Transcript X - Observatoire de la Côte d`Azur

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Yves Rabbia – OCA-UNSA
feb 2010
stellar interferometry :
A glance at basics
Yves Rabbia
Observatoire de la Côte d'Azur
Dpt Fizeau, Grasse 06, France
[email protected]
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purpose of the talk
to recall and to illustrate (hand waving as far as possible)




general framework and some land marks
specific terminology ( and debunking "jargon")
basics of interferometry and aperture synthesis
few things about nulling techniques
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stellar interferometry :
A glance at basics
sections









introduction
science context and motivation
limitations and subsequent needs
basics for interferometry and aperture synthesis
interferometers : principle, production, typology
difficulties in real world (and some remedies)
managing with data and some results
quick-look at some alternative HAR methods
nulling interferometry and coronagraphy
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introduction
stellar interferometry is part of a large body of topics
covering methods and techniques aiming at
High Angular Resolution
( HAR)
the underlying goal is Aperture Synthesis
but other techniques contribute to HAR
eclipsing binaries
lunar occultation
speckle interferometry
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High Angular Resolution :
a matter of "finesse" in exploration
resolution
angular
"finesse" depends on
the instrument used
what does "high" mean ?
it means "responding to current scientific needs"
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self-speaking illustration
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testing your own angular resolution
not much
yet poor
by any other name
not bad
would smell
as sweet
pretty much
W. Shakespeare
really impressive
wooah ! fantastic
A Rose
better
and better
warning :
sensitivity and contrast does matter
(Signal to Noise Ratio)
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resolution and resolution
detector resolution : large number of pixels is not the point
what counts for the astronomer is
the size of pixels over the sky
and this depends
on the instrument
(including observing conditions)
pixel on the sky
jargon : resel,
resolution element
one resel
may cover several pixels
on the detector
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"finesse", exploration lobe, diffraction lobe : the finger
the lobe comes from throwing back to the sky the angular extension
of the image obtained with a point-like source
the so-called Point Spread Function (PSF)
sky
focal plane
telescope
illustration "live"
sky
PSF
PSF
l/ D
result of
the exploration
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science context and motivation
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introduction : the purpose of astro interferometry
describe the brightness angular distribution of a source
and measure morphological parameters (not necessarily images)
Astrophysics
wants

I ( a0 ,W , a  a0 , l ,t , P )
interferometry focuses
primarily on
 l1
I ( a0 ,W , a  a0 , l ,t , P )
then on
then eventually, look for time
evolution and polarisation
in the following , we restrict the topic to stars
d
d0
a0
d
a0
a
sky
coordinates
a
l2
l3
l
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morphology ??? few examples for stars
uniform
disk
symbiotic
limbphotospheric
darkened features
disk
binary
cocoon
disks an
envelops
miras
Betelgeuse
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a gradation of needs
depending on sources, few parameters might be of great help
but some morphologies would requires images
examples
binaries : angular separation (vector)
single stars : angular diameter
limb darkening : radial variation of intensity
multiplicity : number of components, geometry
extended atmosphere : radial structure, schock wave
circumstellar matter : angular diameter, structure
complex objects ( Miras, bipolar jets, disks, .... )
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limitations and subsequent needs
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problem for imaging stars !!
angular dimensions of stars are so terribly small
astronomers (scientific requirements)
need High Angular Resolution
I want milliarcsec level 5 10 - 9
radian
engineers (technical requirements)
hep ! : diffraction sets a limit
l/D
what leads to several tens of meters for D
interferometry aims at breaking the limits of conventional imagery
so as to determine some morphological parameters
without large telescope diameters, (and ultimately produce images)
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immediate limitations
instrumentation
lobe extension l/D and looking for 1 milliarcsec
lead to D = 100 m in the visible (and more in Infrared)
telescope not (yet) available
observing conditions
mainly atmospheric turbulence
PSF degraded , loss of resolution (speckles)
theoretically
the larger the diameter
the finer the lobe
but D small or D large ,
extension of PSF quite the same
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a technical solution ??
conventional imaging does not work !
a new approach is needed to describe the brightness distribution
or at least to obtain some of its parameters
answer is aperture synthesis which is based on interferometry
the idea behind :
fetch spatial information in Fourier Space !
in other words : determine spatial spectrum and then, come back
as far as possible to the angular intensity distribution of the source
conventional imaging : directly provides
a 2-dim representation ( feeding a camera)
with aperture synthesis the imaging process requires
specific methods for observation and computation
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phenomenology_ 1
on the way towards interferometry (pictorial)
basically mainly the peripheral corona of the collector
governs the angular extension of the central peak
yet at the price of increased sidelobes (and less photons, of course)
but ..
not easier to make
large corona
than large diameter !
anneau
so what... ?
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phenomenology_ 2
not easier to make large corona than large diameter !
but ... look
let's take two pieces of the corona
in one direction
the central peak remains
narrow
however
on the perpendicular direction
the peak extension is the one
of the small piece
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how to make it practically ?
B
D
D
do it yourself !
l/B
l/D
where is the gain ?
l/B instead of l/D
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questions appear
extension l/B much finer than l/D
gain B/D example 100m/1m
OK but, the exploration lobe is like a comb
where does it come from ?
what does that mean "exploring with a comb ?"
exploring with a lobe both narrow and large
telescope
some theoretical basis needed
but before let's look again a little more at phenomenology
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exploring and measuring with a "fringed lobe"
point like
and small baseline
enlarging baseline
fringe spacing l/B decreased
binary
fringe patterns
slightly shifted
fringes with
full contrast
(max = 1, min = 0)
yet fringes but
slightly lowered contrast
enlarging baseline
fringe patterns fully overlap
there angular separation
is measured : l/2B
no more fringes, total blurring
contrast fully destroyed
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basics for interferometry
and aperture synthesis
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toolbox and terminology



mathematical tools
Fourier world (reminders)
spatial frequencies & spatial spectra
Fourier optics
linear filtering
fundamentals principles
detection, coherence, VCZ theorem
interferometers
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the very first tool,
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not a joke !
z1 C , z2 C
z1  z2
2
 z1
2
 z2
2
 2. Re z1 . z2 * 
modulus A
in other words
A 1.e
i . 1
Im( Z)
 A2 .e
i . 2 2
phase 
Re( Z)

i .(  1   2 ) 

 A1  A2  2.A 1.A 2. Re e


 A1 2  A2 2  2.A 1.A 2. cos (  1   2 )
2
Z
2
no comments,
see later on
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Fourier formalism
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omnipresent tool : Fourier formalism (reminders)_1
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omnipresent : spatial frequencies, spatial spectra, coherence,
Fourier optics, linear filtering, visibilities, signal processing, data processing, .......
usual definition(s) and usual expression
complex-valued function
fˆ
R C
u  fˆ ( u ) 
"fonctionnelle linéaire"
fˆ
E R  C
( f , u )  fˆ ( u )
E = space of "friendly functions"
RxE  C
linear transformation
F

f ( x ).exp(  i .2 .u .x ).dx
E E
R C
f  F ( f )  fˆ with fˆ
u  fˆ ( u )
ˆf (u)   f ( x ) . e  i .2 .u. x . dx


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Fourier formalism ( reminders) _ 2
side view
f( x )
 "friendly"
functions : no infinity (physical signal)
x
finite definition interval, finite values
possibly complex valued : R C
 scalar product :
f , g  

f ( x ) . g* ( x ) .dx
 Fourier formalism : "friendly" functions can be expressed
as a weighted sum of "base functions" depending on a parameter "u"
e(u,x) = exp( i.2.u.x)
the weight associated to a given "u"
is called Fourier component of "f " for "u"
parameter "u" is a frequency (complex helix)
f( x ) 

weight ( u ) . base( x , u ) .du
Re( e(u,x) )
Im( e(u,x))
x
1/u
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Fourier formalism (reminders) _ 3
wild (and disputable) analogy with vectors

V = weighted sum of "base vectors"

V 

k
ek

Xk . ek
e2
X1= 4
e1
retrieving components (weights) Xk :

V
X2=2
scalar product
 
Xk  V . ek
similar view for "friendly" functions
function f : weighted sum of base functions involving parameter "u"
retrieving weights (components) : scalar product
fˆ ( u )   f , eu  
and we find the usual
definition expression
of the "u" component
f , g  
 f ( x ) . e * ( x , u ). dx
fˆ ( u ) 



f ( x ) . g* ( x ) . dx
f ( x ) . e i .2 .u .x . dx

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practical view and comments_1
a pictorial approach
^
f =
x
f(x)
exp(- i.2.ux)
FT can be seen as a "periodicity sensor" ( period 1/u)
algebra yields the contribution of a periodic-like component
f(x)
1/u
presence of
a periodic feature
x
unmatched frequency
x no response
x
matched frequency
amplitude extracted
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practical view and comments_2
note 1 : all values f(x) needed to have a single value fˆ ( u ) of
fˆ
fˆ ( u )  f ( x ) . e  i .2 .u .x .dx
note 2 : inverse transform
note 3 : frequent use of two-variable transforms (spatial description)

 i .2 .( u .x v .y )
ˆ
f ( u ,v ) 

f ( x , y ) .e
.dx .dy

note 4 : linearity
l .f  g


l .fˆ  ĝ
note 5 : parity real even function  real even FT, odd complex FT
note 6 :
for two extremely simple functions
the definition does not make sense ! (infinity is there)
constant and sine do not match the definition : but Dirac will save us
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Dirac world
d(x,y)
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d(x-a, y-b)
y
b
x
a
crucial relations

d ( x ) .dx 

y
III(x/p)
x
x
p
d ( x  a ) .dx  1
III ( x / p) 
f ( x ) . d ( x  a )  f (a ) . d ( x  a )

