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科技资讯科技资讯S I N &T NOLOGY I NFORM TI ON2008N O .12SC I ENCE &TEC HN OLO GY I NFO RM ATI O N学术论坛1瑞利判断和中心点亮度1.1瑞利判断定义：实际波面与参考球面波之间的最大波像差不超过4/λ时，此波面可看作是无缺陷的。
优点：便于实际应用缺点：不够严密。
适用范围：是一种较为严格的像质评价方法，适用于小像差光学系统。
1.2中心点亮度1)中心点亮度：光学系统存在像差时，其成像衍射斑的中心亮度和不存在像差时衍射斑的中心亮度之比S.D 来表示光学系统的成像质量。
2)斯托列尔准则：当S.D ≥0.8，认为光学系统的成像质量是完善的。
3)适用范围：是一种高质量的像质评价标准，适用于小像差光学系统。
4)缺点：计算相当复杂，很少作为计算评价方法使用。
2分辨率分辨率反映光学系统分辨物体细节的能力，是一个很重要的指标参数，故也可用分辨率作为光学系统的成像质量评价方法。
2.1分辨率基本公式根据衍射理论，光学系统的最小分辨角为Δθ：Δθ=1.22λ/D对不同类型的光学系统，可由上式得到不同的表示形式。
2.2缺点1)只适用于大像差光学系统；2)与实际情况存在差异；3)存在伪分辨现象.故用分辨率来评价光学系统的成像质量也不是一种严格而可靠的评价方法。
2.3优点其指标单一，便于测量，在光学系统像质检测中得到广泛应用。
3点列图3.1点列图定义在几何光学的成像过程中，由一点发出的许多条光线经光学系统成像后，由于像差的存在，使其与像面的交点不再集中于一点，而是形成一个分布在一定范围内的弥散图形，称为点列图。
3.2适用范围适用于大像差光学系统。
照相物镜的像质评价：利用集中30％以上的点或光线所构成的图形区域作为其实际有效的弥散斑，弥散斑直径的倒数为系统的分辨率。
3.3优缺点优点：简便易行，形象直观。
缺点：工作量非常大，只有利用计算机才能实现。
Chapter X Image quality evaluation of optical system(光学系统像质评价)Introduction§10.1 Brightness of center disk and Rayleigh Judgement(中心点亮度和瑞利判断)§10.2 Resolving power(分辨率)§10.3 Spot diagram(点列图)§10.4 Optical transfer function(光学传递函数)IntroductionThe initial preliminary evaluation of the image quality provided by a particular system come from the blur circle formed by all the rays emitted by point objects in different field of view.This blur circle is called usually as diffused disk, spot diagram, point spread function.1. Geometrical aberration:The geometrical aberration is the residual aberration less than the tolerance of aberration or according to the curve of aberration. We determine the dimension of diffused disk.＊ Because of diffraction geometrical aberration can not describe the actualdistribution of energy of a point object.2. Wavefront aberration Rayleigh criterion (ifw 4λ'≤ the image quality is good; if w 10λ'≤the imagequality is perfect) is very simple, it is does not consider the defect area. Only used in small aberration system. (Peak to Valley: maximum wavefront aberration)3. The resolving power of an optical system has been the criterion most used for the quality of the image. It has been expressed by in terms of the maximum number of lines per millimeter. Moreover, the resolving power depends greatly on the contrast in object.4. Optical Transfer Function (OTF)OTF represents the property of an optical system considered as a transmitter of spatial frequence.＊ Even an ideally corrected system can only transmit spatial frequence below some limiting frequence-cutoff frequence.§10.1 Brightness of center disk and Rayleigh Judgement (中心点亮度和瑞利判断)1. Brightness of center diskStrehl Criterion:B r i g h t n e s s o f R a y l e i g h d i s k f o r a b e r r a t io n a l s y s t e m .B r i g h t n e s s o f R a y l e i g h d i s k f o r i d e a l s ys t e m S D =221222(0)1.(0)p ikwp w S D erd rd w πϕϕπϕ≠===⎰⎰Let C o m p l e x A m p l i t u d e a t i m a g e p p x y ϕ- exit pupil radiuslr -eye d istan ce z l '- Get 2120z ikl ik l pz r eerd rd l πωηϕϕλ'--='⎰⎰Expansion series of ikwe212xxex =++()222 .1S D k w w⎡⎤∴≈--⎢⎥⎣⎦12122211, w w rd rd w w rd rd ππϕϕππ==⎰⎰⎰⎰If S.D 0.8 th e system is p erfect,it n o tes th at th e cen ter en erg y o f rayleig h d isk is 0.84.