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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 1 International Young Astronomer School 2010 High Angular Resolution Techniques Diffraction and Fourier Optics Guy Perrin Monday November 1 st, 2010

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 2 Outline 1. Diffraction 2. Fourier transform 3. Fraunhofer diffraction 4. Imaging 5. Images of extended sources 6. The case of interferometry 7. Wavefront distortions, aberrations 8. Sampling theory

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 3 References Optics and diffraction: M. Born and E. Wolf, Principle of Optics, 7 th edition, Cambridge University Press, 2002 E. Hecht, Optics, 4 th edition, Addison Wesley, 2001 Fourier Transform: R.N. Bracewell, Fourier transform and its applications, Mc Graw- Hill, 1986

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P Diffraction

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 5 Wavefront Continous surface of points with the same phase at a given time Case of a wave propagating in a medium with uniform index from a point source: P0P0 P Wavefronts are perpendicular to optical rays.

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 6 Wavefront Expression of the phase of the wave: P0P0 P is the optical path between P 0 and P (Fermat principle): In a medium with uniform index: Wavefronts emitted by a point source are indeed spherical.

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 7 Why diffraction ? A spherical wave emitted by a point source in a medium with uniform refractive index and without obstacles remains spherical: P0P0 P Diffraction is due to the spatial limitation of waves

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 8 Historical discovery of diffraction Optical rays are disturbed by the diaphragm: - the size of the beam projected on the screen decreases with the decreasing diaphragm size; Diaphragm P0P0 Screen

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 9 Diaphragm Screen Historical discovery of diffraction Optical rays are disturbed by the diaphragm: - the size of the beam projected on the screen decreases with the decreasing diaphragm size; P0P0

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 10 Diaphragm Screen Historical discovery of diffraction Optical rays are disturbed by the diaphragm: - the size of the beam projected on the screen decreases with the decreasing diaphragm size; - below a particular value of the diaphragm size, the size of the projected spot increases and diffraction rings appear. P0P0

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 11 P0P0 d D /D The spot increases when: Or when the distance becomes larger than: Rayleigh distance kjjjjj Diaphragm Screen

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 12 The Huygens-Fresnel principle Huygens principle: each point of the wavefront is a source of secondary spherical waves. Huygens-Fresnel principle: diffraction is described by the interferences of propagated secondary spherical waves.

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 13 Why diffraction ? Diffraction is from the latine word diffringere which means to break. Diffraction breaks straight light rays.

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 14 For a monochromatic scalar wave, Maxwell equations take a simple form and waves are described by the following set of equations: where k=2 / is the norm of the wave vector. A solution is the integral form of Helmholtz-Kirchhoff: S is a surface containing P. The wave in the volume is known if it is known on the surface S along with its partial derivatives. Scalar theory

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 15 When distances of P 0 and P to the diaphragm are large with respect to the diaphragm itself much larger than wavelength, the integral becomes: Fresnel-Kirchhoff equation

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P Fourier transform

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 17 f and are Fourier pairs, k are x are conjugated by the Fourier transform General definition FT Direct Fourier transform Inverse Fourier transform FT -1

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 18 Properties of Fourier transforms If f is a real function then its positive and negative spectra are complex conjugated: The Fourier transform of a hermitian function (f(-x)=f(x) * ) is real. The Fourier transform of an anti-hermitian function (f(-x)=-f(x) * ) is imaginary. Particular case: a real symetric function has a symetric FT and direct or inverse FT are equivalent.

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 19 Properties of Fourier transforms Fourier transform of an auto-correlation (Wiener-Khintchine theorem): Fourier transform of a convolution:

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 20 Properties of Fourier transforms The Fourier transform conserves energy:

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 21 Properties of Fourier transforms Scaling theorem: Shifting theorem:

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 22 Examples (cardinal sinus)

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 23 Examples In 2D: J 1 is a Bessel function of the first kind.

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P Fraunhofer diffraction

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 25 When distances of P 0 and P to the diaphragm are large with respect to the diaphragm itself much larger than wavelength, the integral becomes: Fresnel-Kirchhoff equation M

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 26 Fraunhofer diffraction Assumptions: 1. Small diffraction angles. 2. The diaphragm and screen smaller than z: 1/s ≈ 1/z 3. z largeur than the Rayleigh distance: This is the case for astronomical imaging M zy0y0 x0x0 y1y1 x1x1

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P et 2 3

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 28 Angular coordinates are naturally used in astronomy: Angular spatial frequencies or spatial frequencies correspond to the reciprocical of a characteristic scale of variation of the spatial intensity distribution of a source: Spatial frequency coordinates and spatial coordinates have reciprocal dimensions. They are conjugated by the Fourier transform. x1x1 z P u v +1/P-1/P Fourier transform

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 29 Frequency f and time t are conjugated variables. Fourier transform tf

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P (rad -1 or arcsec -1 ) Changing variables to: (rad or arcsec)

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 31 The diffracted wave is (proportional to) the Fourier tranform of the wave in the pupil: In practice, the proportionality factor is not written: - the absolute phase term cannot be measured; - the modulus of the diffracted wave can be adjusted imposing that energy is conserved between pupil and image planes.

