Introduction to Optical/IR Interferometry: history and basic principles
IIntroduction to optical/IR interferometry: history and basicprinciples
Jean
Surdej , ∗ Institute of Astrophysics and Geophysics, Liège University, Allée du 6 Août 19c, 4000 Liège, Belgium
Abstract.
The present notes refer to a 3-hour lecture delivered on 27 September2017 in Rosco ff during the 2017 Evry Schatzman School. It concerns a generalintroduction to optical / IR interferometry, including a brief history, a presenta-tion of the basic principles, some important theorems and relevant applications.The layout of these lecture notes is as follows. After a short introduction, weproceed with some reminders concerning the representation of a field of elec-tromagnetic radiation. We then present a short history of interferometry, fromthe first experiment of Fizeau and Stefan to modern optical interferometers. Wethen discuss the notions of light coherence, including the theorem of Zernicke-van Cittert and describe the principle of interferometry using two telescopes.We present some examples of modern interferometers and typical results ob-tained with these. Finally, we address three important theorems: the fundamen-tal theorem, the convolution theorem and the Wiener-Khinchin theorem whichenable to get a better insight into the field of optical / IR interferometry.
In the absence of the Earth atmosphere above a ground-based telescope equipped with a mir-ror having a diameter D , Figure 1 illustrates the image one would observe from a point-likestar recorded in the focal plane in monochromatic light at a wavelength λ . It is a dot of light,the well-known Airy disk, which angular radius measured in radian is simply given by 1.22 λ/ D . Unfortunately, the Airy disk does not contain any information relative to the star be-ing imaged, irrespective of its size, shape, e ff ective temperature, luminosity, distance, etc. Alarger telescope with a diameter D > D , would similarly lead to a smaller Airy disk (1.22 λ/ D ) of light for the star being imaged (see Fig. 2), providing a slightly better angular res-olution image but with no more specific information related to the star. While observing anextended celestial source (cf. a distant resolved Earth-like planet as shown in Fig. 3), moredetails are seen with the telescope having a larger diameter. The dream of astronomers istherefore to construct always larger telescopes but presently there is a limit (D ∼ ffi cult to construct a single mirror telescope (cf. the ELT, TMT,GMT projects).Fortunately, in 1868 Fizeau and Stephan just realized that ”In terms of angular resolution,two small apertures distant of B are equivalent to a single large aperture of diameter B .” (seeFig. 4). This is actually the subject of the present lecture: to understand how it is possible ∗ e-mail: [email protected] a r X i v : . [ a s t r o - ph . I M ] J u l o reconstruct high angular resolution images of a distant celestial source using modern op-tical / IR interferometers such as VLTI, CHARA, etc. In fact, the image of a distant star thatone would see in the focal plane of a Fizeau-type interferometer is no longer just an Airydisk due to each single telescope aperture but a brighter Airy disk superimposed with a seriesof interference fringes, alternately bright and dark, perpendicularly oriented with respect tothe line joining the two telescopes and with an inter-fringe angular separation equal to λ/ B ,where B is the baseline of the interferometer (see Fig. 5). This naturally leads to the hopethat it will be possible to retrieve along the direction of the baseline having a length B anangular resolution that is equivalent to that of a single dish telescope having a diameter B .As a summary, figure 6 illustrates the improvement expected in angular resolution while ob-serving an extended celestial source with telescopes of increasing size ( D > D ) and withan interferometer composed of two telescopes separated by a distance B > D . Figure 1.
Airy disk of apoint-like star recorded inthe focal plane of atelescope with diameter D .The angular diameter of theAiry disk is 2.44 λ/ D . Figure 2.
As the diameter ofa telescope increases( D > D ), the Airy disk ofa point-like star gets smaller. In mathematical terms, the convolution theorem states that the image I( ζ, η ) we observein the focal plane of an instrument (single dish telescope or interferometer) from a distantextended source as a function of its angular coordinates ζ, η is the convolution product of thereal source image (cf. the extended Earth-like planet), O( ζ, η ) by the point spread functionPSF( ζ, η ) of the telescope (i.e. the Airy disk, see Fig. 7) or of the interferometer (i.e. the Airydisk crossed by the interference fringes).While taking the Fourier transform (FT) of the first expression given in Fig. 7, wefind that FT [ I ( ζ, η )]( u , v ) is simply equal to the natural product of FT [ PS F ( ζ, η )]( u , v ) and FT [ O ( ζ, η )]( u , v ) where u , v represent the angular space frequencies defined as u = B u /λ and v = B v /λ , respectively, where B u and B v correspond to the projected baselines of the interfer-ometer along the directions parallel to the angles ζ, η . One can then expect that by just takingthe inverse Fourier transform FT − of FT [ O ( ζ, η )]( u , v ), it will become possible to retrieve igure 3. While observingan extended celestial object(cf. an Earth-like planet)above the atmosphere, wesee more details as thediameter of the telescopeincreases.
Figure 4.
Fizeau andStephan proposed torecombine the light fromtwo independent telescopesseparated by a baseline B torecover the same angularresolution as that given by asingle dish telescope havinga diameter B . high angular resolution information about the extended source with an angular resolutionequivalent to 1 / u = λ/ B u and 1 /v = λ/ B v , respectively: O ( ζ, η ) = FT − [ FT [ O ( ζ, η )]( u , v )]( ζ, η ) = FT − [ FT [ I ( ζ, η )]( u , v ) FT [ PS F ( ζ, η )]( u , v ) ]( ζ, η ) . (1)The quantity FT [ I ( ζ, η )]( u , v ) can be directly derived from the observation of the extendedsource with the optical / IR interferometer while the other quantity FT [ PS F ( ζ, η )]( u , v ) canbe obtained from the observation of a point-like (unresolved) star. During this lecture, weshall see that the Wiener-Khinchin theorem states that the latter quantity is also merely givenby the auto-correlation function of the distribution of the complex amplitude of the radia-tion field in the pupil plane of the observing instrument being used (single dish telescope orinterferometer). The goal of the present lecture is to establish relations such as Eq. (1). With a few exceptions (cf. the Moon, the Sun, the Andromeda Galaxy, etc.), all the celestialobjects that we see in the sky appear to us, with the naked eye, as point-like objects. Apartfrom their apparent motion with respect to the fixed stars on the celestial sphere, we are noteven able to distinguish between the images of Jupiter, Saturn or even Venus from those of igure 5.
Whenrecombining themonochromatic light of twoindependent telescopes,there results the formation ofa pattern of bright and darkfringes superimposed overthe combined Airy disk. Theangular inter-fringeseparation is equal to λ/ B . Figure 6.
Improvementexpected in angularresolution while observingan extended celestial source(cf. an Earth-like planet)with telescopes of increasingsize ( D > D ) and with aninterferometer composed oftwo telescopes separated bya baseline B > D . ordinary stars. We describe in this course an observation method based on the principle of aFizeau-type interferometer, which allows with just some basic cooking equipment to resolveangularly a planet such as Venus, when it is at its maximum apparent brightness ( V ∼ − . T (measuredperpendicularly to the line-of-sight) and which is incandescent (cf. a star), how to measurethe thickness T of this filament (diameter of the star) not only without breaking the bulb butalso assuming that it is so far away from us, at a distance z , that it is not possible for us toangularly resolve the filament with the naked eye (see Fig. 8a)?Let us now recall that knowledge of the angular radius ( ρ = R / z ) of a star located at adistance z ( z >> R ) and having a linear radius R allows the direct determination of its flux F at the stellar surface from the flux f observed on Earth (as a reminder F = f /ρ , see Fig. 8b).If we can measure the absolute distance z of the star, we can also determine its linear radius R from its angular radius ρ ( R = ρ z ). Moreover, knowledge of the intrinsic flux F of the starallows an immediate determination of its e ff ective temperature T e f f , thanks to the applicationof the Stefan-Boltzmann law ( F = σ T e f f ). It then results that T e f f = ( f /σρ ) / . The igure 7. The image I( ζ, η )we observe in the focalplane of an instrument (cf.single dish telescope) from adistant extended source as afunction of its angularcoordinates ζ, η is theconvolution product of thereal source image (cf. theextended Earth-like planet,O( ζ, η )) by the point spreadfunction PSF( ζ, η ) of thetelescope.
Figure 8.
Resolving theangular diameter of a star(b) is alike trying to estimatethe angular size of thefilament of a light bulb (a). measurements of the angular radius and of the flux of a star measured on Earth thus lead to thedetermination of the e ff ective temperature T e f f of that star. As a reminder, this temperatureis directly involved in the construction of stellar atmosphere models and stellar evolution.We will also show that Fizeau-type stellar interferometry literally allows direct imaging withvery high angular resolution of distant bodies by the method of aperture synthesis. Let us nowproceed with a few theoretical reminders about the description of a field of electromagneticlight radiation. Let us first remind that a beam of light radiation can be assimilated to the propagation of amultitude of electromagnetic waves at the speed of 299,792 km s − in the vacuum. If, forthe sake of simplicity, we assume that we deal with a plane monochromatic wave, linearlypolarized, propagating along the direction of abscissa z , the electric field E at any point inspace and at time t , can be represented by a sinusoidal type function taking for example theshape E = a cos (2 π ( ν t − z /λ )) (2)where = c T = c /ν (3) c , λ , ν , T and a representing the speed of light, the wavelength, the frequency, the periodand the amplitude of the electromagnetic vibrations, respectively (see Figure 9). Figure 9.
Representation of an electromagneticwave.
We know how convenient it is to rewrite the previous equation in complex notation: E = Re { a exp[ i π ( ν t − z /λ )] } (4)where Re represents the real part of the expression between the two curly braces. Thiscomplex representation of an electromagnetic wave has the great advantage that the exponen-tial function can now be expressed as the product of two functions depending separately onthe spatial and temporal coordinates E = a exp( − i φ ) exp( i πν t ) (5)where φ = π z /λ. (6)If we suppose that all the operations that we carry out on the electric field E are linear, itis of course very convenient to use in our calculations its complex representation (see Eq.(5))and to take at the end the real part of the result obtained.We can then rewrite the previous equation as follows: E = A exp( i πν t ) , (7)where A = a exp( − i φ ) (8)with A representing the complex amplitude of the vibration.Because of the extremely high frequencies of electromagnetic waves corresponding tovisible radiations ( ν ∼ Hz for λ = E (the situation is di ff erent in the radio domain).The only measurable quantity is the intensity I , which is the time average of the amountof energy passing through a unit surface element, per unit of time and solid angle, placedperpendicularly to the direction of propagation of the light beam.The intensity I is therefore proportional to the temporal average of the square of theelectric field: (cid:68) E (cid:69) = lim T →∞ T (cid:90) + T − T E dt , (9)hich is reduced to (e.g. replace in the previous relation E by Eq.(2)) (cid:68) E (cid:69) = a , (10)where a is the real amplitude of the electric field.By convention, the intensity of the radiation is defined by the following relation: I = A A ∗ = | A | = a . (11) We recall that, according to Huygens, each point of a wavefront can be considered as beingthe centre of a secondary wave leading to the formation of spherical wavelets, and that themain wavefront, at any subsequent moment can be considered as the envelope of all thesewavelets (see Fig. 10).
