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Optics in Astronomy - Interferometry - Oskar von der Lühe Kiepenheuer-Institut für Sonnenphysik Freiburg, Germany Contents • • • Fundamentals of interferometry – Why interferometry? Concepts of interferometry – Diffraction-limited imaging – A Young‘s interferometry interference experiment Practical – Propagation of light and coherence – Theorem of van Cittert - Zernike September 2002 Optics in Astronomy 2 What is interferometry? • superposition of electromagnetic waves – at (radio and) optical wavelengths – infrared: = 20 µm ... 1µm – visible: = 1 µm ... 0.38 µm • which emerge from a single source • and transverse different paths • to measure their spatio-temporal coherence properties • Topic Why interferometry? of this lecture on interferometry? – to increase the angular resolution in order to – compensate for atmospheric and instrumental aberrations • speckle interferometry • diluted pupil masks – overcome the diffraction limit of a single telescope by coherent combination of several separated telescopes September 2002 Optics in Astronomy 3 Diffraction limited imaging D focal length F telescope aperture September 2002 X 2.44 Optics in Astronomy F D 4 A Young‘s interference experiment X 2.44 x F D diameter D Screen F B baseline B Source Mask wavelength focal length F September 2002 Optics in Astronomy 5 Two-way interferometer I Change of baseline September 2002 Optics in Astronomy 6 Two-way interferometer II Change of element diameter September 2002 Optics in Astronomy 7 Two-way interferometer IIa Change of element diameter September 2002 Optics in Astronomy 8 Two-way interferometer III Change of wavelength September 2002 Optics in Astronomy 9 Dependence on source position Screen Source September 2002 Mask Optics in Astronomy 10 Two-way interferometer IV Change of source position September 2002 Optics in Astronomy 11 Dependence on internal delay Screen delay Source September 2002 Mask Optics in Astronomy 12 Two-way interferometer V Change of internal delay September 2002 Optics in Astronomy 13 Monochromatic e. m. waves Time-dependent, monochromatic e.m. field: Time-independent e.m. field fulfils Helmholz equation: V (r , t ) U (r ) exp j t position in space r time t circular frequency 2 ( 2 k 2 ) U (r ) 0 ; k c 2 U,V C September 2002 Optics in Astronomy 14 Propagation of field r1 r0 r Kirchhoff-Fresnel integral, based on Huyghens‘ principle U (ro ) September 2002 1 1 U ( r ) exp 2j 1 j r Optics in Astronomy r ( ) ds 15 Non-monochromatic waves Spectral decomposition: 1 V (r , t ) U (r ) exp j t d Propagation of a general e.m. wave: r ( ) V (r0 , t ) V r1, t ds t c 2cr Condition of quasimonochromatism: Propagation of a quasimonochromatic wave: September 2002 1 r V (ro , t ) V r1, t ( ) ds j r c Optics in Astronomy 16 Intensity at point of superposition I (r , t ) V1 (r , t ) V2 (r , t ) 2 I1 (r , t ) I 2 (r , t ) 2 Re V1 (r , t )V2* (r , t ) The intensity distribution at the observing screen of the Young‘s interferometer is given by the sum of the intensities originating from the individual apertures plus the expected value of the cross product (correlation) of the fields. September 2002 Optics in Astronomy 17 Concepts of coherence I (terminology from J. W. Goodman, Statistical Optics) (r1, r2 , t1, t2 ) : V (r1, t1 ) V * (r2 , t2 ) mutual intensity: temporally stationary conditions, with = t2 - t1 complex degree of coherence: (r1, r2 , t1, t2 ) (r1, r2 , t1, t1 ) : 12 ( ) 12 ( ) : September 2002 Optics in Astronomy 12 ( ) 11(0) 22 (0) 1/ 2 V r1, t1 V * r2 , t1 2 2 V r1, t1 V r2 , t1 18 Concepts of coherence II self coherence (temporal coherence): 11 : (r1, r1, ) V (r1, t1 ) V * (r1, t1 ) complex degree of self coherence: 11( ) mutual intensity (spatial coherence): J12 12 0 V (r1, t1 ) V * (r2 , t1 ) complex