≥Sample:2If , ()6844p w w λλϕ===22()684.0.8184(0)p p w S D w λϕϕ=====S.D0.8∴≥It corresponds to the Rayleigh criterion.w 4λ≤2. Rayleigh Judgement w 4λ'≤ The image quality is goodw 10λ'≤ The image quality is perfectConsidering only the maximum wavefront aberration, defocusing can improve the wavefront aberration. (Select the best image plane) The ration of the area (w4λ≤) to the whole area is not considered.The advantage is easy to calculate, applied in optical system with small aberration, such as telescope, microscope objective. Zemax: ① analysis ↙wavefront field settin g w avefro n t m ap w avelen g th in t erfero g ram ⎧⎧⎪⎪⎪⎨⎨⎪⎩⎪⎪⎩② windowOpd: determine the defect area§10.2 Resolving power(分辨率)Resolving power of a system will be of the order of 1σ, σ is the mean circlediameter of diffused disk.Rayleigh Judgement: the least resolving separation (LRS) of the two star images is given by the radius of Airy Disk. LRS=1 Airy radius Dove Judgement:LRS=0.85 Airy radius Ideal optical system:Telescope 140Dϕ︒= 120Dϕ︒=Microscope 0.61N A λσ=0.5N A λσ= Photographic 1475L N F=1800Fσ=Resolving power depend onc o n t r a s t o f a n o b j e c tillu im in atin g co n d itio n sen sitivety an d reso lu tio n o f d etecto r ⎧⎪⎨⎪⎩＊ Applied in large aberration system, such as photographic lens.Discussion:⑴ Resolving plate is a square-wave grating which consists of a series of light and darkness at progressively closer spacing.The resolving plate is high contrast, while object is lower contrast.b r i g h t m i n u s d a r k b r i g h t p l u s d a r kK =⑵ Some times it appears pseudo-resolving phenomenon, below cutoff frequence, higher frequence object can be resolved, this is because of phase-inverted.§10.3 Spot diagram(点列图)The entrance pupil is divided into a few equal area rings, equal number of rays pass through each zone. The points distribution in image plane represents the energy distribution.The diameter of the spot less than (0.03~0.01mm) is permitted, and it is depended on the detector.§10.4 Optical transfer function (光学传递函数)The base of OTF is Fourier analysis 1. ConceptAn incoherent optical system (incoherent illumination) can be considered as a low-pass linear filter of spatial frequencies. If sinusoidal intensity distribution (signal) is inputted into optical system, the output is also sinusoidal, but the contrast is reduced (K 1), phase is shifted (ϕ∆). The ration M 1/M and phaseϕ∆ are depends on the frequence.M: inputted object contrast M 1: outputted image contrast＊ The ratio M 1/M is called modulated transfer function (MTF) of an optical system, and ϕ∆ is called phase transfer function (PTF).Inputted:000co s 2(1co s 2)a a x x I I I I f x I f x I ππ=+=+m a x m i n00maxm in 00()()()()a a aa a I II I I I IM I I I I I I I -+--===+++-0 (1c o s 2)xII M f x π∴=+ After passing though optical system0 [1c o s (2)]x I I Mf x πϕ'=++∆a a I I M M'<'∴<M M T F M'=01M T F <<iP T FO T F M T F e-=Why take fine-wave distribution signal as inputted object? Because any object can be seen as be composed by a series of sine patterns with different frequence and different brightness.2. The condition of OTF1) linear condition:Energy distributions of object (image) satisfy linear superposition:● with incoherent illumination ● ground glass● optical system has larger numerical aperture2) space-invariableThere is the same point spread function (PSF) in any region of the image plane-isoplanatic condition.Object (image) is divided into many isoplanatic regions, each