Figure 10.
Illustration of theHuygens-Fresnel principleduring the propagation of aplane or circular wavefrontand di ff raction of lightwhich encounters aconverging lens. Using this model, Fresnel was the first to account for the observed e ff ects of light di ff rac-tion, assuming that secondary wavelets interfere with each other. This combination of theHuygens construction method and the Fresnel interference principle is called the Huygens-Fresnel principle. This is the basis of the concept of the Fourier transform. Let us reminda direct application of this principle when studying the formation of the image of a distantobject at the focus of a telescope having a linear diameter d . Following the di ff raction of thewaves at the passage of the opening of the telescope (as if the waves were trying to spread andbypass the obstacles), we observe a phenomenon of redistribution of the energy of the lightwave: the image of a point-like source produced by a converging circular objective (lens ormirror) is not a point but spreads in a di ff raction pattern called the "Airy disk" (see Fig. 10).The angular diameter of the central spot is (in radian): σ = . λ/ d (12)where λ is the wavelength of light and d is the linear diameter of the aperture.We can resolve an extended source by direct imaging, if and only if, its angular diameter ∆ ( = ρ ) is somewhat larger than σ . For example, our pupil whose approximate diameter variesbetween 1 and 5 mm, allows us to angularly resolve nearby objects separated by more than138” and 28”, respectively. In the visible range, a telescope, with a diameter of 14 cm, willallow us to resolve objects with an angular dimension larger than 1”, and for diameters largerthan 14 cm, their collecting area will naturally be enhanced but their angular resolution willremain limited to (more or less) 1” because of the atmospheric agitation (see Fig. 11). In fact,under the influence of temperature and pressure gradients, a regime of eddies establish itselfn the Earth atmosphere which, at low altitude ( ∼
10 km), have dimensions of the order of 20cm (sometimes only a few cm, sometimes 30 or 40 cm) and evolution periods of the order ofa few milliseconds. Optically, these changes manifest themselves by an inhomogeneity in therefractive index distribution. The amplitude and the relative phase shift of the electromagneticfield in the pupil plane thus get disturbed in a random manner.
Figure 11.
Atmospheric agitation above theobjective of a large telescope causing theseeing e ff ects seen in its focal plane. It follows that if we observe the Moon, Jupiter, etc. either with the largest telescope in theworld with a diameter of 10m ( σ = σ = ff usion indicator of the atmosphere is defined as being the average inclination per-turbation of the wave surfaces. This tilt disturbance reaches values that vary between 1” and10”, depending on the site and the moment. This phenomenon is detected di ff erently accord-ing to the dimensions of the instrument used. The eye, which has an angular resolution closeto the minute of arc, will be sensitive only to variations of amplitude: we then see stars flick-ering. An instrument of 10 to 20 cm in diameter will detect tilt variations and the focal imagewill oscillate around an average position. For larger instruments, a large number of eddieswill, at the same time, be involved in the formation of the focal image. This will thereforehave the dimensions of the di ff usion indicator of the atmosphere. The spatial coherence ofthe entrance pupil will allow, for a point-like source, the realization of interference phenom-ena between the radiations passing through di ff erent points of the pupil. A statistical studymakes it possible to show that the resulting focal image, delimited by the di ff usion indicator,consists of a set of granules (called ’speckles’) which have the size of the Airy disk of theinstrument (see Fig. 11). These granules swarm in the di ff usion spot at the rhythm of thechange of the atmospheric eddies. The stability of the focal image is therefore also of theorder of the millisecond. The technique of speckle interferometry, developed by the Frenchastronomer Antoine Labeyrie, allows to re-construct the images of the stars observed withthe angular resolution given by the true diameter of the telescope. Brief history about the measurements of stellar diameters
In the past, there have been numerous attempts to measure angular diameters of stars, and wewill first recall three of these approaches that clearly show the di ffi culties encountered. A first experimental attempt to measure the angular diameter of stars was made by Galileo(1632). He proceeded as follows: placing himself behind a rigid wire (whose thickness D was known, see Fig. 12) suspended vertically, he determined the distance z to which he hadto move in so that the image of the star Vega ( α Lyrae) of magnitude zero got completelyobscured by the wire. Galileo deduced that the angular diameter of Vega, equal to that of thewire, was about 5”, which was in itself a rather revolutionary result, since the value adopted atthat epoch for the angular diameter of the stars was close to 2’. As we saw earlier, the value of2’ is certainly the result of the low angular resolution of our eye, while the 5” angular diametermeasured by Galileo was the result of the e ff ects of the atmospheric agitation (seeing e ff ects)at the time of his observations. Figure 12.
Experimentalmeasurement by Galileo of theangular diameter of a star (see text).
A theoretical estimate of the angular dimension of a star of magnitude zero was performed byNewton. His approach was as follows: if we suppose that the Sun is a star similar to the starssituated on the celestial sphere and if we place our star of the day at a distance z such thatits apparent brightness V (cid:12) becomes comparable to that of a star of magnitude equal to zero,then its angular diameter ∆ should be of the order of 2 10 − ” (with the current value of thevisual apparent magnitude of the Sun, V (cid:12) = -26.7, we find ∼ − − ”. Theformula to be used to establish this result can be obtained as follows: the angular diameterof the Sun ∆ placed at the distance of Vega ( V =
0) is given by the product of the apparentangular diameter of the Sun ∆ (cid:12) times the factor 10 V (cid:12) / . As a reminder, the apparent diameterof the Sun is about 30’. The third experimental attempt of measuring stellar diameters, based on Fizeau-type interfer-ometry, is in fact the work of prominent scientists such as Young, Fizeau, Stephan, Michelsonand Pease. These last two having measured the first angular diameter of a star in 1920. Al-though other methods of interferometric measurements of stellar angular diameters appearedater (cf. the interferometry in intensity of Brown and Twiss in 1957, speckle interferometryby Antoine Labeyrie in 1970, etc.), we will only describe in detail the Fizeau-type interfer-ometry, which is still the most powerful technique used and the most promising measurementof angular diameters of stars and imagery at very high angular resolution of distant bodies bythe aperture synthesis method.Let us first remind the results obtained in the Young double hole experiment (1803, seeFig. 13).
Figure 13.
The double holeexperiment of Young (seetext).
A monochromatic plane wave coming from a distant point-like source is falling on ascreen drilled with two holes ( P and P ) separated along the x axis by a baseline B . Inaccordance with the Huygens-Fresnel principle, the two holes will emit spherical waves thatwill interfere constructively whenever the di ff erence in their propagation lengths is a multi-ple of λ (see Eq. (13) below), and destructively if it corresponds to an odd number of halfwavelengths.The locus of points P ( x , y, z ) with cartesian coordinates x , y , z (see Fig. 13) where therewill be a constructive interference is thus given by | P P | − | P P | = n λ (13)with n = , ± , ± , etc.Let the points P i ( x i , y i ,
0) in the screen plane and P ( x , y, z ) in the observer plane be suchthat | x i | , | y i | , | x | , | y | << | z | . We then find that | P i P | = (cid:113) ( x − x i ) + ( y − y i ) + z (14)which can be simplified at first order (given the above conditions) as follows: | P i P | = z { + ( x − x i ) + ( y − y i ) z } . (15)Considering the two points P and P in the Young’s screen, Eq. (13) reduces to z { + ( x + B / + y )2 z } − z { + ( x − B / + y z } = n λ (16)and finally xBz = n λ (17)or else Φ = xz = n λ B . (18)Since the angular separation Φ between two successive maxima (or minima) does not dependon the coordinate y , there results a pattern of bright and dark fringes, oriented perpendicularlyith respect to the line joining the two holes, and with an inter-fringe angular separation Φ = λ/ B . In case the two holes are not infinitely small, the observed interference pattern willnaturally overlap the combined Airy disks produced by each single hole. For λ = B = Φ = B between the two holeswas extended. Was there a simple relation between the angular diameter ∆ of the sourceand the spacing B between the two holes corresponding to the disappearance of the fringes?Before establishing such a rigorous relationship, let us try to understand this observationintuitively on the basis of simple geometrical considerations (see Fig. 14). Indeed, if insteadof considering the di ff raction pattern given by Young’s holes for a single point-like source,we consider a composite source made of two incoherent point-like sources separated by anangle ∆ , that is to say between which there is no interference between their light, it will resultin the plane of the observer a superposition of two systems of Young fringes, separated by anangle ∆ . If ∆ ∼ Φ /
2, there will result a total scrambling of the fringes. The bright fringesof one source will overlap the dark fringes of the second one and their contrast will totallyvanish.
Figure 14.
Fizeau experiment: for the case of a single star (left drawing) and for the case of a doublestar with an angular separation ∆ (right drawing, see text). From Fig. 14, it is clear that the visibility of the fringes will significantly decrease when-ever the following condition takes place ∆ > Φ = λ B . (19)A quantity that objectively measures the contrast of the fringes is called the visibility. Itis defined by the following expression: v = (cid:32) I max − I min I max + I min (cid:33) . (20)Whenever a star is not resolved, we have I min =
0, and thus the visibility v =
1. If the star isbeing resolved, I max = I min and thus the resulting visibility v = ffi cient toplace a screen drilled with two elongated apertures at the entrance of a telescope pointedowards a star and to look in the focal plane by means of a very powerful eyepiece the Airydisk crossed by the Young’s fringes and to increase the distance between the two aperturesuntil the visibility of the fringes vanishes.This experiment is attempted in 1873 by Stephan with the 80cm telescope of the MarseilleObservatory. All the bright stars visible in the sky are observed. The two openings at theentrance were actually in the form of crescents but one may demonstrate that the contrastof the fringes is independent of the shape of the two openings if they are identical. Theresult was disappointing: with a maximum base separation of 65cm between the openings,no attenuation of the contrast of the fringes was observed for any star. This proved that nostar could be resolved using that instrument. Stephan concluded that the angular diameter ofthe stars is much smaller than 0.16” (see Fig. 15). From these observations, it is of coursepossible to set a lower limit on the e ff ective temperature T e f f of all those stars (see Section2). Figure 16 illustrates the 80 cm Marseille telescope used by Stephan and Fizeau. Figure 15.
Diagram illustrating the way Fizeau and Stephanproceeded in order to measure the angular diameters of starswith the interferometric technique.
Figure 16.
The 80cm Marseille telescope used by Fizeauand Stephan. c (cid:13)
Michel Marcelin. .4 Home experiments: visualization of the Airy disk and the Young interferencefringes
We propose hereafter two simple experiments that can be carried out at home in order tovisualize with our own eyes the Airy disk and the interference fringe patterns. To do this,take a rectangular piece of cartoon ( ∼ / ∼ Figure 17.
The one hole screen experiment:the small circular hole drilled in thealuminium paper is visible inside the lowerbigger hole perforated in the cartoon screen.When looking through this hole at a distantlight bulb, you perceive a nice Airy disk (cf.right image).
