stellar interferometry : A glance at basics

1. stellar interferometry :A glance at basics Yves Rabbia Observatoire de la Côte d'Azur Dpt Fizeau, Grasse 06, France rabbia@obs-azur.fr

2. purpose of the talk to recall and to illustrate (hand waving as far as possible) • general framework and some land marks • specific terminology ( and debunking "jargon") • basics of interferometry and aperture synthesis • few things about nulling techniques

3. stellar interferometry :A glance at basics • sections • introduction • science context and motivation • limitations and subsequent needs • basics for interferometry and aperture synthesis • interferometers : principle, production, typology • difficulties in real world (and some remedies) • managing with data and some results • quick-look at some alternative HAR methods • nulling interferometry and coronagraphy

4. introduction stellar interferometry is part of a large body of topics covering methods and techniques aiming at High Angular Resolution ( HAR) the underlying goal is Aperture Synthesis but other techniques contribute to HAR eclipsing binaries lunar occultation speckle interferometry

5. resolution angular High Angular Resolution : a matter of "finesse" in exploration "finesse" depends on the instrument used what does "high" mean ? it means "responding to current scientific needs"

6. self-speaking illustration

7. not much yet poor not bad pretty much really impressive wooah ! fantastic A Rose by any other name would smell as sweet W. Shakespeare better and better testing your own angular resolution warning : sensitivity and contrast does matter (Signal to Noise Ratio)

8. detector resolution : large number of pixels is not the point one resel may cover several pixels on the detector pixel on the sky jargon : resel, resolution element resolution and resolution what counts for the astronomer is the size of pixels over the sky and this depends on the instrument (including observing conditions)

9. the lobe comes from throwing back to the sky the angular extension of the image obtained with a point-like source the so-called Point Spread Function (PSF) sky focal plane PSF l / D telescope illustration "live" result of the exploration sky PSF "finesse", exploration lobe, diffraction lobe : the finger

10. science context and motivation

11. Astrophysics wants d d d0 l1 a interferometry focuses primarily on a a0 sky coordinates a0 l2 then on l3 l introduction : the purpose of astro interferometry describe the brightness angular distribution of a source and measure morphological parameters (not necessarily images) then eventually, look for time evolution and polarisation in the following , we restrict the topic to stars

12. limb- darkened disk photospheric features uniform disk binary cocoon symbiotic disks an envelops miras morphology ??? few examples for stars Betelgeuse

13. a gradation of needs depending on sources, few parameters might be of great help but some morphologies would requires images examples binaries : angular separation (vector) single stars : angular diameter limb darkening : radial variation of intensity multiplicity : number of components, geometry extended atmosphere : radial structure, schock wave circumstellar matter : angular diameter, structure complex objects ( Miras, bipolar jets, disks, .... )

14. limitations and subsequent needs

15. astronomers (scientific requirements) need High Angular Resolution I want milliarcsec level 5 10 - 9 radian engineers (technical requirements) hep ! : diffraction sets a limit l / D what leads to several tens of meters for D problem for imaging stars !! angular dimensions of stars are so terribly small interferometry aims at breaking the limits of conventional imagery so as to determine some morphological parameters without large telescope diameters, (and ultimately produce images)

16. instrumentation observing conditions theoretically the larger the diameter the finer the lobe but D small or D large , extension of PSF quite the same immediate limitations lobe extension l/D and looking for 1 milliarcsec lead to D = 100 m in the visible (and more in Infrared) telescope not (yet) available mainly atmospheric turbulence PSF degraded , loss of resolution (speckles)

17. conventional imaging : directly provides a 2-dim representation ( feeding a camera) with aperture synthesis the imaging process requires specific methods for observation and computation a technical solution ?? conventional imaging does not work ! a new approach is needed to describe the brightness distribution or at least to obtain some of its parameters answer is aperture synthesis which is based on interferometry the idea behind : fetch spatial information in Fourier Space ! in other words : determine spatial spectrum and then, come back as far as possible to the angular intensity distribution of the source

18. phenomenology_ 1on the way towards interferometry (pictorial) basically mainly the peripheral corona of the collector governs the angular extension of the central peak yet at the price of increased sidelobes (and less photons, of course) but .. not easier to make large corona than large diameter ! so what... ?

