lecture 2 0:lecture 2.0: optical interferometryoptical...

Lecture 2 0:Lecture 2 0:Lecture 2.0:Lecture 2.0:

Optical InterferometryOptical Interferometry::

A Gentle Introduction to the Theory (or likely a Reminder )A Gentle Introduction to the Theory (or likely a Reminder )A Gentle Introduction to the Theory (or likely a Reminder…)A Gentle Introduction to the Theory (or likely a Reminder…)

В. Г. Турышев В. Г. Турышев Jet Propulsion Laboratory, California Institute of Technology

4800 Oak Grove Drive, Pasadena, CA 91009 USAГосударственный Астрономический Институт им. П.К. Штернберга

Университетский проспект, дом 13, Москва, 119991 Россия

Курс Лекций: «Современные Проблемы Астрономии»для студентов Государственного Астрономического Института им. П.К. Штернберга

7 февраля – 23 мая 2011

INTRODUCTION TO OPTICAL INTERFEROMETRYINTRODUCTION TO OPTICAL INTERFEROMETRY

Overview

• Interferometers:– Interferometers vs telescopes

History and present day– History and present day

• Electromagnetic wave propagation– Wave description– Young’s Double Slit (YDS) interferometer

• Two-element astronomical interferometer– Case of monochromatic radiation– Polychromatic radiation – What interferometer does?

• Fringes and parameters of interestges a d pa a e e s o e es– Van Citter-Zernike theorem– Fringe visibility

Interferometric Imaging– Interferometric Imaging

• Examples…


Ordinary Telescope – Optical, IR, Radio

• An ordinary telescope:– collects electromagnetic radiation, forms an image which is then analyzed

to determine the nature of the object in the field of view. – The larger the collector, the better the resolution and the greater the

energy delivered to the focal plane (assuming no atmosphere).

Telescope Resolution SensitivityTelescope

detector Telescope λ/D ~D

λ = wavelengthλ wavelength

diameter (D) Airy Disk


Michelson Stellar Interferometry

• An interferometer:– combines the light from two or more small telescopes to yield the angular

resolution of a much larger telescope. – BUT… it doesn’t collect as much light as a single collector with the same

diameter as the baseline.

Resolution Sensitivity

Interferometer λ/B ~ √(2dB)

λ = wavelength

Interferometer

λ wavelength

baseline (B)

Telescope (d) Telescope (d)Combiner & detector

FringeFringe Pattern


Telescope and Interferometer Comparison

• A telescope: collects and focuses light for analysis. • An interferometer: collects the light from several small telescopes &

combines its waves for analysis but never forms an image of the objectcombines its waves for analysis but never forms an image of the object.

Resolution Sensitivity

TelescopeTelescope D DInterferometer 2*d*B)

e escope

detector

I t f tInterferometer

Telescope (d) Telescope (d)Combiner & detector

diameter (D) baseline (B)& detector


Interferometry - The Path to High Resolution

• Interferometers can combine light from two or more mirrors as if they were pieces of a single large telescopepieces of a single large telescope.

• Albert Michelson used interferometry to extend the capabilities of the then largest telescope in the orld onlargest telescope in the world on Mt. Wilson in Southern California.


Present Day Telescopes & Interferometers

• Interferometric techniques were adopted by radio astronomers starting i 1946 th k t M ti R l (N b lin 1946, thanks to Martin Ryle (Nobel Prize, 1974) & Joseph Pawsey.

Arecibo Observatory, Puerto Rico

B ll Ob tBracewell Observatory,Stanford University


Changing collector separation changes the fringes …

Michelson measured star sizes by watching their fringes change w/ collector separation:

Small Separation Low Resolution

Large Separation High ResolutionHigh Resolution

The fringes change depending on how close the apertures appear as seen

Fringes Fringesapertures appear as seen by the star. The view from the star shows the different

separations of the t l ( ll t )telescopes (yellow spots)

feeding the combiner.


