Lecture7信息光学

1、#7 Frequency Analysis of Optical Image SystemsGeneralized treatment of imaging systemsAn imaging system is said to be diffraction-limitednThus the terminal property of a diffraction-limited imaging system is that a diverging spherical wave incident on the entrance pupil is converted by the system in

2、to a converging spherical wave at the exit pupil.nFor any real imaging system, this property will be satisfied, at best, over only finite for which this property holds, then the system may be regarded as being diffraction-limited.Effects of diffraction on the imagedd),(),;,(),(oiUvuhvuUdd),(),(),(gi

3、UvuhvuUyxvyuxzjyxPzAvuhiidd)(2exp),(),(dd),(),;,(),(oiUvuhvuUnIncoherent illuminationnCoherent illumination相干照明相干照明:x0BAy0 xiBAyiBlack Box非相干照明非相干照明:x0BAy0 xiBAyiBlack BoxA, B两点光振动相干两点光振动相干, 则引起的以则引起的以A, B为中心的两个分布也相干为中心的两个分布也相干. 应将其应将其干涉图样求出后干涉图样求出后,再作模方求强度。再作模方求强度。A.B 两点在像面上某点引起的复振幅没两点在像面上某点引起的复振幅没

4、有确定的位相关系。观察到的强度是多有确定的位相关系。观察到的强度是多个像点强度的叠加，即非相干叠加。个像点强度的叠加，即非相干叠加。Coherent illuminationnThus a coherent imaging system is linear in complex amplitude.2dd),(),(),(giUvuhvuIIncoherent illuminationnan incoherent imaging system is linear in intensity, and the impulse response of such a system is the squ

5、ared magnitude of the amplitude impulse response.dd),(),(),(2giIvuhvuICompare nUnder coherent illumination, an imaging system is linear in complex amplitude.dd),(),(),(giUvuhvuUnWhen the illumination is perfectly incoherent, an imaging system is linear in intensitydd),(),(),(2giIvuhvuIThe frequency

6、spectra of the input and output are defined as-)(2-)(2dd),(),(dd),(),(vuevuUffGvuevuUffGvfufjiYXivfufjgYXgYXYX-)(2dd),(),(vuevuhffHvfufjYXYXIn addition, define the amplitude transfer function (ATF) H as the Fourier transform of the space-invariant amplitude impulse response h(u,v)coherent illuminati

7、onATFnThus the effects of the diffraction-limited system have been expressed, at least formally, in the frequency domain. nIt now remains to relate H more directly to the physical characteristics of the imaging system itself.dd),(),(),(giUvuhvuU-)(2-)(2dd),(),(dd),(),(vuevuUffGvuevuUffGvfufjiYXivfuf

8、jgYXgYXYX-)(2dd),(),(vuevuhffHvfufjYXYX),(),(),(YXgYXYXiffGffHffGATFFor notational convenience we set the constant A zi equal to unity and ignore the negative signs in the arguments of P (almost all applications of interest to us here have pupil functions that are symmetrical in x and y).),()(dd)()(

9、2exp),(),(YiXiiiiYXfzfzPzAyxyvxuzjyxPzAffHF),(),(YiXiYXfzfzPffHH is the Fourier transform of the amplitude impulse response hh itself is a Fraunhofer diffraction pattern of the exit pupil,i.e. a scaled Fourier transform of the pupil function.Examples of amplitude transfer functions (ATF)(circ),()2(r

10、ect)2(rect),(22wyxyxPwywxyxPFrequency response for diffraction -limited incoherent imagingThe optical transfer function (OTF)dd),(),(),(112giIvuhvuI-)(2-)(2dd),(dd),(),(dd),(dd),(),(vuvuIvuevuIffvuvuIvuevuIffivfufjiYXigvfufjgYXgYXYXGGThe normalized frequency spectra of Ig and Ii are defined byNormal

11、ized transfer function),(),(),(YXgYXYXiffffffGHGThe function H(fX,fY) is know as the optical transfer function (OTF). The modulus |H(fX,fY)| is know as the modulation transfer function (MTF). -2-)(22dd),(dd),(),(vuvuhvuevuhffvfufjYXYXHOTFDefineThe phase of H(fX,fY) is know as the phase transfer func

12、tion (PTF). ),(hffHYXF-22dd),(),(),(vuvuhvuhffYXFH-2d d) , (d d),() , (),(qpqpHqpfqfpHqpHffYXYX-HBased on autocorrelation theorem-FFd d),() , (),(),(dd),(),(22qpfqfpHqpHvuhffHvuyvxuhvuhYXYXand Reyleighs theorem-2-2d d) , (dd),(qpqpHvuvuhChange the variables22YXfqqfpp-2dd),(dd)2,2()2,2(),(qpqpHqpfqfp

13、HfqfpHffYXYXYX-HThus the OTF is the normalized autocorrelation function of the ATF.The equation above will serve as a link between the properties of coherent and incoherent systems.Note: It is entirely valid for system both with and without aberrations.-2d d) , (d d),() , (),(qpqpHqpfqfpHqpHffYXYX-H

14、General properties of the OTFIt should be pointed out that while the OTF is always unity at the zero frequency, this does not imply that the absolute intensity level of the image background is the same as the absolute intensity level of the object background. The normalization used in the definition

15、 of the OTF has removed all information about absolute intensity levels.)0 , 0(),( . 3),(),( . 21)0 , 0( . 1HHHHHYXYXYXffffff-2dd),(dd)2,2()2,2(),(qpqpHqpfqfpHfqfpHffYXYXYX-HThe OTF of aberration-free systemFor coherent systems),(),(YiXiYXfzfzPffH-dd),(dd)2,2()2,2(),(yxyxPyxfzyfzxPfzyfzxPffYiXiYiXiY

16、X-HFor incoherent systemspupil theoutside0pupil theinside1),(yxP),(),(2yxPyxPnThe expression above for H leads to an extremely important geometrical interpretation:qThe numerator represents the area of overlap of two displaced pupil functions, none centered at (-zifX/2, -zifY/2)nSecond centered at (zifX/2, zifY/2), the diametrically op