傅里叶光学C5 Wave-Optics Analysis of Coherent Optical Systems.ppt

1、CHAPTER 5 Wave-Optics Analysis of Coherent Optical Systems,Single lens Complex imaging system Monochromatic illumination,THIN LENS,A lens is said to be a thin lens if a ray entering at coordinates (x, y) on one face exits at approximately the same coordinates on the opposite face,(x,y),0,5.1 Phase t

2、ransform function of a lens,A thin lens simply delays an incident wavefront by an amount proportional to the thickness of the lens at each point,Relation between fields of before and behind the lens,The complex field Ul(x, y) across a plane immediately behind the lens The complex field Ul(x, y) inci

3、dent on a plane immediately in front of the lens,The problem remains to find the mathematical form of the thickness function (x, y),Sign convention,As rays travel from left to right each convex surface encountered is taken to have a positive radius of curvature each concave surface is taken to have

4、a negative radius of curvature,Split the lens into three parts,5.1.1 Thickness function,5.1.2 Paraxial Approximation,5.1.3 Phase transform of the lens,With the approximation, phase transform of the lens is given by,Focal length f is defined by,Phase transform becomes,Generalization of the result,The

5、 sign convention allows the result to be applied to other types of lenses,The physical meaning of the lens transformation,The effect of the lens on a normally incident, unit-amplitude plane wave,Plot of physical meaning of the lens transformation,Converging lens has its inherent ability to perform t

6、wo-dimensional Fourier transforms There are several different configurations for performing the transform operation,5.2 Fourier Transforming Properties of Lenses,Typical configurations,Let a planar input transparency with amplitude transmittance tA(x, y) be placed immediately in front of a convergin

7、g lens of focal length f The input is assumed to be uniformly illuminated by a normally incident, monochromatic plane wave of amplitude A,5.2.1 Input Placed Against the Lens,Field behind the lens,Pupil function P(x, y) defined by,Amplitude distribution behind the lens,Field in the back focal plane,U

8、sing Fresnel diffraction formula yields distribution Uf(u, v) in the back focal plane of the lens,Simplification of Uf(u, v),Substituting (5-12) in (5-13), the quadratic phase factors within the integrand are seen to exactly cancel, leaving,Meaning of Uf(u, v),Complex amplitude distribution of the f

9、ield in the focal plane of the lens is the Fraunhofer diffraction pattern of the field incident on the lens Amplitude and phase of the light at coordinates (u, v) in the focal plane are determined by the amplitude and phase of the input Fourier component at frequencies (fx=u/f, fy=v/f),Incompletenes

10、s of FT,The Fourier transform relation between the input amplitude transmittance and the focal-plane amplitude distribution is not a complete one, due to the presence of the quadratic phase factor that precedes the integral,Intensity on the focal plane,If the intensity distribution in the focal plan

11、e will be measured, and the phase distribution is of no consequence,5.2.2 Input Placed in Front of the Lens,The input, located a distance d in front of the lens, is illuminated by a normally incident plane wave of amplitude A The amplitude transmittance of the input is again represented by tA,Diffra

12、ction from input to lens,Fresnel or paraxial approximation is valid for propagation over distance d Field on the lens:,Field in the back focal plane,For the moment, the finite extent of the lens aperture will be neglected. Let P=1, Eq. (5-14) can be rewritten,Complete form of field,A quadratic phase

13、 factor again precedes the transform integral, but that it vanishes for the very special case d = f,An important case to get exact Fourier transform,Evidently when the input is placed in the front focal plane of the lens, the phase curvature disappears, leaving an exact Fourier transform relation!,V

14、ignetting of the input_1,The shaded area in the input plane represents the portion of the input transparency that contributes to the Fourier transform at (u1,v1),Vignetting of the input_2,The limitation of the effective input by the finite lens aperture is known as a vignetting effect,5.2.3 Input Pl

15、aced Behind the Lens,The input again has amplitude transmittance tA, but it is now located a distance d in front of the rear focal plane of the lens Let the lens be illuminated by a normally incident plane wave of uniform amplitude A Incident on the input is a spherical wave converging towards the b

16、ack focal point of the lens,Geometrical optics approximation,The amplitude of the spherical wave impinging on the object is Af/d The finite extent of the illuminating spot in the input plane is represented by,Field before the input plane,Using a paraxial approximation to the spherical wave that illu