 

d ( x  n. p)
n  
III(x/p, y/q)
f ( x ) . d ( x  a ) .dx  f (a )
1  d( u )
1
cos( 2 .a .x )  . d ( u  a )  d ( u  a ) 
2
d ( x  a )  exp( i .2 .u .a )
y
q
p
 
 
x y
III ( , )  
 d ( x  n. p, y  k .q)
p q
n   k   
x
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usual and frequent functions : notations and physionomy
rectangle (door, pulse,...), shifted rectangle, triangle or Lambda (L),
gaussian, camembert
1
P(
P( x/A)
1
x-a )
A
a
x
A
L( x/A)
1
x
-A
A
2A
P( r / 2R)
exp( - x2/a2)
1
y
1/e
-a
+A
x
a
R
x
r
x
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Fourier Transforms of usual functions _ 1
1
P( x/A)
P(
x
1
A
L( x/A)
x-a )
A
1
x
a
x
-A
+A
A
1/A
1/ 
exp( - . u2)
1/A
1/a
1/e
^
f(u)
u
1/A
1
-1/ 
^
f(u)
u
exp( -.x2)
1
1/e
u
1/ 
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P( r / 2R)
Fourier Transforms of usual functions _ 2
1
camembert
R
^
f(u)
.R2
y
r
x
^
log f(u)
1
0.1
0.01
0.001
1.22 / 2R
P(
r
2R
)
avec




ˆ ( q )   . R 2 .  2 . J 1 ( Z )    . R 2 . Jinc( Z )
 P


Z


Z   . 2 R.q
u
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hand made transforming _ 1
function
base function e(u,x)
u=0
u
u1
u
u2
u
u3
u
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rectangle, door, pulse
integral => one value of TF
a spectrum's component
u
u
u
u
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hand made transforming _2
function
base function e(u,x)
u=0
u
u1
u
u2
u
u3
u
feb 2010
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Dirac !
integral => one value of TF
a spectrum's component
u
u
u
u
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feb 2010
usual functions while rather pathological (but soon friendly)
the definition integral does not converge : Dirac heals
constant
cosinus
p
x
u
x
u
- 1/p
Re (ex,u )
Im (ex,u )
1/f
u
f
x
III(x/p)
III(p.u)
u
x
p
1/p
1/p
f
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complication : introducing FT with 2 variables
"friendly" physical signals again
y
x
what is changing ??
moduli of base functions
longer are "sine lines"
moduli are now something like "tôles ondulées"
(but with negative parts)
having various periods, phases
and orientations
still, a given 2D-signal again is a
weighted sum of 2D- base functions
+
+
+
+
+
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feb 2010
new algebra with two variables
location of a given point of the object
requires two coordinates (x and y) or a vector
 x
X:
y
also, a given frequency is described by two "projected frequencies"
"u" and "v", or a vector
 u
(to account for orientation of the sine-like surface) U :
v
then a given base function ( varying over x and y) will convey (x,y,u,v)
and writes
 
 
e( x , y , u , x )  e ( X ,U )  exp( i . 2 . X .U )  exp[ i . 2 . ( u .x v .y )]
U
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feb 2010
illustrations for 2 variables objects
A sine-like surface in (x,y) space
v
u1,v1
a couple of dirac-distributions
(FT of cosine)
in Fourier space
also named (u,v) plane
the "faster" the oscillation
the farther from origin the diracs
y
0,0
u
u2,v2
x
(higher frequency)
orientation of the couple is perpendicular to the wave crests
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feb 2010
f(t)
another tool : convolution
starting from two functions f(t) and g(t)
we define a new function h(x)
where x is the shifting of f versus g
g(t)
t
g(t)
f(t)

h( x ) 

t
f ( t ) . g ( x  t ) .dt
 
h(x)
reversal
a (bad) current notation ( physicists)
t
x
t
reverse
and shift x
x

f ( x )  g( x ) 

f ( t ) . g ( x  t ) .dt
 
effects of convolution :
smoothing
translating
f ( x )  d( x  a )  f ( x  a )
x
x
=
x
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convolution_3, "live"
f(t)
départ
décalage
f(t)
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g(t)
t
g(x -t)
t
f(t).g(x -t)
h(x)
x = zero
t
t
t
x
x = x1
t
t
t
x
t
t
t
x
t
t
t
x
t
t
t
x
x = x2
x = x3
x<0
global result : convolution decreases stiffness, smoothing
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convolution_3 visual training
f(t)
g(t)
h(x)
t
t
A
A
A
f(t)
x
g(t)
h(x)
t
t
x
a
a
g(t) = d(t – a)
f(t) = P(t)
A
h(x) = P(x – a)
t
t
a
P( x )  d ( x  a )  P ( x  a )
x
a
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convolution_4
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also with two variables
actors : f(t,z) and g(t,z)
shifts : x and y
 
f ( x , y )  g( x , y ) 
illustration


f ( t , z ) . g( x  t , y  z ) .dt .dz
   
starting distribution
this recalls
something
we will come back
to it again
result of
exploration
exploration lobe
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45
some usual theorems , pictorial with a rectangle ditribution
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starting point :
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TF
fˆ ( u)
f ( x) 
translation :
f ( x  a) 
feb 2010
fˆ ( u) . e  i .2 .u.a
similarity
x
f ( )  a . fˆ (a .u)
a
convolution
f ( x )  g( x ) 
autocorrelation
parseval (rayleigh)

fˆ ( u) . gˆ ( u)
f ( x)

2
 f (u)  g * (u)
f ( x ) . g * ( x ) . dx 

fˆ (u) . gˆ * (u) . du
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the first key :
spatial frequencies
feb 2010
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frequencies , spectrum, in the familiar time domain
reminder : a physical signal can be described as a weighted sum
of sinusoidal components (Fourier) of various frequencies
The set of weighting factors (Amplitude, frequency)
is the spectrum of the signal
periodic signal : discrete spectrum
VOOOO.....
non-periodic : continuous spectrum
S(f)
V(t)
t
f
Schboumpf
Driiiiiiii
for very "narrow" signals, a lot of sinusoides needed and high frequencies
thus :
continuous and extended spectrum
limiting case : Dirac
FLAK
superposition
t
f
résultat
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spatial frequencies
time domain : frequency = (1 / time ) , Hz
spatial distribution (2-dim x & y)
spatialfrequency : vector (u,v) each component ( 1 / length)
spatio-angular frequency : vector (u,v), each (1/angle) or (1/radian)
just recalling practical pictures :
"tole ondulée" or "sine surface" (with negative parts)
one direction modulated
+
+
+
all directions modulated
+ ...
+
+
practical training : find spatial frequencies in the room !
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feb 2010
pictorial examples
"sine surface" (x,y) one direction --> couple of diracs in (u,v) plane
u
x
v
y
x
influence of orientation
u
u1,v1
replicated rectangle
u2,v2=0
1/A
x
A
algebra :
(Fourier space )
p
x
x
C ( x )  P ( )  lll ( )
A
p
u
1/p
sin(  .A.u )
Ĉ ( u ) 
.lll ( p .u )
 .A.u
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spatial frequencies : further training
change your zebra for a circus horse
Zebra =
horse + grid
( conspicuous spat. freq.)
remove the grid
y
Z(u,v)
v
x
u
S(u,v)
v
u
circus horse =
horse + double grid
just add
the needed frequencies
Z(u,v)
v
u
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feb 2010
examples of spatial distributions and spatial spectra _ 1
O ( x , y ) or O ( a   )
spatial distribution:
spatial spectrum ( 2-dim Fourier Transform) Ô ( u ,v )
WARNING ! : spatial spectrum is a complex function
only modulus displayed here
camembert
distributions
bessel
y
v
u
x
Gauss
Gauss
x
u
spectra(modulus)
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Yves Rabbia – OCA-UNSA
feb 2010
examples of spatial distributions and spatial spectra _2
distributions
narrow cylinder
+ co-centric corona
cylinder
+ co-centric
hollowed gaussian
spectra (module)
52
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
examples of spatial distributions and spatial spectra _3
couple of diracs
unequal strength
distributions
spectra (module)
O ( x , y ) d( x , y )
h.d( x  r, y )
diracs
unequal strength
oblique orientation
now with
h . d ( x  r , y  )
cylinder +
off-centered
corona
all that pictures can be recasted with
"source" or "object" instead of "distribution"
53
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
living in Fourier Space
distribution
spectrum (module)
narrow extension
large extension
composite
composite (linearity of FT)
A narrow + B large
Â
large  B̂ narrow
but take care of separation of
centers ( complex exponential)
stiff edges, sharp angles
strong high frequencies
smooth shape
faint high frequencies ( Gauss)
dissymetry
spatial modulation, phase effect
(complex exponential, fringes)
to see again later
54
IRAP conferences
just talking
Yves Rabbia – OCA-UNSA
feb 2010
(where are we ?? )
time-signals can be described
either by amplitudes (signal) or by frequencies (spectrum)
similarly
there are two ways to describe brightness distributions :
the direct description, based on coordinates (image)
the other one, based on spatial frequencies (spatial spectrum).
For brightness distributions, when images are not at hand
a "spectrum analyser" must be found
Once available, coming back from the spectrum to the image
could be possible.
This is the goal of aperture synthesis
Thus we have now to conceive and built
such a spectrum analyser
55
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
an important auxiliary :
Fourier optics
56
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
Fourier optics