Keck I and II (and Subaru) on Mauna Kea, HI,


Photogenic Testbed Interferometer

Credit: National Geographic


Palomar Testbed Interferometer (PTI)

Established as a TechnologyTestbed for the Keck Interferometer

First Fringe: July 1995First Sci Pub: August 1998

Ops Through 2006“Th M t S i tifi ll S f l“The Most Scientifically Successful

Interferometer in History”

• PTI is a Near-IR (K & H-band) single-baseline interferometer

– Direct (homodyne) combination

• Single and Dual-Beam Interferometry:– Normalized visibility (V2) measurement

=> modelling (simple morphology like t t )– NS, NW, and SW baselines

– NICMOS array combiner– Point Src Limiting Mag K ~ 6.7

one or two stars)– Simultaneous fringe tracking on two

nearby stars => differential astrometry

– Scientific Limiting Mag K ~ 6.1http://pti.jpl.nasa.gov


Space Interferometry Mission


Electromagnetic Wave Propagation

• Light or Electromagnetic Waves are:1. Solutions to the Maxwell Equations2 M t ll ti ill ti i2. Mutually-supporting oscillations in

electric and magnetic fields3. A mechanism for coupling energy away

from a radiating source

4 H l b t t i l4. How we learn about astronomical sources

5. All of the above

Credit: http://www-istp.gsfc.nasa.gov/Education/wemwaves.html


Electromagnetic Wave Description

• We commonly describe (plane-parallel) EM waves with a simple exponential model

( , ) cos[ ]

Re[exp( [ ])]

E x t E k x t

E i k x t

p p

• We implicitly understand that we take the Real part of the

Re[exp( [ ])]

Re[exp( [ ]) exp( )]

Re[ exp( [ ])]

E i k x t

E i k x t i

E i k x t

we take the Real part of the complex quantity

[ p( [ ])]

exp( [ ])E i k x t

• Pertinent wave descriptors:– Wave vector k (wavenumber k)– Wave frequency and

l h

2 2k n n n

c c

wavelength – Angular frequency – Wave speed c

2

c


Power in an Electromagnetic Wave

• The amount of power in an electromagnetic wave is

2 ( , ) ( , ) ( , )P E x t E x t E x t

electromagnetic wave is proportional to the square of the field strength

( , ) ( , ) ( , )

• In this “complex” model, computing wave power is

*

*

exp[ ( )] exp[ ( )]

exp[ ( ) ( )]

E i k x t E i k x t

E E i k x t i k x t

computing wave power is performed by a conjugation-product operation

*

p[ ( ) ( )]

E E

*~


Young’s Double Slit (YDS) Apparatus

• Young’s Double Slit is a classic physics

i texperiment demonstrating wave properties of light

• Originally due to Thomas Young c 1801Thomas Young, c.1801

• Most of you would have seen this in undergraduate physics

Image Credit: www.ndeepak.info


YDS Geometry in 2-D

Obscuration Screen

• Notional YDS geometry:– Slits S and S separation b obscuration-screen distance a– Slits S1 and S2, separation b, obscuration-screen distance a– Notation taken from Born & Wolf (almost)


YDS – Composite Optical Field

Distant, monochromatic source

1 2 0Re[ exp( )]i t 1 0 1( ) exp( )exp( )x i t ikd

ScreenObscuration

net 1 1( ) ( ) ( )exp( )exp( )x x x

i t ikd

0 1

2 1

exp( )exp( )(1 exp[ ( )])

i t ikdik d d

~ xbd ~da


YDS – Time-Averaged Power

Time-Averaged Power at x proportional to:

net net( ) ~ ( ) ( )P x x x

0 exp( )exp( )(1 exp( )

exp( )exp( )(1 exp( )

kxbi t ikd i akxbi t ikd i

0

20

exp( )exp( )(1 exp( )

2 (1 cos( ) !!!

i t ikd i akxbi a

Fringes! x

Q: How does a photon decideQ: How does a photon decide which slit to go through?


Key Ideas: YDS (Key Ideas #1)

• Young’s slits sample two separate pieces of the incident radiation field

• Separate propagation paths from the slits to an observing screen leads to a superposition of field components with different phases

• These different phases cause the field components to “interfere” (add with different phases), creating interference fringes – variations of (time-p ), g g (averaged) power as a function of position along the observing screen

• Fringes vary in amplitude from “2” (constructive interference) to “0” (destructive interference)(destructive interference).

