哈工大信息光学Chapter6-3

1、幻灯片 Chapter 61 Frequency Analysis of Optical Image SystemsWeimin SunCollege of ScienceHarbin Engineering University幻灯片2幻灯片3幻灯片4Deform：使变形Aberrations and their effects on frequency responsen When wavefront errors exist, we can imagine that the exit pupil is illuminated by a perfect spherical wave, bu

2、t a phase-shifting plate exists in the aperture, thus deforming the wavefront that leaves the pupil.n If the phase error at the point (x,y) is represented by kW(x,y), where k=2p/l and W is an effective path-length error, then the complex amplitude transmittance P(x,y) of the imaginaryphase-shifting

3、plate is given by P (x, y) = P(x, y)e j k W ( x , y)Overlap：交叠P (x, y) = P(x, y)e jkW ( x , y )A2ph(u,v) =lz P (x , y) exp- jlz(ux + vy)dxdyi -iH ( fX , fY ) = P (lzi fX , lzifY ) = P(lzifX , lzi fY )e jkW (lzi f X ,lzi fY )dxdyWhere A(fX,fY) is the area ofH ( fX , fY ) =A ( f X , fY )overlap two sh

4、ifted apeturedxdyA (0 ,0)ejk W ( x+lzi f X,y +lzi f Y) -W ( x-lzi f X,y -lzi fY)dxdy2222H( f X , fY ) = A ( f X , fY )dxdyA (0 ,0)Severe ：严格的，严重的Portion：一部分 Influence of the aberrationsn The aberrations will never increase the MTF (the modulus of the OTF).n Thus aberrations can tincrease the contras

5、t of any spatial-frequency component of the image, and in general will lower the contrast.n The absolute cutoff frequency remains unchanged, but severe aberrations can reduce the high - frequency portions of the OTF to such an extent that the effective cutoff is much lower than the diffraction-limit

6、ed cutoff.幻灯片5 Negative OTFn In addition, aberrations can cause the OTF o have negative values in certain bands of frequency, a result that never occurs for an aberration-free system.n When the OTF is negative, image components at that frequency undergo a contrast reversal; i.e. intensity peaks beco

7、me intensity nulls, and vice versa.Null:零vice versa：反之亦然Undergo：经历幻灯片6Example of a simpleaberrations: a focusing errorn A square aperture is consideredn For a on -axis pointf(x, y) =p(x2 + y2 )lzan where za zin The path-length error is thus given bykW (x, y) =p(x2 + y2 ) -p( x2 + y2 )lz alzin which

8、is seen to depend quadratically on the space variables in the exit pupil.幻灯片7幻灯片8n For a square aperture of width 2 w, the maximum path-length error at the edge of the aperture along the x or y axes, which we represent by Wm , isgiven by W =1(1-1) w2m2zazin The number Wm is a convenient indication o

9、f the severity of the focusing error.n Using the definition of Wm, we can express the path-length error asW (x, y) = Wx2+ y 2w2mW (x, y) = Wx 2+ y2mw2ejkW ( x+lz i fX,y+lz i fY)-W ( x-lz i fX,y-lzi fY)2222dxdyH ( f X , fY ) =A ( fX , fY )dxdyA (0 ,0)H ( f X, fY ) = L(f X)L(f Y)2 fo2 f osinc8Wm(f X)(

10、1 -f X)sinc8Wm(fY)(1 -fY)l 2 fo2 fol 2 fo2 fo幻灯片9Comparison of coherent andincoherent imagingn The OTF of a diffraction -limited system extends to a frequency that is twice the cutoff frequency of the ATF.n It is tempting, therefore, to conclude that incoherent illumination will invariably yield “be

11、tter”resolution than coherent illumination, given that the same imaging system is used in both case.n As we shall now see, this conclusion is in general not a valid one .n Surely any direct comparison of the two systems must be in terms of the same observable quantity, image intensity.幻灯片10幻灯片11Freq

12、uency spectrum of theimage intensityn One simple attribute of the image intensity which can be compared in the two cases is the frequency spectrum.n Whereas the incoherent system is linear in intensity, the coherent system is highly nonlinear in that quantity.Incoherent caseIi=h2 Ig=h2 U g2Coherent

13、caseIi=h U g2n Let the symbol represent the autocorrelation integral.n Instead of in the book.X ( f X , fY ) X ( fX , fY ) = X ( p,q) X * ( p - f X , q - fY )dpdq-n Then we can directly write the frequency spectra of the image intensities in the two cases asIncoherent caseF Ii = (H H )(Gg Gg )Cohere

14、nt caseF Ii = (HG g ) (HG g )幻灯片12nThe results of any such comparison will dependstrongly on both the intensity and phasedistributions across the object. We consider twoobjects with the same intensity transmittance butdifferent phase distributions, one of which can besaid to be imaged better in cohe

15、rent light and theother better in incoherent light.2At(x,h) = cos2pfxtA ( x) = cos2pfxfof fo2B tA (x ) =fo is the cutoff frequency of the ATFcos 2pfx幻灯片13幻灯片14For sample AtA ( x) = cos2pfxt(x2CoherentIncoherent,h) = cos2pfxGg GgGgThe contrast of1/21/21/41/21 /4the image-2fo - fofo2fofX- 2fo- fofo2fo

16、fXintensityHHHdistribution ispoorer forincoherent case-2fo - fofo2fofX- 2fo - fofo2fofXthan for theF Ii = (HGg ) (HGg )F Ii = (H H )(Gg Gg )coherent case1/41/21/41/41/21/4for object A.-2fo - fofo2fofX- 2fo - fofo2fofXFor sample Bt(x2,h) = cos2pfx4 111tA ( x) =cos2pfx=+c o s(4pfx)-cos(6pfx )+Lp21 335

17、CoherentIncoherentGg GgFor coherentGg4/3p2/p4/3p1/41/21/4case novariations of-2fo - fofo2fofX- 2fo- fofo2fofXimage intensityHHHwill be present,while theincoherent-2fo - fofo2fofX- 2fo - fofo2fofXsystem willF Ii = (HGg ) (HGg )F Ii = (H H )(Gg Gg )form the same2/p1/41/21/4image it did forobject A.-2f

18、o - fofo2fofX- 2fo - fofo2fofX幻灯片15幻灯片16Criterion：准则，标准 Two point resolutionn According to the Rayleigh criterion of resolution, two incoherent point sources are barely resolved by a diffraction -limited system with acircular pupil when the center of theAiry intensity pattern generated byIn the nonparaxial caseone point source falls exactly on thed = 0.61lfirst zero of the Airy patternsinqgenerated by the second.= 0.61ln The minimum resolvable separationNAof