Readhead, A. C. S.

Caltech Authors - Main

A cm1PARISON OF THE DETERMINATION OF CLOSL'RE PHASE IN OPTICAL 
:NTERFEROMETRY WITH FL1.LY FILLED APERTURES AND NON-REDUNDANT 
APERTURE MASKS 
1. INTRODUCTION 
A.C.S. Readhead 
Owens Valley Radio Observatory 
California Institute of Technology 
The recent successes in diffraction limited optical imaging from · 
the ground by Weigelt et al. <1986> and Baldwin et al. <1986> rely on 
the use of the closure phase - i.e. the sum of the visibility phases 
around a closed loop of <usually three> baselines <Jennison 1953, 
1958>. The closure phase has been used for the last dozen years in 
VLBI for making diffraction limited images in situations where the 
observed phase of the the incoming wavefront is badly corrupted by 
propagation and instrumental effects <Rogers et al. 1974; Readhead and 
Wilkinson 1978; Readhead 1980; Cornwell and Wilkinson 1981; Schwab 
1980 ; Pearson.and Readhead 1984>. 
The approach adopted by Baldwin et al. is a direct carry-over of 
the methods of VLBI - a non-redundant mask is used to transmit light 
from a few patches on the primary mirror of a large telescope, each 
patch being coherently illuminated and therefore analogous to the 
independent telescopes used in VLBI. The approach of Weigelt et al. 
makes use of the whole aperture. They form the bispectrum, or triple 
product, from speckle images and use this to make the image <Lohmann et 
al. 1983>. The bispectrum is the product of the complex visibility 
function at three points in the <u,v> plane which form a closed set of 
baseline vectors, and the phase of the triple product is therefore the 
closure phase <Cornwell 1987 > plus a noise term which is introduced by 
the redundant baselines, as described below. 
The object structure information obtained by these two approaches 
is therefore the visibility amplitude and the closute phase. What 
makes these methods different and more powerful than previous 
techniques employed in optical interferometry is the use of the closure 
phase. 
Since the closure phase is the basic new piece of information that 
is used in both of these methods, we will avoid the unfortunate and 
misleading apellations "bi-spectrum" and "closure phase" which are 
sometimes used to distinguish between these two approaches. The real 
distinction is clearly between the use of 'fully filled apertures' and 
'non-redundant masks', and we therefore use these terms. 
143 
This paper presents an analysis of the effects of the redundant 
baselines on closure phase measurements in the 'fully filled aperture' 
approach for the case of many photons , i.e. ignoring the effects of 
photon noise, whi ch provides some useful insights into the effects of 
redundant baselines which exist in both the high- a nd low-photon rate 
cases, and which therefore must be taken into account in the crit ical 
case of low photon rates. 
2. THE MEASl:REMENT OF CLOSURE PHASE IN OPTICAL INTERFEROMETRY 
In the following discuss i on we first conside r (in S2.1) the 
measurement of closure phase with a non-redundant 3-hole aperture mask, 
i.e. the technique used by Baldwin et al. We determine the effect on 
the closure phase of integrating for longer than the coherence time. 
We show that the use of the closure phase, in addi tion to making it 
possible to determine the object visibi lity phase, actually extends the 
effective coherence time for individual observations. It is important 
to understand this distinction - the closure phase does, of course, 
effectively enable us to make coherent observations for periods of 
arbitrary duration <typically 12 hours in VLBI> by using the object as 
its own phase reference; in addition the coherence of individual raw 
observations is extended, as shown be low , because of the cancellation 
of some phase terms. 
The results of this analysis in the time domain are then applied 
<in SZ.2> i n the spatial domain to the case of fully filled apertures . 
