南开大学光学工程课件Nov-16th

1、,Fresnel Diffraction,We define a ring-shaped differential area element dS. All of the point sources within dS are coherent and we assume that each radiates in phase with the primary wave. Therefore,Fresnel Diffraction,A is the source strength per unit area. The obliquity factor must vary slowly and

2、may presumed constant over a single Fresnel zone,So,Fresnel Diffraction,We have,The sum of the optical disturbance from all m zones at P is,Fresnel Diffraction,If m is odd, the series can be reformulated in two ways, either as:,Fresnel Diffraction,or,(a),(b),There are now two possibilities: either E

3、lis greater than the arithmetic mean of its two neighbors El-1and El+1, which is a question concerning the rate of change of K(). When,Fresnel Diffraction,equation (a) becomes,From equation (b) we have,Fresnel Diffraction,Since K() goes from 1 to 0 over a great many zones, we can neglect any variati

4、on between adjacent zones, i.e. E1= E2, Em-1= Em. So,We conclude from above that,Fresnel Diffraction,This same result is obtained in the case when,When the last term Em corresponds to an even m, the same procedure leads to,Fresnel Diffraction,Fresnel Diffraction,For a unobstructed wavefront, Em= 0 b

5、ecause K() for mth zone is 0. So,It is approximately equal to one half of the contribution from the first zone, i.e.,Fresnel Diffraction,If the primary wave was simply to propagate from S to P, it is,The two equations must be exactly the same. So we introduce a /2 phase difference between the primar

6、y wave and the secondary wave to make the two equations so.,Fresnel Diffraction,It is necessary to modify the initial statement of Huygens-Fresnel Principle. But, in fairly simply way, the principle give quit good predictions what are in fine agreement with experimental observations.,Circular apertu

7、res Spherical waves,Fresnel Diffraction,Consider that a monochromatic spherical wave incident on a screen which has a small hole.,Fresnels procedure applied to a point source can be used as a semiquantitative method with which to study diffraction at a circular aperture. First we consider the irradi

8、ance arriving a very small sensor placed at point P on the symmetry axis. Let us assume that the sensor at P “sees” an integral number of zones m, filling the aperture.,Fresnel Diffraction,If m is even, since Km0, and each adjacent contribution is nearly equal,Fresnel Diffraction,So I0.,If m is odd,

9、Fresnel Diffraction,which is roughly twice the amplitude of the unobstructed wave. By inserting a screen in the path of the wave, the irradiance at P is increased by a factor of four,We can gain a better appreciation of the actual size of the things we are dealing with by computing the number of zon

10、es in a given aperture. The area of each zone is:,Fresnel Diffraction,If the aperture has a radius R, to a good approxi-mation, the number of zones within it is simply,If m is not an integer, i.e. a fraction of a zone appears in the aperture, the irradiance at P is somewhere between zero and its max

11、imum value.,Fresnel Diffraction,To map the rest of the pattern, we now move the sensor along any line perpendicular to the axis. For off-axial point P, some zones are partially obscured. At certain position, the sum of all partial zones “saw” by P is zero. As P move radically outward, we detects a s

12、eries of relative maxima and minima.,Fresnel Diffraction,Circular obstacles,Fresnel Diffraction,If the opaque obstacle, be it a disk or sphere, obscures the first m zones, then,Emnow approaches zero because Km0. Repeating the same procedure as in circular hole yields,Fresnel Diffraction,There is a b

13、right spot everywhere along the central axis except immediately behind the circular obstacle (as P moves to the disk, increases, Km+10 and the irradiance gradually falls off to zero). The spot is called Poisons Spot. If the disk is large, the (m+1)th zone is very narrow and any irregularities in the

14、 obstacles surface may seriously obscure that zone. For Poisons spot to be readily observable, the obstacle must be smooth and circular.,Fresnel Diffraction,Off the axis, a whole series of concentric bright and dark rings will surround the central spot.,The Fresnel Zone plate If we removed all of ei

15、ther the even or odd zones, a tremendous increase in irradiance at P would be observed. For example, we construct a plate which passes only the first odd zones,Fresnel Diffraction,The irradiance is increased by 1600 times!,A screen which alters the light, either in amplitude or phase, coming from ev

16、ery other half-period zone is called a zone plate,Fresnel Diffraction,The outer edge of the mth zone is marked with the point Am. By definition, a wave which travels the path S-Am-P must arrive out of phase by m/2 with a wave which traverses the path S-O-P.,Fresnel Diffraction,The Fresnel Zone plate

17、,That is,Clearly, Rm is comparatively small, and,Finally we have,The Fresnel Zone plate,The radius of the mth Fresnel zones is given by,So we have the following relation:,The equation has an identical form to that of the thin-lens equation, a fact which is not merely a coincidence since S is actuall

18、y imaged in converging diffracted light at P. Accordingly, the primary focal length is said to be,The Fresnel Zone plate,Discussion The zone plate will show extensive chromatic aberration. Beside the primary focal point, additional higher-order focal points will exist at f1/3, f1/5, f1/7, etc. Advan

19、tages of zone plate as a lens: for UV-light, soft X-rays,The Fresnel Zone plate,The Fresnel Lens,The Fresnel Lens,The Fresnel Lens,Electric field in the near field,What is E(x1,y1) at a distance z from the aperture?,Electric field in the near field,According to Fresnel-Huygens principle,Here we take

20、 the inclination factor as 1.,Approximations,Concern the case that zx0,x1,y0,y1. For a good precision, in the denominator , we can approximate r01 by z. In the numerator:,Fresnel integral,Multiplying out the squares:,This yields:,Fresnel integral,Planar incident wave,Assume that the incident wave is

21、 planar and uniform,In a good approximation,Few Fresnel diffraction problems that can be solved analytically because of the complication in the Fresnel integral,Fresnel diffraction for a semi-infinite opaque screen,Fresnel Diffraction from a single slit,Slit,Fresnel Diffraction from a single slit,Th

22、e Spot of Arago,This irradiance can be quite high and can do some damage!,Interestingly, the hole fills in from the center first!,x0,x1,Stop,Beam after some distance,Input beam with hole,Fresnel diffraction from an array of slits: The Talbot Effect,The Fresnel diffraction pattern of an array of slits alternates between two different fringe patterns.,Cornu Spiral,For analysis, sometimes we use “Cornu Spiral”,Fresnel diffraction for a square aperture,Babinets Principle,The Babinets principle states that the