Figure 18.
The two hole screen experiment:the two small circular holes drilled in thealuminium paper are visible inside the upperbigger hole perforated in the cartoon screen.When looking through these two holes at adistant light bulb, you perceive a nice Airydisk superimposed by a pattern of bright anddark fringes (cf. right image).
Take the second square of aluminium and drill with the thin metallic pin two 0.5 mm(or smaller) circular holes near its centre separated by about 0.5-1 mm (see Fig. 18). Gluenow near the second circular hole (cf. the upper one) inside the folded cartoon this secondaluminium square. After, you should glue the two sides of the cartoon in such a way thatyou can hold it with ease in one hand. Of course you can do this on a single cartoon oron two separate ones. Place at a distance of about 10 m a small light bulb (cf. the light ofa cell phone) and look at it through the single hole drilled in the aluminium. You shouldsee a nice Airy disk which angular diameter is 2 . λ/ D , with D being about 0.5 mm and λ the wavelength of the ambient light ( ∼ λ/ D , where D ∼ Dist be the distance around which the latter totally vanishes.The product of
Dist by λ/ D corresponds to the linear diameter of the light bulb. Instead ofchanging the distance between the light bulb and the milli-interferometer, one could changethe separation between the two holes and determine the separation for which the interferencefringes disappear. Adopting the same approach as Stephan, Abraham Michelson used in 1890the Lick 30cm telescope to resolve the four Galilean satellites of Jupiter. Their angular sizeswere of the order of 0.8” - 1.4” while the resolving power provided by the largest baselinehe used was about 0.5”. An excellent agreement was found for the angular diameters ofthe satellites with the classical measurements made at the same time. To resolve the biggeststars, much longer baselines are needed. Michelson and Pease built a 7m metal beam carryingfour 15cm flat mirrors that they installed at the top of the Mount Wilson telescope, having adiameter of 2.5m (see Fig. 19). Figure 19.
The stellarinterferometer of Michelsonand Pease set on top of the2.5m Mount Wilsontelescope. c (cid:13)
TheObservatories of theCarnegie Institution.
The two mobile exterior mirrors formed the basis of the interferometer and the two fixedinterior mirrors returned the star’s light into the telescope. With a maximum baseline of 7m,the smallest measurable angular diameter was 0.02”. Use of this first stellar interferometerwas very delicate because the visualization of the Young fringes was only possible if thetwo optical paths from the star passing through the two exterior mirrors and reaching theobserver eyes were equal to an accuracy of about 2 microns (see discussion below). Michel-son and Pease finally obtained the first measurement of a stellar diameter during the winterof 1920, that of Betelgeuse ( α Orionis), a red supergiant. They found an angular diameterof 0.047”, that is a linear diameter 400 times larger than that of the Sun, given the distanceof Betelgeuse (650 light years). Five more bright stars were also resolved. Anderson usedthe same observing technique with the 2.5m telescope at Mount Wilson to resolve very tightspectroscopic binaries (cf. Capella). Michelson and Pease did not stop there: they undertookthe construction of a 15m optical beam based on the same principle, and began to use it in1929. Unfortunately, the mechanical vibrations and deformations were such that this instru-ent was too delicate. It was abandoned in 1930, without having reached its limiting angularresolution of 0.01”. It was not until 1956 that optical stellar interferometry was reborn andagain, according to a principle di ff erent from that of Fizeau. Fizeau-type interferometry hadindeed acquired a reputation of great operational di ffi culty. The intensity interferometry bythe two radar manufacturers Hanbury Brown and Twiss (Australia) was then set up, based onan entirely new approach: the measurement of the space correlation of the stellar intensityfluctuations. Their interferometer made it possible to measure the diameter of 32 blue starswith a very high precision ( < / IR interferometry.
When we previously established the relationship between the angular diameter ∆ of a sourceand the separation B between the two apertures of the interferometer for which the inter-ference fringes disappear, we made two approximations that do not really apply to usualconditions of observation. We first assumed that the waves falling on Young’s screen wereplanes, that is to say coming from a very distant point-like source and also that they werepurely monochromatic. In addition, the holes through which light is being scattered shouldhave finite dimensions. Therefore, we shall later take into account the finite dimensions ofthe apertures (see Section 6) but let us first consider the e ff ects due to the finite dimensionof the source, also considering a spectral range having a certain width and to do so, we shallmake use of some elements of the theory of light coherence. This theory consists essentiallyin a statistical description of the properties of the radiation field in terms of the correlationbetween electromagnetic vibrations at di ff erent points in the field. The light emitted by a real source (see Fig. 20) is of course not monochromatic. As in thecase of a monochromatic wave, the intensity of such a radiation field at any point in space isdefined by I = (cid:104) V ( t ) V ( t ) ∗ (cid:105) . (21) Figure 20.
Stars do not emitmonochromatic light. Quasimonochromatic light is assumed tobe emitted at the wavelength λ (resp.the frequency ν ) within thebandwidth ± ∆ λ (resp. ± ∆ ν ). In order to determine the electric field created by such a source, emitting within a certainfrequency range ± ∆ ν , we must sum up the fields due to all the individual monochromaticomponents such that the resulting electric field V ( z , t ) is given by the real part of the follow-ing expression: V ( z , t ) = (cid:90) ν +∆ νν − ∆ ν a ( ν (cid:48) ) exp[ i π ( ν (cid:48) t − z /λ (cid:48) )] d ν (cid:48) . (22)While a monochromatic beam of radiation corresponds to an infinitely long wave train,it can easily be shown that the superposition of multiple infinitely long wave trains, withnearly similar frequencies, results in the formation of wave groups. Indeed, the expressionof the electric field established in Eq. (22) can be reduced as follows. Insert in the integralof Eq. (22) the following factors: exp( i π ( ν t − z /λ )) exp( − i π ( ν t − z /λ )). We then find thatEq. (22) may be rewritten as V ( z , t ) = A ( z , t ) exp[ i π ( ν t − z /λ )] (23)where A ( z , t ) = (cid:90) ν +∆ νν − ∆ ν a ( ν (cid:48) ) exp { i π [( ν (cid:48) − ν ) t − z (1 /λ (cid:48) − /λ )] } d ν (cid:48) . (24)Expression (23) represents that of a monochromatic wave of frequency ν whose amplitude A ( z , t ) varies periodically with a much smaller frequency ∆ ν (cf. beat phenomenon). As anexercise, it is instructive to set a ( ν (cid:48) ) constant in Eq. (24) and establish that indeed A ( z , t ) variesas a function of time with a frequency ∆ ν . This modulation therefore e ff ectively splits themonochromatic wave trains having di ff erent but nearly similar frequencies into wave groupswhose length is of the order of λ / ∆ λ , with ∆ λ = − c ∆ ν/ν and the frequency of the order ∆ ν (see Figure 21). Figure 21.
Superposition oflong wave trains havingquite similar frequencies ν (cid:48) in the range ν ± ∆ ν (resp.wavelengths λ (cid:48) in the range λ ± ∆ λ ) results in thepropagation of long wavetrains with the frequency ν (resp. wavelength λ ) butwhich amplitude A ( z , t ) isvarying with a lowerfrequency ∆ ν (resp. longerwavelength λ / ∆ λ ). What becomes the visibility of the interference fringes in the Young’s hole experiment for thecase of a quasi-monochromatic source having a finite dimension?We can re-write the expression of the intensity I q at point q as indicated below (seeEqs. (25)-(28)). It is assumed that the holes placed at the points P , P in the Young planehave the same aperture size (i.e. V ( t ) = V ( t )) and that the propagation times of the lightbetween P (resp. P ) and q are t q (resp. t q , see Fig. 22) : I q = (cid:68) V ∗ q ( t ) V q ( t ) (cid:69) , (25) igure 22. Assuming anextended source S whichquasi monochromatic lightpasses through the two holes P and P , I q represents theintensity distribution at thepoint q which accounts forthe formation of theinterference fringes. V q ( t ) = V ( t − t q ) + V ( t − t q ) (26)and after a mere change of the time origin V q ( t ) = V ( t ) + V ( t − τ ) (27)where we have defined τ = t q − t q . (28)It follows that Eq. (25) can be easily transformed into (29) where (30) represents the com-plex degree of mutual coherence, and the intensity I = < V V ∗ > = < V V ∗ > . Equation (29)is used to find what is the intensity distribution of the interference fringes in the observationplane. The complex degree of mutual coherence γ ( τ ) (see Eq. (30)) is a fundamental quan-tity whose significance will be highlighted when calculating the visibility of the interferencefringes. By means of (23), this function γ ( τ ) can still be expressed as (31), and if τ << / ∆ ν (i.e. the di ff erence between the arrival times of the two light rays is less than the beat period1 / ∆ ν of the quasi-monochromatic radiation), we can give it the form (32): I q = I + I + I Re [ γ ( τ )] , (29) γ ( τ ) = (cid:10) V ∗ ( t ) V ( t − τ ) (cid:11) / I , (30) γ ( τ ) = (cid:10) A ∗ ( z , t ) A ( z , t − τ ) (cid:11) exp( − i πντ ) / I , (31)and if τ << / ∆ ν γ ( τ ) = | γ ( τ = | exp( i β − i πντ ) . (32)Equation (29) can then be rewritten as (33) and in this case the visibility v of the interfer-ence fringes is | γ ( τ = | (see Eq. (34)), I max and I min representing the brightest and weakestfringe intensities. I q = I + I + I | γ ( τ = | cos ( β − πντ ) (33)and v = (cid:32) I max − I min I max + I min (cid:33) = | γ ( τ = | . (34)We will see in the next section that the module of γ ( τ =
0) is directly related to thestructure of the source that we are observing.e propose hereafter to the reader to answer the two following questions. What is thevalue of | γ ( τ = | in the Young’s holes experiment for the case of a monochromatic wave,two point-like holes and an infinitely small point-like source? And what can we say aboutthe source when | γ ( τ = | = γ ( τ =
0) is for the case we are interested in, namely anextended source emitting quasi-monochromatic light. This leads us directly to study thenotion of the spatial coherence of light.
Let us thus evaluate Eq. (30) for the case τ =
0. We find γ ( τ = = (cid:10) V ∗ ( t ) V ( t ) (cid:11) / I . (35)If V i ( t ) and V i ( t ) represent the electric fields at P and P due to a small surface element dS i on the source S (see Fig. 23), we find that the fields V ( t ) and V ( t ) can be expressed as V ( t ) = N (cid:88) i = V i ( t ) , V ( t ) = N (cid:88) i = V i ( t ) . (36) Figure 23.