key protagonists :
complex amplitude and wavefront

building the tool :
 a very quick look
 Huyghens Fresnel principle
 here comes Fourier formalism

illustrative example(s)
57
IRAP conferences
Yves Rabbia – OCA-UNSA
58
feb 2010
complex amplitude of fields
source S, observation point P
VS(t) = A.exp( i. 2.n.t)
Wave at S
x
S
y
X
Y
Z
P
r(S,P)
z
At P, same behaviour but time-delayed
VP(t,x,y,z) = A.exp( i.2.n.(t- r/c))
VP(t,x,y,z) = A.exp( i.(n. t - r/l) )
for imaging, relevant information is in the phase distribution 2.r/l
at each observation point, so the useful description is given by
the complex amplitude
It describes the shape of the wavefront ( equiphase surface )
y (x,y,z) = A.exp( i.2. r/l)) = A.exp [i.(x,y,z)]
intensity
(power density)
y 2  A2
Re (y)
Im (y)
y
phase
l
r(x,y,z)
IRAP conferences
Yves Rabbia – OCA-UNSA
59
feb 2010
the meaning of "phase of the field" : wavefront
spherical wave
x,y
z
r2(x,y,z) = x2+y2+z2
plane wave (on-axis point-like source at infinity
x,y
z
r (x,y,z) = z
the phase mathematically describes the shape of the wavefront
wavefronts are "equiphase surfaces"
propagation within material medium
( "n" is refractive index)
n1
n2
n3
n1< n2< n3

2
l
image
. index . pathlength 
2
l
. n .r
abberated image
IRAP conferences
Yves Rabbia – OCA-UNSA
Fourier Optics a (very) quick-look
frequent set-up
w : incoming wavefront w
s : diffracting screen
O : observation screen (image)
60
feb 2010
 y(x)
y y(x,y)
x
s
z
x
O
y(x) : amplitude after "s"
y(x,y) : amplitude at O
summary :
if I know y(x0) amplitude transmitted by "s" (pupil plane)
I can calculate y(x,y,z) amplitude over screen O (image plane)
Intensity at O = squared modulus of y(x,h)
tools to do that :
approximations + Huyghens-Fresnel principle
z
IRAP conferences
Yves Rabbia – OCA-UNSA
61
feb 2010
Re (y)
Huyghens-Fresnel principle
Im (y)
complex amplitude
already seen : as the field propagates
the phase increases according to
length of traveled path
r(x,y,z)
l
Huyghens Fresnel principle:
every point Qn within an amplitude distribution
emits a spherical wave, all waves
are synchronous but not necessarily "in phase"
rk
Qk
Qn
P
rn
The field seen by P is the sum of the amplitudes of the spherical waves
(complex numbers)

y( P ) 
 y ( Q ) .exp ( i . l . path ) )
yz ( x , y ) 
n

2
y 0( x , ) . exp ( i .
2
l
x
y
x
. r ( x , , x , y ) ) . dx .d
z
IRAP conferences
Yves Rabbia – OCA-UNSA
62
feb 2010
here comes Fourier formalism
approximations :
scalar field
+ Pythagore :
r ( x , , x , y )  z 
most frequent set-up :
L lens, focal F
y0 : transmitted amplitude w

2.z
s L
y0(x)
 y ( x , )  e
i.
0
l << x,x,y
x  x 2  x   2
x
then (some algebra)
yF ( x , y ) 
z >> x,x,y
F
2
. ( x .x  y . )
l .F
y
x
z
O
yF(x,y)
. dx . d
other writing : WELCOME FOURIER !
y F ( u ,v )  y
ˆ F ( u ,v ) 

y
 i .2 . ( u .x v . )

(
x
,

)
e
. dx . d
0
with u, v spatial frequencies : u  x ,v  y
lF
lF
IRAP conferences
Yves Rabbia – OCA-UNSA
examples
P( r / 2R)
circular
aperture
1
x,y
z
r
( x )  
2
l
F
x
y F ( r )  2.
P ( x ). exp(  i .
x

R
tilted
incoming
wavefront
63
feb 2010
J 1( Z )
Z
2
l
yF(x,y)
with Z  2 . r .R
. path  
2
l
. x .a
x
a
x
.x .a )
 P̂ ( u )  d ( u  a )
warning ! mixed and ill-used notations
a
x,y
z
I(x,y)
IRAP conferences
Yves Rabbia – OCA-UNSA
another example
xP
y(x)
64
feb 2010
double slit as aperture
I
x
x
y 0  P  P  P   d   d  
y F  P̂ .  e i  e i 
xP
y(x)
I  y F  2. I 0 .  1  cos ( )  with
2
I 0  P̂
2
x
x
I
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
other useful auxiliaries :
linear filtering
and
transfer function
65
IRAP conferences
Yves Rabbia – OCA-UNSA
overview
signal
physical system
dirac-like input
physical system
time signals
signal
Ak
Ak+n
tk tk+1
tk+n
2D-objects
response
r(t)
r(t) = s(t)
* h(t)
time
time
object
image
(intensity)
h(x,y) =
I(x,y) = O(x,y) * h(x,y)
response to dirac (impulse)
reponse h(t) =
filter h(t)
Ak+1
response
input-output relationship
is a convolution
syst. phys. = linear filter
s(t)
66
feb 2010
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
bi-dimensional situation
67
object-image relationship in the direct space (coordinates space)
I(x,y) = O(x,y) * h(x,y)
h(x,y) = response to dirac : Point Spread Function ( typical : Airy pattern)
x
P(x)
a
F
opt
syst
description in the frequencies space (Fourier space)
Iˆ ( u, v )  Oˆ ( u, v ) . hˆ ( u, v )  Oˆ ( u, v ) .T ( u, v )
T(u,v) = transfert function = FT of impulse response
shows how frequencies are transmitted : gain (complex)
Also : T(u,v) shaped like autocorrelation of pupilla
(Rayleigh theorem)
IRAP conferences
Yves Rabbia – OCA-UNSA
68
feb 2010
transfer function, mostly pictorial
T ( u)  L (
T(u) = FT of PSF ( Airy)
l^
P(u) l2
2
l .u
D
)
T(u)  L( l.u/D)
1
a
u
+D/l
-D/l
l/D
D/l : cut-off frequency. Consequence : higher frequencies are lost
objet spectrum (source)
transfer function
T(u)
spat freq
image spectrum
image
using T(u)
1
D/l
D/2l
object
u
PSF
IRAP conferences
Yves Rabbia – OCA-UNSA
69
feb 2010
a special one (anticipation) : double slit aperture
x
D
y(x)
T (u)
1
B
1/2
x
D/l
P
pupil
(filter)
x
impulse response: PSF
x
B/l
Transfer function
(FT of PSF)
1
autocorr
x
x
further ? several circular, bi-dim network
B2/l

B2
B3
x B1/l
B1
v
u
B3/l
u
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
fundamental principles :
Coherence
and
the second key : VCZ theorem
70
IRAP conferences
Yves Rabbia – OCA-UNSA
71
feb 2010
defining coherence : a mutual notion
coherence of fields is the ability they might have
to produce observable interferences when mixed
(duration equal or larger than the detector integration time say 10-6 s)
coming back to our first tool
quadratic detection, energy detection (visible and infrared)
incoming field y
2
detection process : output =  y 
 = integration time,
notation < >  means "averaging over  "
"mixing fields" means
 y1  y 2
 y1  y 2
2
2
y
< y 2 >
y1 and y2 arriving together on the detector
   y1
2
  y2
2
  2. Re( y 1 . y *2  )
  I1  I 2  2. Re( y 1 . y *2  )
energy terms
interference term
IRAP conferences
Yves Rabbia – OCA-UNSA
72
feb 2010
how to quantify this coherence ability ? _ 1
the single cosine model is not compatible with observation of interferences
t
V(t)
V(t) = A.exp (i.2.n.t+)
need another model for wave of light,
a relevant model :
a model with random features
train of damped oscillations with
random emission time and phase at origin
length of a wagon : c  1/ spectral Dn
time
y ( t , ) 
y
k
k
with y k  A( t tk ). exp( i 2n ( t tk ) k )
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
73
how to quantify this coherence ability ? _2
 y1  y 2
2
  I1  I 2  2. Re( y 1 . y *2  )
< y1.y2* > is something like a covariance
it could serve but depends on incoming amplitudes (leading to intensities)
better to define
a dimensionless quantity free from differences in intensity
A complex number which modulus varies between 0 and 1 is appropriate
complex degree of coherence
with Ik = <  yk 2 >
 12 
12 can be seen as a complex normalized
 y 1 .y 2 * 
I1 . I 2
covariance
IRAP conferences
Yves Rabbia – OCA-UNSA
74
feb 2010
about addition of fields : generic situations _ 1
incoherent addition
S1
the interference term is destroyed
by averaging over the random phases
2
I   y 1 y 2   
2
I 
y  y
k
1k
2
n
2
2n
y1
y2
S2



