Today. next week. MIT 2.71/ /04/09 wk13-a- 1

Transcription

1 Today Spatially coherent and incoherent imaging with a single lens re-derivation of the single-lens imaging condition ATF/OTF/PSF and the Numerical Aperture resolution in optical systems pupil engineering revisited next week Two more applications of the MTF defocus diffractive optics and holography Multi-pass interferometers: Fabry-Perot optical resonators and Lasers Beyond scalar optics: polarization and resolution 5/4/9 wk3-a-

2 Imaging with a single lens: imaging condition arbitrary complex input transparency gt(x) pupil mask gpm(x ) spatially coherent gillum(x) z z2 input diffraction from the input diffracted after lens + pupil mask wave converging to form the image at the plane The at the plane of this imaging system is (derivation in the supplement to this lecture) We also need to eliminate this quadratic term To eliminate the quadratic term we satisfy the Lens Law of geometrical optics 5/4/9 wk3-a- 2 After eliminating the quadratic, the remaining integral is the Fourier transform of the pupil mask gpm(x, y ).

3 Eliminating the quadratic term gt(x) pupil mask gpm(x ) gillum(x) z z2 Solution : shape gt(x) as a sphere of radius z arbitrary complex input transparency gt(x) gillum(x) pupil mask gpm(x ) collector z z2 Solution 2: attach a lens of focal length z to gt(x) Solution 3: limit gt(x) to ¼ the lateral size of gpm(x ) [see Goodman and Ref. 33] 5/4/9 wk3-a- 3

4 Imaging with a single lens: PSF and ATF arbitrary complex input transparency gt(x) pupil mask gpm(x ) spatially coherent gillum(x) z z2 input diffraction from the input diffracted after lens + pupil mask wave converging to form the image at the plane Assuming one of the three conditions is satisfied and the last remaining quadratic term can be eliminated, the is where is the PSF, i.e. the Fourier transform of the pupil mask scaled so that (x,y )=(λzu,λzv). As in the 4F system, the scaled complex transmissivity of the pupil mask is the ATF 5/4/9 wk3-a- 4

5 Imaging with a single lens: lateral magnification arbitrary complex input transparency gt(x) pupil mask gpm(x ) spatially coherent gillum(x) z z2 input diffraction from the input diffracted after lens + pupil mask wave converging to form the image at the plane If the pupil mask gpm(x, y ) is infinitely large and clear, its Fourier transform is approximated as a δ-function. Therefore, the optical at the plane is phase factor, does not affect the image intensity 5/4/9 wk3-a- 5 replica of the input within the factor of lateral magnification So by ignoring diffraction due to the finite lateral size and, possibly, phase-delay elements inside the clear aperture of the pupil mask gpm(x, y ), we have essentially found that the imaging condition and lateral magnification relationships from geometrical optics remain valid in wave optics as well for the intensity of the optical.

6 spatially coherent Block diagrams for coherent and incoherent linear shift invariant imaging systems thin transparency input cpsf convolution 4F imaging system: Single lens imaging system: Fourier transform ATF multiplication Fourier transform spatially incoherent thin transparency input intensity ipsf intensity convolution 4F imaging system: Fourier transform OTF Fourier transform Single lens imaging system: multiplication 5/4/9 wk3-a- 6

7 4F phase object gt(x,y)=exp{iφ(x,y)} Example: Zernike phase mask objective f pupil mask gpm(x,y ) f2 collector phase contrast image Iout(x,y ) quasi-monochromatic (sp. coherent or incoherent) π/2 phase shift λ=µm f=cm, f2=cm z=cm, z2=.cm SL phase object gt(x,y)=exp{iφ(x,y)} pupil mask gpm(x,y ) f objective phase contrast image Iout(x,y ) quasi-monochromatic (sp. coherent or incoherent) π/2 phase shift 5/4/9 wk3-a- 7

8 ATF and OTF of the Zernike phase mask See also PP3, pp. 6-7 z a=2cm.2cm.5µm T2 air opaque glass, n =.5 opaque g PM [a.u.] b=4cm cm Re(ATF) [a.u.] x 4F Re(ATF) [a.u.] SL phase(g PM ) [rad] Re(g PM ) [a.u.] x [cm] pi pi/2 pi/2 pi x [cm] ATF Im(ATF) [a.u.] u [mm ] u [mm.75 4F Im(ATF) [a.u.] u [mm ] u [mm.75 SL x [cm] OTF OTF [a.u.].5 OTF [a.u.].5 Im(g PM ) [a.u.] /4/9 wk3-a x [cm] u [mm ] u [mm ]

9 4F object gt(x,y) Clear circular aperture objective f pupil mask gpm(x,y )= circ(r /R) f2 collector phase contrast image Iout(x,y ) quasi-monochromatic (sp. coherent or incoherent) radius R (NA)in=R/f (NA)out=R/f2 λ=µm f=cm, f2=cm z=cm, z2=cm SL object gt(x,y) pupil mask gpm(x,y )= circ(r /R) f objective phase contrast image Iout(x,y ) quasi-monochromatic (sp. coherent or incoherent) radius R (NA)in=R/z (NA)out=R/z2 5/4/9 wk3-a- 9

10 ATF, cpsf of clear circular aperture 4F SL 5/4/9 wk3 -a- Common expression for the cpsf

11 Resolution [from the New Merriam-Webster Dictionary, 989 ed.]: resolve v : to break up into constituent parts: ANALYZE; 2 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 2 the action of solving, also : SOLUTION; 3 the quality of being resolute: FIRMNESS, DETERMINATION; 4 a formal statement expressing the opinion, will or, intent of a body of persons 5/4/9 wk3-a-

12 Rayleigh resolution limit Total intensity Source Source 2 Total intensity Source Source I [a.u.].5 I [a.u.] x [µm] x [µm] Two point sources are well resolved if they are spaced such that: (i) the PSF diameter (i) the PSF radius equals the point source spacing equals the point source spacing 5/4/9 wk3-a-2

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