The extendedsource S is assumed to becomposed of a large numberof infinitesimal surfaceelements dS i . It is assumed that the distinct points i of the source are separated by small distances com-pared to the wavelength λ of the light they emit in a mutually incoherent manner. Obtainingthe expression (37) for γ (0) is then immediate γ (0) = N (cid:88) i = (cid:68) V ∗ i V i (cid:69) + N (cid:88) i (cid:44) j (cid:104) V ∗ i V j (cid:105) / I . (37)For an incoherent light source, the second summation appearing in (37) is obviously equalto zero. As a reminder, the contributions V i j ( t ) can be expressed as V i ( t ) = a i ( t − r i / c ) r i exp[ i πν ( t − r i / c )] , V i ( t ) = a i ( t − r i / c ) r i exp[ i πν ( t − r i / c )] (38)where r i and r i respectively represent the distances between the element i of the sourceand the points P and P . The products V ∗ i ( t ) V i ( t ) simplify themselves as ∗ i ( t ) V i ( t ) = | a i ( t − r i / c ) | r i r i exp[ − i πν ( r i − r i ) / c )] , (39)as long as the following condition is verified | r i − r i | ≤ c / ∆ ν = λ / ∆ λ = (cid:96). (40)We thus see how to naturally introduce the coherence length (cid:96) which characterizes theprecision with which we must obtain the equality between the optical paths in order to beable to observe interference fringes (typically 2.5 microns in the visible for ∆ λ = To obtain the mutual intensity due to the whole source, it su ffi ces to insert in the expression(37), the relation (39) using (41). The result is Eq. (42), also known as the Zernicke-vanCittert Theorem I ( s ) ds = | a i ( t − r / c ) | , (41) γ (0) = (cid:90) S I ( s ) r r exp[ − i π ( r − r ) /λ )] ds / I . (42)When the distance between the source and the screen is very large, the expression of thistheorem can be simplified as follows. Let us adopt the orthonormal coordinate system ( x , y , z )shown in Fig. 24 such that the coordinates of the two elements P and P of the interferometerare respectively ( X , Y , 0) and (0, 0, 0) and those of an infinitesimal element dS i of the source( X (cid:48) , Y (cid:48) , Z (cid:48) ). It is then easy to find, by means of a relation analogous to (15), that | r − r | = | P P i − P P i | = | − ( X + Y )2 Z (cid:48) + ( X ζ + Y η ) | (43)where ζ = X (cid:48) Z (cid:48) , η = Y (cid:48) Z (cid:48) (44)represent the angular coordinates of the source measured from the interferometer. Usingthe two last relations, one can easily transform the expression (42) into (45). The X , Y coordinates in the first member of γ (0 , X /λ, Y /λ ) represent the position of one element ofthe interferometer relative to the other. One often defines u = X /λ and v = Y /λ which arequantities having the dimensions of the inverse of an angle, thus of angular space frequencies. Figure 24.
Positions of thetwo elements P and P ofthe interferometer and of theinfinitesimal element P i ofthe source assuming that thedistance Z (cid:48) >> | X (cid:48) | , | Y (cid:48) | , | X | or | Y | . Apart from a multiplicative factor, we thus find that the visibility of the fringes (the func-tion | γ ( τ = | ) is simply the modulus of the Fourier transform of the normalized surfacebrightness I (cid:48) of the source (Eq. (46)). (0 , X /λ, Y /λ ) = exp( − i φ X , Y ) (cid:90) (cid:90) S I (cid:48) ( ζ, η ) exp[ − i π ( X ζ + Y η ) /λ ] d ζ d η (45)with I (cid:48) ( ζ, η ) = I ( ζ, η ) (cid:82) (cid:82) S I ( ζ (cid:48) , η (cid:48) ) d ζ (cid:48) d η (cid:48) . (46)In terms of the angular space frequencies u = X /λ , v = Y /λ , Eq. (45) becomes γ (0 , u , v ) = exp( − i φ u ,v ) (cid:90) (cid:90) S I (cid:48) ( ζ, η ) exp[ − i π ( u ζ + vη )] d ζ d η. (47)By a simple inverse Fourier transform, it is then possible to recover the (normalized)surface brightness of the source with an angular resolution equivalent to that of a telescopewhose e ff ective diameter would be equal to the baseline of the interferometer consisting oftwo independent telescopes I (cid:48) ( ζ, η ) = (cid:90) (cid:90) γ (0 , u , v ) exp( i φ u ,v ) exp[ i π ( ζ u + ηv )] dud v. (48)Equations (47) and (48) thus clearly highlight the power of the complex degree of mu-tual coherence since they make it possible to link the visibility and the normalized intensitydistribution of the source by means of the Fourier transform v = | γ (0) | = | FT [ I (cid:48) ] | , and itsinverse. Aperture synthesis consists in observing a maximum number of visibilities of thesource, thus trying to cover as well as possible the ( u , v ) plane from which we shall try, some-times with some additional assumptions, to determine the structure of the source from theinverse Fourier transform (48) in which the integrant is not the visibility (i.e. the module ofthe complex degree of mutual coherence) but the complex degree of mutual coherence itself,within the factor exp ( i φ x ,y ). It is now good to remind some specific properties of the Fouriertransform. Let us remind that the Fourier transform of the function f ( x ), denoted FT [ f ( x )]( s ), where x ∈ (cid:60) , is the function FT [ f ( x )]( s ) = (cid:90) ∞−∞ f ( x ) exp( − i π sx ) dx (49)where s ∈ (cid:60) . The functions f and FT [ f ] form a Fourier pair. The function FT [ f ]exists if the function f ( x ) is bounded, summable and has a finite number of extrema anddiscontinuities. This does not necessarily imply that the inverse Fourier transform, denoted FT − [ FT [ f ]] transform is f . For the Fourier transformation to be reciprocal, f ( x ) = (cid:90) ∞−∞ FT [ f ]( s ) exp(2 i π xs ) ds , (50)it su ffi ces f to be of summable square, i.e. that the following integral exists (cid:90) ∞−∞ | f ( x ) | dx . (51)he definition of FT can be extended to the distributions. The FT of a distribution is notnecessarily of summable square. Let us also note that the functions f and FT [ f ] can be realor complex.We can generalize the FT to several dimensions, by defining f on the space (cid:60) n . Let r , w ∈ (cid:60) n , we then have FT [ f ]( w ) = (cid:90) ∞−∞ f ( r ) exp( − i π wr ) d r . (52)As a reminder, if f ( t ) designates a function of time, FT [ f ]( s ) represents its content as afunction of time frequencies. Similarly, if f ( r ) is defined on (cid:60) , where (cid:60) represents a two-dimensional space, the function FT [ f ]( w ) represents the space frequency content of f ( r ),where w ∈ (cid:60) .Among the interesting properties of the Fourier transform, let us remind: FT [ a f ] = a FT [ f ] , (53)with the constant a ∈ (cid:60) , FT [ f + g ] = FT [ f ] + FT [ g ] . (54) The considerations of symmetry are very useful during the study of the Fourier transform.Let P ( x ) and I ( x ) be the even and odd parts of f ( x ) such that f ( x ) = P ( x ) + I ( x ) , (55)we find that FT [ f ]( s ) = (cid:90) ∞ P ( x ) cos(2 π xs ) dx − i (cid:90) ∞ I ( x ) sin(2 π xs ) dx . (56)From this result, we can deduce for instance that if f ( x ) is real, the real part of FT [ f ]( s )will be even and its imaginary part will be odd whereas if f ( x ) is complex, the imaginary partof FT [ f ]( s ) will be even and its real part will be odd. The relationship of similarity is the following one FT [ f ( x / a )]( s ) = | a | FT [ f ( x )]( as ) (57)where a ∈ (cid:60) , is a constant. The dilation of a function causes a contraction of its Fouriertransform. This very visual property is very useful to understand that a function whose sup-port is very compact, has a very spread transform. In the analysis of temporal frequencies,one would state that a pulse of very short duration results in a very broad frequency spectrum,that is to say, contains frequencies all the higher as the pulse is brief. This is the classical re-lation of the spectrum of a wave packet, according to which the knowledge of the propertiesof a signal cannot be arbitrarily precise both in time and in frequency. .5.4 Translation: The translation relation is written as FT [ f ( x − a )]( s ) = exp( − i π as ) FT [ f ( x )]( s ) . (58)A translation of the function in its original space corresponds to a phase rotation of itsFourier transform in the transformed space. The door function, denoted Π ( x ), is defined by (see Fig. 25) Π ( x ) = x ∈ [ − . , . , and Π ( x ) = . (59)It is easy to find that FT [ Π ( x )]( s ) = sinc ( s ) = sin ( π s ) π s . (60)Applying the similarity relation, we also find that FT [ Π ( x / a )]( s ) = | a | sinc ( as ) = | a | sin ( π as ) π as . (61)The door function is also sometimes called the window function or simply window. Figure 25.
The doorfunction and its Fouriertransform (cardinal sine).
The Dirac distribution, also called Dirac peak, is noted δ ( x ). It is defined by the followingintegral, which exists only in the sense of the distributions δ ( x ) = (cid:90) ∞−∞ exp(2 i π xs ) ds . (62)Its Fourier transform is therefore 1 in the interval ] −∞ , + ∞ [ since δ ( x ) appears above asthe inverse Fourier transform of 1. .5.7 Applications: We propose hereafter several astrophysical applications that make use of the previous remark-able properties of the Fourier transform.Let us first consider the case of a double star which two point-like components are equallybright and separated by an angle 2 ζ . Making use of Eqs. (34), (46) and (47), one may easilyestablish using the properties (58) and (62) that the normalized intensity I (cid:48) ( ζ ) takes the simpleform I (cid:48) ( ζ ) = δ ( ζ − ζ ) + δ ( ζ + ζ )2 (63)and that the visibility v measured with an interferometer composed of 2 telescopes sepa-rated by the baseline X is given by the expression v = | γ (0) | = | cos (2 πζ u ) | (64)where u = X /λ .A second nice application consists in deriving the visibility of the interference fringesmeasured with the same interferometer of a 1 − D Gaussian star which intensity I ( ζ ) distribu-tion is given by the following expression I ( ζ ) = exp( − ζ FWHM ) (65)where FWHM represents the angular full width at half maximum of the 1 − D Gaussianstar. The expression of the corresponding visibility is then easily found to be v = | γ (0 , u ) | = exp (cid:32) − π u FWHM (cid:33) , (66)and we notice that narrower is the angular size of the star, broader is its visibility contentin angular space frequencies.In the third proposed application, we ask to establish the expression of the visibil-ity of a 2 − D uniformly bright square star which each angular side is ζ , i.e. I ( ζ ) = Cte Π ( ζ/ζ ) Π ( η/ζ ).The expression to be derived is the following one v = | γ (0 , u , v ) | = | sin ( πζ u ) πζ u sin ( πζ v ) πζ v | . (67)Finally, a generalization of the previous application consists in deriving the visibility of astar which is seen as a projected 2 − D uniform circular disk which angular radius is ρ UD andits angular diameter θ UD .Due to the circular symmetry of the problem, it is convenient to make use of polar coor-dinates in Eq. (45) as follows: u = X /λ = R cos( ψ ) /λv = Y /λ = R sin( ψ ) /λ, (68)where R denotes the baseline between the two telescopes of the interferometer, and ζ = θ cos( φ ) η = θ sin( φ ) . (69)q. (45) then transforms into | γ (0 , R /λ, ψ ) | = | πρ UD (cid:90) ρ UD θ (cid:90) π exp [ − i πθ R /λ (cos( φ ) cos( ψ ) + sin( φ ) sin( ψ ))] d φ d θ | . (70)Making use of the additional changes of variables z = πθ R /λ Φ = π/ − φ + ψ, (71)Eq. (70) becomes | γ (0 , R /λ, ψ ) | = | ( λ π R ) πρ UD (cid:90) πρ UD R /λ θ (cid:90) π/ + ψ − π/ + ψ cos( z sin( Φ )) d Φ d θ | . (72)Reminding the definition of the zero order Bessel function J ( x ) J ( x ) = π (cid:90) π cos[ x sin( θ )] d θ, (73)and the relation existing between J ( x ) and the first order Bessel function J ( x ), namely xJ ( x ) = (cid:90) x (cid:48) J ( x (cid:48) ) dx (cid:48) , (74)Eq. (72) successively reduces to | γ (0 , R /λ ) | = | ( λ π R ) πρ UD π (cid:90) πρ UD R /λ zJ ( z ) dz | (75)and | γ (0 , R /λ ) | = | J (2 πρ UD R /λ )2 πρ UD R /λ | . (76)We thus find that the expression (34) of the fringe visibility for the case of a star seen as aprojected 2 − D uniform circular disk with an angular dimater θ UD = ρ UD is v = (cid:32) I max − I min I max + I min (cid:33) = | γ (0 , u ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J ( πθ UD u ) πθ UD u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (77)where we have set u = R /λ . As a reminder, the Bessel function has the following proper-ties J ( x = . ... ) = x → J ( x ) x = / , (78)which allow us to easily understand the behavior of the visibility function illustrated inFig. 26.One could then wonder whether it is possible to observe interferometric fringes from ournearest star, i.e. the Sun? Figure 27 illustrates such fringes in while light obtained on 9thof April 2010 using a micro interferometer consisting of 2 holes with a diameter of 11.8 µ separated by a baseline of 29.4 µ . This micro-interferometer was placed in front of the igure 26. Visibilityfunction expected for a starconsisting of a uniformlybright circular disk with anangular diameter θ UD . objective of an EOS 5D Canon camera. Since the picture was taken in white light, it ispossible to see the e ff ects due to color dispersion. It is then easy to get an estimate of thefringe visibility, using Eq. (77), assuming that the Sun is a uniform disk with an angulardiameter of 30’. Figure 27.
