DIFFRACTION OPTICS - University of Arizona

1 ECE 425 CLASS NOTES 2000 DR. ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax) 207 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) ECE 425 CLASS NOTES 2000 DR. ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax) 208 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 where J 1 is the Bessel function of the first kind and the normalized radius r is given by,PSF r'()2J1r'()r'--------------2= ECE 425 CLASS NOTES 2000 DR.

2 ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax) 209 where The first zero occurs at , or Compare to the course notes on 2-D Fourier transformsr' D f-------r r N--------==Daperture diameter=ffocal length=Nf-number= wavelength of light= N==r' = ECE 425 CLASS NOTES 2000 DR. ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax) radius r'2-D view (contrast-enhanced) Airy pattern central bright region, to first-zero ring, is called the Airy disk radial profile ECE 425 CLASS NOTES 2000 DR. ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax) 211 Example calculation of PSF size system specs: D = 1cm f = 50mm N = f/D = 5 = m (green) radius of PSF = N = m very small!

3 ECE 425 CLASS NOTES 2000 DR. ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax) 212 DIFFRACTION -Limited OTF Incoherent light, circular aperture where the normalized radial spatial frequency is given by, and the cutoff frequency is given by, where OTF '()2 --- '()acos '1 '2 = ' c = cD f------1 N--------==Daperture diameter=ffocal length=Nf-number= wavelength of light= ECE 425 CLASS NOTES 2000 DR. ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax) 213 In 2-D spatial frequency space, the OTF is nearly a cone functionOTF is a low-pass filter of spatial spatial frequency ( / c)uvOTF = c ECE 425 CLASS NOTES 2000 DR.

4 ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax) 214 Example calculation of OTF cutoff frequency system specs: D = 1cm f = 50mm N = f/D = 5 = m (green) cutoff frequency = 1/ N = m = 363cycles/mm very high! (the human vision system has a cutoff frequency of about 10cycles/ mm (object scale) at normal viewing distance) ECE 425 CLASS NOTES 2000 DR. ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax) 215 Modulation Transfer Function (MTF) Amplitude part of complex OTF signal modulation modulation is a measure of signal contrast for sinewave, , 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 ) for sinewave input (object) to LSI system the output (image) is MTF u v,()OTF u v,()output signal modulationinput signal modulation------------------------------ ------------------------------==signal modulationmax min maxmin+--------------------------=AB2 uox()sin+input x()AB2 uox()sin+=output x()AMTF uo()B2 uox()

5 Sin+=ECE 425 CLASS NOTES 2000DR. ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax)216and the image modulation at spatial frequency uo is image modulationMTF uo()BA--- =MTFu1inputoutputu02u04u0u = u0u = 2u0u = typical MTFfor imaging systemECE 425 CLASS NOTES 2000DR. ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax)217 DIFFRACTION IMAGING EXAMPLESSine-wave imaged by circular aperture, DIFFRACTION -limited ECE 425 CLASS NOTES 2000DR. ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax)218opticscutoff-frequency, ucspatial frequency, uInput patternOutput patternSystem response(image)(object)(PSF)convolutionE CE 425 CLASS NOTES 2000DR.

6 ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax)219 Square-wave imaged by circular aperture, DIFFRACTION -ECE 425 CLASS NOTES 2000DR. ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax)220limited opticscutoff-frequency, ucspatial frequency, uInput patternOutput patternSystem response(image)(object)(PSF)convolutionE CE 425 CLASS NOTES 2000DR. ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax)221 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 Use edge source ( knife edge) to further increase energy > LSFxx()PSF x y,()yd =LSFyy()PSF x y,()xd =ECE 425 CLASS NOTES 2000DR.

7 ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax)222 Edge Spread Function (ESF) Integrate LSF up to position of ESF measurement Equivalent to step response in electronic system ESF is a monotonic function of position LSF is the derivative of the ESF, Both LSF and ESF are orientation-dependent, if the PSF is not isotropicESFxx()LSFx () d x =ESFyy()LSFy () d y =LSFxx()xddESFxx()=ECE 425 CLASS NOTES 2000DR. ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax)223 Must measure LSF and ESF at multiple orientations to reconstruct full 2-D PSFECE 425 CLASS NOTES 2000DR.

8 ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax)224 RELATIONSHIPS BETWEEN PSF, LSF, ESF AND OTF For example, the relationships in the x-direction are:PSF(x,y)OTF(u,v)LSFx(x)2-Done-sided1 -DESFx(x)1-Dderivative1-DFourierTransfor mTransformFourierOTF(u,0)profileone-side d 1-D integrationintegration1-DintegrationECE 425 CLASS NOTES 2000DR. ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax)225 SECTION III INTEGRATED IMAGING SYSTEM ANALYSISS canningImage QualitySystem SimulationECE 425 CLASS NOTES 2000DR. ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax)226 SCANNINGI nstantaneous image on focal plane detector(s)Scan the field-of-view Object-space scanning (mirror moves) Show that the object-space scan angle is twice that of the mirror scan angleobject 2 detectorECE 425 CLASS NOTES 2000DR.

9 ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax)227 Image-space scanning (detector moves) Translation scanning (camera or object moves) objectdetectorobjectdetectorECE 425 CLASS NOTES 2000DR. ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax)228 Examples from airborne or satellite remote sensing systemsWhat types of scanning are these examples?1-D arraywhiskbroom scannerGFOVFOV cross-trackin-trackline scannerwavelengthdispersion2-D arraypushbroom scannerpushbroom scannerFOV: Field-Of-View (radians)GFOV: Ground-projected FOV (km)(aka swath width )ECE 425 CLASS NOTES 2000DR.

10 ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax)229 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 undersampledWhy 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 smearECE 425 CLASS NOTES 2000DR. ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax)230 PSF of linear image smear is rect(x/S), where S is the amount of smear during the integration time: 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) if S = W, image smear is equal to one detector width.