y 1k    y 2n   2. Re   y 1k .y *2n 
k
n

k

n
all y 1k .y *2n  convey  exp i ( k  n )   and averaging results in zero
I = I1 + I2 + nothing !
consequence : introducing
a non-coherence relationship
two distinct sources cannot produce
an observable interference term
y ( a ) .y * (  )    y ( a ) 2 . d ( a   )
S1
S2
a

y1
y2
IRAP conferences
Yves Rabbia – OCA-UNSA
75
feb 2010
about addition of fields : generic situations _ 2
coherent addition
let us consider a machine performing
addition of field collected at P1 and P2
from a unique point-like source
and nearly equal paths r1 and r2
S
r2
r1
y2
y1
P2
+
P1
 2 r  i . 
k 
 l 1 or 2

each "wagon" is like : A( t tk ). exp i .
and interference term conveys components like :
r
r
2
 A( t tk  1 ).A( t tk  2 ). exp  i .
. ( r1  r2 )  i .( k  k )  
c
c
 l

now averaging over  has no effect, and
interference term remains, in spite of the effect of statistics. zero
However it is non-zero only if "wagons" are overlaping
what requires that (r1-r2) must be smaller than the length of a wagon
( subsequently named : coherence length)
IRAP conferences
Yves Rabbia – OCA-UNSA
76
feb 2010
path difference must be small enough
P2
Sk
P1
strong
Dt
<
faint
y 1.y
2
*>
nearly zero
IRAP conferences
Yves Rabbia – OCA-UNSA
77
feb 2010
a third situation : synthesis of the two previous ones _1
addition partially coherent
several sources S1,S2,…Sn
and the addition machine
rn2 yn2
S2 S1
Sn
rn1
yn1
P2
+
P1
phenomenology
all sources Sk yield an observable interférence term (situation 2),
independantly one to another, each term depends on (rk1 – rk2 )
in other words depends on its location
Sk
P2
+
P1
the resulting value for the interference term depends
on the ( angular) extension of the distribution of point-like sources
IRAP conferences
Yves Rabbia – OCA-UNSA
78
feb 2010
a third situation : synthesis of the two previous ones_2
O(a) = continuous distribution of point-like sources
field from direction "a" noted y(a)
such as <ly(a)l2> = O(a)
field incoming on P1 noted y1 and y2 for P2
P2
distance P1P2 is B
O(a)
a
y(a)
B
+
P1
On P1 and P2 comes the sum (over a) of fields from directions (a)
respective paths are r1(a) and r2(a)
y1 
2
 y ( a ) . exp( i . l . r ( a )) .da
1
y2 

y ( a ) . exp(  i .
2
l
. r2( a )) . da
then we have (classical expression for a product of integrals)
 y 1 .y 2*  
look !

y ( a ) .y * (  )  . exp(  i .
2
l
. r1( a )  r2(  ) ) . da .d
IRAP conferences
Yves Rabbia – OCA-UNSA
79
feb 2010
so what ?
 y 1 .y 2*  

2
y ( a ).y * (  )  . exp(  i .
l
. r1( a )  r2(  ) ) . da .d
and thanks to non-coherence relation

 y 1 .y 2*  
O ( a ).d ( a   ) . exp(  i .
2
l
. r1( a )  r2(  ) ) . da .d
and thanks to dirac properties
 y 1 .y 2*  

O ( a ) . exp(  i .
and so and so ?
 y 1 .y 2*  

isn't that beautiful ?
2
l
. r1( a )  r2( a ) ) . da
 d ( a   ).d
=1
O ( a ) . exp(  i .
2
l
. B .a ) . da
Fourier strikes back
 y 1 .y 2*   Ô (
B
l
)
IRAP conferences
Yves Rabbia – OCA-UNSA
80
feb 2010
last step towards complete happiness
a by-product:
2
y 1 .y *1    y 1   I 1 
same for y2
 O (a ) . da
= Ô (u=0)
let us then recast the degree of coherence
B
 12
 y 1 .y 2*  Ô ( l )


I1 . I 2
Ô ( 0 )
in other words
with P1P2 = B and with l
the degree of coherence is the FT at frequency B / l
of the angular brightness distribution of the source
normalized on its FT at origin
this simply is the Van Cittert & Zernike theorem
good grief !
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
coherence volume: in the guts !
Dl decreased
tc coherence time
increased
apparent dimension decreased
coherence area increased
81
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
Long Baseline Interferometry
functional features
82
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
interferometry : measuring by means of interferences
among several compact descriptions
the two main features here to handle are :


interferometer = machine performing coherent addition
interferometer = filter for spatial frequencies
but the main feature for science concerns is :
(simply a consequence of coherent addition)
interferometer =
here we look at
the academic case
(Fizeau configuration)
instrument making observable
the degree of coherence
83
IRAP conferences
Yves Rabbia – OCA-UNSA
84
feb 2010
coherent addition of fields : observed intensity
incoming field y(x)
collecting pupil : P(x  B/2)  P(x  B/2)
x
x
y(x)
x
D
B/2
collected field:
Q(x) = y1 . P( x  B/2)  y2 . P( x  B/2)
amplitude at focal plane (FT) :
B
B
 i .2 . .u
 i .2 . .u 

2
2
ˆ ( u ) y1 . e
Q̂ ( u )  P

y
.
e
2


2
intensity at focal plane ( squared modulus )
2
with I 0   y 1    y 2 
2
ˆ
and
Airy  P

  y 1 .y *2 
B .x  


I ( x )  2.Airy ( x ) . I0 .  1  Re 
. exp( i .2 .
) 
I0
l.F  



IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
85
interferometer makes observable the degree of coherence

  y 1 .y *2 
B .x  
I ( x )  2.Airy . I 0 .  1  Re 
. exp( i .2 .
) 
I0
l .F  


 12 might be complex
we write :  12 
then :
 y 1 .y *2 
I0
here is 12(B/l)
. exp( i12 )  V . exp( i12 )
B .x


I ( x )  2.Airy . I 0 . 1  V . cos( 2 .
 12 ) 
l .F


warning :
generally with two apertures, only V can be extracted from data
12 is mixed with random spurious phases and is not available
So the observation only yields V = modulus of 12
also called : amplitude of fringes
x
note : with point-like source V = 1
IRAP conferences
Yves Rabbia – OCA-UNSA
86
feb 2010
interferometer as filter for spatial frequencies
point-like source yields impulse response
B x


I ( x )  2.I 0 .Airy .  1  cos ( 2. . .
)   h( x )
l F


transfer function (normalized FT of h(x))
1

T ( u )  Â iry ( u )  d ( u ) 
2

1
ˆ iry(u)
A
u
B
B  

. d ( u  )  d ( u  )  
l
l  

1
T (u)
1/2
D/l
B/l
any source yields : I ( x )  O ( x )  h( x ) and Î ( u )  Ô ( u ) .T ( u )
interferometer allows sensing the source spectrum
at frequencies as high as B/l
note : two telescopes, one baseline, one spatial frequency
u
IRAP conferences
Yves Rabbia – OCA-UNSA
87
feb 2010
back to our "fringed lobe" : what is it doing ?
the lobe questions the source about the presence of a particular
spatial frequency (the one born by the fringes : B/l)
it makes a measure of this "presence" , it looks into the spatial
spectrum
conceptually the source answer is the
"visibility" of the source
at frequency B/l
The visibility is given par by the
modulation rate of the observed fringes
on the camera
u1
u1,
how much ?
u2
u2,
how much ?
thanks to VanCittert & Zernike,
we now know that this visibility is the
degree of coherence of the source
at frequency B/l
note : one baseline, one frequency,
each time a little piece of information
(a component of the spatial spectrum)
IRAP conferences
Yves Rabbia – OCA-UNSA
88
feb 2010
listen to the source
base
freq
answer
B1
B2
B3
B4
u1
u2
u3
u4
very much
pretty much
nearly nothing
faint presence
answers are compiled
on a graph
answer
vm
pm
so, baseline after baseline
is progressively built a so-called
visibility curve
fp
nn
question
u1
u4
which is assumed to reproduce
a cut in the spatial spectrum of the source
As announced , we have picked up information in the frequency space
IRAP conferences
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feb 2010
Long Baseline Interferometry
the machine
89
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feb 2010
90
principles for operation




