光电学部分答案刘旭版.ppt

Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,光电子学作业参考答案,作业,1,Fundamentals of Photonics,（括号内为中文教材上题目的位置）：,1.4-1,1.4-4,3.1-2(P41-2),3.1-3(P41-3),3.2-1(P41-4),作业,1,1.4-1 Ray-Transfer Matrix of a Lens System.,Determine the ray-transfer matrix for an optical system made of a thin convex lens of focal length f and a thin concave lens of focal length-f separated by a distance f.Discuss the imaging properties of this composite lens.,f,-f,f,作业,1,注意矩阵相乘的顺序,作业,1,1.4-4 4X4 Ray-Transfer Matrix for Skewed Rays.,Matrix methods may be generalized to describe skewed paraxial rays in circularly symmetric systems,and to astigmatic(non-circularly symmetric)systems.A ray crossing the plane z=0 is generally characterized by four variables-the coordinates(x,y)of its position in the plane,and the angles()that its projections in the x-z and y-z planes make with the z axis.The emerging ray is also characterized by four variables linearly related to the initial four variables.The optical system may then be characterized completely,within the paraxial approximation,by a 4X4 matrix.,(a)Determine the 4X4 ray-transfer matrix of a distance d in free space.,(b)Determine the 4X4 ray-transfer matrix of a thin cylindrical lens with focal length f oriented in the y direction(Fig.P1.4-4).The cylindrical lens has focal length f for rays in the y-z plane,and no focusing power for rays in the x-z plane.,可列式求解：,作业,1,(a),过长度为,d,的自由空间，光线角度不变，位置改变,(,近轴光线,),作业,1,(b),薄透镜厚度,d 0,y-z,面上为焦距为,f,的薄透镜,作业,1,3.1-2 Beam Identification by Two Widths.,A Gaussian beam of wavelength=10.6 pm(emitted by a CO,2,laser)has widths W,1,=1.699mm and W,2,=3.38mm at two points separated by a distance d=10 cm.Determine the location of the waist and the waist radius.,借助公式：,可求解，计算较麻烦,作业,1,由方程组可得,（舍去）,由,(2),得,:,解得：,所以 或者,作业,1,3.1-3 The Elliptic Gaussian Beam.,The paraxial Helmholtz equation admits a Gaussian beam with intensity in the z=0 plane,with beam waist radii W,0 x,and W,0y,in the x and y-directions respectively.The contours of constant intensity are therefore ellipses instead of circles.Write expressions for the beam depth of focus,angular divergence,and radii of curvature in the x and y directions,as functions of W,0 x,W,0y,and the wavelength.If W,0 x,=2W,0y,sketch the shape of the beam spot in the z=0 plane and in the far field(z much greater than the depths of focus in both transverse directions).,X,和,Y,两方向独立求解,作业,1,(a),(1),扩散角：,(2),焦深：,(3),曲率半径：,作业,1,(b),当,W,0 x,=2W,0y,时,:,(1)z=0,(2)z=Mz,0,作业,1,3.2-1 Beam Focusing.An argon-ion laser produces a Gaussian beam of wavelength=488 nm and waist radius W,0,=0.5 mm.Design a single-lens optical system for focusing the light to a spot of diameter 100 um.What is the shortest focal-length lens that may be used?,作业,1,由,(3.2-5),及,(3.2-9),得：,求导得：,作业,1,故,z=0.1z,0,即,f=161.75mm,时候，,f,取得最小值,作业,2,Fundamentals of Photonics,（括号内为中文教材上题目的位置）：,1.3.1-1(P41-1),2.3.2-2(P41-5),3.3.2-3(P41-6),4.9.1-1(P42-10),5.9.1-2(P43-11),作业,2,3.1-1 Beam Parameters.,The light from a Nd:YAG laser at wavelength 1.06 um is a Gaussian beam of 1-W optical power and beam divergence 2=1mrad.Determine the beam waist radius,the depth of focus,the maximum intensity,and the intensity on the beam axis at a distance z=100cm from the beam waist.,本题求解激光（高斯）光束的束腰半径，焦深，最大光强和在光轴上,z=100cm,处的光强,作业,2,由发散角公式,得,由,得：,作业,2,由光功率公式：,得：,由光轴上的光强分布公式,得：,作业,2,3.2-2 Spot Size.,A Gaussian beam of Rayleigh range z,0,=50cm and wavelength=488nm is converted into a Gaussian beam of waist radius using a lens of focal length f=5cm at a distance z from its waist,as illustrated in Fig.3.2-2.Write a computer program to plot Wd as a function of z.Verify that in the limit z-fz,0,(3.2-10)and(3.2-12)hold;and in the limit zz,0,时,该近似在,z400,后与原图像差别不大,作业,2,(2),当,zz,0,时,该近似在,zphotonselectrons,作业,3,11.1-5 Levitation by Light Pressure.,Consider an isolated hydrogen atom of mass 1.66x10,-27,kg.,(a)Find the