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

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

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

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

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.

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.

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

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

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

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

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

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.

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.

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

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

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

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

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.

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:

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

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

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

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.

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

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

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

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

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

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 

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)

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.

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

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:

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:

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

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

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

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"

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

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

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

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

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 )

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

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

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

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

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

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

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

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

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

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

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 /

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

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

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

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

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

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.

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:

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 )

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

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 tt

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

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: ff

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

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:

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