Presentation on theme: "Today • Diffraction from periodic transparencies: gratings"— Presentation transcript:
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 Spatial frequency: 1/ΛA spherical wave is generatedat each hole;we need to figure out how theperiodically-spaced spherical wavesinterfereincidentplanewave
3 Diffraction from periodic array of holes Spatial frequency: 1/ΛInterference is constructive in thedirection pointed by the parallel raysif the optical path differencebetween successive raysequals an integral multiple of λ(equivalently, the phase delayequals an integral multiple of 2π)incidentplanewaveOptical path differences
4 Diffraction from periodic array of holes Spatial frequency: 1/ΛFrom the geometrywe findTherefore, interference isconstructive iff
5 Diffraction from periodic array of holes incidentplanewaveGrating spatial frequency: 1/ΛAngular separation between diffracted orders: Δθ ≈λ/Λ2nd diffractedorder1st diffracted“straight-through”order (aka DC term)–1st diffractedseveral diffracted plane waves“diffraction orders”
6 Fraunhofer diffraction from periodic array of holes
13 Prism dispersion vs grating dispersion Blue light is refracted atlarger angle than red:normal dispersionBlue light is diffracted atsmaller angle than red:anomalous dispersion
14 The ideal thin lens as a Fourier transform engine
15 Q: how can we “undo” the convolution optically? Fresnel diffractionRemindercoherentplane-waveilluminationThe diffracted field is the convolution of the transparency with a spherical waveQ: how can we “undo” the convolution optically?
16 Fraunhofer diffraction ReminderThe “far-field” (i.e. the diffraction pattern at a largelongitudinal distance l equals the Fourier transformof the original transparencycalculated at spatial frequenciesQ: is there anotheroptical element whocan perform aFouriertransformationwithout having to gotoo far (to ∞ ) ?
17 The thin lens (geometrical optics) f (focal length)object at ∞(plane wave)Ray bending is proportionalto the distance from the axispoint objectat finite distance(spherical wave)
30 Spherical – plane wave duality plane wave orientedpoint source at (x,y)amplitude gin(x,y)towards... of plane wavescorresponding topoint sourcesin the objecteach output coordinate(x’,y’) receives ...... a superposition ...
31 Spherical – plane wave duality produces a spherical wave converginga plane wave departingfrom the transparencyat angle (θx, θy) has amplitudeequal to the Fourier coefficientat frequency (θx/λ, θy /λ) of gin(x,y)produces a spherical wave convergingtowardseach output coordinate(x’,y’) receives amplitude equalto that of the correspondingFourier component
32 ConclusionsWhen 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 wavefrontThe lens produces at its focal plane the Fraunhofer diffraction pattern of the transparencyWhen the transparency is placed exactly one focal distance behind the lens (i.e., z=f ), the Fourier transform relationship is exact.