Solar fringesphotographed with an EOS5D Canon camera in front ofwhich was set amicro-interferometerconsisting of two holeshaving a diameter of 11.8 µ separated by a baseline of29.4 µ . One of the most respected sanctuaries of optical interferometry is located on the plateau ofCaussols, north of Grasse, in the south of France. The I2T (in French, ”Interféromètre à 2Télescopes”), made of 2 telescopes with an aperture of 26cm each and separated by a baselineof up to 144m was characterized by an angular resolution Φ ∼ . (cid:48)(cid:48) attainable for objectswith an apparent magnitude brighter than V lim ∼ igure 28. First fringes obtained with the I2T on Vega (Labeyrie et al. 1975, c (cid:13)
Observatoire de la Côted’Azur ).
Figure 29.
Use of opticaldelay lines to compensatefor the continuous change inthe lengths of the two lightpaths as the Earth rotates.
The GI2T (in French, ”Grand Interféromètre à 2 Télescopes”) composed of two 1.5mtelescopes was subsequently used by the same team. The two big telescopes could in principlebe set 2 km apart, corresponding to an angular resolution Φ ∼ . (cid:48)(cid:48) (see Fig. 32).Since the beginning of the 21 st century, the modern sanctuary of stellar interferometryand aperture synthesis is undoubtedly the Very Large Telescope Interferometer (VLTI) ofESO (Southern European Observatory), located in Chile on Mount Paranal (see Fig. 33). TheVLTI is a European interferometer that can re-combine the signal from 2, 3 or 4 telescopes igure 30. View inside the opticaldelay line tunnel of the VLTI atESO, Paranal, Chile. c (cid:13)
ESO.
Figure 31.
Zoom on one of theoptical delay lines used in the tunnelof the VLTI at ESO, Paranal, Chile.c (cid:13)
ESO. depending on the instrument used. It has 4 telescopes of 8.2m and 4 mobile telescopes of1.8m. Only telescopes of the same size can be re-combined together. The auxiliary telescopesof 1.8m can be easily moved allowing a better coverage of the u , v plane. The maximum baselength of this interferometer is about 200m.CHARA is another very performing interferometer located on the heights of Los Angeles,California (see Fig. 34). It is installed on the historic observatory of Mount Wilson. Remem-ber that it was with the 2.5m telescope of this observatory that the first measurement of astellar diameter was made by Michelson and Pease by installing a beam of 7m at the top ofthe telescope. The CHARA interferometric array, operational since 1999 is composed of 6telescopes of 1m in diameter. These 6 telescopes can be either re-combined by 2, by 3 since2008 and recently the 6 together. The maximum base length of this interferometer is 330mallowing to achieve an angular resolution of 200 µ arcsec.It is mainly used for angular diameter measurements but also for the detection and char-acterization of tight binary stars as well as for the detection of exo-zodiacal clouds (cloudsof dust gravitating around the stars). Another famous optical / IR interferometer is the Keck igure 32.
The GI2Tconstructed by AntoineLabeyrie and his closecollaborators on the plateauof Caussols, north of Grasse,near Nice (France,c (cid:13)
Observatoire de la Côted’Azur ). Figure 33.
The Very LargeTelescope Interferometer(VLTI) at the top of Paranal(Chile, c (cid:13)
ESO).
Interferometer made of two 10m telescopes separated by a fixed baseline of 85m (see Fig. 35)on top of Mauna Kea (Hawaii, USA).
When we previously established the relation existing between the structure of a celestialsource and the visibility of the fringes observed with an interferometer (Sections 3.3 and4.4), we implicitly assumed that the size of the apertures was infinitely small (pinhole aper-tures). Use of the fundamental theorem allows one to calculate the response function ofan interferometer equipped with finite size apertures. This theorem actually formalizes, inmathematical terms, the physical connection existing between the focal plane and the pupilplane of an optical instrument (telescope, interferometer, grating, etc.). Use of the convo-lution theorem will then enable us to establish the relation between a celestial source thatis extended and its observed image in the focal plane of an optical instrument. Finally, the igure 34.
The CHARAinterferometer composed ofsix 1m telescopes at MountWilson Observatory(California, USA). c (cid:13)
TheObservatories of theCarnegie Institution.
Figure 35.
The Keckinterferometer on top ofMauna Kea (Hawaii, USA).c (cid:13)
Ethan Tweedie.
Wiener-Khinchin theorem establishes the relation between the frequency content of the pointspread function of an optical instrument and its pupil plane characteristics.
The fundamental theorem that we shall demonstrate here merely stipulates that given a con-verging optical system which can be assimilated to the lens or to the mirror of a telescope,or of an optical interferometer, the complex amplitude distribution a ( p , q ) of the electromag-netic field of radiation in the focal plane is the Fourier transform of the complex amplitudedistribution A ( x , y ) of the electromagnetic field in the pupil plane, i.e. a ( p , q ) = (cid:90) R A ( x , y ) exp [ − i π ( px + q y )] dxd y, (79)or in a more compact form a ( p , q ) = FT [ A ( x , y )]( p , q ) , (80)with = x (cid:48) λ f and q = y (cid:48) λ f , (81)where x (cid:48) , y (cid:48) refer to the Cartesian coordinates in the focal plane, λ to the wavelengthof the monochromatic light under consideration and f to the e ff ective focal length of theconverging system.Figure 36 represents a convergent optical system, its focal point F (cid:48) , its principal planes P , P (cid:48) and its principal points H and H (cid:48) . The latter degenerate with the optical center in thecase of a thin lens or with the bottom of the dish in the case of a single mirror. The twoorthonormal coordinate systems ( O , x , y , z ) and ( F (cid:48) , x (cid:48) , y (cid:48) , z (cid:48) ) make it possible to locate theinput pupil plane and the image focal plane of the optical system. The term ’pupil plane’serves as the support for the definition of the vibration state at the entrance of the collectorwhile the ’focal plane’ serves as the support for the definition of the image that the collectorgives of the source located at infinity. Defining the action of the collector is thus to establishthe transformation that it operates on the radiation between these two planes. Figure 36.
