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1、2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,一.Optical Fibers,1.光纤的结构,SMF:2a=4-10m,2b=125理论分析中，可以认为包层是无限大的MMF:(阶跃多模光纤)2a=50m,2b=125,2.阶跃光纤和渐变光纤 Step-Index Fiber&Graded-Index Fiber,3.光纤制作 reading:P15-P19,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,二.几何光学 Geometrical Optics,4.When all
2、 apertures are much larger than a wavelength,we can model the light as ray traveling in straight lines perpendicular to the wavefront,1.Speed of light in a medium of refractive index n:v=c/n2.Law of reflection:i=r3.Snells law:ntsin t=nisin i,oy:normal line.o-xz:boundary,2023/9/22,Optical fiber commu
3、nications,第二章 光导纤维的传输原理,三.Uniform plane wave 均匀平面波,A:Maxwell Equations(fundamental equations),B:在稳态简谐条件下，线性各项同性非色散，不导电媒质中Maxwell Equations,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,C.Helmholtz方程,1.时间参量:角频率，周期T，频率f。,2.空间参量:纵向相移常数或角波数kz，波长z，波数1/z.,where,2023/9/22,Optical fiber communication
4、s,第二章 光导纤维的传输原理,3.空间与时间的联系:相速度 phase velosity,均匀平面波的纵向相移常数等于空间相移常数,其相速度等于光速,4.电磁联系:波阻抗,对于均匀平面波,真空中,均匀平面波是电磁波的最基本的形式，它是沿固定方向传播的等幅行波，垂直于传播方向的平面是等相面或称波阵面。其等相位面同时是等振幅面，其电场和磁场都垂直于传播方向且互相垂直，故称为横电磁波(TEM:Transverse Electromagnetic wave)波。同时是等幅波，即振幅不随传播距离而增加或衰减，等幅均匀平面波中电场与磁场同相，波阻抗为实数。,2023/9/22,Optical fiber
5、 communications,若在空间垂直于电场的面放置两块平行的导体平板，则因在导体表面切向电场为0，两导体板不影响平面波的传播。这就形成双平板传输线中的TEM模式的行波，它是双平板传输线中的主模，或称最低模。,TE:transverse electric wave.TM:transverse magnetic wave,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,Plane Waves at a Dielectric Interface,1.Consider a monochromatic plane wave inciden
6、t on a dielectric interface described by the surface normal S,2.The plane wave takes the form:,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,4.Maxwells Equations tell us that the tangential component of the electric field must be continuous at the interface,so we have:,Plane Waves at a Dielec
7、tric Interface,3.The corresponding reflected and transmitted fields are:,where r and t are phase constants.Non-zero values of the phase constants can be interpreted as a spatial shift.,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,Laws of Refraction,Similarly we find from the second of the tw
8、o equations:,It follows that kt is also in the incident plane,and,This is Snells law of refraction.,1.The laws of reflection and refraction forms the basis for Geometrical optics,2.Geometrical optics also assumes collimated optical beams(rays),which are unphysical,3.Geometrical optics is nevertheles
9、s very useful for modeling a large number of optical devices and phenomena,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,Lenses Ray Picture,The rays are deflected at the air-lens interface due to the higher index of the lens.,The effect of the ray deflection is that all the rays pass through
10、the focus,Collimated(parallel)optical beam,If the lens is thin,we consider both deflections to take place at the center plane of the lens.,1.The operation of lenses can be understood by tracing individual rays through the lens,2.The rays all bend in different ways at the air-lens interface,3.The sha