instrumental functions of an interferometer
various configurations and associated constraints
encoding information
extraction of information (academic case)
IRAP conferences
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feb 2010
91
functions of a stellar interferometer
as a scientific tool
the interferometer is sampling the incident wavefront (multi-aperture)
so as to built (thanks to V anCittert and Zernike)
a sampling of the spatial spectrum of the source
as an optical set-up
the interferometer has (at least) 2 functions :
collection :
pick up the fields at P1 and P2
corrélation : make available <y1.y2*>, covariance of the fields
in addition Optical Paths Differences must be kept much less than
the coherence length c.tc = l2/Dl
telescopes
collection
entrance
beam guiding
recombination
exit
correlation
detection
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feb 2010
92
questions of pupils
entrance pupil (or input pupil):
the set of collecting apertures ("collection" stage)
exit pupil ( or output pupil) :
a set of images (through the optical set-up) of the
collecting apertures ("recombination" stage)
warning :
" exit pupil"
does not mean
"image of the entrance pupil", thoug it can be the case
rather a "re-mapping" of the map of collecting apertures
two schemes :
Fizeau configuration (homothetic mapping)
Michelson configuration (non-homothetic mapping)
IRAP conferences
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Fizeau configuration
93
feb 2010
L. H. Fizeau
1819-1896
input
Fizeau , (it is the "do it yourself" already seen)
the telescope performs, just by construction
both collection AND corrélation
exit
B
maximum baseline is limited by the telescope's diameter
the mappings of entrance and exit distribution of pupils
are homothetic
b
telescope
sampling of the spatial spectrum is made
by changing the baseline (distance between apertures)
note : spatial period changes with baseline
I (x)
I (x)
I (x)
x
x
x
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feb 2010
94
looking at interferences fringes
source
artificial star
eye
or camera
with
dispersion
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feb 2010
domestic set-up and observation
95
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Michelson configuration
collection and correlation
are well separated and
work independantly
non-homothetic mapping
spatial period of the fringe pattern
is given by "b"
and does not change
with baseline B
the measured
spectral component is
at spatial frequency B/l
not at b/l
cut-off frequency
not limited by the
telescope 's diameter
96
feb 2010
A. A. Michelson
1852-1931
B
b
B
d
b
IRAP conferences
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feb 2010
various input and output configuration
Fizeau configuration
homothetic mapping
input
output
Michelson configuration
non-homothetic mapping
97
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feb 2010
extension of the Michelson configuration
going farther in the separation of collectors :
independant telescopes
Radioastronomy, Labeyrie
both configurations F et M,
intrinsically achieve
Optical Paths balance
No longer the case here
because of an astronomical misbalance
introducing
severe metrological constraints
equal optical paths, automatically
kept well balanced
whatever pointing direction is
98
IRAP conferences
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feb 2010
extended Michelson : mostly the usual scheme for years
99
100
metrological constraints when using separated telescopes
IRAP conferences
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feb 2010
an extra optical path "da" must be taken into account to
achieve the balance. Equating paths must be control permanently
Balance is made by
an adjustable Optical Delay Line
continuously moving during observation
and inserting optical path "dr"
required accuracy
a small fraction
of coherence length l2/Dl
da
relative accuracy regime :
coherence length/baseline dynamical nanometrology
dr
optical
delay line
IRAP conferences
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feb 2010
101
ground-based and space-based situations
plan d'onde
incident
T
ground-based Labeyrie
1
labo focal
(recombinaison)
space : platform
T2
space :
free-flyer
formation
IRAP conferences
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feb 2010
Long Baseline Interferometry
using the machine
102
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feb 2010
information encoding : recombination stage
up to know only the "fringe pattern" has been mentioned
other schemes exist
here below : some combinations and denominations
information
in pupil plane
fringes
filtered light
information
in image plane
flat tint
dispersed light
we consider here only the two main encoding schemes
103
IRAP conferences
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104
feb 2010
fringe pattern in filtered and in dispersed light
fringe pattern
filtered light, (x,y) mode
T1
T2
d
d
diffraction
grating
camera
sensitive area
in image plane
spatial
multi-axial
mode : x-l
l
Dl
x x
dispersed
Young's fringes
dl
l
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Yves Rabbia – OCA-UNSA
modulated flat-tint, pupil plane
superimposition of output pupils
d
d
ring pattern
1
reminder :
laboratory Michelson interferometer
nearly just the same
_1
T1
separation is made ZERO
mono-axial mode
105
feb 2010
images of pupils
onto the detector
(l/4)
detector
2
BS
d = d1 - d2
measure
signal s(d)
time modulation
of Optical Path Difference (l/4)
OPD =l/2
T2
detector only sees the center
of the ring pattern
where bright/dark alternate
IRAP conferences
Yves Rabbia – OCA-UNSA
106
feb 2010
modulated flat-tint, pupil plane _2
how measure a fringe contrast ?
encoding using time modulation of optical paths,
yielding intensity modulation
S(d)
collector
correlator
Detector
l/2
Demod
modulation
demodulation
synchronous detection
signal : modulated energy
modulated optical path
time
d
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Yves Rabbia – OCA-UNSA
extracting information
x
107
feb 2010
academic case_1
interfr. l/B
x
D
B
z
x
l/D
F
intensité observee : figure de franges
info = taux de modulation = VISIBILITE ou contraste
B x 

I ( x )  Airy( x ) . 2. I .  1  V . cos ( 2. . . ) 
l F 

V 
Oˆ ( B / l )
Oˆ (0)
illusoire de faire C = (Imax-Imin)/(Imax+Imin), on passe par TF
TF
1/2
- B/l
1
D/l
B/l
f  x/l
IRAP conferences
Yves Rabbia – OCA-UNSA
extracting information
academic case_2
B x 

I ( x )  Airy( x ) .  1  V . cos ( 2. . . ) 
l F 

V
ˆ iry( f ) 
Iˆ ( f )  A
d
(
f
)

*
2

108
feb 2010
Oˆ ( B / l )
V 
Oˆ (0)
B
B  

. d ( f  )  d ( f  ) 
l
l  

ˆ iry( f )  V . A
ˆ iry( f  B )  V . A
ˆ iry( f  B )
ˆI ( f )  A
2
l
2
l
pic BF
pic HF
pic HF
f
V 
2 .  picHF . df
 picBF . df

V=
energie _ coherente
energie _ incoherente
x
TF
l
f
IRAP conferences
Yves Rabbia – OCA-UNSA
extracting information
feb 2010
109
modulated flat-tint
two approaches :
modulation l/2 and sequence of interferograms
s(d)
s(d)
d< l/2
modulation l/2 :
d: off coherence
average "s"
d
l/2
s max
energie _ coherente
s min
V 

energie _ incoherente
s moyen
Q.
Q = shape factor
sequence of interferograms (sausage pattern)
d
amplit modulation
time
IRAP conferences
Yves Rabbia – OCA-UNSA
managing with data
feb 2010
110
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feb 2010
data produced with a 2-telescope interferometer_1
111
an interferometer does not measure angular diameters
a
2-T- interferometer measures raw-Visibilities (modulus of )
after a long and tedious process for
calibrating the interferometer response and unbiasing of data,
it gives an estimate of the true-visibility
(one component of the spatial spectrum of the source)
the job of the interferometer is to sample the spatial spectrum
of the brightness distribution of the source
one baseline , one component
V(B/l)
baseline after baseline is built a "visibility curve"
or a "visibility surface"
u= B/l
when baselines of various orientations are used
frequently said : interferometer samples the u-v plane
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
data produced with a 2-telescope interferometer_2
what to do with visibility curves or surfaces ?
112
V(B/l)
from a model, expected for the source
a family of theoretical visibility surfaces
are calculated according to parameters of the model
(examples : angular diameter, angular separation of binaries)
u= B/l
a model-fitting process yields
parameters of the selected model,not the ones of the star
a typical visivility curve
(model : uniform disk, UD)
model-fitting
V(u)
V(u)
first
lobe
u
second
lobe
u
best fit ( in blue)
gives the
diameter of the UD
(not a star has a UD)
IRAP conferences
Yves Rabbia – OCA-UNSA
113
feb 2010
a pictorial for some couples (source,visibility)
visibility
curve


V(B/l)
visibility
curve

a
a
a
V(B/l)
1
V(B/l)
1
B/l
B/l
B/l
B/l
B/l
source
1
1
1
1
V(B/l)
V(B/l)
V(B/l)
V(B/l)
a
a
a
a
source




1
B/l
B/l
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Yves Rabbia – OCA-UNSA
114
feb 2010
a non-realistic but illustrative example
q1
O(a) = O*(a) + Oenv(a)
q 2 internal
q 3 external
TF(O*)
B/l
TF(Oenv)
q3
camembert 3 – camembert 2
V(B/l)
1
Penv/Ptotal
P*/Ptotal
q2
here visibility curve yields an estimate of P*/Penv
B/l
IRAP conferences
Yves Rabbia – OCA-UNSA
visibility playground : phenomenology
V(B/l)
1
lobe 1 : features larger or equal to diameter
separations, enveloppes, diameters
115
feb 2010
1
2
3
B/l
lobe 2, 3,... : features smaller and smaller than diameter
limb darkening and rotation (oblateness), photospheric features
baseline orientation: asymmetries
wavelength :
change of resolution scale
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
116
examples of visibilities from observation
visibility (log)
courtesy Steph. Sacuto
preliminary fit
with model
Uniform Disk
lobe 1
lobe 2
spatial frequencies (1/arcsec)
departure from UD model
look for
limb darkening ?
enveloppe ?
Surface Brightness Asymetries ?
IRAP conferences
Yves Rabbia – OCA-UNSA
117
feb 2010
spectro interferometry
up to now, only one wavelength used
observing fringes in dispersed light provides chromatic visibilities
possible chromatic morphology can be exhibited
Visibility ( l, B / l )
observed source
morphology
in spectral
continuum
morphology
at a given
wavelength
l0
lowered visibility at l0
l
l0
B/l
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more than 2 telescopes,
118
feb 2010
more than one baseline
N telescopes  N.(N-1)/2 baselines
N.(N-1)/2 spatial frequencies simultaneously
B23
v