region has the same effect of diffraction and aberration.3. Expression of OTFIn order to calculate OTF the object (image) break down many points, mathematically is (,)x y δ. Object distribution:11111(,)(,)(,)o x y o x y x x yy d x d yδ∞=--⎰⎰Image distribution:(,)(,)(-,-)(,)(-,-)x y y I x y o x y h x x y y d xd y o x y h x x y y xββββ∞''''=⎰⎰''=*h – PSF (point spread function)Method 1: (,){(,)}{(,P S F x y P x y P x y *''''''=ℑℑ(,)(,), I t i s c a l l e d e x i t p u p i l f u n c t i o n.i k w x y P x y E e ''-''=(,)w a v e f r o n t a b e r r a t io n w x y w ηξ''''-即 -2(-){(,)}(,)i xx yy w x y w x y edxdyπ''''ℑ=⎰⎰ 2(-){(,)}(,)i xx yy w x y w x y e dxdyπ''*''ℑ=⎰⎰(,){(,)}x y O T F f f P SF x y ''=ℑMethod 2:(,)(,)(,)I x y o x y h x x y y ''''=*--Fourier transform(,)(,)(,)x y x y x y I f f o f f H f f =(,) .x y H f f O T F -Method 1: measured by interferometer; Method 2: measured by MTF instrument. Sample: Using linear spread function (LSF) LSF:()(,)h x h x y d y∞-∞=⎰Similarly:()()(,)x I x o x h x x d xβ∞-∞''=*⎰Fourier transform()()()x x x I f o f H f =OTF ：2()()x x i f xH f h x ed xπ∞-∞=⎰, (x xf fλ=)- c o s s i nOT F : ()()c o s 2()s i n 2()()i x x x x r x veH f hxf xdx i h xf xdx H f iH f θθθππ∞∞-∞-∞=-∴=-=-⎰⎰MTF:()x H f =PTF: 1()()()x v x x rH f f tgH f φ-=Linear spread function (LSF) can be analogized by scanning slit.4. Example Let000()cos 2a x o x I I f x π=+Image distribution00000000000()()(co s 2) ()()co s 2a x x I x h x x I I f x d x I h x x d x I h x x f x d x ππ+∞-∞+∞+∞-∞-∞''=-+''=-+-⎰⎰⎰Suppose:000x x x d x d x x x x '=-⎫⇒=⎬'=-⎭Result:000()()()co s 2() {1()co s[2()]}x x x x I x I h x d x I h x f x x d xI H f f x f ππφ+∞+∞-∞-∞''=+-'=+-⎰⎰5. Application of OTF ① AnalysisLimiting frequence:curve Ⅱ better than Ⅰ.Low frequence:curve Ⅰbetter than Ⅱ.It means that lens Ⅰ has better image quality for low frequence object than lens Ⅱ.② Integrate the curve, the larger the area is, the better the lens quality is. The area represents the transmitted information magnitude.③ The advantage of using MTF as quality criterion is the ability to cascade the MTF curves of a lens and film by multiplying together the MTF values.l e n sf iM T F M T F M T F =⨯ ＊ Notes that:If the illumination is perfectly coherent, the resolution drops to half, butthe contrast at the low frequence is great improved.6. The calculation of exit pupil function1(,)(,)(,)x y GO T F f f Px x y y P x y d xd y C*''=++⎰⎰OTF is the autocorrelation of exit pupil function,x y x y f f RRλλ''==R – Distance from ideal point to center of exit pupil. C – Constant 2(,)P x y d xd y=⎰⎰, used for normalized OTF.(,)0(,)i k w x yP x y E e-=E - Constant (amplititude), can be discarded.(,)w x y - wavefront aberration.-[(,)(,)](,)(,)0, o u t o f su p erp o sitio nik w x x y y w x y e P x x y y P x y *''++-⎧⎪''++=⎨⎪⎩'y zT T ww δδηζ''∂∂=='∂∂,x y ζη''-- i f ζ'∴= Tangential: (0,)(0,)co s 2sin 2yy y y T T P y y P y f i f πδπδ*'''+=+i fη'=Sagittal: (,0)(,0)cos 2sin 2z z z z T T P x x P x f i f πδπδ*'''+=+附一c o s xx x f f θλλ==Normally 1R f '==, x yx f y f λλ''∴==For geometrical optics, assume that λ tends towards zero, so we can write:(0,)(0,)y yw w y f w y f yλλ∂+-=∂(0,)(0,)2y y y ik w y f w y i f T ee λπδ'⎡⎤-+--⎣⎦∴=[](,0)(,0)2c o s 2s i n 2x x z ik w x f w x i f T x z x z eef T f T λπδπδπδ'-+--''∴==+＊ 这里的坐标(,)x y 即像差中出瞳(),ξη''。
