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会计学1高斯光速过透镜高斯光速过透镜wkcmzkkksz/),(smkkvz第1页/共17页vFvvviwtFaE)exp(Re2va0VFsmzVdkdk第2页/共17页zmkkGkkimkkGkkskmkkGkkskzFztzzzzV),(42)1,(41)1,(41),(kkkkkrFrFdvimzikkJmkkGy ixkkks smmzzvvztmzzt/)()()()()exp()exp()(),(2/)(22正交条件第3页/共17页For the remainder of this section,we will only consider monochromatic beams propagating in the positive z direction.we take as a mode number instead of ,and we reduce set of mode numbers by tkzk smttkdsmk),()2/()()()(tants smmtttconszkkkrFrFdS第4页/共17页)2/exp(2fik)2/(exp(222ffikp0zA ideal parabolic lens would be represented by a phase factor In the plane of the lens FFfkidSkkrFkrFFFztoutoin002)2exp(2)()(),(第5页/共17页If we approximate the incoming beam by a circularly polarized(lowest-order)Gaussian beam with Rayleigh range inz1inkz)2exp(),(2inozkF)2exp()2exp()(22001intztmzkfkikJdkskkkk)2exp(21kkkskkkkktztm)4/exp(21)exp()(2200 xxdxxJ第6页/共17页)1(exp()exp()()1,()22exp()22exp()2exp()2exp()()1,(41)0exp()2exp()2exp(22)2exp(2)()()2exp(),(),(42)1,(41)1,(41),(1221220012200022mizikkJmkkGikzfkfzikzffzkskkkikzfkfzikzffzkskkkzkfkikJdkskkkmkkGkkskizkfkidkFFfkidSkkrFkrFzkFFzmkkGkkimkkGkkskmkkGkkskzFztmzintinininztintinininztmintztmzzintztoutinoinztzzzzV第7页/共17页22202220fzfzzfzzfzizzininininRRWhen the paraxial limit is valid for the outgoing beam,i.e.,when,and correspond.But also outside the paraxial limit,the focused light beam is characterized by the two parameters and .1RkzRz0z0zRz)2exp(21kkkskkkkktztm第8页/共17页)exp()()2exp(22)exp()(41)2exp(2)exp()(41)2exp()1,(41)2exp()()()()1(exp()exp()()1,(02222002222002110211001zikkJkkkkkkkdkzikkJkkkkkkkdkzikkJkkskkkkskkkkdkmkkGkkskkkkskkkkdkrFkdkrFkrFmizikkJmkkGzttttktzttttktztztztmsktzztztmsktmsktoutztmz第9页/共17页The largest component of the output field is the component,which is given by )exp()(2exp()(2202022200kikizzkkkJkkkkkdkizzFzRttttktRFFWhen the paraxial approximation is valid for the outgoing beam,we may take out a factor extend the integration limit to infinity)exp(zikkkkktz2/2use defining)(0zzizzRw第10页/共17页)2exp()exp()()2exp()(2)exp(2202000wwRwttttRzkzikzizzkzkkJkkdkikzizzF)2exp()(2 1)2()exp(2exp()(2)2exp()21(222)exp(2022232202220222213210kzkkJkkkkkdkFzkkkkikzkkJkkkkkdkFzkzkkzzFFFFizzikzFwttttkttzwttttktwwwwR第11页/共17页Outside the paraxial limit,the focal plane is no longer at z=z0 but moves towards the lens by several wavelengths,furthermore,unlike in the paraxial approximation,the shape of the field is not just determined by the value of but depends on as well Rz0zThe plots of the transverse-mode profiles show that beyond a certain point the width of the field no longer decreases with stronger focusing.One cannot focus down a laser field to below a certain limit,roughly about half a wavelength,with the idea lens factor,no matter how small becomes.Moreover,no one notes the asymmetry of the outgoing beam around the focal plane,in contrast to a paraxial beam that is symmetry under reflections in the focal plane,this fact should not be surprising.Rz第12页/共17页F/)(0zzZ100f1000z,103,101,103,101/5433inz30/,3/10,100z/FIG 1(a)field strength of strongly focused Gaussian beams on the z axis as a function Note that maximum strength of the incoming field is 1.The lens is located at z=0 and characterized by ,so that The incoming Gaussian beam has increasing calues of respectively,for the bottom to top curves.This implies,for the outgoing beam,decreasing values of (b)field strength in the focal plane as a function of the transverse coordinate.第13页/共17页第14页/共17页Fig.2.Lommel field,which describes the focus of a lens with uniform illumination in the paraxial limit for the case NA=0.1 and f=1000 .(a)Amplitude u()in the focal plane z=0 according to relation(9).(b)Intensity distribution in the(,z)plane according to relation(8).The intensity()contours indicate intensities of ,.;the intensity in the geometrical focus(0,0)is normalized to 1.2|u2|u2110,10第15页/共17页In case of an incoming higher-order LG beam,the incoming field distribution can be written as(again assuming its focal plane coincides with the plane of the lens)oF)2exp()2exp()(2)exp()2exp(221020,220intinztmininzkfkikJzdkskkkky ixzizkF)4/exp(4)exp()(22200 xxdxxJ)2exp(22220,2kkzkskkkkktinztm第16页/共17页
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