11、pe of the lens surfaces is chosen such that each ray passes through the focus,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,Lenses Wave Picture,Collimated(parallel)incident optical beam,The lens is thinner on the periphery,so the peripheral parts of the beam are delay less than the center par
12、t,resulting in curved phase fronts,The curvature of the wavefronts focus the beam to a point,1.we can also consider the wavefronts of the light passing through the lens to understand its operation,2.The lens is delaying the center part of the beam with respect to the sides such that all parts of the
13、 beam arrive at the focus in phase,3.In other words,all parts of the beam are interfering constructively at the focus,which leads to high intensity at this point,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,Diffraction limits resolution,In the ray-optics(geometrical-optics)approximation,the
14、focus is a point,In reality,diffraction limits the focusing ability of the lens,1.Ray optics is a useful approximation for many considerations,but the fact that it does not consider diffraction gives it some major flaws,2.The main one is that it predicts that a lens focuses light to a mathematical p
15、oint,3.This is not possible(even for a perfect lens)with wave phenomena of finite wavelength,4.The central spot of the focus of a perfect,uniformly illuminated lens is d=2.44f/D,where is the wavelength,f is the focal length,and D the lens diameter,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理
16、,Ray Tracing in Ideal Lenses,To trace rays through ideal lenses we need only two simple facts:,1.Each ray goes through the focal plane at the same point as its parallel central ray,2.Central rays are not deflected,This is very useful for first order analysis of lens systems,2023/9/22,Optical fiber c
17、ommunications,第二章 光导纤维的传输原理,Imaging,Thin lens equation:1/a+1/b=1/f,Magnification:b/a,a diverging beams,no imaging,a=f=collimation,f Imaging according to the lens equation with M1,a=2f=Imaging according to the lens equation with M=1,a2f=Imaging according to the lens equation with M1,2023/9/22,Optical
18、 fiber communications,第二章 光导纤维的传输原理,Fresnel Law,A:,是入射光，反射光和透射光的矢量复数振幅,反射系数R传递系数T,B:,模表示强度，幅角表示相位,C:水平偏振波 TE横电波,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,D:垂直偏振波 TM 横磁波,Fresnel Law,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,Tatal Internal Reflection,A:n1n2&c i900.c=arcsin(n2/n1)
19、,B:waves in medium one,|R|=1,2R=,在临界状态下，ci，2R0，反射不引起相位突变，i900，cosi 0，2R1800，有半波损失。,TE wave:,TM wave:,|R|=1,2R=,Attension:全反射时，TE，TM的位相变化是不相同的,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,Wave in medium 1:TE wave,A:Incident wave takes the form as,入射波在x方向的相位常数,入射波的纵向相移常数,B:The corresponding re
20、flected wave will take the form:,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,C:讨论:波的状态,Z方向：,行波，相位常数,X方向：,驻波,x,z,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,Wave in medium 2:TE wave,由上式可以看出传输系数的模值及其幅角。,令,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,Z方向:z方向是行波状态，且相位常数与介质1中的相同
