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) 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-2 Coherent imaging as a linear, shift-invariant system Thin transparency illumi nation impulse response convolution output amplitude Fourier transform Fourier transform (plane wave spectrum) transfer function multiplication transfer functionaka pupil function

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

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

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

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

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

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

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

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

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

MIT 2.71/2.710 Optics 11/10/04 wk10-b-14 Incoherent imaging as a linear, shift-invariant system Thin transparency illumi nation incoherent impulse response convolution output intensity 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-15 Incoherent imaging as a linear, shift-invariant system Thin transparency illumi nation incoherent impulse response convolution output intensity Fourier transform Fourier transform (plane wave spectrum) transfer function multiplication transfer function of incoherent system:optical transfer function (OTF)

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

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

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

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

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

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

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

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

MIT 2.71/2.710 Optics 11/10/04 wk10-b-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 Fourier plane radial image plane (unit magnification)

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

MIT 2.71/2.710 Optics 11/10/04 wk10-b-29 Numerical Aperture and Speed (or F–Number) medium of refr. index n half-angle subtended by the imaging system from an axial object 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

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

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

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

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

MIT 2.71/2.710 Optics 11/10/04 wk10-b-34 The two–point resolution problem Imaging system intensity pattern observed (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-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 resolving 2the 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-36 Resolution in optical system

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

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

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

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

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

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

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

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