1、,2.6 Fundamental Gaussian Beams in a Lenslike Medium,Lecture 3,2.6 Fundamental Gaussian Beams in a Lenslike Medium,So that, w (z) is the beam spot size and R its radius of curvature, as in the case of a homogeneous medium.,The ABCD Law,I. A spherical wave,A. Free space,B. Thin lens,The ABCD Law,The
2、ABCD Law,Ray Matrices,ABCD express,So, for the spherical wave, we have:,The ABCD Law,II. A Gaussian beam,高斯光束是一个非均匀的、曲率中心不断改变的球面波，也具有类似与球面波的曲率半径R这样的参量，其传播规律与球面波R的表述完全类似。而这一参量就是我们前面讲到的q参数。,A. Free space,The ABCD Law,Compare to a spherical wave:,The ABCD Law,B. Thin lens,(1) 由 Boyd-Gordon 理论可知高斯光束通过薄透
3、镜后仍为高斯光束。,(2),q is complex radius of curvature of the Gaussian beam,The ABCD Law,C. Two thin lenses sequence,First Lens,Second Lens,D. Above conclusion can be expanded to the multiple optical elements,Exercise: Gaussian Beam Focusing,Characteristic the beam focusing status. The smaller the z0 , the
4、stronger the focusing, but less depth of focusing,Exercise: Gaussian Beam Focusing,Focus at focus point,Spot size independent on input beam size, just a point spot.,Diffraction limit:,Geometric Optics,Gaussian Optics,Physical Optics,Depth of foucs,2.7 A Gaussian Beam in Lens Waveguide,2.7 A Gaussian
5、 Beam in Lens Waveguide,The condition for the confinement of the Gaussian beam by the lens sequence is that be real; otherwise, the sine function will yield growing exponentials. This condition becomes:,or,The same as Ray case,2.8 High-order Gaussian Beam Modes in a Homogeneous medium,2.8 High-order
6、 Gaussian Beam Modes in a Homogeneous medium,Longitudinal phase shift:,When l=m=0, it reduces to the results of fundamental Gaussian mode,Transverse variation:,Well-known harmonic oscillator wavefunction, Hl is the Hermite polynomial of order l,Transverse phase shift:,The same as fundamental Gaussia
7、n mode,2.8 High-order Gaussian Beam Modes in a Homogeneous medium,Note: the node points and the peak intensity variations at different l,2.8 High-order Gaussian Beam Modes in a Homogeneous medium,x direction has (l+1) bright spotsor l dark spots,y direction has (m+1) bright spotsor m dark spots,TEMl
8、m,2.8 High-order Gaussian Beam Modes in Quadratic index media,In previous sections, we treated the propagation of a circularly symmetric Gaussian beam in lenslike media. Here we extend the treatment to higher-order modes and limit our attention to steady state (q(z)=const.) solutions in medium whose
9、 index refraction can be described by:,I. The modes of Gaussian Beam,2.8 High-order Gaussian Beam Modes in Quadratic index media,Well-known harmonic oscillator Schwinger equation.,Eigen value:,Eigen function:,2.8 High-order Gaussian Beam Modes in Quadratic index media,Eigen value:,Eigen function:,Sp
10、ot size:,2.8 High-order Gaussian Beam Modes in Quadratic index media,Unlike the homogenous medium, the mode spot size is independentof z; focusing action counteracts the natural spread (n2 0), however, no confined solution exist for n20B. The propagation constant b depends on mode indices l, m cause
11、s the different modes to have phase velocities vl,m=w/bl,m and group (vg)l,m=dw/dbl,m velocities that depend on l and m.,Notes:,2.8 High-order Gaussian Beam Modes in Quadratic index media,II. Modal Dispersion of Group Velocity,2.8 High-order Gaussian Beam Modes in Quadratic index media,III. Pulse Sp
12、reading in Quadratic Index Glass Fibers,Glass fibers with quadratic index profiles are excellent channels for optical communication system. The information is coded into trains of optical pulse and the channel information capacity is thus fundamentally limited by the number of pulses that can be tra
13、nsmitted per unit time.,A. Modal Dispersion,The maximum pulses number per second,2.8 High-order Gaussian Beam Modes in Quadratic index media,Students Exercise:,B. Group Velocity Dispersion,2.8 High-order Gaussian Beam Modes in Quadratic index media,Single mode, but with some spectral width,2.8 High-
14、order Gaussian Beam Modes in Quadratic index media,Depends on the laser spectral linewidth or pulse duration width, in most case laser linewidth is smaller,2.9 Propagation in Media with Quadratic Gain Profile,Gain is a strong function of position:The radial distribution of energetic electrons in the
15、 plasma region of gas lasers;The variation of pumping intensity in solid state lasers;The dependence of the degree of gain saturation on the radial position of the beam.,2.9 Propagation in Media with Quadratic Gain Profile,For steady-state solution:,For non-steady-state solution:,Homework,章节后习题2.8, 2.9和2.12。,