21、，相位常数相同意味着相速相同，这是因为介质1中的波和介质2中的波是一个统一波形的两部分，它们必须有相同的传播速度。,X方向:说明波的幅度随离开界面的距离按指数形式减小，减小的快慢以参数决定，叫做x方向的衰减常数，它是由介质参数n1，n2及入射角i决定,媒质2中存在透射场，透射场沿平行与界面的方向z为行波，而沿垂直与界面的x方向为衰减场。场在界面处最强，沿离开界面的法线方向逐渐减弱，在无穷远处消失。这种场称为“渐消场”，这种振幅沿横向渐消，沿纵向传播的行波，它的场集中早表面附近，故称为表面波。,场沿法向的逐渐减弱并非由于媒质的吸收，只是场的指数渐弱分布：衰逝波,渐消场中，Hx与Ey相差900，即
22、无界面法线方向即x方向的有功功率流。,有效穿透深度,渐消场的出现说明，不能简单的认为全反射介质2中完全不存在波场，实际上在界面附近波长量级的厚度内自然有场，入射波的能量不是在严格的界面上全部反射，而是穿透到介质2内一定的深度。,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,复习,TE,TM,B:n1n2&c i900.c=arcsin(n2/n1),C:TE wave:,D:Wave in medium 1:TE wave,E:Wave in medium 2:TE wave,A:,2023/9/22,Optical fiber co
23、mmunications,第二章 光导纤维的传输原理,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,Total Internal Reflection,The transmission angle must be/2,No transmission,but instead Total Internal Reflection(TIR)if:nisini=ntsintnt=sinTIRnt/ni,=No transmission,Note the Goos-Hanchen shift,The G-H shift is important
24、in many optical components,including waveguides and fibers,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,Ray picture of waveguides,We find the numerical aperture of the fiber or waveguide by requiring that the internal angle,i,must exceed t
25、he critical angle for Total Internal Reflection,实际光纤中传输模型比较复杂，按上式确定的NA必有差异，数值孔径往往由测试值确定,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,Modal Dispersion Ray Picture,The different path lengths through an optical fiber give rise to pulse spreading,This effect is referred to as modal dispersion(in
26、termodal dispersion in the book),Modal dispersion,as well as material and waveguide dispersion(both referred to as chromatic dispersion in the book),limits the useful bandwidth of optical fibers,It is often repeated mistake to assume that the modal dispersion can be found by calculating the differen
27、ce in path length using a simple geometrical optics approach,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,Modal Dispersion with GH shift,A.The wave picture of optical fiber propagation shows that significant parts of the optical power propagated in the cladding,B.In the geometrical-optics pi
28、cture,this effect is modeled as the Goos-Hanchen shift,C.Close to cut-off most of the optical power propagates in the cladding so that the efective index is lower than for the fundamental mode,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,Diffraction Invalidates Ray Picture,A.As the optical b
29、eam propagates,it spreads(diffraction),B.At each reflection from the core-cladding interface there will be parts of the beam that is beyond the TIR angle,C.The beam will therefore spread and lose energy until only the part of the incoming beam that can be expanded on he guided mode(s)is left,2023/9/
30、22,Optical fiber communications,第二章 光导纤维的传输原理,Wave picture of waveguides,A.The step-index profile provide focusing just like lenses and GRIN materials,B.The guides modes of the fiber are those that propagate without changing their profile,C.The guided modes are those intensity profiles,for which the
