1. Overview from last week Optical systems act as linear shift-invariant (LSI) filters (we have not yet seen why) Analysis tool for LSI filters: Fourier transform – decompose arbitrary 2D functions into superpositions of 2D sinusoids (Fourier transform) – use the transfer function to determine what happens to each 2D sinusoid as it is transmitted through the system (filtering) – recompose the filtered 2D sinusoids to determine the output 2D function (Fourier integral, aka inverse Fourier transform) MIT 2.71/2.710 Optics 11/01/04 wk9-a-1

2. Today Wave description of optical systems Diffraction – very short distances: near field, we skip – intermediate distances: Fresnel diffraction expressed as a convolution – long distances (∞): Fraunhofer diffraction expressed as a Fourier transform MIT 2.71/2.710 Optics 11/01/04 wk9-a-2

3. Space and spatial frequency representations SPACE DOMAIN 2D Fourier transform 2D Fourier integral Aka inverse 2D Fourier transform SPATIAL FREQUENCY DOMAIN MIT 2.71/2.710 Optics 11/01/04 wk9-a-3

4. 2D linear shift invariant systems MIT 2.71/2.710 Optics 11/01/04 wk9-a-4 inputoutput convolution with impulse response multiplication with transfer function Fourier transform inverse Fourier

5. Wave description of optical imaging systems MIT 2.71/2.710 Optics 11/01/04 wk9-a-5

6. Thin transparencies MIT 2.71/2.710 Optics 11/01/04 wk9-a-6 coherent illumination: plane wave Transmission function: Field before transparence: Field after transparence: assumptions: transparence at z=0 transparency thickness can be ignored

7. Diffraction: Huygens principle MIT 2.71/2.710 Optics 11/01/04 wk9-a-7 incident plane Wave Field at distance d: contains contributions from all spherical waves emitted at the transparency, summed coherently Field after transparence:

8. Huygens principle: one point source MIT 2.71/2.710 Optics 11/01/04 wk9-a-8 incoming plane wave opaque screen spherical wave

9. Simple interference: two point sources MIT 2.71/2.710 Optics 11/01/04 wk9-a-9 incoming plane wave opaque screen intensity

10. Two point sources interfering: math… MIT 2.71/2.710 Optics 11/01/04 wk9-a-10 intensity Amplitude: (paraxial approximation)

11. Diffraction: many point sources MIT 2.71/2.710 Optics 11/01/04 wk9-a-11 incoming plane wave opaque screen many spherical waves tightly packed

12. Diffraction: many point sources,attenuated & phase-delayed MIT 2.71/2.710 Optics 11/01/04 wk9-a-12 incoming plane wave thin transparency

13. Diffraction: many point sources attenuated & phase-delayed, math MIT 2.71/2.710 Optics 11/01/04 wk9-a-13 incoming plane wave field Transmission function Thin transparency continuous limit wave

14. Fresnel diffraction MIT 2.71/2.710 Optics 11/01/04 wk9-a-14 The diffracted field is the convolution of the transparency with a spherical wave amplitude distribution at output plane transparency transmission function (complex te iφ ) spherical wave @z=l (aka Green’s function) FUNCTION OF LATERAL COORDINATES: Interesting!!! CONSTANT: NOT interesting

15. Example: circular aperture MIT 2.71/2.710 Optics 11/01/04 wk9-a-15 input field

16. Example: circular aperture MIT 2.71/2.710 Optics 11/01/04 wk9-a-16 input field

17. Example: circular aperture MIT 2.71/2.710 Optics 11/01/04 wk9-a-17 input field

18. Example: circular aperture MIT 2.71/2.710 Optics 11/01/04 wk9-a-18 (from Hecht, Optics, 4thedition, page 494) input field Image removed due to copyright concerns

19. Fraunhofer diffraction MIT 2.71/2.710 Optics 11/01/04 wk9-a-19 propagation distance lis “very large” approximaton valid if

20. Fraunhofer diffraction MIT 2.71/2.710 Optics 11/01/04 wk9-a-20 The“far-field” (i.e. the diffraction pattern at a large longitudinal distance l equals the Fourier transform of the original transparency calculated at spatial frequencies

21. Fraunhofer diffraction MIT 2.71/2.710 Optics 11/01/04 wk9-a-21 spherical wave originating at x plane wave propagating at angle –x/l ⇔ spatial frequency –x/(λl)

22. Fraunhofer diffraction MIT 2.71/2.710 Optics 11/01/04 wk9-a-22 superposition of spherical waves originating at various points along superposition of plane waves propagating at corresponding angles –x/l ⇔ spatial frequencies –x/(λl)

23. Example: rectangular apertureinput MIT 2.71/2.710 Optics 11/01/04 wk9-a-23 input field far field free space propagation by

24. Example: circular aperture MIT 2.71/2.710 Optics 11/01/04 wk9-a-24 also known as Airy pattern, or free space propagation by input field far field