 2 (1984) 14. [ 7 ] F. Roddier, Progress in Optics 19 ( 1981 ) 315. [8] D. Korff, G. Dryden and M.G. Miller, Optics Comm. 5 (1972) 187. [9] J.E. Baldwin, C.A. Haniff, C.D. Mackay and P.J. Warner, Nature 320 (1986) 595.