• Adjacent fringes are separated on the screen by x = a/b, in angle (from the obscuration) by = x/a = /b

Q: How does the presence of the slits change the propagation of the original wave, creating the interference pattern?, g p


A Two-Element Astronomical InterferometerDistant

• Sampling of the radiation field

Incident

Distant Source

• Transport to a common location

Phasefronts

• Compensation for geometric delay (s*B)

• Combination ofthe beams A2

x2

A2x2

Aperture 1

• Detection of theresulting output

Metrology

DelayLine 1

d

DelayLine 2

d2Beamd12

(s*B)/2BeamCombiner


A Two-Element Astronomical InterferometerDistant

• Telescopes located at x1, x2. Incident

Distant Source

• Baseline B = (x2-x1).

P i ti di ti

Phasefronts

• Pointing directionis S.

• Geometric delay(s*B)

• Geometric delayis ŝ·B, whereŝ = S/|S|.

A2x2

A2x2

Aperture 1

• Optical paths along two arms are d1 and d2.

Metrology

DelayLine 1

d

DelayLine 2

d2Beamd12

(s*B)/2BeamCombiner


Key Ideas #2

• Functions of an interferometer: Sampling incident radiation field– Sampling incident radiation field

– Beam propagation & optical path matching– Combination of electromagnetic fields– Detection of combined fields

• Nomenclature:Nomenclature:– Baseline– Pointing direction

G t i d l– Geometric delay


Distant

Collected Monochromatic Radiation

Incident

Distant Source

Phasefronts

2 2

1

exp[ ( )]

exp[ ( ( ) )]

A i k x t

A i k B x t

(s*B)

1 1exp[ ]( )A i k x t

1

1

exp[ ]

exp[ ]

ik B

iks B

A2x2

A2x2

Aperture 1

Metrology

DelayLine 1

d

DelayLine 2

d2BeamRecall: d12

(s*B)/2BeamCombiner

Recall:k = 2/ = /c


Monochromatic Output of a 2-element Interferometer (i)

• Propagated to combination, the E-fields from the two apertures can be described as:– 1 = A exp[ikd1] exp[-it] and 2 = A exp[-ik(ŝ B - d2)] exp[-it]

net = 1 + 2 = A [exp(ikd1)] + exp[-ik(ŝ B - d2)] exp[-it]

• Summing these at the detector we get:

* (exp[ ik(ŝ*B d )] + exp[ikd ])]

• The time averaged intensity, *, will be given by:

(exp[-ik(ŝ B - d2)] + exp[ikd1])]

(exp[ik(ŝ*B - d2)] + exp[-ikd1])

2 + 2 cos[k (ŝ B + d1 - d2)]2 2 cos[k (ŝ B d1 d2)] 2 + 2 cos (kD)

Note, here D [ŝ B + d1 - d2].This is a function of the path lengths, d1 and d2, the pointing direction s, and the baseline B.


Monochromatic Output of a 2-Element Interferometer (ii)

Detected power, P = * 2 + 2 cos (k [ŝ B + d1 - d2]) 2 + 2 cos (kD), where D [ŝ B + d1 - d2]

– The output varies ( ) i i d ll(co)sinusiodallywith D.

– Adjacent fringepeaks areseparated byd1 or 2 = ororŝ = /B.


Key Ideas #3

• The output of the interferometer is a time-averaged intensity.

It h i id l i ti th th “i t f• It has a cosinusoidal variation - these are the “interference fringes” – variations of (time-averaged) power as a function of pathlength difference

• The cosinusoidal variation is a function of kD, which in turn can depend on many things:

The wavenumber (k = 2/)– The wavenumber (k = 2/).

– The baseline, B.

– The pointing directions, ŝ.

– The optical path difference between the two arms of the interferometer, d1-d2.


What – Sources are polychromatic?!?