B12/l
B31
B12
x
B23/l
u
B31/l
moreover :
as soon as we have 3 telescopes combined
some information pertaining to the phases of nk can be recovered
from the composite fringe pattern
namely : observation yields a phase closure ( a number)
which is clos = 12  23  31
IRAP conferences
Yves Rabbia – OCA-UNSA
phase closure _ 1
119
feb 2010
preliminary comment : the context
2 telescopes, 1 baseline :
sampling u-v plane along a line
several orientations :
sampling along several lines
N telescopes , N.(N-1)/2 baselines used independantly :
as many (u,v) components
there, no information regarding the phase of
exception in some cases via spectral dispersion
 (complex)
N telescopes used simultaneously : composite fringe pattern
sampling N.(N-1)/2 (u,v) within u-v plane in one snapshot
phases respective to individual baselines yet not available
BUT
as soon as 3 telescopes are combined
the sum of phases can be extracted from fringes
this sum named "phase closure" provides constraints on
models to fit on the spatial spectrum
IRAP conferences
Yves Rabbia – OCA-UNSA
phase closure _2
feb 2010
120
constraint on what ?
phase closure allows diagnosis on asymetries
in short :
centro-symmetry of the brightness distribution :
all phases are zero , phase closure is zero
departure from symmetry, phase closure NOT ZERO
example (academic) : source : O ( a ), spectrum Ô ( u )  Ô ( u ) . exp( i  )

O ( a )  d ( a )  h . d ( a  r )  d ( a  r ) 
a Ô ( u )  1  h . exp( i .2 .u . r )  exp(  i .2. .u . r ) 
Ô ( u )  1  2.h . cos ( 2 .u . r ) REAL ! , phase ZERO , u

a
O ( a )  d( a ) h.d( a  r )
Ô ( u )  1  h . exp( i .2 .u . r ) a priori COMPLEX
phase non ZERO
hep !, simply Fourier parity properties
IRAP conferences
Yves Rabbia – OCA-UNSA
phase closure _3
121
feb 2010
OK, a little short !
how to perform ?
and what, as for using it ?
B23
for individual baselines,
atmospheric turbulence corrupts phase information B31
but for a closed network this corruption
is eliminated and we can extract (measure) :
h
x
B12
clos  12  23  31
any asymmetry results in a double photometric barycenter,
say separation r (vector)
2   
h




Arg
1

h
.
exp(
i
.
.
B
.
r
)

B ,l
so we find


l
B ,l
involving l and vectors B and r
1

WARNING :
clos actually
is not a single number, it is a function of
wavelength and time (baseline network and source evolution)
2
l
 
. B .r
IRAP conferences
Yves Rabbia – OCA-UNSA
122
feb 2010
V (1/l)
phase closure _4
even with a centro symmetric object
phase closure is not always zero
it is 0 or  depending on l
phase of  is present
in UD visibility curve
chromatic phase closure with UD
3 baselines (3 visibility curves)
change of sign occurs
when crossing a "zero visibility"
phase closure = sum of 0 and ,
varying along the spectral interval
steps from 0 to  trace pure UD
=0

0.8
Base1
0.6
Base2
Base3
0

0.4
0.2
phasclos
0
0.2
l
1/l
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
phase closure _5
departing from steps (0,)
indicates complex structure
various assumptions depend on which visibility lobe
is considered
123
IRAP conferences
Yves Rabbia – OCA-UNSA
124
feb 2010
using phase closure : (very short)
numerical simulations and observations (thanks Steph Sacuto)
TW Oph, second lobe
10
100
phase
closure
UD
degrees
model
nearing second lobe
phase
closure
-50°
degrees
spot
0°
spot
1.5
100°
with help
of observations
2
-100°
l (mm)
time effect
same object
same network
different dates
1.5
2
-150°
l (mm)
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
VLA, Soccoro desert, New Mexico, USA
27 antennas
356 spatial frequencies simultaneously
125
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
difficulties in real world
and some remedies
126
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
127
some difficulties
prime goal : reliability of data
accuracy
no bias
precision
small error bars
sensitivity
reaching faint visibilities
reproducibility
robust measures
three key points
stability : any departure from nominal degrades measure
calibration : mandatory compensation for degradations
Signal to Noise Ratio : reliability of measures
several regime of difficulties
methodology, operation, data exploitation for science
IRAP conferences
Yves Rabbia – OCA-UNSA
illustrations
V(u)
1
O(a)
accuracy
qmesur = too big
a
q
precision
O(a)
1
a
O(a)
sensitivity
128
feb 2010
O(a)
?
a
a
1
V(u)
V(u)
l/qmesur
u
l/q
binarity : unseen
u
structure : undetected
V min reachable
u
IRAP conferences
Yves Rabbia – OCA-UNSA
129
feb 2010
causes and nature of degradations in V measured
deterministic causes
non-nominal adjustment of configuration parameters
alignements optiques, aberrations
échantillonnage des franges (spatio-temporel)
metrologie ( maintien à zero, de la ddm)
désequilibre photometrique (I1  I2)
uncompensated effects of external causes
Vmesur = q.V
with q <1
but q stable
refraction differentielle atmospherique
conjugaison de pupilles (entrée / sortie)
orientation des polarisations (vecteurs champs non \\)
calage de la base
random causes
any instability of instrumental parameters
But atmosphere is the main cause for ground-based
piston, tip-tilt, speckles
Here q is random,
specific processing (statistics) needed
Vmesur = q.V
with q <1
but q random
IRAP conferences
Yves Rabbia – OCA-UNSA
130
feb 2010
the major random cause : atmosphere
f(x)
atmosphere !!
random phase screen
involving space and time
P : piston
phase noise
P
fringes
constantly and
randomly moving
tél.1
x
tél.2
TT
TT : tip-tilt
uncomplete superimposition of images
Speckles :
r0
D
PSF rather like this
and change every ms
l / r0
IRAP conferences
Yves Rabbia – OCA-UNSA
piston, tilt : V
under-estimated
x
x
(t)
I(x,l,D)
131
feb 2010
a.F
2.D(t) /l
@1