coherence factor: September 2002 12 12 (0) Optics in Astronomy 19 Concepts of coherence III • Coherence is a property of the e.m. field vector! • It is important to consider the state of polarisation of the light as orthogonal states of polarisation do not interfere (laws of Fresnel-Arago). • Optical designs of interferometers which change the state of polarisaiton differently in different arms produce instrumental losses of coherence and therefore instrumental errors which need calibration. September 2002 Optics in Astronomy 20 Temporal coherence S1 2R 2Z 1 2 S2 1 c : 11( ) 2 d coherence time: refined monochromatic condition: September 2002 c 2 lc : c c Optics in Astronomy 21 Two-way interferometer VI Extended spectral distribution of a point source September 2002 Optics in Astronomy 22 Two-way interferometer VII Extended spectral distribution source and delay errors September 2002 Optics in Astronomy 23 Transport of coherence 1 r1 S1 x1 r1 r2 1 J ( x1, x2 ) 2 x2 2 (1 ) (2 ) 2 J ( r , r ) exp j ( S S ) S S 1 2 2 1 d 1d 2 1 2 1 September 2002 S 2 x2 r2 x1 1 Optics in Astronomy 24 Extended sources Screen Source September 2002 Mask Optics in Astronomy 25 Extended, incoherent sources Most astronomical sources of light have thermal origin. The processes emitting radiation are uncorrelated (incoherent) at the atomic level. Mutual intensity at the surface of an incoherent source: 2 J (r1, r2 ) I (r1 ) (r1 r2 ) Transport of mutual intensity to the interferometer: J ( x1, x2 ) 1 (1 ) (2 ) 2 I (r1 ) exp j ( S 2 S1 ) d 1 S1S 2 1 September 2002 Optics in Astronomy 26 Theorem of van Cittert - Zernike Celestial sources have large distances S compared to their dimensions. Differences in S can be expanded in first order to simplify the propagation equation. Inclination terms are set to unity. Linear distances in the source surface are replaced by x apparent angles . The transport equation can then be simplified: J ( x1, x2 ) September 2002 2 I ( ) exp j x y d source 1 x1 x2 I ( ) exp 2 j d source 1 Optics in Astronomy 27 Intensity distribution in the Young‘s intererometer I (r ) I1 (r ) I 2 (r ) 2 Re V1 (r , t )V2* (r , t ) 12 Br I1 (r ) I 2 (r ) 2 ReJ x1 , x2 cos 2 F B r 2 A(r ) 1 12 cos 2 F with B B I exp 2j d I d The complex coherence factor µ is often called the complex visibility. It fulfils the condition 1 September 2002 Optics in Astronomy 28 Two-way interferometer VII Spatially extended source - double star September 2002 Optics in Astronomy 29 Two-way interferometer VII Spatially extended source - limb darkened stellar disk September 2002 Optics in Astronomy 30 Extended sources - not unique? September 2002 Optics in Astronomy 31 van Cittert - Zernike theorem Source intensity Response to a point source in direction of B x I x Re I exp 2j d z Bx B Reexp 2j I exp 2j d z Observed Intensity Instrumental cosine term September 2002 2D Fourier transform of source intensity at angular frequency B/ (visibility function) Optics in Astronomy 32 What does the vCZT mean? An interferometer projects a fringe onto the source‘s intensity distribution The magnitude of the fringe amplitude is given by the structural content of the source at scales of the fringe spacing The phase of the fringe is given by the position of the fringe which maximises the small scale signal September 2002 Optics in Astronomy 33 What have we learned? • An astronomical interferometer measures spatio-temporal coherene properties of the light emerging from a celestial source. • The spatial coherence properties encodes the small scale structural content of the intensity distribution in celestial coordinates. • The temporal coherence properties encodes the spectral content of the intensity distribution. • The measured interferometer signal depends on structural and spectral content. September 2002 Optics in Astronomy 34