gravitational force on this hydrogen atom near the surface of the earth(assume that at sea level the gravitational acceleration constant g=9.8m/s,2,).,(b)Let an upwardly directed laser beam emitting 1-eV photons be focused in such a way that the full momentum of each of its photons is transferred to the atom.Find the average upward force on the atom provided by one photon striking it each second.,(c)Find the number of photons that must strike the atom per second and the corresponding optical power for it not to fall under the effect of gravity,given idealized conditions in vacuum.,(d)How many photons per second would be required to keep the atom from falling if it were perfectly reflecting?,光子动量表达式和动量守恒,作业,3,(a),氢原子重力：,G=mg=1.66*10,-27,kg*9.8m/s,2,=1.63*10,-26,N,(b),光子动量变化,每秒有一个光子撞击氢原子时产生的平均作用力,(c),理想情况下,(d),当发生完全弹性碰撞时：,作业,4,Fundamentals of Photonics,（括号内为中文教材上题目的位置）：,12.2-1(P152-22),12.3-1(P153-24),12.3-2(P153-25),作业,4,12.2-1Comparison of Stimulated and Spontaneous Emission.,An atom with two energy levels corresponding to the transition(=0.7um,t,sp,=3ms,=50GHz,Lorentzian lineshape)is placed in a resonator of volume V=100cm,3,and refractive index n=1.Two radiation modes(one at the center frequency and the other at)are excited with 1000 photons each.Determine the probability density for stimulated emission(or absorption).If N,2,such atoms are excited to energy level 2,determine the time constant for the decay of N,2,due to stimulated and spontaneous emission.How many photons(rather than 1000)should be present so that the decay rate due to stimulated emission equals that due to spontaneous emission?,作业,4,(a),离散的模式，受激辐射（吸收）的概率密度,(1),根据题意该光有两个辐射模式,作业,4,(2),发射截面值,(3),结果,P,st,=2.98*10,-7,s,-1,作业,4,(b),类似于,故总时间常数为,(c),由第二步，得：,所需要的光子数为：,作业,4,12.3-1 Rate Equations for Broadband Radiation.,A resonator of unit volume contains atoms having two energy levels,labeled 1 and 2,corresponding to a transition of resonance frequency v,0,and linewidth .There are N,1,and N,2,atoms in the lower and upper levels,1 and 2,respectively,and a total of photons in each of the modes within a broad band surrounding v,0,.Photons are lost from the resonator at a rate as a result of imperfect reflection at the cavity walls.Assuming that there are no nonradiative transitions between levels 2 and 1,write rate equations for N,2,and.,作业,4,（,1,）对于上能级粒子数,N,2,N,2,上粒子自发辐射损耗：,N,2,上粒子受激辐射损耗：,N,1,上粒子受激吸收使得,N,2,增加,:,为光子数损耗因子，与,N,2,无关,若为洛伦兹线形有,作业,4,忽略其他无辐射跃迁，则,N,2,减少量等于光子增加量,光子数损耗因子，三维态密度,故,若为洛伦兹线形：,作业,4,12.3-2 Inhibited Spontaneous Emission.,Consider a hypothetical two-dimensional black-body radiator(e.g.,a square plate of area A)in thermal equilibrium at temperature T.,(a)Determine the density of modes M(v)and the spectral energy density(i.e.,the energy in the frequency range between v and v+dv per unit area)of the emitted radiation (see Sec.9.1C).,(b)Find the probability density of spontaneous emission P,sp,for an atom located in a cavity that permits radiation only in two dimensions.,作业,4,(a),设二维正方形腔边长为,d,，则,A=d,2,在,Z,Y,方向上，模式间隔为,所以：,故在倒易空间中，一个模式面积为,假设在,v,时有,N,个态，则,作业,4,一种模式两种偏振：,作业,4,（,b,）由：,作业,5,Fundamentals of Photonics,（括号内为中文教材上题目的位置）：,13.1-2(P212-15),13.1-3(P212-16),13.2-1(P212-17),作业,5,13.1-2 Laser Amplifier Gain and Population Difference.,A H-cm-long rod of Nd3+:glass used as a laser amplifier has a total small-signal gain of 10 at A,=1.06pm.Use the data in Table 13.2-1 on page 480 to determine the population difference N required to achieve this gain(Nd3+ions per cm3).,由表得,:,作业,5,固体加宽线形比较复杂，不是洛伦兹线形,作业,5,13.1-3 Amplification of a Broadband Signal.,The transition between two energy levels exhibits a Lorentzian lineshape of central frequency with a linewidth .The population is inverted so that the maximum gain coefficient =0.1 cm,-1,.The medium has an additional loss coefficient =0.05 cm,-1,which is independent of v.Approximately how much loss or gain is encountered