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MIT 2.71/2.710 Optics 11/10/04 wk10-b-1

Today. • Review of spatial filtering with coherent coherent illumination • Derivation of the lens law using wave optics • Point-spread function of a system with incoherent incoherent illumination • The Modulation Transfer Function (MTF) and Optical Transfer Function (OTF)

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MIT 2.71/2.710 Optics 11/10/04 wk10-b-1

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  1. Today • Review of spatial filtering with coherent coherent illumination • Derivation of the lens law using wave optics • Point-spread function of a system with incoherent incoherent illumination • The Modulation Transfer Function (MTF) and Optical Transfer Function (OTF) • Comparison of coherent and incoherent imaging • Resolution and image quality – The meaning of resolution – Rayleigh criterion and image quality MIT 2.71/2.710 Optics 11/10/04 wk10-b-1

  2. Coherent imaging as a linear, shift-invariant system Thin transparency output amplitude impulse response convolution illumi nation Fourier transform Fourier transform transfer function (≡plane wave spectrum) multiplication transfer function aka pupil function MIT 2.71/2.710 Optics 11/10/04 wk10-b-2

  3. The 4F system with FP aperture object plane Fourier plane: aperture-limited Image plane: blurred (i.e. low-pass filtered) MIT 2.71/2.710 Optics 11/10/04 wk10-b-3

  4. Single-lens imaging condition lens object image • Imaging condition • (akaLens Law) Derivation using wave optics ?!? lateral Magnification MIT 2.71/2.710 Optics 11/10/04 wk10-b-4

  5. Single-lens imaging system lens object image spatial “LSI” system“ MIT 2.71/2.710 Optics 11/10/04 wk10-b-5

  6. Single-lens imaging system Impulse response (PSF) spatial “LSI” system“ Ideal PSF: Diffraction- -Limited PSF: MIT 2.71/2.710 Optics 11/10/04 wk10-b-6

  7. Imaging with incoherent light MIT 2.71/2.710 Optics 11/10/04 wk10-b-7

  8. Two types of incoherence temporal incoherence spatial incoherence matched paths point source Young interferometer Michelson interferometer poly-chromaticlight (=multi-color, broadband) mono-chromaticlight (= single color, narrowband) MIT 2.71/2.710 Optics 11/10/04 wk10-b-8

  9. Two types of incoherence temporal incoherence spatial incoherence matched paths point source • waves with equal paths • but from different points • on the wavefront • do not interfere • waves from unequal paths • do not interfere MIT 2.71/2.710 Optics 11/10/04 wk10-b-9

  10. Coherent vs incoherent beams Mutually coherent: superposition field amplitude is described by sum of complex amplitudes Mutually incoherent: superposition field intensity is described by sum of intensities (the phases of the individual beams vary randomly with respect to each other; hence, we would need statistical formulation to describe them properly —statistical optics) MIT 2.71/2.710 Optics 11/10/04 wk10-b-10

  11. Imaging with spatially incoherent light simple object: two point sources narrowband, mutually incoherent (input field is spatially incoherent) MIT 2.71/2.710 Optics 11/10/04 wk10-b-11

  12. Imaging with spatially incoherent light incoherent: adding in intensity ⇒ MIT 2.71/2.710 Optics 11/10/04 wk10-b-12

  13. Imaging with spatially incoherent light Generalizing: thin transparency with sp. incoherent illumination intensity at the output of the imaging system MIT 2.71/2.710 Optics 11/10/04 wk10-b-13

  14. Incoherent imaging as a linear, shift-invariant system Thin transparency output intensity incoherent impulse response convolution illumi nation Incoherent imaging is linear in intensity with incoherent impulse response (iPSF) where h(x,y) is the coherent impulse response (cPSF) MIT 2.71/2.710 Optics 11/10/04 wk10-b-14

  15. Incoherent imaging as a linear, shift-invariant system Thin transparency output intensity incoherent impulse response convolution illumi nation Fourier transform Fourier transform transfer function (≡plane wave spectrum) multiplication transfer function of incoherent system: optical transfer function (OTF) MIT 2.71/2.710 Optics 11/10/04 wk10-b-15

  16. The Optical Transfer Function normalized to 1 real real max max max max MIT 2.71/2.710 Optics 11/10/04 wk10-b-16

  17. some terminology ... • Amplitude transfer function • (coherent) • Optical Transfer Function (OTF) • (incoherent) Modulation Transfer Function (MTF) MIT 2.71/2.710 Optics 11/10/04 wk10-b-17

  18. MTF of circular aperture physical aperture filter shape (MTF) MIT 2.71/2.710 Optics 11/10/04 wk10-b-18

  19. MTF of rectangular aperture physical aperture filter shape (MTF) MIT 2.71/2.710 Optics 11/10/04 wk10-b-19

  20. Incoherent low–pass filtering MTF Intensity @ image plane MIT 2.71/2.710 Optics 11/10/04 wk10-b-20

  21. Incoherent low–pass filtering MTF Intensity @ image plane MIT 2.71/2.710 Optics 11/10/04 wk10-b-21

  22. Incoherent low–pass filtering MTF Intensity @ image plane MIT 2.71/2.710 Optics 11/10/04 wk10-b-22

  23. Diffraction-limited vs aberrated MTF real ideal thin lens, finite aperturez realistic lens finite aperture & aberrations max max MIT 2.71/2.710 Optics 11/10/04 wk10-b-23

