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MIT 2.71/2.710 Optics 11/03/04 wk9-b-1 Today Diffraction from periodic transparencies: gratings Grating dispersion Wave optics description of a lens: quadratic phase delay Lens as Fourier transform engine

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-2 Diffraction from periodic array of holes 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 incident plane wave

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-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

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-4 Diffraction from periodic array of holes Period: Λ Spatial frequency: 1/Λ From the geometry we find Therefore, interference is constructive iff

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-5 Diffraction from periodic array of holes incident plane wave Grating spatial frequency: 1/Λ Angular separation between diffracted orders: Δ λ/Λ 2 nd diffracted order 1 st diffracted order straight-through order (aka DC term) –1 st diffracted order several diffracted plane waves diffraction orders

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-6 Fraunhofer diffraction from periodic array of holes

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-7 Sinusoidal amplitude grating

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-8 Sinusoidal amplitude grating incident plane wave Only the 0 th and ±1 st orders are visible

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-9 Sinusoidal amplitude grating one plane wave three plane waves far field three converging spherical waves +1 st order 0 th order –1 st order diffraction efficiencies

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-10 Dispersion

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-11 Dispersion from a grating

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-12 Dispersion from a grating

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-13 Prism dispersion vs grating dispersion Blue light is refracted at larger angle than red: normal dispersion Blue light is diffracted at smaller angle than red: anomalous dispersion

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-14 The ideal thin lens as a Fourier transform engine

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-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?

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-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 ) ?

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-17 The thin lens (geometrical optics) f (focal length) object at (plane wave) point object at finite distance (spherical wave) Ray bending is proportional to the distance from the axis

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-18 The thin lens (wave optics) incoming wavefront a(x,y) outgoing wavefront a(x,y) t(x,y)e iφ(x,y) (thin transparency approximation)

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-19 The thin lens transmission function

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-20 The thin lens transmission function this constant-phase term can be omitted whereis the focal length

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-21 Example: plane wave through lens plane wave: exp{i2πu0x} angle θ 0, sp. freq. u 0 θ 0 /λ

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-22 Example: plane wave through lens back focal plane spherical wave, converging off–axis wavefront after lens : ignore

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-23 Example: spherical wave through lens front focal plane spherical wave, diverging off–axis spherical wave (has propagated distance ) : lens transmission function :

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-24 Example: spherical wave through lens front focal plane spherical wave, diverging off–axis wavefront after lens ignore plane wave at angle

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-25 Diffraction at the back focal plane thin transparency g(x,y) thin lens back focal plane diffraction pattern gf(x,y)

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-26 Diffraction at the back focal plane 1D calculation Field before lens Field after lens Field at back f.p.

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-27 Diffraction at the back focal plane 1D calculation 2D version

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-28 Diffraction at the back focal plane spherical wave-front Fourier transform of g(x,y)

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-29 Fraunhofer diffraction vis-á-vis a lens

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-30 Spherical – plane wave duality point source at (x,y) amplitude gin(x,y) plane wave oriented towards... a superposition...... of plane waves corresponding to point sources in the object each output coordinate (x,y) receives...

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-31 Spherical – plane wave duality 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 produces a spherical wave converging

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-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.

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