by a light wave in 1 cm if it has a uniform power spectra1 density centered about with a bandwidth?,作业,5,总增益为,泰勒展开，取一阶,所以,作业,5,13.2-1 The Two-Level Pumping System.,Write the rate equations for a two-level system,showing that a steady-state population inversion cannot be achieved by using direct optical pumping between levels 1 and 2.,图,13.2-3,：外加泵浦下能级粒子的跃迁示意图,作业,5,对于,2,能级系统,如上图 ，,R,1,=R,2,=R,令,R=(N,1,-N,2,)W,，光泵浦对受激吸收和发射的作用相同,作业,5,平衡时,所以,不可能有稳定的粒子数反转,作业,6,Fundamentals of Photonics,（括号内为中文教材上题目的位置）：,14.2-1(P209-1),14.2-2(P209-2),13.3-1(P209-19),13.3-2(P209-20),作业,6,14.2-1 Number of Longitudinal Modes.An Ar,+,-ion laser has a resonator of length 100 cm.The refractive index n=1.,(a)Determine the frequency spacing v,F,between the resonator modes.,(b)Determine the number of longitudinal modes that the laser can sustain if the FWHM Doppler-broadened linewidth is,=3.5 GHz and the loss coefficient is half the peak small-signal gain coefficient.,(c)What would the resonator length d have to be to achiev operation on a single longitudinal mode?What would that length be for a CO,laser that has a much smaller Doppler linewidth,v,D,=60MHz under the same conditions?,作业,6,(a)v,F,=c/2dn=3*10,8,/(2*1*1)=1.5*10,8,Hz,(b),题中,B=,Mb=B/v,F,=23,(c)As B v,F,=c/(2d),对于,Ar,+,激光器,B=3.5GHz 0dc/2B=0.043m=4.3cm;,对于,CO,2,激光器,B=60MHz 0dc/2B=2.5m;,作业,6,14.2-2Frequency Drift of the Laser Modes.A He-Ne laser has the following characteristics:,(1)A resonator with 97%and 100%mirror reflectances and negligible internal losses;(2)a Doppler-broadened atomic transition with Doppler linewidth=1.5 GHz;and(3)a small-signal peak gain coefficient=2.5*10,-3,cm,-1,.While the laser is running,the frequencies of its longitudinal modes drift with time as a result of small thermally induced changes in the length of the resonator.Find the allowable range of resonator lengths such that the laser will always oscillate in one or two(but not more)longitudinal modes.The refractive index n=1.,作业,6,忽略其他内部损耗，仅有境内反射损耗,多普勒加宽为高斯线形,令 ，得,作业,6,只允许,1,到,2,个纵模振荡时,带入已知条件则,令,t=2d,；,则,解超越方程得,:,作业,6,13.3-1 Significance of the Saturation Photon-Flux Density.In the general two-level atomic system of Fig.13.2-3,represents the lifetime of level 2 in the absence of stimulated emission.In the presence of stimulated emission,the rate of decay from level 2 increases and the effective lifetime decreases.Find the photon-flux density at which the lifetime decreases to half its value.How is that flux density related to the saturation photon-flux density,?,Fig.13.2-3,作业,6,自发辐射时：,受激辐射时：,当第二能级寿命减小到无受激辐射时的一半时：,所以,W=,由 ，得：,作业,6,(2),由,得：,作业,6,13.3-2 Saturation Optical Intensity.Determine the saturation photon-flux density and the corresponding saturation optical intensity,for the homogeneously broadened ruby and Nd,3+,:YAG laser transitions provided in Table 13.2-1.,Table 13.2-1.,红宝石为三能级系统，,Nd,3+,:YAG,为四能级系统,作业,6,（,1,）,ruby,：,3,能级系统,=1/(6.0ms*2*10,-20,cm,2,)=8.3*10,21,s,-1,cm,-2,(2)Nd,3+,:YAG,：,4,能级系统,=1/(1.2ms*4*10,-19,cm,2,)=2.083*10,21,s,-1,cm,-2,I,s,=390.3Js,-1,cm,-2,2-3(P41-6),4-4 4X4 Ray-Transfer Matrix for Skewed Rays.,699mm and W2=3.,2-2 Spot Size.,There are N1 and N2 atoms in the lower and upper levels,1 and 2,respectively,and a total of photons in each of the modes within a broad band surrounding v0.,w0=abs(f/(z-f)/sqrt(1+(z0/(z-f)2)*sqrt(r*z0/pi);,=1/(6.,(d)当发生完全弹性碰撞时：,(c)Find the number of photons that must strike the atom per second and the corresponding optical power for it not to fall under the effect of gravity,given idealized conditions in vacuum.,题中 ，故,Write a computer program to plot Wd as a function of z.,作业,7,Fundamentals of Photonics,（括号内为中文教材上题目的位置）：,EXERCISE 15.2-1,15.1-1,15.1-2,16.3-2,Enter,作业,8,Fundamentals of Photonics,（括号内为中文教材上题目的位置）：,16.1-2,16.2-1,16.3-2,16.3-3,Enter,作业,9,Fundamentals of Photonics,（括号内为中文教材上题目的位置）：,18.1-1,18.1-2,18.2-1,Enter,Thank You!,PS,：能力有限，无法解答的部分请在下节课答疑时问李导,