Today Review of spatial filtering with coherent illumination Derivation of the lens law using wave optics Point-spread function of a system with 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.7/.70 Optics /0/04 wk0-b-
Coherent imaging as a linear, shift-invariant system Thin transparency t, y ( y) g, illumi nation ( ) g g (, y) (, y) t(, y) impulse response g (, y ) convolution g output amplitude 3 (, y) h(, y) Fourier transform Fourier transform ( plane wave transfer function G spectrum) 3( u, v) G ( u, v) multiplication G (, ) (, ) u v H u v MIT.7/.70 Optics /0/04 wk0-b- transfer function H(s,s y ): aka pupil function
The 4F system with FP aperture f f f f ( u v) G, θ ( y) g, object plane MIT.7/.70 Optics /0/04 wk0-b-3 G y, f f r circ R Fourier plane: aperture-limited ( g h) y f f, f Image plane: blurred (i.e. low-pass filtered) f
Single-lens imaging condition object s lens s image + s m lateral s MIT.7/.70 Optics /0/04 wk0-b-4 f s s Imaging condition (aka Lens Law) Magnification Derivation using wave optics?!?
Single-lens imaging system object s lens s image g in (,y) spatial LSI system g out (,y ) MIT.7/.70 Optics /0/04 wk0-b-5
Single-lens imaging system Impulse response (PSF) g in (,y) spatial LSI system g out (,y ) Ideal PSF: h (, y;, y ) δ ( m) δ ( y my) Diffraction- -limited PSF: h (, y;, y ) R jinc s s + y s y s MIT.7/.70 Optics /0/04 wk0-b-6
MIT.7/.70 Optics /0/04 wk0-b-7 Imaging with incoherent light
Two types of incoherence temporal incoherence spatial incoherence r r r point source d d matched paths Michelson interferometer poly-chromatic light (multi-color, broadband) Young interferometer mono-chromatic light ( single color, narrowband) MIT.7/.70 Optics /0/04 wk0-b-8
Two types of incoherence temporal incoherence spatial incoherence r r r point source d d matched paths MIT.7/.70 Optics /0/04 wk0-b-9 waves from unequal paths do not interfere waves with equal paths but from different points on the wavefront do not interfere
Coherent vs incoherent beams a a e iφ Mutually coherent: superposition field amplitude is described by sum of comple amplitudes a a + a a e iφ + a e iφ a a e iφ I a a + a I Mutually incoherent: superposition field intensity is described by sum of intensities I I + I MIT.7/.70 Optics /0/04 wk0-b-0 I (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)
Imaging with spatially incoherent light f f f f simple object: two point sources narrowband, mutually incoherent (input field is spatially incoherent) MIT.7/.70 Optics /0/04 wk0-b-
Imaging with spatially incoherent light f f f f 0 incoherent: adding in intensity ( ) h( ) + h( ) I + 0 0 MIT.7/.70 Optics /0/04 wk0-b-
Imaging with spatially incoherent light f f f f I ( ) Generalizing: thin transparency with sp. incoherent illumination MIT.7/.70 Optics /0/04 wk0-b-3 I ( ) I( ) h( ) d intensity at the output of the imaging system
Incoherent imaging as a linear, shift-invariant system Thin transparency t, y ( y) I, illumi nation ( ) I I (, y) (, y) t(, y) incoherent impulse response I (, y ) convolution I 3 (, y) output intensity h(, y) Incoherent imaging is linear in intensity with incoherent impulse response (ipsf) ~ h (, y) h(, y) where h(,y) is the coherent impulse response (cpsf) MIT.7/.70 Optics /0/04 wk0-b-4
Incoherent imaging as a linear, shift-invariant system Thin transparency t, y ( y) I, illumi nation ( ) I I (, y) (, y) t(, y) incoherent impulse response I (, y ) convolution I 3 (, y) output intensity h(, y) Fourier transform Fourier transform ( plane wave MIT.7/.70 Optics /0/04 wk0-b-5 spectrum) I ˆ ( u, v) transfer function multiplication ~ transfer function of incoherent system: H (, ) s s y Iˆ 3( u, v) ˆ ~ I ( u, v) H ( u, v) optical transfer function (OTF)
The Optical Transfer Function ~ H { } ( u, v) I h(, y) * H ( u, v ) H ( u u, v v) H ( u, v ) du dv normalized to du dv real(h) real( H ~ ) u ma u ma u ma u ma MIT.7/.70 Optics /0/04 wk0-b-6
some terminology... ( u v) H, Amplitude transfer function (coherent) ~ H, ( u v) ~ H, ( u v) Optical Transfer Function (OTF) (incoherent) Modulation Transfer Function (MTF) MIT.7/.70 Optics /0/04 wk0-b-7
MTF of circular aperture f 0cm 0.5µm MIT.7/.70 Optics /0/04 wk0-b-8 physical aperture filter shape (MTF)
MTF of rectangular aperture f 0cm 0.5µm MIT.7/.70 Optics /0/04 wk0-b-9 physical aperture filter shape (MTF)
Incoherent low pass filtering f 0cm 0.5µm MIT.7/.70 Optics /0/04 wk0-b-0 MTF Intensity @ image plane