19. phenomenology_ 2 not easier to make large corona than large diameter ! but ... look let's take two pieces of the corona in one direction the central peak remains narrow however on the perpendicular direction the peak extension is the one of the small piece

20. D B D do it yourself ! l/B l/D where is the gain ? l/B instead of l/D how to make it practically ?

21. where does it come from ? what does that mean "exploring with a comb ?" exploring with a lobe both narrow and large telescope questions appear extension l/B much finer than l/D gain B/D example 100m/1m OK but, the exploration lobe is like a comb some theoretical basis needed but before let's look again a little more at phenomenology

22. point like and small baseline enlarging baseline fringe spacing l/B decreased binary fringe patterns slightly shifted yet fringes but slightly lowered contrast enlarging baseline fringe patterns fully overlap no more fringes, total blurring contrast fully destroyed exploring and measuring with a "fringed lobe" fringes with full contrast (max = 1, min = 0) there angular separation is measured : l/2B

23. basics for interferometry and aperture synthesis • toolbox and terminology • mathematical tools Fourier world (reminders) spatial frequencies & spatial spectra Fourier optics linear filtering • fundamentals principles detection, coherence, VCZ theorem • interferometers

24. in other words Im( Z) Z modulus A phase f Re( Z) no comments, see later on the very first tool, not a joke !

25. Fourier formalism

26. complex-valued function "fonctionnelle linéaire" E = space of "friendly functions" RxE  C linear transformation omnipresent tool : Fourier formalism (reminders)_1 omnipresent : spatial frequencies, spatial spectra, coherence, Fourier optics, linear filtering, visibilities, signal processing, data processing, ....... usual definition(s) and usual expression

27. "friendly" functions : no infinity (physical signal) finite definition interval, finite values possibly complex valued : R C • scalar product : • Fourier formalism : "friendly" functions can be expressed • as a weighted sum of "base functions" depending on a parameter "u" • e(u,x) = exp( i.2p.u.x) Re( e(u,x) ) Im( e(u,x)) x 1/u Fourier formalism ( reminders) _ 2 side view the weight associated to a given "u" is called Fourier component of "f " for "u" parameter "u" is a frequency (complex helix)

28. X2=2 e2 = weighted sum of "base vectors" ek X1= 4 e1 retrieving components (weights) Xk : scalar product similar view for "friendly" functions function f : weighted sum of base functions involving parameter "u" retrieving weights (components) : scalar product and we find the usual definition expression of the "u" component Fourier formalism (reminders) _ 3 wild (and disputable) analogy with vectors

29. a pictorial approach = x FT can be seen as a "periodicity sensor" ( period 1/u) algebra yields the contribution of a periodic-like component f(x) exp(- i.2p.ux) f(x) ^ presence of a periodic feature f 1/u x unmatched frequency no response x matched frequency amplitude extracted x practical view and comments_1

30. note 2 : inverse transform note 1 : all values f(x) needed to have a single value of note 3 : frequent use of two-variable transforms (spatial description) note 4 : linearity note 5 : parity real even function  real even FT, odd complex FT practical view and comments_2 note 6 : for two extremely simple functions the definition does not make sense ! (infinity is there) constant and sine do not match the definition : but Dirac will save us

31. III(x/p) x p III(x/p, y/q) crucial relations y d(x,y) d(x-a, y-b) y q b y p x a x x Dirac world

32. P( ) P( x/A) x - a 1 1 A L( x/A) x a x 1 A A -A +A x 2A P P ( ( / 2R) / 2R) r r exp( - x2/a2) 1 y 1/e r r x x x a -a R R usual and frequent functions : notations and physionomy rectangle (door, pulse,...), shifted rectangle, triangle or Lambda (L), gaussian, camembert

33. L( x/A) P( ) P( x/A) x - a 1 1 1 exp( -p.x2) A x a x x 1 -A +A 1/e A A x 1/ 1/ -1/ ^ ^ p p p f(u) f(u) u 1/A exp( -p. u2) u 1 1/A 1/e 1/A u 1/a Fourier Transforms of usual functions _ 1