2.1 Non-Redundant Masks 
We begi n by considering the f ormation of an intensity fringe 
pattern by interference of light from three elementary areas i,j,k. Let 
the amplitudes and phases of the intensity patterns corresponding to 
the three intersecting fringe patterns thus formed be represented by 
the vectors A1jei+1j, Ajkei+jk and Ak 1ei•ki on baselines ij,jk and ki, 
respectively. ' 
Consider an observation for which the integration time, ~. is 
significantly longer than the coherence time, ~coh · For simplicity, and 
without loss of generality, we assume that the fringe pattern is frozen 
for intervals ~0 ~ ~coh• with random phase changes of order one radian 
at each of the three apertures between each interval ~0 • and that ~ = 
n~0 . We will examine the intensity pattern obtained in such a 
situation . The Fourier Transform of the intensity pattern arising from 
baseline ij for an observation of duration ~ is simply the vector sum: 
A ·· ei<•1J· ). 1Jn n ' 
where the subscripts 1 , 2 n refer to the 1st, 2nd . . nth 
144 
interval of duration 1:0 • 
and ki is: 
The triple product o n the baselines ij, jk 
(Aij 1ei$ij 1 + .. + Aijnei4>ijn) (Ajk 1ei$jk1 + .. + Ajknei4>jkn)x 
(Aki1 ei4>ki1 + .. + Akinei4>kin) .. .. 
This contains terms of three types <Readhead et al . 1987> : 
( 1) 
1 > There are n combinations of complex visibilities over the same 
interval, which give us the closure phase, and which we will call 
'terms of the first kind'. 
2 > There are 3nC n-1 > combinations involving cross terms from two 
intervals, which we call 'terms of the second kind'. These terms of the 
second kind can be combined in complementary pairs for which the 
resultants either have phase equal to the closure phase or they differ 
from the closure phase by 11. Combining 3n< n-1 > /2 of these pairs 
results in a vector of rms length [3n<n-1>J1/2 at angle 4>ijk or cl>ijk+ 
r . When the terms of the second kind are added to those of the first 
kind they therefore produce no change in the closure phase or they 
produce a change of n. 
3 > There are nC n- 1 > < n-2 > combinations of three intervals. These terms 
have random phases , and therefore they add incoherently. They introduce 
a noise ~erm into the triple product, and corrupt the closure phases of 
the triple p~oduct . We call these 'terms of the third kind'. 
For the sake of the present discussion we will assume that all the 
amplitudes, Aij• are equal and set these equal to unity, i.e. we assume 
that the scintillation is negligible. The bispectrum then consists of 
a coherent <constant phase> term of length n, and phase 4>iJ·k; a term of 1/ 2 phase 4>ijk or 4>ijk+ n and rms length [3n<n-1>J ; and a term of random 
phase a nd rms length [n <n-1><n-2 >J 112 . Note that if only two intervals 
are considered, i.e. if n = 2, there are no terms of the third kind, so 
that the noise in the closure phase is not increased by the longer 
integration time, but it will be wrong by TI 217. of the time. 
Apart from this possible uncertainty of n in the 'closure phase , the 
signal-to-noise-ratio is n/[n(n-1><n-2 >J112 , so that for a signal-to-
noise-ratio of unity, corresponding roughly to an rms error in the 
measured closure phase of 1 r adian, nc = 3 . 41. While for an rms 
closure phase error of 104°, th'e rms value of a random variable with 
uniform distribution between -11 and n, nc ,., 6. Thus the use of the 
closure phase effectively extends the coherence time by a significant 
factor, the reason for this extension in coherence time being the 
cancellation of the terms of the second kind. 
In order to me asure the closure phase we have to take many 
exposures and aaa the bispectra. Since the closure phase is preserved, 
the terms of the first kind add coherent ly, while those of the second 
145 
kind add incoherently and affect the amplitude of the coherent signal. 
Thus, after summing N exposures we have the situation illustrated in 
Figure 1. Note that the amplitude of the fixed phase term is Nn ± 
[N3n< n-1> J 112; while the rms amplitude of the incoherent random phase 
term is [Nn<n-1><n- 2>J 112 . This shows that it is possible to extract 
closure phase information from observations which extend to much longer 
than ~coh• but that the number of frames needed to reduce the rms error 
in the closure phase to 1/x radians is N = xZ<n-3+2/n>. Ideally, of 
course, one would observe for ~ < ~coh and so eliminate these noise 
terms altogether. 