Fourier transform by a focusing optical system represented by its main planes P and P (cid:48) . Forthe case of a thin lens, the latter would be degenerated into a single plane passing through its center. The hypotheses underlying this theorem are:H1. The optical system is free from any geometric aberration.H2. The edges of the diaphragm do not disturb the electromagnetic field of radiation, that isto say that the diaphragm behaves as an ”all (1) or nothing (0)” function with respect to thisfield. This is equivalent to assume that the dimensions of the collecting aperture(s) are largewith respect to the wavelength of the light.H3. No disturbance, other than those imposed by the optical system, intervenes between thepupil and the focal planes. The optical elements are thus assumed to be perfectly transparentor reflective.H4. The light source is located at an infinite distance from the optical system and can thusbe considered to be point-like.H5. The disturbances occurring between the source and the pupil plane are weak and havevery long evolution times relative to the period (i.e. T = /ν = λ/ c ) of the radiation.6. The radiation is monochromatic and has a fixed polarization plane. Theorem statement:
Within a multiplicative coe ffi cient of the variables, the amplitude distribution in the focalplane is the Fourier transform of the amplitude distribution in the pupil plane. Demonstration:
Consider the di ff erent points ( x , y ) of the pupil plane. H6 (i.e. the previous hypothesis 6)makes it possible to represent the electrical component of the electromagnetic field by thereal part of the vibration distribution A ( x , y ) exp ( i πν t ) , (82)with the very general representation of the expression of the complex amplitude A ( x , y ) A ( x , y ) = A ( x , y ) exp ( i Φ ( x , y )) P ( x , y ) , (83)where A ( x , y ) and Φ ( x , y ) represent the amplitude and phase of the electric field and P ( x , y )the input pupil function which is 1 inside the pupil and 0 outside (in agreement with H2 andH3).In agreement with the Huygens-Fresnel principle, we will consider in the following thatevery point reached by a wave can be considered as a secondary source re-emitting a vibrationwith the same amplitude, the same frequency ν , the same polarization and the same phase(within a constant phase shift of π/
2) as those of the incident vibration at this point. Thepoint N ( x (cid:48) , y (cid:48) ) of the focal plane will thus receive vibrations emitted by all the points of thepupil plane. The laws of geometrical optics, deduced from the Fermat principle, make itpossible to write that the rays which, after the optical system, converge at the point N of theimage focal plane, were, before the optical system, parallel to H (cid:48) N . Having assumed thatthe source is at infinity (in agreement with H4), the amplitude will be preserved between thepupil plane and the focal plane. From the point M ( x , y ) of the pupil plane, the point N ( x (cid:48) , y (cid:48) )of the focal plane will thus receive the vibration A ( x , y ) exp ( i πν t + i Ψ ) . (84)Let us take as the zero phase shift reference that of the ray passing through the point O alongthe direction OJN . The phase shift Ψ can then be expressed using the di ff erence between theoptical paths δ = d ( MIN ) − d ( OJN ) , (85)where d () refers to the distance along the specified path, and the relation Ψ = πδ/λ. (86)If the point K corresponds to the orthogonal projection of M onto OJ , M and K belong tothe same wave plane which, after the optical system, will converge at the N point of the focalplane.The Fermat principle, according to which the optical path between a point and its image isconstant (rigorous stigmatism) or extremum (approximate stigmatism) makes it possible towrite that the di ff erence in optical path ( MIN ) − ( K JN ) behaves in the neighborhood of zeroas an infinitely small second order with respect to the d ( I , J ) and thus also with respect to d ( O , M ) and d ( O , K ), which are of the same order as d ( I , J ). As a result (see Fig. 36), δ = − d ( O , K ) = −| ( OM u ) | , (87) designating the unit vector along the direction H (cid:48) N and ( OM u ) the scalar productbetween the vectors OM and u . If the angle that H (cid:48) N makes with the optical axis is small,the vector of components ( x (cid:48) / f , y (cid:48) / f , 1) is the vector director of H (cid:48) N and has a norm closeto 1 (at first order because f >> | x (cid:48) | , | y (cid:48) | ). Moreover, OM has for components ( x , y , 0). UsingEq. (87) in (86), the expression (84) becomes A ( x , y ) exp ( i πν t − xx (cid:48) /λ f − yy (cid:48) /λ f ) . (88)Choosing as new variables in the focal plane those defined in (81), we get A ( x , y ) exp ( − i π ( xp + y q ) exp ( i πν t ) . (89)The resulting vibration at the point N will be the resultant of the vibrations transmittedtowards N by all the points of the pupil plane.The equi-phase wave surfaces which reach the pupil plane are not planes if the radiation hasbeen disturbed between the source and the entrance pupil. But the hypotheses H H ffi rm that the pupil plane is spatially coherent, that is to say that at thetime scale of the vibration periods, the relative phase shift of its di ff erent points is constant.Consequently, to calculate the resulting vibration at the point N ( p , q ) of the focal plane, it isnecessary to sum the amplitudes that N receives from the di ff erent points of the pupil plane.The amplitude distribution a ( p , q ) in the focal plane then becomes a ( p , q ) = (cid:90) R A ( x , y ) exp ( − i π ( xp + y q ) dxd y, (90)that is, the complex amplitude distribution in the focal plane a ( p , q ) is the Fourier trans-form of the complex amplitude distribution A ( x , y ) in the pupil plane, i.e. a ( p , q ) = FT [ A ( x , y )]( p , q ) . (91) Considering first the case of a single square aperture as depicted in Fig. 37 (left) and a point-like source perfectly located at zenith, i.e. the plane wavefronts arrive parallel to the aperturewith a constant and real amplitude A ( x , y ) = A , we find that the calculation of the amplitudein the focal plane is very simple a ( p , q ) = A FT [ Π ( x / a ) Π ( y/ a )]( p , q ) . (92)Making use of the separation of the variables x , y and of the relation (61), Eq. (80) suc-cessively transforms into a ( p , q ) = A FT [ Π ( x / a )]( p ) FT [ Π ( y/ a )]( q ) , (93) a ( p , q ) = A a sin ( π ap ) π ap sin ( π aq ) π aq . (94)This is the impulse response, in amplitude, for a square pupil and in the absence of anyexternal disturbance. Adopting the definition (11) for the intensity of the vibrations, we findthat (see Fig. 37, at right) i ( p , q ) = a ( p , q ) a ∗ ( p , q ) = | a ( p , q ) | = i a [ sin ( π ap ) π ap ] [ sin ( π aq ) π aq ] . (95) igure 37. Complex amplitude distribution A ( x , y ) in the plane of a single square aperture (left) andresulting response function in intensity i ( p , q ) (right). Defining the angular resolution Φ of an optical system as being the angular width of theresponse function in intensity inside the first minima, we obtain for the values of π pa = ± π (resp. π qa = ± π ), i.e. p = ± / a (resp. q = ± / a ) and with the definition (81) for p , qx (cid:48) λ f = ± a (resp . y (cid:48) λ f = ± a ) , (96) Φ = ∆ x (cid:48) f = ∆ y (cid:48) f = λ a . (97)The angular resolution is thus inversely proportional to the size a of the square aperture,and proportional to the wavelength λ . Working at short wavelength with a big size aperturethus confers a better angular resolution.Up to now, we have considered that the source S , assumed to be point-like and locatedat an infinite distance from the optical system, was on the optical axis of the instrument.Suppose now that it is slightly moved away from the zenith direction by a small angle. Let( b / f , c / f , 1) be the unit vector representing the new direction of the source, the previous onebeing (0, 0, 1). The plane wavefront falling on the square aperture will not have anymorea constant amplitude A because each point of the pupil touched by such a wavefront willexperience a phase shift given by the angle Ψ = πδλ = π ( xb / f + y c / f ) λ (98)and consequently the correct expression of the complex amplitude A ( x , y ) to be insertedin Eq. (80) becomes A ( x , y ) = A Π ( x / a ) Π ( y/ a ) exp[ i π ( xb / f + y c / f ) λ ] . (99)Proceeding as previously, we easily find that a ( p , q ) = A FT [ Π ( xa )]( p − b λ f ) FT [ Π ( y a )]( q − c λ f ) (100)nd finally i ( p , q ) = a ( p , q ) a ∗ ( p , q ) = | a ( p , q ) | = i a sin ( π a ( p − b λ f )) π a ( p − b λ f ) sin ( π a ( q − c λ f )) π a ( q − c λ f ) . (101)The resulting intensity response function in the focal plane is nearly the same as the onepreviously calculated for the case b = , c =
0. It is being merely translated by a linearo ff set ( b , c ) in the x (cid:48) , y (cid:48) focal plane and implies the invariance of the response function for areference star that is being slightly o ff set from the optical axis of the system. Considering now a circular aperture with radius R , the complex amplitude A ( x , y ) in thepupil plane may be represented as a circular symmetric distribution, i.e. A ( ρ, ϕ ) = A for ρ < R , ϕ ∈ [0 , π ] and A ( ρ, ϕ ) = ρ > R (see Figure 38, at left). We naturally expect thedistribution of the complex amplitude in the focal plane to be also circular symmetric, i.e. a ( ρ (cid:48) ) = FT [ A ( ρ, ϕ )]( ρ (cid:48) ) . (102)It is here interesting to note that performing the above Fourier transform is quite alikederiving the expression of the visibility of a 2 − D uniform circular disk star which angulardiameter is θ UD (see the last application in Section 4.5). We may just make use of the result(77) with appropriate changes of the corresponding variables. We easily find that a ( ρ (cid:48) ) = A π R [2 J ( 2 π R ρ (cid:48) / ( λ f )2 π R ρ (cid:48) / ( λ f ) )] . (103)The resulting intensity response function is thus given by i ( ρ (cid:48) ) = a ( ρ (cid:48) ) = A ( π R ) [2 J ( 2 π R ρ (cid:48) / ( λ f )2 π R ρ (cid:48) / ( λ f ) )] . (104)This is the very expression of the famous Airy disk (see Fig. 38, at right). Figure 38.
The Airy disk: complex amplitude distribution A ( ρ, ϕ ) = A in the plane of a circularaperture (left) and the resulting response function in intensity i ( ρ (cid:48) ) (right). nowing that the first order Bessel function J ( x ) = x ∼ .
96, it is easy to deducethat the angular resolution Φ of a telescope equipped with a circular objective which diameteris D = R is given by Φ = ∆ ρ (cid:48) f = . λ D . (105) Figure 39 (upper left) illustrates the principle of optically coupling two telescopes. Such asystem is equivalent to a huge telescope in front of which would have been placed a screenpierced with two openings corresponding to the entrance pupils of the two telescopes. Thepupil function A ( x , y ) of this system is shown in that same Figure for the case of two squareapertures. Figure 39.
The two telescope interferometer: distribution of the complex amplitude for the case of twosquare apertures (upper left) and the corresponding impulse response function (lower right).
Let us now calculate the impulse response function a ( p , q ) of such a system. Representingthe distribution of the complex amplitude over each of the individual square apertures bymeans of the function A ( x , y ) and assuming that the distance between their optical axes is D,we find that a ( p , q ) = FT [ A ( x + D / , y ) + A ( x − D / , y )]( p , q ) . (106)Making use of the relation (58), the previous equation reduces to a ( p , q ) = [exp( i π pD ) + exp( − i π pD )] FT [ A ( x , y )]( p , q ) , (107) a ( p , q ) = cos ( π pD ) FT [ A ( x , y )]( p , q ) (108)and finally ( p , q ) = a ( p , q ) a ∗ ( p , q ) = | a ( p , q ) | = cos ( π pD ) { FT [ A ( x , y )]( p , q ) } . (109)Particularizing this intensity distribution to the case of two square apertures, or circularapertures, and making use of relations (95) or (104) leads to the respective results i ( p , q ) = A (2 a ) [ sin ( π pa ) π pa ] [ sin ( π qa ) π qa ] cos ( π pD ) (110)or i ( p , ρ (cid:48) ) = A (2 π R ) [ 2 J (2 π R ρ (cid:48) / ( λ f ))2 π R ρ (cid:48) / ( λ f ) ] cos ( π pD ) . (111)Figure 39 (lower right) illustrates the response function for the former case. We see thatthe impulse response of each individual telescope is modulated by the cos(2 π pD ) functionand that the resulting impulse response function shows consequently a more detailed structurealong the p axis, leading to a significantly improved angular resolution Φ along that direction.The angular width Φ of the bright central fringe is equal to the angular width separating thetwo minima located on its two sides. We thus find successively π pD = ± π , (112) p = ± D , (113) ∆ p = D (114)and making use of relation (81) Φ = ∆ x (cid:48) f = λ D . (115)The angular resolution of the interferometer along the direction joining the two telescopesis approximately equivalent to that of a single dish telescope which diameter is equal to thebaseline D separating them, and not any longer to the diameter of each single telescope (seeEqs. (97) or (105)). When establishing the expression for the response function of an interferometer composedof two single square or circular apertures (see Section 6.1.3, Eqs. (110-111)), we implicitlyassumed that the exit pupil perfectly matched the entrance pupil (see Figs. 14, 15, 18 and 40).The baseline B between the two entrance pupil apertures was indeed equal to the baseline B (cid:48) between the two exit pupil apertures.This type of recombination of the two beams is referred to as the Fizeau-type or homo-thetic one. As we have seen in Section 3.4, Michelson and Pease have used another typeof beam recombination, known as the Michelson Stellar Interferometer or still, the densifiedrecombination type (see Fig. 41).When the two exit beams are being superimposed, resulting in the baseline B (cid:48) =
0, therecombination is referred to as being co-axial, or the Michelson Interferometer type (seeFig. 42). igure 40.
