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Diffraction Optics

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Diffraction Optics ECE 425 CLASS NOTES – 2000 DIFFRACTION OPTICS Physical basis • Considers the wave nature of light, unlike geometrical optics • Optical system apertures limit the extent of the wavefronts • Even a “perfect” system, from a geometrical optics viewpoint, will not form a point image of a point source • Such a system is called diffraction-limited Diffraction as a linear system • Without proof here, we state that the impulse response of diffraction is the Fourier transform (squared) of the exit pupil of the optical system (see Gaskill for derivation) 207 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 • impulse response called the Point Spread Function (PSF) • image of a point source • The transfer function of diffraction is the Fourier transform of the PSF • called the Optical Transfer Function (OTF) Diffraction-Limited PSF • Incoherent light, circular aperture ()2 J1 r' PSF() r' = 2-------------- r' where J1 is the Bessel function of the first kind and the normalized radius r’ is given by, 208 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 D = aperture diameter πD πr f = focal length r' ==------- r -------- where λf λN N = f-number λ = wavelength of light λf • The first zero occurs at r = 1.22 ------ = 1.22λN , or r'= 1.22π D Compare to the course notes on 2-D Fourier transforms 209 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 2-D view (contrast-enhanced) “Airy pattern” central bright region, to first- zero ring, is called the “Airy disk” 1 0.8 0.6 PSF 0.4 radial profile 0.2 0 0246810 normalized radius r' 210 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 • Example calculation of PSF size • system specs: D = 1cm f = 50mm N = f/D = 5 λ = 0.55µm (green) • radius of PSF = 1.22λN = 3.36µm • very small! 211 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 Diffraction-Limited OTF • Incoherent light, circular aperture ()ρ 2 ()ρ ρ ρ 2 OTF ' = π--- acos ' – '1– ' where the normalized radial spatial frequency is given by, ρ ρρ⁄ ' = c and the cutoff frequency is given by, D = aperture diameter D 1 f = focal length ρ ==------ -------- where c λf λN N = f-number λ = wavelength of light 212 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 1 0.8 0.6 OTF 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 ρ/ρ normalized spatial frequency ( c) In 2-D spatial frequency space, the OTF is nearly a “cone” function OTF OTF is a low-pass filter of spatial frequencies v ρ ρ u = c 213 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 Example calculation of OTF cutoff frequency • system specs: D = 1cm f = 50mm N = f/D = 5 λ = 0.55µm (green) • cutoff frequency = 1/λN = 0.363cycles/µm = 363cycles/mm • very high! (the human vision system has a cutoff frequency of about 10cycles/mm (object scale) at normal viewing distance) 214 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 Modulation Transfer Function (MTF) • Amplitude part of complex OTF output signal modulation MTF() u, v ==OTF() u, v ------------------------------------------------------------ input signal modulation • signal modulation max– min signal modulation = -------------------------- max+ min • modulation is a measure of signal contrast ()π • for sinewave, AB+ sin 2 uox , modulation = B/A • Use MTF to predict image contrast, given object and system • output modulation(u,v) = input modulation(u,v) x MTF(u,v) () ()π • for sinewave input (object) to LSI system input x = AB+ sin 2 uox () () ()π the output (image) is output x = A+ MTF uo Bsin 2 uox 215 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 B and the image modulation at spatial frequency u is image modulation = MTF() u ⋅ --- o o A MTF 1 “typical” MTF 0.8 for imaging system 0.5 0.1 u u0 2u0 4u0 u = u0 u = 2u0 u = 4u0 input output 216 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 DIFFRACTION IMAGING EXAMPLES Sine-wave imaged by circular aperture, diffraction- limited 217 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 optics spatial frequency, u Input pattern (object) System response convolution (PSF) Output pattern (image) cutoff-frequency, uc 218 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 Square-wave imaged by circular aperture, diffraction- 219 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 limited optics spatial frequency, u Input pattern (object) System response convolution (PSF) Output pattern (image) cutoff-frequency, uc 220 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 Line and Edge Spread Functions • PSF is often difficult to measure because of insufficient energy • Use a line source (slit) to increase energy –> Line Spread Function (LSF) • Integrate PSF along direction of the line source ∞ () (), LSFx x = ∫ PSF x y dy –∞ ∞ () (), LSFy y = ∫ PSF x y dx –∞ • Use edge source (knife edge) to further increase energy –> 221 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 Edge Spread Function (ESF) • Integrate LSF up to position of ESF measurement x () ()αα ESFx x = ∫ LSFx d –∞ y () ()αα ESFy y = ∫ LSFy d –∞ • Equivalent to step response in electronic system • ESF is a monotonic function of position • LSF is the derivative of the ESF, d LSF ()x = ESF ()x x dx x • Both LSF and ESF are orientation-dependent, if the PSF is not isotropic 222 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 • Must measure LSF and ESF at multiple orientations to reconstruct full 2-D PSF 223 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 RELATIONSHIPS BETWEEN PSF, LSF, ESF AND OTF • For example, the relationships in the x-direction are: PSF(x,y) ESFx(x) one-sided 1-D integration one-sided 2-D 1-D 1-D integration Fourier derivative Transform 1-D integration OTF(u,v) OTF(u,0) LSFx(x) profile 1-D Fourier Transform 224 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 SECTION III – INTEGRATED IMAGING SYSTEM ANALYSIS Scanning Image Quality System Simulation 225 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 SCANNING Instantaneous image on focal plane detector(s) Scan the field-of-view • Object-space scanning (mirror moves) θ detector 2θ object Show that the object-space scan angle is twice that of the mirror scan angle 226 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 • Image-space scanning (detector moves) detector object • Translation scanning (camera or object moves) detector object 227 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 Examples from airborne or satellite remote sensing systems whiskbroom scanner 1-D array line scanner pushbroom scanner in-track FOV cross-track GFOV FOV: Field-Of-View (radians) 2-D array wavelength dispersion pushbroom scanner GFOV: Ground- projected FOV (km) (aka “swath width”) What types of scanning are these examples? 228 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 Scanning is equivalent to convolution • At each instant, the detector integrates a small area of the image • Integration time at each sample determines number of photons collected • Generally, a scanned image is sampled at an interval equal to detector size • undersampled Why is one sample/detector element undersampled? • PSF of scanning detector is rect(x/W,y/W), where W is the detector size • If sample integration time is not negligible, there is also image smear 229 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 • PSF of linear image smear is rect(x/S), where S is the amount of smear during the integration time: S = image velocity x integration time = vimgtint • Image smear sometimes used to increase integration time/ pixel, and therefore SNR (even though image is blurred) • Total scanning PSF • Combine scanning detector and image smear during pixel integration (normalize by volume PSF) rect() x⁄ W, yW⁄ * rect() x⁄ S PSF ()xy, = --------------------------------------------------------------------------- scan volumePSF []rect() x⁄ W * rect() x⁄ S rect() y⁄ W = --------------------------------------------------------------------------------------------- volumePSF • if S = W, i.e. image smear is equal to one detector width: tri() x⁄ W rect() y⁄ W PSF ()xy, = ---------------------------------------------------- scan volumePSF 230 DR.
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