2.2 Fully Filled Apertures 
The above analysis can also be applied in the spatial domain. 
Consider the intensity pattern produced by a fully filled aperture. 
There are many identical triangles of elementary coherence areas, and 
the triple product of the complex visibility of these redundant 
triangles is again given by <1>. In the case of 'frozen turbulence', 
i.e. under the Taylor hypothesis, there is no difference between 
combining the interference fringes from a single triplet of elementary 
areas over a period of time and combining the fringes from a number of 
spatially distinct identical triplets at a single instant. 
The presence of ihstantaneously redundant baselines thus gives rise 
to a noise term very similar to that which is found in integrations 
which exceed the coherence time and which can be eliminated only by 
covering the aperture with a non-redundant aperture mask. This is 
because the point spread function of a non-redundant mask is unity, but 
the presence of redundant baselines destroys the coherence forcing the 
point spread function to depart from unity. It is clear from this 
analysis that the full aperture has some very undesirable properties 
which can be avoided by using masks, and this is therefore the 
preferred approach in observations with high photon rates. It is not 
quite so clear which approach is best in the case of low photon rates, 
but as this is not the subject of the present paper it will be 
discussed fully elsewhere . 
In most situations ther are some baselines which are not matched 
one for one with baselines on the other two sides of the triangle. 
Suppose that there are n redundant triangles, and in addition there are 
p baselines iJ, q baselines jk, and r baselines ki which are not 
matched on the other two sides. Then the triple product is: 
+ A· . eitijn + A·. eitijn+i + ... 1 Jn 1 Jn+1 + 
.+ Aijn+peitijn+p ) (Ajk1ei•jk1 + .. + Ajknei•jkn + Ajkn+1 ei•jkn+1 
+ 
.+ Ajkn+qei¢jkn+q )x (Aki 1ei¢ki 1 + . . + Akinei•kin + 
146 
Thus now, in addition to the terms of the first three kinds discussed 
above, there are terms of the kind 
A;J· AJ·k Aki ei<<l>ijn+s+ <l>jkq+ <l>kit>, etc. which we shall call 
- n+s q t 
'terms of the fourth kind'. There are <p+q+r)n2 + <pq+qr+rp>n + pqr 
terms of the fourth kind. The inco herent or noise term in the triple 
product t hus has an rms amplitude of 
[n3+<p+q+r-3) n2 + <pq+qr+rp+2>n + pqrJ112, and in order to measure the 
closure phase to within an rms error of 1/x radians we require 
N > x2[n + <p+q+r-3> + <pq+qr+rp+2>/n + pqr/n2J. 
The bi-spectrum approach works because the closure phase is 
preserved by the terms of the first kind . It is only because there are 
some closed triang~es that any information about the phase is 
recoverable. 
In the case of a filled aperture there are triangles for which 
n >> p and triangles for which p >> n, etc., so the number of exposures 
required for . a given signal- to-noise-ratio can be set by eithe r of 
these. 
We consider a square ape r ture and assume, for convenience, that the 
coherent patches are also square <see Figure 2>, and we also assume, to 
begin with , that the phase d i str ibution across the aperture can be 
approximated by a uniform grid of elementary areas of size r 0 , each of 
which has a constant phase, with the value of the phase being selected 
randomly from the range -11 ~ 11. This imposes much more r egularity than 
the real situation, but leads to the correct qualitative results. The 
more realist i c approach is most easily addressed by computer 
simulations, but some of the results which can be seen directly are 
discussed below. We also discuss , without loss of generality, only 
triangles for which two sides are parallel to the sides of the 
aperture . It is possible t o cover the full complement of Z<m-1 )2 
independent closure phases in this way. In this case r = 0. Suppose 
that there are m2 coherent patches over the whole aperture. The number 
of identical triangles i s n = <m-t. +1 > <m-£ +1), where t 1 , t 2 are the lengths of the sides of the triang'1e parahel to the sides of the 
aperture . We will discuss just two examples of triangles in such an 
aperture. These are marked "a" and "b" in Figure 2. Then we can 
calculate the quantities in Tabl e I. 