31、 focusing,due to the index profile,exactly matches the diffraction,D.In the core is small,only one such mode exists(single mode fiber),2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,导行波 Guided Traveling Wave,A.全反射时介质1和介质2中的波时一个统一体，是一个波形的两个部分，沿纵向z方向，它们都是以相同的相位常数k1z在传播。沿x方向，介质1中的电磁波的幅度按三角函数规律变化，
32、是驻波分布。介质2 中的电磁波幅度按指数规律衰减，是衰减波，渐逝场。如果衰减常数足够大，介质2中的波将只存在于介质1表面。由于电磁能量集中于介质1及其表面附近，因而把这种波叫做表面波。又因为它是与界面平行的方向传播，是由介质表面导行的，因而也叫导行波，简称导波。,B:相位常数及传播速度 z向:k1z=k0n1sini,全反射:c i900 n2/n1sin i1.K0n2 K0n1 sin I K0n1 或者k2 k1,导波在传播方向的相位常数是介于两介质的波数之间的，比第二介质的波数k2大，比第一介质的波数k1小。导行波的传播速度为v=/,因而导行波的速度/k1v/k2,导行波的传播速度介于
33、两介质中平面波的传播速度之间。由于其速度小于第二介质中的平面波的相速度，所以也称慢波。,等相位面:导波相位只沿z方向变化，令z为常数，就得到等相位面，所以等相位面是与传播方向垂直的一系列平面。,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,C:幅度变化规律,在无损介质中，导波幅度沿z方向是不变的，但沿下方向却按一定的规律变化。将电场幅度用Em表示，则在介质一中：Em正比于cos(k1xx+1).显驻波分布。,1.d与i有关,1=c,=0,d=.i=900,2.=n1-n2有关，,在界面x=0处,Em正比于cos(1),k1x是x方向的
34、相位常数,x=2/k1x 是x方向的波长.入射角i越小,k1x越大,x越小,驻波节点越密集。,在介质2中，幅度按exp(-x)规律衰减，其幅度变化规律在x=0处，幅度曲线是连续的，衰减速度取决于衰减常数,越大，衰减越快.,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,穿透深度d,1.与入射角I有关.临界状态时，1=c,=0,d=.不衰减。i=900,2.与两介质的折射率有关，差别越大，衰减越快。,3.与工作波长0有关，0越小，越大，d越小。对电磁波集中越有利,等振幅面:x为常数,导波的等振幅面与等相位面时相互垂直的，这与均匀平面波不同
35、。,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,D.辐射波的概念:radiated wave,平面波入射到两介质分界面上，当不满足全反射条件时，就产生了部分反射，一部分能量反射回介质1中，只一部分能力返回到介质2中，这种波叫辐射波。对于介质波导来说，这种波远没有导波重要。,对于TE波,K2x为实数，故沿x方向产生相移而不产生衰减，与导波不同=k2z=k1z=k0n1sinI 对于辐射波sinIv2v1 快波。,TM波与此类似,Summary 当平面波由光密到光疏介质时,I c 导行波 能量集中于光密介质及其界面附近,I c 辐射波
36、有一部分能量辐射的光疏介质中,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,薄膜波导 Step Index Slab Waveguides,下界面:薄膜与衬底的交界面，全反射临界角,上界面:薄膜与敷层的交界面，全反射临界角,The typical slab waveguide consists of a substrate,a thin dielectric film,and a cladding layer.The substrate and cladding are assumed to both be infinitely th
37、ick,while the film(core)has a finite thickness,h.,A:,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,B:导波,条件:平面波在上，下界面都要产生全反射，,或:,如果n1和n2差别不大，I将只限于接近900的很小角度范围内，即平面波的传播方向与z轴几乎是平行的。,C:辐射模,当形成导波的条件被破坏时，将有能量自模中辐射出去。,下边界的全反射条件被破坏,上下边界的全反射条件被破坏,We are searching for solutions to the wave equation wit
38、hamplitudes that are independent of z,i.e.solutions of the,where is the longitudinal wave vector,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,薄膜波导中的导波,A:导波的波形TE波和TM波,TE波:横电波 电场强度矢量E在波导的横截面上，在传播方向上只有磁场强度分量M,TM波:横磁波,TE波,TM波,B:导波的特征方程,1.1c.在此范围内的任一都可以形成导波？,2.特征方程,2023/9/22,Optical fiber communi
39、cations,第二章 光导纤维的传输原理,射线ABCD与ABCD中，BB，CC分别在不同的等相面上，由B到C和由B到C各经历了不同的相位变化，但两射线相位变化之差应为2的整数倍。,射线ABCD:由B到C所经历的相位变化为,为在BC路程上所引起的相位变化，22，23分别为在上，下界面处全反射引起的相位突变,射线ABCD:由B到C所经历的相位变化为,相位差为,由图:,所以,各参数的意义。当波导和波长给定时，上式时关于I的方程，它确定了形成波导的入射角i的条件，因而叫薄膜波导的特征方程，有时也叫薄膜波导的色散方程，特征方程有很重要的意义，它是讨论导波特性的基础,2023/9/22,Optical
40、fiber communications,第二章 光导纤维的传输原理,对于TE波,对于TE波,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,C:特征方程的另一种形式,薄膜中波矢量在x方向的分量，为横向相位常数。,代入特征方程:,或者:,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,导波的模式,A:波指数 m 当m0,1,2,3可得到TE0,TM0;TE1,TM1;TE2,TM2;TE3,TM3模。M表明各模式的特点，叫做波指数。,B:模参数,纵向相位常数,横向相位常数,衬底衰