• (Of course), real sources radiate over a range (spectrum) of wavelengths• And real detectors are sensitive over a range of wavelengths…• So we must consider (add) all the monochromatic contributions toSo we must consider (add) all the monochromatic contributions to

compute the polychromatic responseSIM passband PTI passband


Response to Polychromatic Light

• To compute the polychromatic response we can just add (integrate) the previous (monochromatic) result over a range of wavelengths:– e.g for a uniform passband of 0 /2 (i.e. 0 /2) we obtaing 0 ( 0 )

0 / 2

0

0

/ 2/ 2

[2 2cos( )]

2 [1 cos(2 / )]

P kD d

D d

0 / 2

2 [1 cos(2 / )] D d


Response to Polychromatic Light (2)

• …evaluating the previous result over a range of wavelengths:– e.g for a uniform bandpass of 0 /2 (i.e. 0 /2) we obtain:

0

0

0

/ 2

/ 2/ 2

[2 2cos( )] P kD d

0

0

/ 2

/ 2

2 [1 cos(2 / )] D d

20

020

sin /1 cos/

D k DD

0sin /1 cos

/coh

coh

D k DD

So, the fringes are modulated with an envelope with a characteristic width given by the coherence length, coh = 2

0/.


Key Ideas #4

• The response for a polychromatic source is given by adding (integrating) the intensity response for each color. (Q: Why?)

This modifies the monochromatic interferometric response mod lating the• This modifies the monochromatic interferometric response, modulating the fringes with a “coherence envelope”:

– The “correct” (maximum) response is only when k [ŝ B + d1 - d2]) = 0.

– This path matching is the so called “white-light” condition.

This is the primary motivation for matching the optical paths in an interferometer and correcting for the external geometric delayinterferometer and correcting for the external geometric delay.

• The narrower the range of wavelengths detected, the smaller is the effect of this modulation:

– We describe the envelope size with the coherence length, coh = 20/.

– Narrower passband => less fringe modulation

– Narrower passband => less light!


Fringes and Parameters of Interest

• From an interferometric point of view the key features of any interference fringes are their modulation and their location with respect to some reference pointrespect to some reference point.

• In particular we can identify:

[ImaxImin]V =

• The fringe visibility:

[Imax+Imin]

(Michelson visibility)

• The fringe phase:– The location of the white-

light fringe as measured fromsome reference (radians).


Key Ideas #5

• The parameters of interference fringes that we are usually interested in are:interested in are:– The fringe amplitude or contrast (excluding finite bandwidth effects).

– The fringe phase.

• We are usually not interested in:– The fringe period (Q: why?)

• The questions you should all be asking now are:– Why are these the parameters of interest?y p

– And what do they tell us?


Interferometric Resolution: the Intuitive Picture

• Go back to YDS…

• A “point” source creates fringes with unit visibility (i e minimawith unit visibility (i.e. minima go to zero)

• Additional sources create f i t diff t itifringes at different positions on the screen

• The sum of these fringe patters Obscuration

S1 S2

are fringes that have less than unit (normalized) visibility

• This is what it means to say

Obscuration

This is what it means to say that an interferometer “resolves” a scene – the (normalized) fringe visibility is Po

wer

less than one Screen


Heuristic Operation of an Interferometer


Heuristic Operation of an Interferometer

• The resulting fringe• The resulting fringe pattern has a modulation depth that is reduced with respect to that from eachrespect to that from each source individually.

• The positions of the sources are encoded in the fringe phase.

Q: Why can we superposeQ: Why can we superposeresponses from sources?


Back to Photons Making Choices…Distant

• So how was it that the photon decides which aperture to enter? Incident

Distant Source

aperture to enter?

• A: It doesn’t – it goes through both!

Phasefronts

• The interference we see are photons interfering with themselves!!! (s*B)

• Time-averaged correlation between different photons is zero

A2x2

A2x2

Aperture 1

p(incoherent source)

• This is (generally) true even on a single source

Metrology

DelayLine 1

d

DelayLine 2

d2Beameven on a single source d12

(s*B)/2BeamCombiner


Key Ideas #6

• A general source can be described as a superposition of point sources

• Each of these point sources produces its own interference pattern

• The sum of the interference patterns is measured by the interferometer:– Technically this is known as the “spatially-incoherent sourceTechnically this is known as the spatially incoherent source

approximation”, and it is a very good approximation

– This all works because we measure the interference one photon at a time – different photons are mutually incoherent (in a time-averaged sense)– different photons are mutually incoherent (in a time-averaged sense)

– [The photon is NOT a particle, it is a “quantized excitation of the electromagnetic field”.]