  24. Imaging with polychromatic light Monochromatic, spatially incoherent response at wavelength λ0: Polychromatic (temporally and spatially incoherent) response: MIT 2.71/2.710 Optics 11/10/04 wk10-b-24

  25. Comments on coherent vs incoherent • Incoherent generally gives better image quality: – no ringing artifacts – no speckle – higher bandwidth (even though higher frequencies are attenuated because of the MTF roll-off) • However, incoherent imaging is insensitive to phas objects • Polychromatic imaging introduces further blurring due to chromatic aberration (dependence of the MTF on wavelength) MIT 2.71/2.710 Optics 11/10/04 wk10-b-25

  26. Resolution MIT 2.71/2.710 Optics 11/10/04 wk10-b-26

  27. Connection between PSF and NA Monochromatic coherent on-axis illumination • object plane • impulse Fourier plane circ-aperture • image plane • observed field • (PSF) Fourier transform radial coordinate @ Fourier plane radial coordinate @ image plane (unit magnification) MIT 2.71/2.710 Optics 11/10/04 wk10-b-27

  28. Connection between PSF and NA Monochromatic coherent on-axis illumination NA: angle of acceptance for on–axis point object Fourier plane circ-aperture image plane Numerical Aperture (NA) by definition: MIT 2.71/2.710 Optics 11/10/04 wk10-b-28

  29. Numerical Aperture and Speed (or F–Number) half-angle subtended by the imaging system from an axial object medium of refr. index n Numerical Aperture Speed(f/#)=1/2(NA) pronounced f-number, e.g. f/8 means (f/#)=8. Aperture stop the physical element which limits the angle of acceptance of the imaging system MIT 2.71/2.710 Optics 11/10/04 wk10-b-29

  30. Connection between PSF and NA MIT 2.71/2.710 Optics 11/10/04 wk10-b-30

  31. Connection between PSF and NA lobe width MIT 2.71/2.710 Optics 11/10/04 wk10-b-31

  32. NA in unit–mag imaging systems Monochromatic coherent on-axis illumination in both cases, Monochromatic coherent on-axis illumination MIT 2.71/2.710 Optics 11/10/04 wk10-b-32

  33. The incoherent case: MIT 2.71/2.710 Optics 11/10/04 wk10-b-33

  34. The two–point resolution problem intensity pattern observed Imaging system (e.g. with digital camera) object: two point sources, mutually incoherent (e.g. two stars in the night sky; two fluorescent beads in a solution) • The resolution question[Rayleigh, 1879]: when do we cease • to be able to resolve the two point sources (i.e., tell them apart) • due to the blurring introduced in the image by the finite (NA)? MIT 2.71/2.710 Optics 11/10/04 wk10-b-34

  35. The meaning of “resolution” [from the New Merriam-Webster Dictionary, 1989 ed.]: resolve v: 1to break up into constituent parts: ANALYZE; 2to find an answer to : SOLVE; 3DETERMINE, DECIDE; 4to make or pass a formal resolution resolution n: 1the act or process of resolving2the action of solving, also: SOLUTION; 3the quality of being resolute : FIRMNESS, DETERMINATION; 4a formal statement expressing the opinion, will or, intent of a body of persons MIT 2.71/2.710 Optics 11/10/04 wk10-b-35

  36. Resolution in optical system MIT 2.71/2.710 Optics 11/10/04 wk10-b-36

  37. Resolution in optical system MIT 2.71/2.710 Optics 11/10/04 wk10-b-37

  38. Resolution in optical system MIT 2.71/2.710 Optics 11/10/04 wk10-b-38

  39. Resolution in optical system MIT 2.71/2.710 Optics 11/10/04 wk10-b-39

  40. Resolution in optical system MIT 2.71/2.710 Optics 11/10/04 wk10-b-40

  41. Resolution in optical system MIT 2.71/2.710 Optics 11/10/04 wk10-b-41

  42. Resolution in noisy optical systems MIT 2.71/2.710 Optics 11/10/04 wk10-b-42

  43. “Safe” resolution in optical system MIT 2.71/2.710 Optics 11/10/04 wk10-b-43

  44. Diffraction–limited resolution (safe) • Two point objects are “just resolvable” (limited by diffraction only) • if they are separated by: Two–dimensional systems (rotationally symmetric PSF) One–dimensional systems (e.g. slit–like aperture) Safe definition: (one–lobe spacing) Pushy definition: (1/2–lobe spacing) • You will see different authors giving different definitions. • Rayleigh in his original paper (1879) noted the issue of noise • and warned that the definition of “just–resolvable” points • is system–or application –dependent MIT 2.71/2.710 Optics 11/10/04 wk10-b-44

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