高斯光速過透鏡[優(yōu)選課堂]

1、Strongly focused light beams interacting with single atoms in free space 1簡易輔導(dǎo) 1 Strongly focused Gaussian beams We use a set of eigenmodes of four commuting operators corresponding to the following four physical quantites: energy angular momentum in z direction momentum in the z direction helicity

2、The modes are thus characterized by the four numbers wkc m z k kks z / ),(smkkv z 2簡易輔導(dǎo) The complete orthogonal set of modes is defined such that the free electric field (the solution of the source-free Maxwell equations) can be expanded in this set as is arbitrary complex amplitudes. The function i

3、s transverse, the summation over v is a short-hand notation for v F v vv iwtFaE)exp(Re2 v a 0 V F sm z V dkdk 3簡易輔導(dǎo) zmkkG k k imkkG k ksk mkkG k ksk zF z t z z z z V ),( 4 2 ) 1,( 4 1 ) 1,( 4 1 ),( kkkkkrFrFdv imzikkJmkkG y ix kkk s smmzzvv ztmz zt /)()()()( )exp()exp()(),( 2/ )( 22 正交條件 4簡易輔導(dǎo) For t

4、he 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 t k z k sm t t kd smk ),( )2/()()()( tan ts smmtt tconsz kkkrFrFdS 5簡易輔導(dǎo) The action of the lens is modeled here

5、by assuming that the field distribution of the incoming field is multiplied by a local phase factor )2/exp( 2 fik )2/(exp( 222 ffik p 0z A ideal parabolic lens would be represented by a phase factor In the plane of the lens FF f k idSkk rFkrF FF z t out oin 0 0 2 ) 2 exp(2 )()( ),( 6簡易輔導(dǎo) If we appro

6、ximate the incoming beam by a circularly polarized (lowest-order) Gaussian beam with Rayleigh range in z 1 in kz ) 2 exp(),( 2 in o z k F ) 2 exp() 2 exp()( 22 0 01 in t z tm z k f k ikJd k skk kk ) 2 exp( 2 1 k k k skk k k k tzt m )4/exp( 2 1 )exp()( 22 0 0 xxdxxJ 7簡易輔導(dǎo) ) 1(exp()exp()() 1,( ) 22 ex

7、p( ) 22 exp( ) 2 exp() 2 exp()( ) 1,( 4 1 )0exp() 2 exp() 2 exp(22 ) 2 exp(2 )()( ) 2 exp(),( ),( 4 2 ) 1,( 4 1 ) 1,( 4 1 ),( 1 2 2 1 22 0 01 22 0 0 0 2 2 mizikkJmkkG ikzf kfz ikzf fz k skk k ikzf kfz ikzf fz k skk k z k f k ikJd k skk k mkkG k ksk i z k f k idk FF f k idSkk rFkrF z k FF zmkkG k k i

8、mkkG k ksk mkkG k ksk zF ztmz in tin in inz t in tin in inz tm in t z tm z z in t z t out in oin z t z z z z V 8簡易輔導(dǎo) 22 2 0 22 2 0 fz fz z fz zf z izz in in in in R R When the paraxial limit is valid for the outgoing beam, i.e., when , and correspond. But also outside the paraxial limit, the focused

9、 light beam is characterized by the two parameters and . 1 R kz R z 0 z 0 zR z ) 2 exp( 2 1 k k k skk k k k tzt m 9簡易輔導(dǎo) )exp()() 2 exp( 2 2 )exp()( 4 1 ) 2 exp( 2 )exp()( 4 1 ) 2 exp( ) 1,( 4 1 ) 2 exp( )( )()( ) 1(exp()exp()() 1,( 0 2 2 22 0 0 2 2 22 0 0 2 1 1 0 2 1 1 0 0 1 zikkJ k k k kk k k dk zi

10、kkJ k k k kk k k dk zikkJ k ksk k k k skk k k dk mkkG k ksk k k k skk k k dk rFkdk rFkrF mizikkJmkkG zt ttt k t zt ttt k t zt ztzt m s k t z ztzt m s k t ms k t out ztmz 10簡易輔導(dǎo) The largest component of the output field is the component, which is given by )exp()( 2 exp()( 2 2 0 2 0 2 22 0 0 kikizz k

11、k kJ k kk k k dk izz F zR t t tt k t R FF When the paraxial approximation is valid for the outgoing beam, we may take out a factor extend the integration limit to infinity )exp(zik kkkk tz 2/ 2 use defining )( 0 zzizz Rw 11簡易輔導(dǎo) ) 2 exp( )exp()( ) 2 exp()( 2 )exp( 2 2 0 2 0 0 0 ww R wt t t t R z k z

12、ikzizz k zk kJ k k dkikz izz F ) 2 exp()( 2 1) 2 ()exp( 2 exp()( 2 ) 2 exp() 2 1 ( 22 2 )exp( 2 0 2 22 3 22 0 2 22 0 2 22 2 1 321 0 k zk kJ k kk k k dkF z k k kki k zk kJ k kk k k dkF z k z k kzz F FFF izz ikzF wt t tt k t t z wt t tt k t wwww R 12簡易輔導(dǎo) Outside the paraxial limit, the focal plane is

13、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 R z 0 z The plots of the transverse-mode profiles show that beyond a certain point the width of

14、 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

15、to a paraxial beam that is symmetry under reflections in the focal plane, this fact should not be surprising. R z 13簡易輔導(dǎo) F / )( 0 zzZ 100f 100 0 z ,103 ,101 ,103 ,101/ 5433 in z 30/, 3/10,10 0 z / FIG 1 (a) field strength of strongly focused Gaussian beams on the z axis as a function Note that maxim

16、um 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 funct

17、ion of the transverse coordinate . 14簡易輔導(dǎo) 15簡易輔導(dǎo) 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 |u 2 |u 21 10,10 16簡易輔導(dǎo) In case of an incoming higher-order LG beam, the incoming field distribution can be writ