The two beams of light rays, represented with blue dashed lines, collected by the twoentrance pupil apertures are separated by a baseline B which is identical to the baseline B (cid:48) between thetwo apertures in the exit pupil plane of the main converging lens. Figure 41.
Sketch of the Michelson Stellar Interferometer. The baseline B between the two entrancepupil apertures is much larger than the baseline B (cid:48) between the two apertures in front of the recombininglens. The 45 ◦ inclined black lines symbolize reflective plane mirrors. In the case of the Michelson-Peaseexperiment, these four mirrors were set on a 7m beam just above the 2.5m Wilson telescope (see Fig.19). A more general model of beam recombination, that includes the three previously de-scribed ones, is illustrated in Fig. 43. Two main collectors receive the light beams from adistant celestial source. While passing through the beam reducers, the beams are compressedby a magnification factor M, corresponding to the ratio between the focal lengths of the twolenses of the focal reducers. The two compressed beams are then relayed by means of a setof 4 mirrors, just like in the Michelson Stellar Interferometer. Before entering the exit pupil igure 42.
The Michelson Interferometer. In this case, the beam recombination is co-axial corre-sponding to the exit pupil baseline B (cid:48) = of the recombining lens, their separation or baseline is B (cid:48) < B . To calculate the responsefunction of such an interferometer, we just need to apply the fundamental theorem to thissecondary Fizeau-type interferometer with a baseline B (cid:48) , taking into account the correct ex-pression for the distribution of the complex amplitude of the electric field over the two exitpupil apertures. Figure 43.
General case of beam recombination. The two beams of parallel light rays from a distantcelestial source are first collected by two unit telescopes having a diameter D . The beams are thencompressed by a magnification factor M. They are subsequently relayed by a system of plane mirrorsto the exit pupil of the recombining lens. At that stage, their separation (baseline) is B (cid:48) < B . onsidering a point-like celestial source emitting a plane wave making an angle θ withrespect to the line joining the two telescopes, the angle between the outcoming beam - com-pressed in size by the magnification factor M ( = f in / f out ) - and the main axis of the opticalsystem is M θ (since sin[ θ ] (cid:39) θ , given that θ <<
1, see Fig. 44). The resulting complexamplitude in the focal plane of the recombining lens is along the p direction, i.e. along theline joining the two exit pupil apertures (see Fig. 45 and Eq. (79)) a ( p ) = FT [ A ( x )]( p ) + FT [ A ( x )]( p ) (116)where A ( x ) and A ( x ) represent the distribution of the complex amplitude in the two exitpupil apertures along the x axis. Figure 44.
Propagation of an incoming plane wave from a distant celestial object with an inclinationangle θ through a beam reducer. The beam size is being reduced by the magnification factor M = f in / f out while the outcoming direction of the beam has changed into M θ . We subsequently find that FT [ A ( x )]( p ) = (cid:90) − ( B (cid:48) − D / M ) / − ( B (cid:48) + D / M ) / M exp[ − i π ( px )] exp[2 i π M sin[ θ ]( x − ( B / M − B (cid:48) ) / /λ ] dx , FT [ A ( x )]( p ) = exp[2 i π ( d /λ )] (cid:90) ( B (cid:48) + D / M ) / B (cid:48) − D / M ) / M exp[ − i π ( px )] exp[2 i π M sin[ θ ]( x + ( B / M − B (cid:48) ) / /λ ] dx . (117)In this expression, we have taken into account the fact that most of existing interferome-ters are equipped with a delay line and we have assumed here that an extra length d a ff ects thepath of the second beam. This explains the origin of the factor exp[2 i π ( d /λ )] in the expres-sion of FT [ A ( x )]( p ). The limits of integration are straightforward to establish (see Fig. 45,Level 3). The presence of the factor M merely accounts for the fact that when a beam is com-pressed, its constant amplitude is being multiplied by M (and the intensity i ( p ) by M in orderto preserve energy conservation). The factor exp[ − i π ( px )] merely accounts for the pupil-to-image relationship from Fourier optics (cf. the fundamental theorem). Since for the case ofa co-phased array, the path di ff erences a ff ecting the arrival of the plane waves at the centresof the two apertures at Level 1 in Fig. 45 are + δ and − δ ( = ± ( B /
2) sin[ θ ] (cid:39) ± ( B / θ ), the igure 45. Propagation of an incoming plane wave from a distant celestial object with an inclinationangle θ through two beam reducers (Level 1 - Level 2). When arriving in the exit pupil plane (Level 3),the delays ± δ of the plane waves near the centres of the two apertures are the same but their inclinationis now M θ . latter remain una ff ected when reaching the centres of the two apertures in the exit pupil plane(Level 3). Nevertheless, their relative inclination has changed from θ to M θ . Therefore,we easily understand the origin of the two factors exp[2 i π M sin[ θ ]( x − ( B / M − B (cid:48) ) / /λ ]and exp[2 i π M sin[ θ ]( x + ( B / M − B (cid:48) ) / /λ ] appearing in the two previous equations. Afterseveral successful changes of variables (see Appendix), Eq. (116) reduces to a ( p ) = D exp[ i π ( d /λ )] sin[( π D / M )( p − M sin[ θ ] /λ )]( π D / M )( p − M sin[ θ ] /λ ) cos[ π ( B (cid:48) p + ( d − B sin[ θ ]) /λ )] . (118)The corresponding expression for the intensity i ( p ) = | a ( p ) | becomes i ( p ) = D [ sin[( π D / M )( p − M sin[ θ ] /λ )]( π D / M )( p − M sin[ θ ] /λ ) ] [cos[ π ( B (cid:48) p + ( d − B sin[ θ ]) /λ )]] . (119)The previous equations describe the response function of any interferometer having itsentrance and exit baselines such as 0 ≤ B (cid:48) ≤ B .In the absence of an internal delay d , the previous expression for i ( p ) can be rewritten as i ( p ) = D [ sin[( π D / M )( p − M sin[ θ ] /λ )]( π D / M )( p − M sin[ θ ] /λ ) ] [cos[ π B (cid:48) ( p − B sin[ θ ] / ( B (cid:48) λ )]] . (120)Some nice features become outstanding: we first notice that the width of the envelopefunction is governed by the factor π D / M which is related to the size of the beam after com-ression. The angular separation of the fringes ( λ/ B (cid:48) ) is essentially determined by the exitpupil baseline B (cid:48) . It does neither depend on the main baseline B nor on the magnification (orbeam compression) M . This last equation also reveals that for the response function to befield invariant, we must have M = B / B (cid:48) . In that case, the centre of the main envelope (cf.Airy disk for the case of a circular aperture) will always coincide with the central fringe peak,whatever the position ( θ ) of the source in the field of view.Let us now consider the case of Fizeau-type interferometry for which we have d = M = B (cid:48) = B , also sin[ θ ] (cid:39) θ , Eq. (120) thenreduces to i ( p ) = D [ sin[( π D )( p − θ /λ )]( π D )( p − θ /λ ) ] [cos[ π ( B ( p − θ /λ ))]] . (121)Posing θ = b / f in the latter equation, we simply recover the result previously establishedfor the case of Fizeau interferometry (see Eqs. (101) and (110)). We also note here that theresponse function of a Fizeau-type interferometer is field invariant.Finally, the response function of a co-axial interferometer is easily derived by insertingthe value B (cid:48) = i ( p ) = D [ sin[( π D / M )( p − M θ /λ )]( π D / M )( p − M θ /λ ) ] [cos[ π ( d − B θ ) /λ ]] . (122)We note here that the cos factor is only a function of d and θ , and not any longer of p .In conclusion, we have established in this section a very general expression (see Eq. (119))for the response function of an interferometer composed of two similar apertures separatedby a baseline B and which beams have been compressed by a magnification factor M . In theexit pupil plane, the new baseline between the two beams is B (cid:48) (0 < B (cid:48) < B ) such that thefringe separation is essentially governed by the latter term. The fundamental theorem has allowed us to take into account the finite size of the aperturesof an optical system instead of considering that the apertures are made of pinholes. However,we have considered that the source is point-like. To treat the case of an extended source, weshall make use of the convolution theorem.The convolution theorem states that the convolution of two functions f ( x ) and g ( x ) isgiven by the following expression f ( x ) ∗ g ( x ) = ( f ∗ g )( x ) = (cid:90) R f ( x − t ) g ( t ) dt . (123)Figure 46 illustrates such a convolution product for the case of two rectangular functions f ( x ) = Π ( x / a ) and g ( x ) = Π ( x / b ) having the widths a and b , respectively.Every day when the Sun is shining, it is possible to see nice illustrations of the convolutionproduct while looking at the projected images of the Sun on the ground which are actuallyproduced through small holes in the foliage of the trees (see the illustration in Fig. 47). It isa good exercise to establish the relation existing between the observed surface brightness ofthose Sun images, the shape of the holes in the foliage of the trees, their distance from theground and the intrinsic surface brightness distribution of the Sun. igure 46. Convolution product of two 1-D rectangular functions. (a) f ( x ), (b) g ( x ), (c) g ( t ) and f ( x − t ). The dashed area represents the integral of the product of f ( x − t ) and g ( t ) for the given x o ff set,(d) f ( x ) ∗ g ( x ) = ( f ∗ g )( x ) represents the previous integral as a function of x . Figure 47.
Projected imagesof the Sun on the groundactually produced throughsmall holes in the foliage oftrees (bamboo trees atIUCAA, Pune, India, June2016). These imagesactually result from theconvolution of the intrinsicSun intensity distributionand the shapes of the holesin the trees.