17、minates the input, the amplitude of the wave transmitted by the input may be written,Field in the focal plane,Assuming Fresnel diffraction from the input plane to the focal plane, we get the field in the focal plane,Characteristics of the configuration,Up to a quadratic phase factor, the focal-plane

18、 amplitude distribution is the Fourier transform of that portion of the input subtended by the projected lens aperture. An extra flexibility has been obtained in the present configuration; namely, the scale of the Fourier transform is under the control of the experimenter,Case of illumination with s

19、pherical wave,Point source S at finite distance,do,di,S,tA,Field behind the lens tl,It is the product of sphere wave, tA and lens transmittance tl,Field on the observation screen Ui,It is Fresnel diffraction of tl,Simplification of Ui,If light source and observation screen are conjugate,The Ui becom

20、es,Meaning of the result,On the conjugate plane of light source is the Fourier transform plane of the object This is true for both above cases (point source locates at finite and infinite distances) fx=xi/(di), fy=yi/(di),5.2.4 Example of an Optical Fourier Transform,Transparent character 3, which i

21、s placed in front of a positive lens and illuminated by a plane wave,5.3 Image formation: monochromatic illumination,First we restrict attention to a positive, aberration-free thin lens that forms a real image Second, we consider only monochromatic illumination Both of these restrictions will be rem

22、oved in Chapter 6,5.3.1 The Impulse Response of a Positive Lens,Our purpose is to find the conditions under which the field distribution Ui can reasonably be said to be an image of the object distribution Uo,Linear relation of object and image,We can in all cases express the field Ui by the followin

23、g superposition integral:,where h(u,v;,) is the field amplitude produced at coordinates (u, v) by a unit amplitude point source applied at object coordinates (,).,(5-23),Condition of imaging,If the optical system is to produce high-quality images, then Ui must be as similar as possible to Uo Equival

24、ently, the impulse response h(u,v;,) should closely approximate a Dirac delta function,where K is a complex constant, M represents the system magnification, and the plus and minus signs are included to allow for the absence or presence of image inversion,Finding the impulse response h_1,Let the obje

25、ct be a function (point source) at coordinates (,) Incident on the lens will appear a spherical wave diverging from the point,Finding the impulse response h_2,After passage through the lens (focal length f), the field distribution becomes,Using the Fresnel diffraction equation to account for propaga

26、tion over distance z2, we have,Finding the impulse response h_3,Combining (5-25), (5-26), and (5-27), yields the formidable result,(5-28),Necessity to simplify,Equations (5-23) and (5-28) now provide a formal solution specifying the relationship that exists between the object Uo, and the image Ui Th

27、e result is difficult to determine the conditions under which Ui can reasonably be called an image of Uo, unless further simplifications are adopted,5.3.2 Eliminating Quadratic Phase Factors: The Lens Law,Quadratic Phase Factors depending on the lens coordinates: The Lens Law If,then,=1,Note that th

28、is relationship is precisely the classical lens law of geometrical optics,Simplification_1,This term can be ignored under either of two conditions It is the intensity distribution in the image plane that is of interest The image is measured on a spherical surface, centered at the point where the opt

29、ical axis pierces the thin lens, and of radius z2,Quadratic Phase Factors depending on the image coordinates (u,v),Simplification_2,Quadratic Phase Factors depending on the object coordinates (,),There are three different conditions under which this term can be neglected Condition 1 The object exist

30、s on the surface of a sphere of radius z1 centered on the point where the optical axis pierces the thin lens,Condition 2,The object is illuminated by a spherical wave that is converging towards the point where the optical axis pierces the lens,Condition 3,The phase of the quadratic phase factor chan

31、ges by an amount that is only a small fraction of a radian within the region of the object that contributes significantly to the field at the particular image point (u, v),Possibility of the three conditions,The first of these conditions rarely occurs in practice. The second can easily be made to oc

32、cur by proper choice of the illumination The response of the system to an impulse at object coordinates should extend over only a small region of image space,where M=-z2/z1 is the magnification of the system,Illustration of condition 3,Simplified expression for the impulse response,Final simplified form for the impulse response,Defining the magnijication of the system by,Explanation of impulse response,If the lens law is satisfied, the impulse response is seen to be g