34. 1 0.1 0.01 0.001 P( r / 2R) 1 log y r p.R2 x R ^ ^ f(u) f(u) u 1.22 / 2R avec Fourier Transforms of usual functions _ 2 camembert

35. u=0 u u u1 u u u2 u u u3 u u hand made transforming _ 1 rectangle, door, pulse function base function e(u,x) integral => one value of TF a spectrum's component

36. u=0 u u u1 u u u2 u u u3 u u hand made transforming _2 Dirac ! function base function e(u,x) integral => one value of TF a spectrum's component

37. constant x u cosinus u x - 1/p 1/p p Re (ex,u ) Im (ex,u ) u x 1/f f III(x/p) III(p.u) u x f p 1/p usual functions while rather pathological (but soon friendly) the definition integral does not converge : Dirac heals

38. y "friendly" physical signals again x what is changing ?? moduli of base functions longer are "sine lines" moduli are now something like "tôles ondulées" (but with negative parts) having various periods, phases and orientations + + + still, a given 2D-signal again is a weighted sum of 2D- base functions + + + complication : introducing FT with 2 variables

39. location of a given point of the object requires two coordinates (x and y) or a vector also, a given frequency is described by two "projected frequencies" "u" and "v", or a vector (to account for orientation of the sine-like surface) then a given base function ( varying over x and y) will convey (x,y,u,v) and writes U new algebra with two variables

40. A sine-like surface in (x,y) space a couple of dirac-distributions (FT of cosine) in Fourier space also named (u,v) plane u1,v1 v y 0,0 u u2,v2 x illustrations for 2 variables objects the "faster" the oscillation the farther from origin the diracs (higher frequency) orientation of the couple is perpendicular to the wave crests

41. starting from two functions f(t) and g(t) we define a new function h(x) where x is the shifting of f versus g f(t) g(t) f(t) g(t) x t t t t reverse and shift x h(x) reversal x effects of convolution : smoothing translating a (bad) current notation ( physicists) = x x x another tool : convolution

42. f(t) g(t) départ t t décalage f(t) g(x -t) f(t).g(x -t) h(x) x t t t x t t t x t t t x t t t x t t t convolution_3, "live" x = zero x = x1 x = x2 x = x3 x < 0 global result : convolution decreases stiffness, smoothing

43. f(t) g(t) h(x) t x t A A A A f(t) g(t) h(x) t x t a a g(t) = d(t – a) f(t) = P(t) h(x) = P(x – a) t x t a a convolution_3 visual training

44. actors : f(t,z) and g(t,z) shifts : x and y illustration starting distribution this recalls something we will come back to it again result of exploration exploration lobe convolution_4 also with two variables

45. TF translation : similarity convolution autocorrelation parseval (rayleigh) some usual theorems , pictorial with a rectangle ditribution starting point :

46. the first key : spatial frequencies

47. VOOOO..... S(f) V(t) periodic signal : discrete spectrum non-periodic : continuous spectrum t f Driiiiiiii Schboumpf for very "narrow" signals, a lot of sinusoides needed and high frequencies thus : continuous and extended spectrum limiting case : Dirac FLAK f t frequencies , spectrum, in the familiar time domain reminder : a physical signal can be described as a weighted sum of sinusoidal components (Fourier) of various frequencies The set of weighting factors (Amplitude, frequency) is the spectrum of the signal

48. + ... one direction modulated all directions modulated + + + + + spatial frequencies time domain : frequency = (1 / time ) , Hzspatial distribution (2-dim x & y) spatialfrequency : vector (u,v) each component ( 1 / length) spatio-angular frequency : vector (u,v), each (1/angle) or (1/radian) just recalling practical pictures : "tole ondulée" or "sine surface" (with negative parts) practical training : find spatial frequencies in the room !

49. "sine surface" (x,y) one direction --> couple of diracs in (u,v) plane (Fourier space ) x influence of orientation u p replicated rectangle x 1/A u A 1/p v y x algebra : u u1,v1 u2,v2=0 pictorial examples

50. y Zebra = horse + grid ( conspicuous spat. freq.) x remove the grid Z(u,v) v S(u,v) Z(u,v) v u v u u spatial frequencies : further training change your zebra for a circus horse circus horse = horse + double grid just add the needed frequencies