41、减常数,敷层衰减常数,导波模式的纵向传播规律，表明了它们的传播特性,导波模式在薄膜内的横向驻波规律,导波在衬底中的横向衰减规律,导波在敷层中的横向衰减规律,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,导波模式的横向分布规律:按驻波规律分布,跨过薄膜的宽度为d，其相位变化为k1xd=k0n1cosi,由特征方程得k1xd=m+2+3.,B:,当m0,TE0模,TM0模 00k1xd=2+3,因此其场沿x方向得变化不足半个驻波,当m1,TE1模,TM1模 k1xd=2+32,因此其场沿x方向得变化不足两个“半个驻波”,m表示导波场沿薄膜
42、横向出现得完整得半驻波个数，m越大，导波得模次越高,A:,TE0,TE1,TE2,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,Zero Order Mode,Plane wave picture of the fundamental mode of a slab waveguide.The fundamental mode has the smallest transversal wavevector,and only one maximum in the field pattern created by the interfere
43、nce of the two planes waves,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,First Order Mode,Plane wave picture of the first order mode on an optical waveguide.This mode has a larger transversal wavevector than the fundamental mode,and two maxima,as well as one null,in the field profile.,2023/9
44、/22,Optical fiber communications,第二章 光导纤维的传输原理,m=0时，薄膜中横向完整得“半驻波”个数为0，m1时，薄膜中得完整得“半驻波”数为1。故m表示了导波场沿薄膜横向出现得完整得半驻波数(波节数),m越大，导波得模数越高。,特征方程,在其它条件不变得情况下，m越大，cosi越大，i越小,高次模是由入射角i较小得平面波构成得。当i越小时，平面波得射线倾斜得比较严重，其横向相位常数k1x大，驻波密集。,在衬底和敷层中，以衰减系数2，3按指数规律衰减，在穿透深度d21/2和d31/3处，电磁场衰减为表面处得1/e.,Q:高阶模的衰减情况,由于高次模的i小,值也
45、小，因而衰减慢，能量集中差，TE0，TM0模的模次最低，值最大，能量集中最好。,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,导波模式的轴向相位常数,轴向相位常数,对于给定的模式，有确定的i值，因而也有确定的相位常数。由特征方程求出i值，就可确定各模式的值。但特征方程时超越方程，得不到解析解，因而只能确定性讨论得一些变化规律。,对给定得波导和工作波长，模次越高，i越小，其值越小，在所有得模式中，TE0，TM0模额模次最低，因而其值最大。,对给定得模式，其值随工作波长(或频率)而变，由特征方程可以看出，当m给定时，工作波长越长，k0越小
46、，i值越小，故值也越小。,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,下图给出了几个较低模式得-曲线，它表明了的变化范围及变化规律，不能大于k0n2，否则将给出辐射模，也不能大于k0n1。故对于导波，其时限制在k0n1-和k0n2-两条直线所夹的区域中。,m=0,1,2时导波的-曲线,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,特征方程,m=0,1,2时导波的-曲线,m增大，i减小，值减小。,m=0,一定，m增大,i减小,值减小,所有模式中,TE0,TM0模次最低,最大,
47、m一定，增大，增大0,截止波长,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,导波的截止波长,Q:在金属波导中，什么情况下导波截止。,Q:在介质波导中，什么情况下平面波截止。,A:导波截止的临界状态 c12=i,sin i=n2/n1,B:截止波长,Q:如何求截止频率或波长。,A:特征方程,A:=0,A:全反射条件破坏,导波截止的临界状态 c12=i,sin i=n2/n1,下边界处于全反射的临界状态,对TE和TM波有20，上边界处于全反射状态3,TE:,TM:,2023/9/22,Optical fiber communicatio
48、ns,第二章 光导纤维的传输原理,TE:,TM:,讨论：,1，各模式的截止波长完全取决也波导参数n1，n2，n3，和d。,2，不同模式有不同的截止波长,3，波指数m相同的TEm模和TMm模的截止波长不同,4，TE0模的截止波长最长,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,单模传输 Single Mode Operation,对每个模式来说，只有当工作波长小于其截止波长时才能在波导中传输。TE0模的截止波长最长，它的截止条件最容易满足。,A:基模：TE0,B:单模传输:如果设计波导结构或选择工作波长，使只允许TE0模传输，而其它高
49、次模都截止，则可实现单模传输。,C:单模传输的条件：c(TM0)0c(TE0),D:当n1和n3差别不大时，TM0和TE0模的截止波长非常接近，实际上很难分开。在介质光波导中，单模传输的概念常常不是那么严格，因而，就是在TM0和TE0模同时存在时可使用单模传输的术语,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,对称薄膜波导,A:n2=n3,3=2,B:特征方程：,C:截止波长 20,D:TEm和TMm是兼并模，具有相同波指数的TE和TM模具有相同的截止波长,E:TE0和TM0的截止波长。基模的截止波长为无穷大，此时无截止现象。,题目:,平板波导n1=1.5,n2=1.45,n3=1.4,d=5m,1，求出其最低的3个模式的截止波长。2，作出其-曲线。3，对0=1 m的光波，画出其最低的3个模式的场分布。4，用作图法求出区特征方程,2023/9/22,Optical fiber communications,第二章 光导纤维的传输原理,Ray Picture of Optical Fibers(N.A),A:Numerical Aperture:反映了光纤搜集射线的能力,B:,C:实际光纤中传输模型比较复杂，按上式确定的NA必有差异，数值孔径往往由测试值确定,
