Today

1. Today • Diffraction from periodic transparencies: gratings • Grating dispersion • Wave optics description of a lens: quadratic phase delay • Lens as Fourier transform engine

2. Diffraction from periodic array of holes incident plane wave Period: Λ Spatial frequency: 1/Λ A spherical wave is generated at each hole; we need to figure out how the periodically-spaced spherical waves interfere

3. Diffraction from periodic array of holes Period: Λ Spatial frequency: 1/Λ Interference is constructive in the direction pointed by the parallel rays if the optical path difference between successive rays equals an integral multiple of λ (equivalently, the phase delay equals an integral multiple of 2π) incident plane wave Optical path differences

4. Period: Λ Spatial frequency: 1/Λ From the geometry we find Therefore, interference is constructive iff Diffraction from periodic array of holes

5. incident plane wave Grating spatial frequency: 1/Λ Angular separation between diffracted orders: Δθ ≈λ/Λ 2nd diffracted order 1st diffracted order “straight-through” order (aka DC term) –1st diffracted order several diffracted plane waves “diffraction orders” Diffraction from periodic array of holes

6. Fraunhofer diffractionfrom periodic array of holes

7. Sinusoidal amplitude grating

8. Sinusoidal amplitude grating incident plane wave Only the 0th and ±1st orders are visible

9. Sinusoidal amplitude grating three plane waves far field one plane wave three converging spherical waves +1st order 0th order –1st order diffraction efficiencies

10. Dispersion

11. Dispersion from a grating

12. Dispersion from a grating

13. Prism dispersion vs grating dispersion Blue light is diffracted at smaller angle than red: anomalous dispersion Blue light is refracted at larger angle than red: normal dispersion

14. The ideal thin lensas a Fourier transform engine

15. Fresnel diffraction Reminder coherent plane-wave illumination The diffracted field is the convolution of the transparency with a spherical wave Q: how can we “undo” the convolution optically?

16. Fraunhofer diffraction Reminder 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 Q: is there another optical element who can perform a Fourier transformation without having to go too far (to ∞ ) ?

17. The thin lens (geometrical optics) f (focal length) object at ∞ (plane wave) Ray bending is proportional to the distance from the axis point object at finite distance (spherical wave)

18. The thin lens (wave optics) outgoing wavefront a(x,y) t(x,y)eiφ(x,y) incoming wavefront a(x,y) (thin transparency approximation)

19. The thin lens transmission function

20. The thin lens transmission function this constant-phase term can be omitted where is the focal length

21. Example: plane wave through lens plane wave: exp{i2πu0x} angle θ0, sp. freq. u0≈ θ0 /λ

22. Example: plane wave through lens back focal plane spherical wave, converging off–axis wavefront after lens : ignore

23. Example: spherical wave through lens front focal plane spherical wave, diverging off–axis spherical wave (has propagated distance ) : lens transmission function :

24. Example: spherical wave through lens front focal plane spherical wave, diverging off–axis plane wave at angle wavefront after lens ignore

25. Diffraction at the back focal plane back focal plane diffraction pattern gf(x”,y”) thin lens thin transparency g(x,y)

26. Diffraction at the back focal plane 1D calculation Field before lens Field after lens Field at back f.p.

27. Diffraction at the back focal plane 1D calculation 2D version

28. Diffraction at the back focal plane spherical wave-front Fourier transform of g(x,y)

29. Fraunhofer diffraction vis-á-vis a lens

30. Spherical – plane wave duality plane wave oriented point source at (x,y) amplitude gin(x,y) towards ... of plane waves corresponding to point sources in the object each output coordinate (x’,y’) receives ... ... a superposition ...

31. Spherical – plane wave duality produces a spherical wave converging a plane wave departing from the transparency at angle (θx, θy) has amplitude equal to the Fourier coefficient at frequency (θx/λ, θy /λ) of gin(x,y) produces a spherical wave converging towards each output coordinate (x’,y’) receives amplitude equal to that of the corresponding Fourier component

32. Conclusions • When a thin transparency is illuminated coherently by a monochromatic plane wave and the light passes through a lens, the field at the focal plane is the Fourier transform of the transparency times a spherical wavefront • The lens produces at its focal plane the Fraunhofer diffraction pattern of the transparency • When the transparency is placed exactly one focal distance behind the lens (i.e., z=f ), the Fourier transform relationship is exact.