@2
a
x
Ecoh(t)
time
Eincoh
I ( x)
V(t)
x
IRAP conferences
Yves Rabbia – OCA-UNSA
132
feb 2010
illustration : intensity in presence of speckles
randomly changing
time-scale : millisecond
a
I(a)
I(a)
D
l/B : interfringe
a
l/D
l/r0
IRAP conferences
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feb 2010
133
degradation of measured visibility
the measured visibility is n ot the true (astrophysic) visibility
bad estimation of spatial spectrum
each cause "k" of degradation is traced by a degradation factor "qk"
varying between 0 and 1 (never 1,actually)
a relation must be considerer :
Vmeasure = (q1.q2…..qK). Vtrue
Vmeasure = (response to visibility).Vvraie
challenge to face :
Vtrue must be recovered and the qk
must be made as close as possible to 1
Note :
the "response to visibility" varies with time (observing conditions unstabilities)
and not only because of atmosphere
IRAP conferences
Yves Rabbia – OCA-UNSA
some remedies
calibration of data
adaptive optics
fiber linked telescopes
feb 2010
134
IRAP conferences
Yves Rabbia – OCA-UNSA
adaptative optics NACO VLT
let's go to movies
feb 2010
135
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
monomode optics fibers
even with adaptive optics wavefronts have residual distorsions
it is possible to make the wavefront quasi perfect in shape
by sending the beams through monomode optics fibers
that performs spatial filtering
the price is "less photons" but the ones you keep are
the "efficiently interfering" photons
136
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
quick-look at
some technical responses
137
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
speckle interferometry - 1
the problem : theoretical resolving power destroyed
short exposure ( 1 ms)
long exposure ( > 100 ms)
classic imaging
here we can hopefully
retrieve some
theoretical resolution
here we cannot
how can we fight ?
*
*
=
138
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
139
how can we restore the theoretical resolution ?
optimistic (thus too much candid with interferometry) :
perform a FT on the instantaneous image
too bad ! frames look like this 
adding images ? (again exceedingly candid)
everything is moving
We have a randomly unstable
Transfert function beyond
a (low) frequency set by the
atmospheric turbulence
high frequencies lost
so what ?
adding what remains stable in spite of turbulence
autocorrelation or squared modulus of the FT
then , when summing, random features do not mutually cancel
and some high frequency information is saved
D/l
IRAP conferences
Yves Rabbia – OCA-UNSA
140
feb 2010
Occultations d'étoiles par la Lune - 1
le phénomène :
la lune dérive
par rapport au grand manège,
elle vient cacher des étoiles
20 h
à cause de la diffraction de Fresnel
la lumière de l'étoile projette sur Terre
des franges de bord d'écran
21 h
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
141
Occultations d'étoiles par la Lune - 2
l'observation du phénomène :
les franges se déplacent
devant le telescope
vitesse de l'ordre de 700 m/s
l'observable est une courbe de lumière (bruitée) :
signal photométrique
donnant la
puissance collectée
par le télescope
au cours du passage
des franges
(durée typique
du passage
0.1 seconde en visible)
S(t)
t
IRAP conferences
Yves Rabbia – OCA-UNSA
142
feb 2010
Occultations d'étoiles par la Lune_3
un interféromètre grand comme ça !
l'exploitation du phénomène
lune
le profil des franges est donné par la convolution
Ifranges (a) = O(a) * R(a)
=
*
lune
*
=
l'information HRA est dans la forme
de la courbe (contraste et enveloppe des franges)
on peut dire qu'on a un interféromètre
l'interféromètre d'Augustin Fresnel
base de
l'interferometre
telescope
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
143
Occultations d'étoiles par la Lune_3
interféromètre grand comme ça ..... et instrument petit comme ça ?
le profil des franges est donné par la convolution Ifranges (a) = O(a) *
R(a)
plus grosse l'étoile, plus faible est le contraste
convolution  lissage
*
*
=
=
le profil enregistré est donné par la convolution avec T(a) (telescope)
Itelescope(a) = Ifranges(a) * T(a)
alors plus petit le diamètre, plus fidèle l'enregistrement ?
*
=
peut être, mais plus faible le rapport signal à bruit !!!
moins fiable l'exploitation des franges, moins bon l'interféromètre
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
nulling interferometry
and coronagraphy
144
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145
feb 2010
definitions and broad lines of the topic
nulling techniques belong to the more general domains
High Angular Resolution
Very High Contrast Imaging
what ?
a tentative definition for VHCI
set of instrumental methods and devices dedicated to
the study (morphology)
of faint emitting sources
in the close environment of a point-like source
"Nulling Techniques" are those methods based on
the coherence of light and destructive interferences
alternative approaches to the same goal exist
and will only be evoked
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Very High Contrast Imaging _ 1
the morphology paradygm to tackle
an unresolved source with closely surrounding matter
which morphology is looked for
and which emitted flux is largely fainter than the source flux
galaxies
stars
planeto
AGNs,...
circumstellar features (mass loss)
binaries with faint companions
exoplanets
protoplanetary or post planetary disks
multiple asteroïds
unfortunately
central source is blinding
and prevents detection of
surrounding sources
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feb 2010
Very High Contrast Imaging _ 2 the problem : "see around"
typical target : central point-like source on-axis, and features around
I(a)
a
examples of surrounding features
a
a
a
here we will focus on : star + faint companion
and even more specifically : star + planet
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
VHCI _ 3 a first need : rejection on-axis
T
must manage to remove the starlight from the image,
occulting lobe or
Inner Working Angle
T
rejector
system
camera photometric dynamics
saturated
by starlight
Dyn
rejection of starlight =>
recovering photometric dynamics
for faint features
to overshoot noisy background
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VHCI_4
Yves Rabbia – OCA-UNSA
feb 2010
a second need : angular resolution
such interesting features as planets
or ejected matter (mass loss tracing)
are very close to the central source :
occulting lobe must be narrow enough to
avoid removing light from planet
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non-direct / direct
feb 2010
:
150
what does that mean ??
non-direct detection
the planet is revealed
by its effects on the light coming
from the parent-star
and this light
is the one to analyse
direct detection
what we want now is
the light from the planet itself !!
and especially
NOT AT ALL the one from the star
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
non-direct methods : a quick-look
151
assessing the presence of a non-visible companion
we observe the parent star

analysis of motion :
variable position : perturbated proper motion, astrometry
variable radial velocity : spectral lines motion, velocimetry
pulsar : perturbation of pulse periodicity,
timing

monitoring of brightness
occultations / transits
micro-effect of gravitational lensing
all that, rather out of the scope of the talk
photometry
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
direct methods
no other light wanted than the one from the planet
here is the
BIG problem
requiring to conceive, develop and operate
specificically dedicated methods and instruments
among them
coronography and nulling interferometry
strange terminology, we come back later to that
152
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feb 2010
observations : what to expect ? short overview
assume we have planetary photons !
why did we want them ?
why can we do with them ?
flux and spectrum (depending wavelength domain)
provide pieces of information pertaining
to (among others) such questions as
 temperature
 presence of an atmosphere
 search for life
 chemistry, physics and climate diagnostics
 albedo
 seasonal changes
 ....
153
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feb 2010
154
immediate constraints to get planetary photons
photometric dynamics ( very high contrast)
need to tackle very large flux ratio (star/planet)
angular resolution
need to separate star and planet, very close objects
photometric sensitivity
planet = very faint object, few photons
very large ?, very close ?, very faint ?
what does it mean ?
need numbers !!!
IRAP conferences
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feb 2010
photometric dynamics (contrast)
key parameter :
ratio Rflux = flux star/flux planet
"hot" jupiters
exoplanets of Pegasides type
Rflux 104, 105
exo-earths
mid IR, : millions 106, 107
visible : billions 109, 1010
9
6
IRAP conferences
Yves Rabbia – OCA-UNSA
angular resolution : example earth-sun
1 arcsec : 5.10-6 rad
a green pea seen at
distance 1 km
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feb 2010
1 arcsec
1 U.A.
obs
1 parsec

3 A.L.
O.1 arcsec
1 U.A.
obs
10 parsec
O.01 arcsec = 10 marcsec
obs
1 U.A.
100 parsec
Rayleigh criterium : separation must exceed  l /DiamTel
l
mm
requested
diameter
for 100 parcsec
0.6

10
2.2

40
11

200
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feb 2010
photometric sensitivity
example : Sun-earth at d = 10 psc
(about 30 AL)
collected power from planet (l) = 10-n .Stel.Dl.transm. Fluxstar(l)
visible n = 9,
thermal IR n = 6
V band lvis = 0.55 mm
telescope
D (m)
N band
Number photons/sec
S (m2)
Sun 10 psc
V
10
3.6
lir = 10.2 mm
N
Earth
V
100
1010
1.5 109
10
10
109
1.5 108
1
short-cut :
106 flux ratio magnitude gap 15
109 flux ratio  magnitude gap 22 to 23
N
1500
150
warning
atmosph
radiation
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
158
technical answers to science requirements for exo-earths _1
very high contrast :
dedicated instruments : coronagraphs, nulling interfeometry
to be seen later on
key parameter :
rejection = ratio residual on-axis energy / collected energy
better performance :
go to space (no atmosphere : no background in mid IR, no turbulence ) !
(though ground-based projects remain under study) :
photometric sensitivity : large collectors, high quality detectors, ...
spectral coverage : again go to space
and achromatic rejection (large working bandwidth)
IRAP conferences
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feb 2010
technical answers to science requirements for exo-earths _2
angular resolution :
need : narrowest extinction lobe , smallest l / D
spectral domain dependancy
visible : largest single aperture, 10 m class OK
mid infrared : interferometers required
interferometer: 2 or N telescopes which outcoming beams
are combined (superimposed on a same detector)
resolution is now l / B (instead of l / D)
significant gain, even with small telescopes (2m, 4m class)
note : accomodation of
several small telescopes
in space launcher
is currently feasible
B
combiner
IRAP conferences
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feb 2010
material focal mask : stellar Lyot coronagraph
telescope transmission is corrupted by a focal mask
so as to make an occulting lobe
example : Lyot-type coronagraph (originally for Sun, 1936)
_1
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
nulling coronagraphy
161
incident wave is divided in two "sub-waves" (beamsplitter : BS)
extinction of star (on-axis) is obtained
by destructive interference between recombined sub-waves (BS again)
destructive interference obtained by inserting  -phase shift
what about the planet ?
it escapes this "nulling process" because the incoming wavefront is tilted
and induce an extra phase shift (see later on)
image
S

path 1
R
path 2
S : separation by amplitude "splitting"
R : recombination for interference
the "transmission map" set an occulting lobe (IWA) in center of field
we have a non-material mask, or a "coherence" mask
IRAP conferences
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feb 2010
nulling interferometer ( 2 telescopes, Bracewell concept)
Tel 1
Tel 2