Thus, for example, fo~ m = 10 the signal- to-noise-ratio for 
triangles of both types is 0.10. Not e that we have only included the 
"noise" term due to the effect of redundant baselines on the closure 
phase, no terms due to photon noise, or other sources of error, have 
been included. 
Experience with optical interferometry on the Sm telescope shows 
that we can use apertures of 20 cm diameter. In this case m = 25 and 
the signal-to-noise-ratio for tri.angles of both types is 0.040. Thus 
about 600 frames are needed to reduce the rms noise in the closure 
phase to below 1 radian. These errors are random and the closure 
147 
phases are unbiassed so that good images can be made provided that 
enough closure phases are measured. There are 2<m-1>2 independent 
closure phases in an aperture consisting of m2 elemental areas, so that 
in the case of the Sm telescope with 20 cm diameter coherence patches 
about 1000 independent closure phases can be measured. 
length of side £1 
length of side t 2 
Number < = n > 
of similar triangles 
number ( = p > 
of unmatched, redundant 
baselines, ij. 
number < = q > 
of unmatched, redundant 
baselines, jk. 
number < = r > 
of unmatched, redundant 
baselines, ki. 
Signal-to-noise-ratio 
J 
= 
1 \ 
\ [n+p+q-3+<pq+2)/n]112f 
SNR for large m 
Triangles similar 
to "a" 
2 
2 
<m-1)2 
m-1 
m-1 
0 
a: 1/m 
Triangles similar 
to "b" 
m 
m 
1 
m-1 
m-1 
0 
a: 1/m 
The signal-to-noise-ratio for large m is given approximately by 
[<m-t 1 Hm-t 2 >+m2/<m- t 1 +1><m-t +1>r112. It is easy to show that 
this is approximately proportionaf to 1/m for most triangles. In fact 
only those with 00. 9m deviate from this dependence by more than 107. 
for values of m > 20. There are very few of these, and so it is a good 
approximation to assume that the rms noise in the closure phases is 
proportional to m. 
There are many ways of choosing the independent triangles. One 
could, for example, adopt a strategy in which the number of long 
baselines was minimized; or one in which the number of short baselines 
was minimised. The actual optimum strategy depends on the brightness 
distribution of the object under study. We have neglected the other 
effects, and these could make the signal-to-noise-ratio very small in 
148 
some cases, for example on long baselines with heavily resolved 
objects . 
We have seen that if m is large the rms closure phase error « m, 
thus we can calculate the variation of sensitivity with m. We can 
think of the error in the closure phases arising from errors J3 times 
smaller on each baseline . . We then have a situation similar to that in 
conventional aperture synthesis, in which the rms noise on the map is 
proportional to the rms noise on individual baselines, and inversely 
proportional to the total area <m2) of the aperture. This shows that 
in the present example the noise will decrease in proportion to m, i.e. 
the sensitivity will increase in proportion to · the diameter of the 
aperture, instead of the ~sual case in which it increases in proportion 
to the area . This is the sens itivity dependence that we get for 
incoherent, or post-detector, integration. 
In the real situation we do not have a regular grid but a random 
distribution in two dimensions. In this case there are fewer 
identical triangles, and the probability, E, of having triangles of a 
certain size and shape is a constant per unit aperture area which is 
determined by the characteristics of the scattering medium. The ratio 
of the coherent signal to the redundant baseline noise is: 
( En + <p+q+r-3> + (pq+qr+rp+2)/(En) + pqr/<En)2 ) 112 , for £n ~ 1 
and 0 for £n < 1. If En » 1 the signal-to-noie-ratio for large m is 
therefore still « 1/m and we gain in sensitivity at most in proportion 
to m as the aperture size is increased. In practice it will often be 
the case that En - 1 so that the gain is even less. 