Incoherent low pass filtering f 0cm 0.5µm MIT.7/.70 Optics /0/04 wk0-b- MTF Intensity @ image plane
Incoherent low pass filtering f 0cm 0.5µm MIT.7/.70 Optics /0/04 wk0-b- MTF Intensity @ image plane
Diffraction-limited vs aberrated MTF ideal thin lens, finite aperture real( H ~ ) realistic lens, finite aperture & aberrations u ma u ma MIT.7/.70 Optics /0/04 wk0-b-3
Imaging with polychromatic light Monochromatic, spatially incoherent response at wavelength 0 : ( ) ( ) ( ) y y y h y I y I d d ;, ;, ;, 0 0 0 Polychromatic (temporally and spatially incoherent) response: ( ) ( ) ( ) ( ) 0 0 0 0 0 d d d ;, ;, d ;,, y y y h y I y I y I MIT.7/.70 Optics /0/04 wk0-b-4
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 phase objects Polychromatic imaging introduces further blurring due to chromatic aberration (dependence of the MTF on wavelength) MIT.7/.70 Optics /0/04 wk0-b-5
MIT.7/.70 Optics /0/04 wk0-b-6 Resolution
Connection between PSF and NA f f f f monochromatic coherent on-ais illumination R g in object plane impulse (, y) δ ( ) δ ( y) r + y r + y MIT.7/.70 Optics /0/04 wk0-b-7 radial coordinate @ Fourier plane radial coordinate @ image plane H Fourier plane circ-aperture r R (, y ) circ jinc I Fourier transform (.,.) J image plane observed field (PSF) R r π f R r π f (unit magnification)
Connection between PSF and NA f f f f monochromatic coherent on-ais illumination R NA: angle of acceptance for on ais point object Fourier plane circ-aperture image plane R R y jinc, f f J R r π f R r π f J r π ( NA) r NA π ( ) MIT.7/.70 Optics /0/04 wk0-b-8 Numerical Aperture (NA) by definition: ( NA) R f
Numerical Aperture and Speed (or F Number) medium of refr. inde n θ θ: half-angle subtended by the imaging system from an aial object Numerical Aperture (NA) n sinθ Speed (f/#)/(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.7/.70 Optics /0/04 wk0-b-9
Connection between PSF and NA h (, y ) J r π ( NA) r NA π ( ) null @ r 0.6 ( NA) MIT.7/.70 Optics /0/04 wk0-b-30
Connection between PSF and NA h (, y ) J r π ( NA) r NA π ( ) lobe width r. ( NA) MIT.7/.70 Optics /0/04 wk0-b-3
NA in unit mag imaging systems f f f f monochromatic coherent on-ais illumination R ( NA) ( NA) monochromatic coherent on-ais illumination R f R f in both cases, PSF h (, y ) h( r ) jinc ( NA) f R f r MIT.7/.70 Optics /0/04 wk0-b-3
The incoherent case: ~ h (, y ) h(, y ) ~ h (, y ) J r π ( NA) r π ( NA) null @ r 0.6 ( NA) MIT.7/.70 Optics /0/04 wk0-b-33
The two point resolution problem imaging system object: two point sources, mutually incoherent (e.g. two stars in the night sky; two fluorescent beads in a solution) intensity pattern observed (e.g. with digital camera) MIT.7/.70 Optics /0/04 wk0-b-34 The resolution question [Rayleigh, 879]: 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)?
The meaning of resolution [from the New Merriam-Webster Dictionary, 989 ed.]: resolve v : to break up into constituent parts: ANALYZE; to find an answer to : SOLVE; 3 DETERMINE, DECIDE; 4 to make or pass a formal resolution resolution n : the act or process of resolving the action of solving, also : SOLUTION; 3 the quality of being resolute : FIRMNESS, DETERMINATION; 4 a formal statement epressing the opinion, will or, intent of a body of persons MIT.7/.70 Optics /0/04 wk0-b-35
r 3.0 > 0.6 ( NA) ( NA) Resolution in optical systems.5 ( ) h ~ + NA.5 ( ) h ~ NA MIT.7/.70 Optics /0/04 wk0-b-36
r 3.0 > 0.6 ( NA) ( NA) Resolution in optical systems ~.5 h + + ( NA) ~.5 ( ) + h NA MIT.7/.70 Optics /0/04 wk0-b-37
r 0.4 < 0.6 ( NA) ( NA) Resolution in optical systems 0. ( ) h ~ + NA 0. ( ) h ~ NA MIT.7/.70 Optics /0/04 wk0-b-38
Resolution in optical systems r 0.4 < 0.6 ( NA) ( NA) ~ 0. ~ 0. ( ) ( ) h + + h NA NA MIT.7/.70 Optics /0/04 wk0-b-39
r 0.6 ( NA) Resolution in optical systems 0.305 ( ) h ~ + NA 0.305 ( ) h ~ NA MIT.7/.70 Optics /0/04 wk0-b-40
r 0.6 ( NA) Resolution in optical systems ~ 0.305 ~ 0.305 ( ) ( ) h + + h NA NA MIT.7/.70 Optics /0/04 wk0-b-4
r 0.6 ( NA) Resolution in noisy optical systems MIT.7/.70 Optics /0/04 wk0-b-4
r. ( NA) Safe resolution in optical systems ~ 0.6 ~ 0.6 ( ) ( ) h + + h NA NA MIT.7/.70 Optics /0/04 wk0-b-43
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: (/ lobe spacing) r. ( NA) r 0.6 ( NA) 0.5 ( NA) ( NA) MIT.7/.70 Optics /0/04 wk0-b-44 You will see different authors giving different definitions. Rayleigh in his original paper (879) noted the issue of noise and warned that the definition of just resolvable points is system or application dependent