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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

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**Diffraction from periodic array of holes**

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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**Diffraction from periodic array of holes**

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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**Diffraction from periodic array of holes**

Spatial frequency: 1/Λ From the geometry we find Therefore, interference is constructive iff

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**Diffraction from periodic array of holes**

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

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**Fraunhofer diffraction from periodic array of holes**

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**Sinusoidal amplitude grating**

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**Sinusoidal amplitude grating**

incident plane wave Only the 0th and ±1st orders are visible

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**Sinusoidal amplitude grating**

three plane waves far field one plane wave three converging spherical waves +1st order 0th order –1st order diffraction efficiencies

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Dispersion

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**Dispersion from a grating**

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**Dispersion from a grating**

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**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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**The ideal thin lens as a Fourier transform engine**

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**Q: how can we “undo” the convolution optically?**

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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**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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**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)

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**The thin lens (wave optics)**

incoming wavefront a(x,y) outgoing wavefront a(x,y) t(x,y)eiφ(x,y) (thin transparency approximation)

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**The thin lens transmission function**

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**The thin lens transmission function**

this constant-phase term can be omitted where is the focal length

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**Example: plane wave through lens**

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

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**Example: plane wave through lens**

back focal plane spherical wave, converging off–axis wavefront after lens : ignore

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**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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**Example: spherical wave through lens**

front focal plane spherical wave, diverging off–axis plane wave at angle wavefront after lens ignore

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**Diffraction at the back focal plane**

diffraction pattern gf(x”,y”) thin transparency g(x,y) thin lens

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**Diffraction at the back focal plane**

1D calculation Field before lens Field after lens Field at back f.p.

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**Diffraction at the back focal plane**

1D calculation 2D version

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**Diffraction at the back focal plane**

spherical wave-front Fourier transform of g(x,y)

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**Fraunhofer diffraction vis-á-vis a lens**

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**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 ...

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**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

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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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