We have previously seen that for the case of a point-like source having an intrinsic surfacebrightness distribution O ( p , q ) = δ ( p ) δ ( q ), there results the formation of an image e ( p , q )in the focal plane which is the impulse response e ( p , q ) = i ( p , q ) = | a ( p , q ) | of the opticalinstrument (see Eqs. (95), (104), (110), (111) for the case of a single square aperture, a singlecircular aperture, an interferometer composed of two square or circular apertures, respec-tively). Considering now an extended source represented by its intrinsic surface brightnessdistribution O ( p , q ), application of the convolution theorem in two dimensions directly leadsto the expression of its brightness distribution e ( p , q ) in the focal plane of the optical system e ( p , q ) = O ( p , q ) ∗ | a ( p , q ) | (124)or more explicitly e ( p , q ) = (cid:90) R O ( r , s ) | a ( p − r , q − s ) | drds . (125)Since the Fourier transform of the convolution product of two functions is equal to theproduct of their Fourier transforms, we find that T [ e ( p , q )] = FT [ O ( p , q )] FT [ | a ( p , q ) | ] (126)and also that the inverse Fourier transform of FT [ O ( p , q )] leads to the result O ( p , q ) = FT − [ FT [ O ( p , q )]] = FT − [ FT [ e ( p , q )] FT [ | a ( p , q ) | ] ] , (127)namely, that it should be possible to recover interesting information on the intrinsic sur-face brightness distribution of the source O ( p , q ) at high angular resolution provided that weget su ffi cient information at high frequencies in the u , v plane on the object FT [ e ( p , q )] itselfas well as on a reference point-like object FT [ | a ( p , q ) | ]. Considering the case of a symmetric source around the Y axis observed by means of aninterferometer composed of two square apertures which size of their sides is d separatedalong the X axis by the baseline D , we find by means of Eqs. (95), (110) and (124) that e ( p ) = d (cid:32) sin( π pd ) π pd (cid:33) (cid:104) O ( p ) ∗ cos ( π pD ) (cid:105) . (128)Making use of the relation cos(2 x ) = ( x ) −
1, Eq. (128) reduces to e ( p ) = d (cid:32) sin( π pd ) π pd (cid:33) (cid:34) (cid:90) R O ( p ) d p + O ( p ) ∗ cos(2 π pD ) (cid:35) . (129)Since the function O ( p ) is real, the previous relation may rewritten in the form e ( p ) = A (cid:34) B + Re [ O ( p ) ∗ exp( i π pD )] (cid:35) (130)where A = d (cid:32) sin( π pd ) π pd (cid:33) and B = (cid:90) R O ( p ) d p . (131)Given the definition of the convolution product (cf. Eq. (123)), relation (130) can berewritten as e ( p ) = A (cid:34) B + Re [ (cid:90) R O ( r ) exp( i π ( p − r ) D ) dr ] (cid:35) , (132)or e ( p ) = A (cid:34) B + cos (2 π pD ) FT [ O ( r )]( D ) (cid:35) , (133)because O ( p ) being real and even, its Fourier transform is also real. The visibility of thefringes being defined by (see Eq. (20)) v = | γ ( D ) | = (cid:32) e max − e min e max + e min (cid:33) , (134)we obtain = | γ ( D ) | = FT [ O ( r )2 B ]( D ) = FT [ O ( r ) (cid:82) O ( p ) d p ]( D ) . (135)We have thus recovered the important result (see Eq. (47), i.e. the Zernicke-van CittertTheorem), first established for the case of two point-like apertures, according to which thevisibility of the fringes is the Fourier transform of the normalized intensity distribution of thesource. This result can be generalized to the case of a source that is not symmetric. Finally, the Wiener-Khinchin theorem allows one to easily figure out what is the space fre-quency content of the point spread function for a given entrance pupil of an optical instrument.We may then directly find out which information is recoverable in terms of space frequencywhen observing an extended source.The Wiener-Khinchin theorem merely states that the Fourier transform of the responsefunction of an optical system, i.e. the Fourier transform of the Point Spread Function in ourcase, is given by the auto-correlation of the distribution of the complex amplitude in the pupilplane. In mathematical terms, the theorem can be expressed as follows FT [ | a ( p , q ) | ]( x , y ) = FT [ i ( p , q )]( x , y ) = (cid:90) + ∞−∞ (cid:90) + ∞−∞ A ∗ ( x (cid:48) + x , y (cid:48) + y ) A ( x (cid:48) , y (cid:48) ) dx (cid:48) d y (cid:48) . (136)When establishing the expression (127), we wrote that the quantity FT [ | a ( p , q ) | ] ap-pearing in its denominator could be retrieved from the observation of a point-like star. TheWiener-Khinchin theorem states that it can also be retrieved from the auto-correlation of thedistribution of the complex amplitude A ( x , y ) in the pupil plane. Figure 48 illustrates the ap-plication of this theorem to the case of an interferometer composed of two circular apertureshaving a diameter a and separated by the baseline b . We see that the autocorrelation of aninterferometer gives access to high space frequencies. Figure 48.
Diagramrepresenting theautocorrelation functionversus the space frequency,for a two telescopeinterferometer, each havinga diameter a , separated bythe baseline b . A simple demonstration of the Wiener-Khinchin theorem (136) is given below.We may successively establish that FT [ i ( p , q )]( x , y ) = FT [ | a ( p , q ) | ]( x , y ) = FT [ a ∗ ( p , q ) a ( p , q )]( x , y ) , (137) T [ i ( p , q )]( x , y ) = (cid:90) (cid:90) exp[ − i π ( px + q y )] (cid:90) (cid:90) A ∗ ( x (cid:48)(cid:48) , y (cid:48)(cid:48) ) exp[2 i π ( px (cid:48)(cid:48) + q y (cid:48)(cid:48) )] dx (cid:48)(cid:48) d y (cid:48)(cid:48) (cid:90) (cid:90) A ( x (cid:48) , y (cid:48) ) exp[ − i π ( px (cid:48) + q y (cid:48) )] dx (cid:48) d y (cid:48) d pdq , (138) FT [ i ( p , q )]( x , y ) = (cid:90) (cid:90) exp[(2 i π { p [ x (cid:48)(cid:48) − ( x (cid:48) + x )] + q [ y (cid:48)(cid:48) − ( y (cid:48) + y )] } )] (cid:90) (cid:90) A ∗ ( x (cid:48)(cid:48) , y (cid:48)(cid:48) ) dx (cid:48)(cid:48) d y (cid:48)(cid:48) (cid:90) (cid:90) A ( x (cid:48) , y (cid:48) ) dx (cid:48) d y (cid:48) d pdq (139)and taking into account the definition (62) of the Dirac distribution FT [ i ( p , q )]( x , y ) = (cid:90) (cid:90) (cid:90) (cid:90) δ [ x (cid:48)(cid:48) − ( x (cid:48) + x )] δ [ y (cid:48)(cid:48) − ( y (cid:48) + y )] A ∗ ( x (cid:48)(cid:48) , y (cid:48)(cid:48) ) A ( x (cid:48) , y (cid:48) ) dx (cid:48) d y (cid:48) dx (cid:48)(cid:48) d y (cid:48)(cid:48) . (140)We finally find that FT [ i ( p , q )]( x , y ) = (cid:90) (cid:90) A ∗ ( x (cid:48) + x , y (cid:48) + y ) A ( x (cid:48) , y (cid:48) ) dx (cid:48) d y (cid:48) dx (cid:48)(cid:48) d y (cid:48)(cid:48) = (cid:90) + ∞−∞ (cid:90) + ∞−∞ A ∗ ( x (cid:48) + x , y (cid:48) + y ) A ( x (cid:48) , y (cid:48) ) dx (cid:48) d y (cid:48) dx (cid:48)(cid:48) d y (cid:48)(cid:48) , (141)i.e. the quoted result, namely that the Fourier transform of the impulse response function ofan optical system can be represented by the autocorrelation of the distribution of the complexamplitude A ( x , y ) in the pupil plane.These lecture notes are based upon lectures on the same subject delivered by the authorin French at the Liège University during the past ten years (see [1]). To get deeper into thefield of interferometry, we highly recommend the following books: [2], [3], [4], [5].Finally, I wish to thank the organizers of the 2017 Evry Schatzman School (Dr. N.Nardetto, Prof. Y. Lebreton, Dr. E. Lagadec and Dr. A. Meilland) for their invitation togive these lectures and for the warm hospitality and nice atmosphere in Rosco ff during thatevent. In this appendix, we detail the calculations leading from Eqs. (116)-(117) to Eq. (118).First of all, we proceed with the following change of variables in the expression of FT [ A ( x )]( p ): y = − x , d y = − dx . We then replace y by x and d y by dx . Putting the factorexp[ i π d /λ ] in evidence, the summation of FT [ A ( x )]( p ) and FT [ A ( x )]( p ) leads to ( p ) = exp[ i π ( d /λ )] { exp[ − i π ( d /λ )] M exp[ − i π M sin[ θ ]( B / M − B (cid:48) ) /λ ] · (cid:90) ( B (cid:48) + D / M ) / B (cid:48) − D / M ) / exp[2 i π x ( p − M sin[ θ ] /λ )] dx + exp[ i π ( d /λ )] M exp[ i π M sin[ θ ]( B / M − B (cid:48) ) /λ ] · (cid:90) ( B (cid:48) + D / M ) / B (cid:48) − D / M ) / exp[ − i π x ( p − M sin[ θ ] /λ )] dx } (142)and subsequently a ( p ) = M exp[ i π ( d /λ )] (cid:90) ( B (cid:48) + D / M ) / B (cid:48) − D / M ) / { exp[ i π [2 x ( p − M sin[ θ ] /λ ) − ( d + M sin[ θ ]( B / M − B (cid:48) )] + exp[ − i π [2 x ( p − M sin[ θ ] /λ ) − ( d + M sin[ θ ]( B / M − B (cid:48) )] dx } , (143) a ( p ) = M exp[ i π ( d /λ )] (cid:90) ( B (cid:48) + D / M ) / B (cid:48) − D / M ) / cos[ π [2 x ( p − M sin[ θ ] /λ ) − ( d + M sin[ θ ]( B / M − B (cid:48) )] dx . (144)Let us now make use of the change of variables z = π [2 x ( p − M sin[ θ ] /λ ) − ( d + M sin[ θ ]( B / M − B (cid:48) )] such that dx = dz / [2 π ( p − M sin[ θ ] /λ )],Eq. (144) then transforms into a ( p ) = M exp[ i π ( d /λ )]2 π ( p − M sin[ θ ] /λ ) { sin[ π { ( B (cid:48) + D / M )( p − M sin[ θ ] /λ ) − ( d + M sin[ θ ]( B / M − B (cid:48) )) /λ } ] − sin[ π { ( B (cid:48) − D / M )( p − M sin[ θ ] /λ ) − ( d + M sin[ θ ]( B / M − B (cid:48) )) /λ } ] } , (145)and still a ( p ) = D exp[ i π ( d /λ )] π D [ p − M sin[ θ ] /λ ] / M { sin[ Γ + Λ ] − sin[ Γ − Λ ] } , with Γ = π { B (cid:48) (p − M sin[ θ ] /λ ) − (d + M sin[ θ ](B / M − B (cid:48) )) /λ } , and Λ = π D(p − M sin[ θ ] /λ ) / M . (146)Making use of the well known relation sin( Γ + Λ ) − sin( Γ − Λ ) = Γ ) sin( Λ ), theprevious equation reduces to a ( p ) = D exp[ i π ( d /λ )] sin[ π D [( p − M sin[ θ ] /λ )] / M ] π D [( p − M sin[ θ ] /λ )] / M · cos[ π { B (cid:48) ( p − M sin[ θ ] /λ ) − ( d + M sin[ θ ]( B / M − B (cid:48) )) /λ } ] (147)and since π { B (cid:48) ( p − M sin[ θ ] /λ ) − ( d + M sin[ θ ]( B / M − B (cid:48) )) /λ } = π { B (cid:48) p + ( d − B sin[ θ ]) /λ } , (148)e finally obtain a ( p ) = D exp[ i π ( d /λ )] sin[ π D [( p − M sin[ θ ] /λ )] / M ] π D [( p − M sin[ θ ] /λ )] / M · cos[ π { B (cid:48) p + ( d − B sin[ θ ]) /λ } ] (149)which is the same result as that quoted in Eq. (118). References
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