• The resolution of a source by the interferometer is manifested as a reduction in the (normalized) fringe visibility amplitude

The modulation and phase of the resulting fringe pattern encode informationThe modulation and phase of the resulting fringe pattern encode information about the source structure (albeit in an apparently complicated way)


Response to a Distributed Source

• Consider looking at an incoherent source whose brightness on the sky is described by I(ŝ). This can be written as I(ŝ0+s), where ŝ0 is the pointing direction and s is a vector perpendicular to thisthe pointing direction, and s is a vector perpendicular to this.

• The detected power will be given by:

0

1 2

ˆ ˆ( , ) ( ) 1 cos

ˆ ˆ ( ) 1 cos ( . )

P s B I s kD d

I s k s B d d d

1 2

0 1 2

( ) ( )

ˆ ˆ ( ) 1 cos ([ ]. )

ˆ ˆ( ) 1 ( )

I s k s s B d d d

I k B B d d d

0 1 2 ( ) 1 cos ( . . )

( ) 1 cos ( . )

I s k s B s B d d d

I s k s B

'd


The van Cittert-Zernike theorem (i)

• Consider now adding a small phase delay, , to one arm of the interferometer. The detected power will become:

0( , , ) ( ) 1 cos ( ) '

( ) ' cos ( )cos ( ) '

P s B I s k d

I s d k I s k d

s B

s B

sin ( )sin ( ) 'k I s k d s B

• We now define something called the complex visibility V(k,B):g p y ( )

so that we can write our interferometer output as:

( , ) ( )exp[ ] 'V k B I s ik d s B

so that we can write our interferometer output as:

0( , , ) ( ) ' cos Re[ ] sin Im[ ]P s B I s d k V k V 0( ) Re exp[ ]lP s B I V ik 0( , , ) Re exp[ ]totalP s B I V ik


What is this Complex Visibility thing?

• Lets assume ŝ0 = (0,0,1) and s is (,,0), with and small angles (measured in radians)

( , ) ( )exp[ ] '

( exp[ ( )] x y

V k B I s ik d

I ik B B d d

s B

wvu

( exp[ 2 ( )] I i u v d d

• Here, u (= Bx/) and v (= By/) are the projections of the baseline onto a plane perpendicular to the pointing direction.

– These are usually referred to as spatial frequencies and have units of rad-1These are usually referred to as spatial frequencies and have units of rad .

– Their values define the scale of modulation where the interferometer is sensitive (*u ~ O(1) for significant power detected by interferometer).

So, the complex visibility is the Fourier Transform of the source brightness distribution.


The van Cittert Zernike theorem (ii)

• We can put this all together as follows:

• Our interferometer measures ( ) Re exp[ ]P s B I V ik • Our interferometer measures

• So, if we make measurements with, say, two value of = 0 and /4,

0( , , ) Re exp[ ]totalP s B I V ik

this recovers the both real and imaginary parts of the complex visibility

• And since the complex visibility is nothing more than the Fourier• And, since the complex visibility is nothing more than the Fourier transform of the brightness distribution, we have our final result:

The output of an interferometer measures the Fourier transformThe output of an interferometer measures the Fourier transform (spatial coherence function) of the source brightness distribution.

This is the van Cittert-Zernike theorem.

Q: Why FT, and not some other transform?


Physical Interpretation of Interferometric Visibility

• Interferometric Visibility as Fourier Transform of brightness distribution

2 ( )1( , ) ( , ) i u v

total

V u v d d I eI

• Visibility is Complex (Phasor)– Real/imaginary parts

correspond to brightnesscorrespond to brightness components symmetric/antisymmetric WRT phase center

• u and v Spatial Frequency Coordinates (~B / )

• Source structure scale must match spatial frequency scale for significant power detection

• Instantaneous Information fromInstantaneous Information from Single Point in u-v Plane


Tangent: SIM OPD Measurements

0( , , ) Re exp[ ]totalP s B I V ik

• The SIM beam combiner disperses the radiation as a function of wavelength (k)