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P Imaging

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 33 pupil P image plane Image of a point source at infinity The source is point-like and at infinity on the optical axis. The normalized field in the pupil is: with P(x,y) the pupil function equal to 1 in the pupil and 0 outside. In the image plane, the normalized field is: Parseval-Plancherel theorem to scale the field in the image plane to conserve energy:

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 34 pupil P image plane Image of a point source at infinity Optical detectors are sensitive to intensity: Applying the Wiener-Khintchine theorem yields:

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 35 Point Spread Function The Point Spred Function (PSF) is the image of a point source I( , ). The PSF is the Fourier transform of the autocorrelation of the pupil function. Its normalized Fourier transform is the Optical Transfer Function (OTF). Pupil planeImage plane FT

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 36 Pupil function: OTF: cutt-off frequency: D/ PSF (Airy pattern) : FWHM: ≈ /D Example: uniform & circular pupil f

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 37 f Definition 1 : characteristic scale equal to the reciprocal of the OTF cut-off frequency = /D Definition 2 : FWHM of the PSF ≈ /D Definition 3 (Rayleigh criterion) : first zero of the Airy pattern: 1.22 /D Definition(s) of angular resolution

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 38 Two sources are separated by the optical system if their angular separation is > /D Example: - Hubble Space Telescope (D=2,4m), =0,5 µm => R= 0,042 " Image HST 0,6"

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 39 Image plane f D

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 40 Image plane f D <

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 41 Image plane f D >

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P Images of extended sources

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 43 Image of an off-axis point-like source The image of a point source located at ( 0, 0 ) is: I’( , )=I( 0, 0 ) And therefore: I’( , )=PSF( 0, 0 )

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 44 Image of an (uncoherent) extended source Assuming the spatial intensity distribution of the source: O( 0, 0 ) And the object is spatially uncoherent (waves from two different points are uncorrelated). A point-like source at ( 0, 0 ) produces the intensity: Summing individual contributions in the image plane yields: pupil P Image plane The image is the convolution of the object intensity spatial distribution by the PSF

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 45 Image spectrum: The optical system acts as a low-pass filter whose cut-off frequency is D/ Spatial frequency contents of the image Object x FT Source spectrum = Image spectrum

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P The case of interferometry

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 47 The same theory applies to interferometers which are just a particular case of telescopes. Band-pass filtering and interferometry +D/ -D/ +D/ -D/ -B/ +B/ 1 1/2 PupilOptical Transfer Function +D/ -D/ -B/ +B/ 1 Spatial spectrum of the object Measured visibility For an extended source: D B

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 48 < Unresolved source Image plane f D Angular resolution: > Resolved source

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 49 Image plabe f D Angular resolution: Interferometry

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 50 Image plane f B Angular resolution: Interferometry D

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 51 Image plane f B Angular resolution : Interferometry D

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 52 Image plane f B Angular resolution: Interferometry D

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 53 Synthesized pupil Beam combination and detection D B Naive approach of aperture synthesis

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 54 Efficient use of telescopes +D/ -D/ What matters is to measure as many spatial frequencies as possible and not to fill the entrance pupil what matters is the synthesized OTF. +D/ -D/ -B max / +B max /

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 55 Non redundant configurations A pupil is non redundant when two different pairs of sub-pupils yield different spatial frequencies. Pupil OTF Golay 6 Spatial frequencies 15 spatial frequencies

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 56 Example with Keck I Pupil mask

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 57 Spatial frequencies Example with Keck I Image Pupil mask

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 58 Masquage de pupille avec Keck 2.27 m Tuthill, Monnier & Danchi (1999) 160 u.a. = 100 mas Wolf-Rayet star WR 104

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P Wavefront distortions, aberrations

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 60 Wavefront and aberrations Static (optics defects) or dynamic (turbulence) aberrations induce non-flat wavefronts in the pupil plane: Wave in the pupil plane for a point source at infinity: Rayleigh criterion on wavefront or image quality: The image is diffraction limited if the peak-to-valley the wavefront is less than: /4 Caveat: this criterion was historically established for the spherical aberration.

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 61 Wavefront and aberrations Static (optics defects) or dynamic (turbulence) aberrations induce non-flat wavefronts in the pupil plane:

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 62 Case of a tilted wavefront The source is point-like and at infinity angularly shifted by ( 0, 0 ) with respect to the optical axis: The wave in the pupil plane writes: And the image is shifted accordingly (shows up in the autocorrelation): I( , )=PSF( 0, 0 )

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P Sampling theory

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 64 1D signal: i(t). I(t) is i(t) after sampling at t i =i. t t is the sampling step and f=1/ t is the sampling frequency. I(t) writes: Sampling of a real-life signal i(t) t I(t) t tt

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 65 The Shah function or Dirac comb is stable by Fourier transform: = Multiplying by the Shah function samples the signal. The effet of convolving the spectrum by the Shah function is to periodically replicate the spectrum. Spectrum of the sampled signal

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 66 Spectrum of the sampled signal Assuming the spectrum of i has a final support [-f 0,f 0 ] f t f After sampling The signal If f ≥ 2f 0 and if then And eventually: ff

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 67 Sampling theorem If the spectrum of i has a finite support [-f 0,f 0 ] no information is lost on i if it is sampled with a frequency f larger than 2f 0. i can be rebuilt from I by convolution: î(f) f Î(f) f t

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 68 Case of 2D diffration-limited imagers pix The image is well sampled if:

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IYAS 2010 on HAR techniques November 1 st 2010 Diffraction & Fourier Optics G.P. 69 THE END

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