MIT 2.71/2.710 Optics11/03/04 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

MIT 2.71/2.710 Optics11/03/04 wk9-b-2

Diffraction from periodic array of holesPeriod: Λ

Spatial frequency: 1/Λ

A spherical wave is generatedat each hole;

we need to figure out how theperiodically-spaced spherical waves

interfere

incidentplanewave

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Diffraction from periodic array of holesPeriod: Λ

Spatial frequency: 1/Λ

Interference is constructive in thedirection pointed by the parallel rays

if the optical path differencebetween successive rays

equals an integral multiple of λ(equivalently, the phase delay

equals an integral multiple of 2π)

incidentplanewave

Optical path differences

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Diffraction from periodic array of holesPeriod: Λ

Spatial frequency: 1/Λ

From the geometrywe find

Therefore, interference isconstructive iff

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Diffraction from periodic array of holes

incidentplanewave

Grating spatial frequency: 1/ΛAngular separation between diffracted orders: Δθ≈λ/Λ

2nd diffractedorder

1st diffractedorder

“straight-through”order (aka DC term)

–1st diffractedorder

several diffracted plane waves“diffraction orders”

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Fraunhofer diffractionfrom periodic array of holes

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Sinusoidal amplitude grating

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Sinusoidal amplitude grating

incidentplanewave

Only the 0th and ±1st

orders are visible

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Sinusoidal amplitude grating

oneplanewave

threeplanewaves

far field threeconverging

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 atlarger angle than red:

normal dispersion

Blue light is diffracted atsmaller angle than red:

anomalous dispersion

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The ideal thin lensas a Fourier transform engine

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Fresnel diffraction

Reminder

coherentplane-waveillumination

The diffracted field is the convolution of the transparency with a spherical waveQ: how can we “undo” the convolution optically?

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Fraunhofer diffraction

Reminder

The “far-field” (i.e. the diffraction pattern at a largelongitudinal distance l equals the Fourier transform

of the original transparencycalculated at spatial frequencies

Q: is there anotheroptical element who

can perform aFourier

transformationwithout having to go

too far (to ∞) ?

MIT 2.71/2.710 Optics11/03/04 wk9-b-17

The thin lens (geometrical optics)

f (focal length)

object at ∞(plane wave)

point objectat finite distance(spherical wave)

Ray bending is proportionalto the distance from the axis

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The thin lens (wave optics)

incoming wavefronta(x,y)

outgoing wavefronta(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 lensfront focal plane

spherical wave,divergingoff–axis spherical wave (has propagated distance ) :

lens transmission function :

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Example: spherical wave through lensfront focal plane

spherical wave,divergingoff–axis

wavefront after lens

ignore

plane wave

at angle

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Diffraction at the back focal plane

thintransparency

g(x,y)

thinlens

back focal planediffraction pattern

gf(x”,y”)

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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

sphericalwave-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

point source at (x,y)amplitude gin(x,y)

plane wave oriented

towards

... a superposition ...

... of plane wavescorresponding to

point sourcesin the object

each output coordinate(x’,y’) receives ...

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Spherical – plane wave duality

a plane wave departingfrom the transparency

at angle (θx, θy) has amplitudeequal 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 equ

alto that of the corresponding

Fourier component

produces a spherical wave converging

MIT 2.71/2.710 Optics11/03/04 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.