3. CONCLUSION 
The above analysis suggests t hat the advantage of fully filled 
apertures over non-redundant masks is not as great as might at first be 
supposed . For example, on the Sm telescope we may well be able to use 
a dozen 25 cm apertures. If these 12 apertures comprised a single 
aperture of equivalent s ize 87 cm, then the advantage of going to the 
full S m aperture would be less than a factor ten in sensitivity. In 
fact the non-redundant approach has no errors due to redundancy, so the 
gain will be even less. In addition, as we have seen, the irregularity 
of the phase variations across the aperture reduces the gains still 
further. 
An alternative approach which should be explored would be to use 
masks with some redundancy a nd impose a ' minimum redundancy' condition 
which max1m1ses the number of independent closure phases while 
restricting the number of redundant baselines. We know that we can 
compensate very well for non·-uniform coverage of the aperture plane, 
and t his is not exploited i n the full aperture speckle approach. 
149 
4. REFERENCES 
Baldwin, J.E., Haniff, C.A., Mackay, C.D., Warner, P.J. 1986. Nature 
320, 595. 
Cornwell, T.J., Wilkinson, P.N. 1981. MNRAS 196, 1067. 
Cornwell, T.J. Astron. Astrophys. 1987 <in press> . 
Jennison, R.C. 1953. Ph.D. Thesis, University of Manchester. 
Jennison, R.C. 1958 . MNRAS 118, 276. 
Lohmann, A.W., Weigelt, G. and Wirnitzer, B. 1983. Applied Optics. 24, 
4028 
Pearson, T.J., Readhead, A.C.S. 1984. ARM 22, 97. 
Readhead, A.C.S., Wilkinson , P.N. 1978. Ap . J . 223, 25. 
Readhead, A.C.S . , Walker, R.C., Pearson, T. J., Cohen, M.H. 1980. 
Nature 28~, 137. 
Readhead, A.C.S., Nakajima, T.S., Pearson, T.J., Neugebauer, G., Oke, 
J. B. and Sargent , W. L.W. 1987. Ap.J. <submitted> 
Rogers, A.E.E., Hinteregger, H.F., Whitney, A.R., Counselman, C.C., 
Shapiro, I.I. et al. 1974. Ap.J. 193, 293. 
Schwab, F . R. 1900 . Proc . Soc. Photo-Opt. Instrum. Eng. 231, 18 . 
Weigelt, G., K. -H. Hofmann and Reinheimer, T. 1986. Proc . of the ESO 
Conf. "ESO's VEry Large Telescope II", Eds . D'Odorico, S . and 
Swings, J.-P. 
5. FIGURE CAPTIONS 
Figure 1. The effects of redundant baseline noise . The relative 
magnitudes of the contributions from terms of the first three kinds in 
the triple product are shown. Terms of the first kind add coherently. 
Terms of the second and third kinds add incoherently. In this example 
n = 10, i.e. the integration time is ten times the coherence time, or 
there are ten identical triplets of apertures contributing 
instantaneously to t he triple product; and N = 100, i.e. the triple 
product has been averaged over 100 frames. 
Figure 2. Coherent phase patches over a fully filled aperture . It 
is assumed that each elementary square , of area r 0 2 , has a constant 
phase, and that the phases change randomly between patches . This 
imposes more regularity than obtains in the real situation, but gives 
results which ar e qualitatively correct <see text>. 
150 
____ [N3n(n-n]''2 
[Nn(n-l)(n-2 >] 1/2 
151 
(/) 
-c: Q) 
E 
Q) 
a; 
E 
~-----m elements-----_.... 
~ '/~ ~ ~  
~ ~
• i+ro-
t 
~  ~  
152 


A Comparison of the Determination of Closure Phase in Optical Interferometry with Fully Filled Apertures and Non-Redundant Aperture Masks

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A Comparison of the Determination of Closure Phase in Optical Interferometry with Fully Filled Apertures and Non-Redundant Aperture Masks

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