• Fringe power oscillates as function of kof k

• FT(exp(-ik)) => (d=)

• Discrete realization from spectrometer pixelsspectrometer pixels

– FT(Vi(ki)) est

– est + internal metrology = SIM delay measurement (astrometric quantity)measurement (astrometric quantity)

– d = ŝB + C


Key Ideas #7

• The complex visibility is also known as the “spatial coherence” function.• Since the FT is a linear transform, if we know the complex visibility we can

recover the source brightness distribution.g• Received power can be modeled as inner product of a grating function on the

sky; this is the acceptance function established by the interferometer.• The visibility function is complex it has an amplitude and a phaseThe visibility function is complex, it has an amplitude and a phase.• The amplitude and phase of the interference fringes we spoke of earlier, are

actually the amplitude and phase of the complex visibility.Alternatively real/imaginary parts represent response to symmetric/antisymmetric– Alternatively, real/imaginary parts represent response to symmetric/antisymmetric parts of brightness distribution (around a given pointing direction)

• To measure these quantities we have to adjust D (or adjust k).A t f i l i t f t b li i t f• A measurement from a single interferometer baseline gives a measurement of one value of the FT of the source brightness distribution.

• Long interferometer baselines measure small structures on the sky, and short b li l t tbaselines, large structures.


Example Visibility Calculations

• In the next few slides we will work through some “toy” visibility calculations on some simple (but useful) source morphologies:p ( ) p g

– These demonstrate the application of the theory

– Believe it or not – these actually provide some astrophysically useful results

• Here we will be using a power-normalized form of the visibility function:function:

2 ( )1( , ) ( , ) i u v

total

V u v d d I eI


Visibility Functions of Simple Sources (i)

• Point source of strength A1 and located at angle 1 relative to the optical axis.

Power-Normalized Visibility!2 ( )( ) ( ) / ( )i uV u I e d I d g 1 g 1 p

1

2 ( )1 1 1 1

2 ( )

( ) ( ) / ( )

i u

i u

V u A e d A d

e

• The visibility amplitude is unity u.

• The visibility phase varies linearly with u (= B/).

• Since |V| is unity the interference• Since |V| is unity, the interferencefringes have high contrast.

PTISIM


Visibility Functions (ii)

• A double (binary) source comprising point sources of strength A1 and A2located at angles 0 and relative to the optical axis

2 ( )( ) ( ) / ( )i uV u I e d I d

located at angles 0 and2 relative to the optical axis.

2

2 ( )1 2 2 1 2 2

2 ( )1 2 1 2

( ) [ ( ) ( )] / [ ( ) ( )]

[ ] /[ ]

i u

i u

V u A A e d A A d

A A e A A

• The visibility amplitude and phaseoscillate as functions of u.

1 2 1 2[ ] [ ]

• To identify this as a binary, baselines from 0 /2 are required.

• The modulation of the visibility function tells us the separation and brightness ratio of thePTISIM and brightness ratio of the components.


Visibility Functions (iii)

• An on-axis uniform disc source of diameter

2 ( )( ) ( , ) / ( , )i u vV u I e d d I d d

An on axis uniform disc source of diameter .

Homework: Derive

/ 2

0

1

( ) (2 )

2 ( ) ( )r r

r r

V u J u d

J u I u

1( ) ( )r r

• To identify this as a disc requiresbaselines from 0 / at least.

• The visibility amplitude falls rapidlyas ur increases.

Information on scales smaller than

PTISIM

• Information on scales smaller than the disc diameter correspond to values of ur where V << 1, where the interference fringes have very l t tlow contrast.


Key Ideas #8

• Unresolved, sources have visibility functions that remain high, giving produce high contrast fringes for all baseline lengths.

• Resolved sources have visibility functions that fall to low values at long baselines, giving fringes with very low contrast.

Fringe parameters for resolved sources will be difficult to measure.

• Imaging with many resolution elements generally needs measurements g g y g ywhere the fringe contrast is both high and low (to pick out large scale and small scale features respectively).

• To usefully constrain an unknown source, the visibility function must be measured adequately. Measurements on a single, or small number of, baselines are normally not enough for unambiguous image recovery.


