Outline

1. International Young Astronomer School 2010High Angular Resolution Techniques Diffraction and Fourier OpticsGuy PerrinMonday November 1st, 2010

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

3. References Optics and diffraction: • M. Born and E. Wolf, Principle of Optics, 7th edition, Cambridge University Press, 2002 • E. Hecht, Optics, 4th edition, Addison Wesley, 2001 Fourier Transform: • R.N. Bracewell, Fourier transform and its applications, Mc Graw-Hill, 1986

4. 1. Diffraction

5. P P0 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: Wavefronts are perpendicular to optical rays.

6. Wavefront Expression of the phase of the wave: P P0 is the optical path betweenP0 and P (Fermat principle): In a medium with uniform index: Wavefronts emitted by a point source are indeed spherical.

7. P P0 A spherical wave emitted by a point source in a medium with uniform refractive index and without obstacles remains spherical: Why diffraction ? Diffraction is due to the spatial limitation of waves

8. Historical discovery of diffraction Screen P0 Diaphragm Optical rays are disturbed by the diaphragm: - the size of the beam projected on the screen decreases with the decreasing diaphragm size;

9. Historical discovery of diffraction Screen P0 Diaphragm Optical rays are disturbed by the diaphragm: - the size of the beam projected on the screen decreases with the decreasing diaphragm size;

10. Historical discovery of diffraction Screen P0 Diaphragm 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.

11. l/D The spot increases when: Or when the distance becomes larger than: Rayleigh distance kjjjjj Screen P0 D d Diaphragm

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.

13. Why diffraction ? Diffraction is from the latine word diffringere which means to break. Diffraction breaks straight light rays.

14. Scalar theory For a monochromatic scalar wave, Maxwell equations take a simple form and waves are described by the following set of equations: where k=2p/l 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.

15. Fresnel-Kirchhoff equation When distances of P0and P to the diaphragm are large with respect to the diaphragm itself much larger than wavelength, the integral becomes:

16. 2. Fourier transform

17. Direct Fourier transform FT Inverse Fourier transform FT-1 General definition f and areFourier pairs, k are x are conjugated by the Fourier transform

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.

19. Properties of Fourier transforms Fourier transform of a convolution: Fourier transform of an auto-correlation (Wiener-Khintchine theorem):

20. Properties of Fourier transforms The Fourier transform conserves energy:

21. Properties of Fourier transforms Shifting theorem: Scaling theorem:

22. Examples (cardinal sinus)

23. Examples In 2D: J1 is a Bessel function of the first kind.

24. 3. Fraunhofer diffraction

25. Fresnel-Kirchhoff equation M When distances of P0and P to the diaphragm are large with respect to the diaphragm itself much larger than wavelength, the integral becomes:

26. y0 y1 x0 x1 Fraunhofer diffraction M z • 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

27. 1 et 2 3

28. x1 a z  v u  -1/P +1/P P 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. Fourier transform

29. Fourier transform Frequency f and time t are conjugated variables. -1/T t f n=1/T

30. 3 (rad-1 or arcsec-1) Changing variables to: (rad or arcsec)

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.

32. 4. Imaging

33. b a 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:

34. b a pupil P image plane Image of a point source at infinity Optical detectors are sensitive to intensity: Applying the Wiener-Khintchine theorem yields:

35. Point Spread Function The Point Spred Function (PSF) is the image of a point source I(a,b). The PSF is the Fourier transform of the autocorrelation of the pupil function. Its normalized Fourier transform is the Optical Transfer Function (OTF). Pupil plane Image plane FT FT

36. f Example: uniform & circular pupil Pupil function: OTF: cutt-off frequency: D/l PSF (Airy pattern) : FWHM: ≈ l/D

37. f Definition(s) of angular resolution Definition 1 : characteristic scale equal to the reciprocal of the OTF cut-off frequency = l/D Definition 2 : FWHM of the PSF ≈ l/D Definition 3 (Rayleigh criterion) : first zero of the Airy pattern: 1.22 l/D

38. Two sources are separated by the optical system if their angular separation is > l/D Example: - Hubble Space Telescope (D=2,4m), l=0,5 µm => R= 0,042 " Image HST 0,6"

39. Image plane D f

40. < l Image plane D f

41. > l Image plane D f

42. 5. Images of extended sources

43. Image of an off-axis point-like source The image of a point source located at (a0,b0) is: I’(a,b)=I(a-a0,b-b0) And therefore: I’(a,b)=PSF(a-a0,b-b0)

44. Image of an (uncoherent) extended source b a pupil P Image plane Assuming the spatial intensity distribution of the source: O(a0,b0) And the object is spatially uncoherent (waves from two different points are uncorrelated). A point-like source at (a0,b0) produces the intensity: Summing individual contributions in the image plane yields: The image is the convolution of the object intensity spatial distribution by the PSF

45. Spatial frequency contents of the image Image spectrum: The optical system acts as a low-pass filter whose cut-off frequency is D/l Source spectrum Image spectrum Object FT x =

46. 6. The case of interferometry

47. -D/l +D/l 1 B 1/2 -B/l -D/l +D/l +B/l Spatial spectrum of the object 1 Measured visibility -B/l -D/l +D/l +B/l Band-pass filtering and interferometry Pupil Optical Transfer Function D The same theory applies to interferometers which are just a particular case of telescopes. For an extended source:

48. > l Resolved source < l Unresolved source Angular resolution: Image plane D f

49. Angular resolution: Image plabe Interferometry D f

50. Image plane Interferometry D B f Angular resolution: