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Today • Diffraction from periodic transparencies: gratings • Grating dispersion • Wave optics description of a lens: quadratic phase delay • Lens as Fourier transform engine MIT 2.71/2.710 Optics Diffraction from periodic array of holes incident plane wave MIT 2.71/2.710 Optics Period: Λ Spatial frequency: 1/ A spherical wave is generated at each hole; we need to figure out how the periodically-spaced spherical waves interfere Diffraction from periodic array of holes Period: Λ incident plane wave 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) Optical path differences MIT 2.71/2.710 Optics Diffraction from periodic array of holes Period: Λ Spatial frequency: 1/ From the geometry we find Therefore, interference is constructive iff MIT 2.71/2.710 Optics Diffraction from periodic array of holes incident plane wave Grating spatial frequency: 1/ Angular separation between diffracted orders: λ/ 2nd diffracted order 1st diffracted order “straight-through” order (aka DC term) –1st diffracted order MIT 2.71/2.710 Optics several diffracted plane waves “diffraction orders” Fraunhofer diffraction from periodic array of holes MIT 2.71/2.710 Optics Sinusoidal amplitude grating MIT 2.71/2.710 Optics Sinusoidal amplitude grating incident plane wave Only the 0th and ±1st orders are visible MIT 2.71/2.710 Optics Sinusoidal amplitude grating one plane wave three plane waves far field +1st order three converging spherical waves 0th order –1st order diffraction efficiencies MIT 2.71/2.710 Optics Dispersion MIT 2.71/2.710 Optics Dispersion from a grating MIT 2.71/2.710 Optics Dispersion from a grating MIT 2.71/2.710 Optics Prism dispersion vs grating dispersion Blue light is refracted at larger angle than red: Blue light is diffracted at smaller angle than red: normal dispersion anomalous dispersion MIT 2.71/2.710 Optics The ideal thin lens as a Fourier transform engine MIT 2.71/2.710 Optics 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? MIT 2.71/2.710 Optics 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 MIT 2.71/2.710 Optics Q: is there another optical element who can perform a Fourier transformation without having to go too far (to ) ? The thin lens (geometrical optics) f (focal length) object at (plane wave) point object at finite distance (spherical wave) MIT 2.71/2.710 Optics Ray bending is proportional to the distance from the axis The thin lens (wave optics) incoming wavefront a(x,y) outgoing wavefront a(x,y) t(x,y)eiφ(x,y) (thin transparency approximation) MIT 2.71/2.710 Optics The thin lens transmission function MIT 2.71/2.710 Optics The thin lens transmission function this constant-phase term can be omitted where MIT 2.71/2.710 Optics is the focal length Example: plane wave through lens plane wave: exp{i2πu0x} angle θ0, sp. freq. u0θ0 /λ MIT 2.71/2.710 Optics Example: plane wave through lens back focal plane wavefront after lens : ignore MIT 2.71/2.710 Optics spherical wave, converging off–axis Example: spherical wave through lens front focal plane spherical wave, diverging off–axis spherical wave (has propagated distance ) : lens transmission function : MIT 2.71/2.710 Optics Example: spherical wave through lens front focal plane spherical wave, diverging off–axis plane wave at angle wavefront after lens ignore MIT 2.71/2.710 Optics Diffraction at the back focal plane thin transparency g(x,y) MIT 2.71/2.710 Optics thin lens back focal plane diffraction pattern gf(x”,y”) Diffraction at the back focal plane 1D calculation Field before lens Field after lens Field at back f.p. MIT 2.71/2.710 Optics Diffraction at the back focal plane 1D calculation 2D version MIT 2.71/2.710 Optics Diffraction at the back focal plane spherical wave-front MIT 2.71/2.710 Optics Fourier transform of g(x,y) Fraunhofer diffraction vis-á-vis a lens MIT 2.71/2.710 Optics Spherical – plane wave duality point source at (x,y) amplitude gin(x,y) plane wave oriented towards ... of plane waves ... a superposition ... corresponding to point sources in the object MIT 2.71/2.710 Optics each output coordinate (x’,y’) receives ... Spherical – plane wave duality produces a spherical wave a plane wave departing produces a spherical wave converging from the transparency towards converging at angle (θx, θy) has amplitude equal to the Fourier coefficient each output coordinate at frequency (θx/λ, θy /λ) of (x’,y’) receives amplitude gin(x,y) equal to that of the corresponding Fourier component MIT 2.71/2.710 Optics 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. MIT 2.71/2.710 Optics