Introduction to Interferometric Imaging (one slide!)

• The visibility function, V(u, v) is the Fourier transform of the source brightness distribution:

( ) ( [ 2 ( )]V I i d d ( , ) ( exp[ 2 ( )] V u v I i u v d d • So the idea is to measure V for as many values of u and v as practical

& perform an inverse FT:& perform an inverse FT:

But since what we measure is a sampled version of V(u v) what

( , )exp[ 2 ( )] (normV u v i u v du dv I • But since what we measure is a sampled version of V(u, v), what

we actually recover is the so-called “dirty map”:

( ) ( ) [ 2 ( )] (S V i d d I

Bdi t (l m) is the Fourier transform of the sampling distribution and is

( , ) ( , ) exp[ 2 ( )] (

( (

dirty

norm dirty

S u v V u v i u v du dv I

I B

Bdirty(l,m) is the Fourier transform of the sampling distribution, and is known as the “dirty beam”.


Dirty (& Corrected) Interferometric Images

• Correcting an interferometric map for the Fourier plane sampling• Correcting an interferometric map for the Fourier plane sampling function is known as deconvolution (CLEAN, MEM, WIPE).


A Real Astronomical Example

K-band image of IRC+10216. Image courtesy of Peter Tuthill

d J hand John Monnier.


u-v Plane Coverage

• Interferometer i f tiinformation currency visualized as u-v tracks

• u-v tracks are a funny way of visualizing thevisualizing the Modulation Transfer Function (MTF) f th(MTF) of the interferometer


More Sample u-v Coverage

NPOI u-v coverage

No Free Lunch!

5 Mmas

Notional filled-apertureu-v coverage

(Keck Telescope)


Key Ideas #9

• Imaging with an interferometer => measuring the visibility function for a wide range of baselines.

It also => measuring its amplitude and phase– It also => measuring its amplitude and phase– The atmosphere makes this harder than it sounds…

• The map you get will ONLY contain information corresponding to the baselines you measured.– Of course this applies to conventional imaging as wellpp g g

• We can represent the information contained in an interferometric d l b it ti l f ( ) t tmodel by its spatial frequency (u-v) content.

• It is impossible to realize the “correct” image.– All images are finite-information approximations to the true scene


I t f t hi t k i t f f i ( d

Lecture Summary

• Interferometers are machines to make interference fringes (and measure their properties).– Interference fringes are variations in measured power as function of

l ti thl th ( ith th h d l b )relative pathlength (either through delay or wavenumber)• The fringe parameters (modulation/amplitude and phase) tell you

about what you are looking at:– In particular, fringe phase encodes object position information– Fringe amplitude encodes information on object size

• More precisely, these measure the amplitude and phase of the FT (spatial coherence function) of the source brightness distribution.

• A measurement with a given interferometer baseline measures one value of the FT of the source brightness distribution.value of the FT of the source brightness distribution.

• Multiple baselines are required to synthesize an image; fewer may be required if additional information are used (e.g. binary star RV orbit).

• Once many visibility measurements are made an inverse FT delivers• Once many visibility measurements are made, an inverse FT delivers a representation of the source that may (or may not) be useful!


Summary

• Interferometers measure fringes.

• The fringe modulation and phase are the quantities of interest• The fringe modulation and phase are the quantities of interest.

• These measure the amplitude and phase of the FT of the source brightness distribution (the visibility function).

• Any given interferometer baseline responds to a single spatial frequency in the source brightness distribution.

• Multiple baselines are obligatory to build up an image.Multiple baselines are obligatory to build up an image.

• Once many visibility measurements are made, an inverse FT delivers a representation of the source that may (or may not) be useful!a representation of the source that may (or may not) be useful!


Next Lecture

• Next lecture: February 21, 2011 (from Pasadena)• Will talk about

– Palomar tested Interferometer (PTI)– Space Interferometry Mission (SIM)

• Let me know your comments, suggestions! • Contact information:

– Class website: http://lnfm1.sai.msu.ru/~turyshev

– My email: [email protected]

• See you in a week!• See you in a week!

lecture 2 0:lecture 2.0: optical interferometryoptical...

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