encoded
information
R
R
differences with interfero coronagraphs,

 the interferometer explores the sky with a grid
(non connex occulting lobe, fringed lobe)
 resolution (spacing of the grid) now is l / B instead of l / D
occulting
open
occulting
open
occulting
coronagraph
conventional interferometry
nulling interferometry
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
163
nulling interfero ? a provocative short-cut
basically
the interferometer simply built
by destructive interferences
an appropriate transmission map
this map is used like a sift,
through which only planetary photons
are (ideally) allowed to pass
nulling interferometry is but
while
photometry through a sift !
stellar photons (ideally)
are blocked
utterly null !
IRAP conferences
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feb 2010
additional constraints
exo-zodiacal matter :
planet immersed in comparatively bright exo-zodiacal cloud
appropriate modulation needed (discriminate contributions)
free flying telescopes :
formation flight to be controled
relative positions : needed accuracy few millimiters (laser auxiliaries)
BUT at recombination, nanometer accuracy needed
(not the job of flyers)
achromaticity of  phase shift :
highly accurate phase shift ( < 10 –3 rad ) needed for 10-6 rejection
and this over the whole Dl
to allow large working bandwidth (more photons from planet)
achromatic phase shifters are the heart of the instrument
several technical constraints to be added (see later)
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feb 2010
Bracewell : sensing along one_direction only
where is the planet ?
for exo-earths we may have
an idea of the angular separation star-planet
from the "habitable zone" constraint
but orientation of the couple is unknown
?
?
need to explore all orientations
for example by making the interferometer
rotate as a whole around pointing axis
(Bracewell concept)
extra interest : modulation of planet signal
( synchronous detection)
not really appropriate
time and energy consuming, technically not easy
and problem with exo-zodiacal matter remains
courtesy
Olivier Absil
IRAP conferences
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feb 2010
166
Bracewell concept : contamination of exo-earth signal
modulation of planetary signal by global rotation
problem : exo-zodi also modulated
heavy contamination
detected signal
p
ez
z
instrum
time
possible solutions
 more than 2 telescopes
( non symmetrical configuration)
 internal modulation
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
a schematic example of internal modulation (no rotation)
just "hand waving" approach
two Bracewell Nullers with perpendicular baselines
and so
two fringed maps working together
periodic  phase shift added in one nuller
one grid remains untouched, the other is reversed
so as to
keep star in dark zone
planet periodically seen and unseen
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more than 2 telescopes



BUT
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feb 2010
interest and difficulties
stiff edges of transmission profile (star leakage)
internal modulation (no rotation needed)
non_symmetrical configuration (exo-zodi)

several phase shifts on interferometer respective arms,

flight formation control (mm accuracy, baselines hundreds of meters)
not simply , but fractions of 2 (technical challenge)
1
R
T(a)
2
..
N
..
q*
a
( sky)
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
configurations, transmission maps, modulation
169
both map and modulation processes depend on the number of telescopes
and on
their relative positions
modn
IRAP conferences
Yves Rabbia – OCA-UNSA
key system : the APS's
170
feb 2010
Achromatic Phase Shifters
function : inserting the appropriate phase shift
in each interferometer arm
1
N
2
R
..
..
example for a 2-telescope interferometer
need :  nominal phase shift
science requirements
targeted null depth 10
–5
, stability level 10
–6
over....days
R

subsequent specifications :
 phase shift accuracy at 0.001 radian over large bandwidth
 intensity relative mismatch at 0.001
 optical paths balance at less than few nanometers
 Wavefront quality at recombination at few nanometers level
only achievable bu using spatial filtering ( optical fibers)
IRAP conferences
Yves Rabbia – OCA-UNSA
key system : the APS's
171
feb 2010
Achromatic Phase Shifters
function : inserting the appropriate phase shift
in each interferometer arm
1
N
2
R
..
..
example for a 2-telescope interferometer
need :  nominal phase shift
science requirements
targeted null depth 10
–5
, stability level 10
–6
over....days
R

subsequent specifications :
 phase shift accuracy at 0.001 radian over large bandwidth
 intensity relative mismatch at 0.001
 optical paths balance at less than few nanometers
 Wavefront quality at recombination at few nanometers level
only achievable bu using spatial filtering ( optical fibers)
IRAP conferences
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feb 2010
172
dedicated instrumentation : an example ESA-Darwin
possible instrumental options in response to science needs
target : direct detection and spectroscopy of exo-earths
signatures of bio-activity
(carbon chemistry)
very small angular separation (star-planet) , range < 0.01 arcsec
flux ratio : flux star/flux planet tremendously high (> 10^6)
thermal InfraRed
large bandwidth ( 6-18 mm)
interferometry
"nulling" techniques
ESA choice : nulling interferometry in thermal InfraRed
IRAP conferences
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feb 2010
173
ESA-Darwin : a sequence of configurations
key-word MONEY ! !
proposal July 2000
proposal Oct 2004
Darwin estimated cost (2007):
over 650 Meuros
the most recent (2007)
X_Emma (Lady Darwin)
proposal to
ESA Cosmic Vision
a discarded precursor :
Pegase
proposal
to CNES for
flight formation
and Science
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
other projects for
coronagraphy and nulling interferometry in space
NASA
 James Webb Space Telescope
 Terrestrial Planet Finder- Coronagraph
 Terrestrial Planet Finder- Interferometer
Europ
TPF corono
174
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175
what is done ?
over roughly the last decade, several institutes and major level integrators
have built teams, studies and "know-how" via consortia.
USA : NASA TPF-I
Europ : ESA-Darwin
Jet Propulsion Laboratory, .....
ThalesAleniaSpace, Keyser Threde, MPI (Heidelberg),
Institut d'Astrophysique Spatiale (Orsay),
ONERA ( Chatillon), Obs Paris, Obs Côte d'Azur, .......
moreover : Darwin and TPF-I initial designs have merged
in a common concept, mixing teams is considered,
cost sharing mandatory new partnerships foreseen (Japan, India, ...)
on-going Research & Development (up to recently)
achromatic phase shifting :
test benches : nulling depth , large bandwidth and stability
flight formation control :
simulation with air-sustented vehicles, (JPL)
metrology : laser link formation control (OCA)
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feb 2010
R&D : achromatic phase shifters
generic schemes : Young'sType or Mach-Zehnder test-benches
detector
DL

DL
0
source
DL

DL
0
DL = delay lines
monomode optic fibers mandatory ( when existing at working l )
for spatial coherence of the source and clean outcoming wavefronts
Thales : 10-5, 5% at 1.55 mm,
stability > 1 hour at s =10-7
(2006)
Planet Detection Testbed
JPL : 10-5, 32% at 10 mm,
stability 2 hours
(2008)
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feb 2010
R&D : flight formation control
modulated laser link ( OCA)
under way
telescope
emiter/
receiver
recombiner
specifications
longitudinal resolution
~ 1 mm (absolute) per link
~ 1 nm (relative between interferometric arms)
global: ~ 1 mm lateral shift , angular error box ~ 10 mas
air-sustended vehicles (JPL)
operational since 2007
Formation Control Testbed
corner
cube
IRAP conferences
Yves Rabbia – OCA-UNSA
feb 2010
today and later on
ESA Cosmic Vision 2007 did not select Darwin for next slot
launch postponed, no new decision foreseen before 2015 (?)
funding for both TPF-I and Darwin is stopped
R&D continuing ? techno test bench PERSEE (France)
envisioned collaboration still under way ?
return to techno precursors ? Pegase, proba-3 ?
after meeting in Barcelona, 14-18 sept 2009
not yet official
Nulling Interferometry in Space is rather frozen
not to be considered before years
but maintening R&D seems favourably considered by ESA (?)
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titre
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feb 2010
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supersynthesis
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feb 2010
u-v coverage
during observation the baseline, as seen by the star, is changing
and so do the explored spatial frequencies (u,v)
since they are determined by
baseline projected over the sky coordinates
this effect, extending the range of explored spatial frequencies,
is named "super synthesis"
To know how the u-v plane is sampled it is necessary to know
the variation of the measured spatial frequencies
100
maps of this sampling along time
can be calculated
(currently nowadays
by using software packages)
declin
 20
lat  70
B  100
v
X   46.985
50
Y  81.38
Z  34.202
0
50
0
50
u
100
-6<H<+6
d = 90° - lat = 20°
 = - 120°
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feb 2010
u-v coverage overview_1
the geometry
of the baseline-star
system
D

u and v
are components of
the projected baseline
onto the u-v plane
w is only concerned
by optical path in excess
which is compensated
within the instrument
toward
celestial
North pole
a
earth
N
w
local
South
u-v plane is
perpendicular
to D
B/l
v
u
East
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feb 2010
u-v coverage overview_2
N
components (X,Y,Z) of the baseline
are refered to equatorial coordinates
B/l
d
X 
 cos d .cos h 
B
 


Y

.

cos
d
.
sin
h
  l 

Z 


sin d
S
components (u,v,w)
of the projected baseline are refered to
equatorial coordinates, via a matrix product
involving (X,Y,Z) and star coordinates
sin H
u  
  
v    sin d . cos H
w   cos d . cos H
cos H
sin d . sin H
 cos d . sin H
0  X 
  
cos d  . Y 
sin d  Z 
latitude of observatory is involved
and is related to "d"
H
X d
h Y
celestial equator
E
N
d
w
d
H
v
Y
H
X
u
E
IRAP conferences
imaging..... ?
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183
interferometric imaging requires to come back
from a sampled u-v plane to a brightness distribution
in other words : from Fourier space to normal space
the currently used approach still is model-fitting
(exceptions for binaries)
individual phases would be of great help
phase closure also is indirectly helping
the straight way (inverse Fourier from sampled spectrum)
is difficult and unsafe in optical interferometry
(sparse sampling, uncertainties on visibility,
uncomplete phase information, "non-regular data (?)", ...)
this approach is somewhat claimed to be an "ill-posed inverse problem"
radio interferometrists have done much work in image recovery
(theory and results) and have achieved great success with VLA