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1、Chapter 2 Dielectric Waveguides and Optical Fiber,2.1 Symmetric planar dielectric Slab waveguide2.2 dispersion in the planar waveguide 2.3 Step index fiber2.4 Numerical aperture2.5 Dispersion in single mode fibers2.6 Dispersion,electrical,and optical bandwidth2.7 The graded index optical fiber2.8 Li
2、ght absorption and scattering2.9 Attenuation in optical fibers2.10 Fiber manufacture,“The introduction of optical fiber systems will revolutionize the communications network.The low-transmission loss and the large bandwidth capability of the fiber systems allow signals to be transmitted for establis
3、hing communications contacts over large distance with few or no provisions of intermediate amplification.”Charles K.Kao,光纤系统的引入将使通信网络产生革命。光纤系统的低传输损耗和巨大的带宽能力允许信号在很少经过或者不经过中间放大的情况下实现超远距离的传输，建立通信联系。查尔斯 高锟,Terminology,Waveguide:波导Further Phase Change of：附加相位；Optical Path Length（OPL）：光程；Constructive Inte
4、rference：相长干涉，相干加强；Destructive Interference：相消干涉，相干减弱；Phase Difference(PD)：相位差；Optical Path Difference(OPD):光程差；Polarization State of Light Wave：光波偏振态State of Polarization（SOP）:偏振态Field Pattern：场图Resultant Wave：合成波；Standing Wave：驻波；,Terminology,Wavevector：波矢Single Mode Waveguide(SM waveguide):单模波导Mu
5、ltimode Waveguide(MM waveguide):多模波导Normalized Thickness:归一化膜厚Normalized Frequency:归一化化频率V-number:V 数Cut-off Wavelength:截至波长Transverse Electric field mode:横电场模式Transverse Magnetic field mode:横磁场模式Mode Field Width(MFW):模场宽度Mode Field Diameter(MFW):模场宽度Constant phase wavefront：等相位面,To understand the g
6、eneral nature of light wave propagation in optical waveguides,we first consider the planer dielectric slab waveguide shown in Figure 2.1,which is the simplest waveguide in terms of tractable analysis.A slab of dielectric of thickness 2a and refractive index n1 is sandwiched between two semi-infinite
7、 regions both of refractive index n2(n2 n1).The region of higher refractive index(n1)is called core and the region of lower refractive index n2 sandwiching the core is called the cladding.,理解传输在光波导中光波的基本性质，我们首先需要考虑平面介质波导，如图2.1所示，用于分析的最简的波导。厚度为2a和折射率为n1的介质平板被反射率为n2的两个半无限大区域的包起来。折射率较高的区域成为芯，包裹芯的折射率较低的
8、区域称为包层。,Figure 2.1,A light ray can readily propagate along such a waveguide,in a zigzag fashion,provided it can undergo total internal reflection(TIR)at the dielectric boundaries.It seems that any light wave that has an angle of incidence greater than the critical angle for TIR,will be propagated.Th
9、is,however,is true only for a very thin light beam with a diameter much less than the slab thickness,2a.We consider the realistic case when the whole end of the waveguide is illuminated,as depicted in Figure 2.1.To simplify the analysis,we will assume that light is launched from a line source in a m
10、edium of refractive n1.In general,the refractive index of the launching medium will be different than n1,but this will affect only the amount of light coupled into the guide.,光线很容易沿着这样的波导传输，以z字形，如果在介质的边界上发生全反射的话。看样子似乎是，任何入射角只要大于临界角发生全反射的光波，都会被传输。然而，只有当光束的直径远小于波导厚度2a的光才能真正存在。我们考虑实际的情况，光由波导的一端注入，如图2.1
11、所示。为了简化问题的分析，我们假设在折射率为n1中光从线光源出射。通常，发光媒质的折射率将与n1不同，但是它只影响耦合入波导中光的多少。,As shown in Figure 2.2,We will take the electric field E to be along x,parallel to the interface and perpendicular to z.The ray is guided in a zigzag fashion along the guide axis z by reflections from the core-cladding(n1/n2)bounda
12、ries.The result is the effective propagation of the electric field E along z.The figure also shows the constant phase wavefronts,normal to direction of propagation,on this ray.This particular ray is reflected at B and then at C.Just after the reflection at C,the wavefront at C overlaps the wavefront
13、 at A on the original ray.The wave interferes with itself.Unless these wavefronts at A and C in phase,the two will interfere destructively and destroy each other.Only certain reflection angles give rise to the construc-tive interference and hence only certain waves can exist in the guide.,如图2.2所示，我们
14、让电场E沿着x轴，平行于交界面，垂直于z轴。光线沿着波导z轴，通过在芯与包层边界上的反射，形成z字形的传导。结果是电场E沿着z轴有效的传输。图中表示等相位面，为垂直于光线的传输方向。光线先在B点反射，然后在C点。在C点反射之后，C点的波阵面与原光线上A点的波阵面交叠。光波发生自干涉。除非A点播阵面与C点是同相的，否则两波将相干减弱，互相破坏。只有确定的反射角导致相干加强，因此在波导中只能传输角度的光波。,It may be thought that the treatment above is somewhat artificial as we took a narrow angle for.
15、It turns out that Eq.(3)can be derived as a general waveguide condition for guided waves whether we use a narrow or a wider angle,one or multiple rays.We can derive the same condition if we take two arbitrary parallel rays entering the guide as in Figure 2.3.The rays 1 and 2 are initially in phase,a
16、nd represent the same“plane-wave”.Ray 1 then suffers two reflections at A and B,and is then again traveling parallel to ray 2.Unless the wavefront on ray 1 just after reflection at B is in phase with the wavefront at B on ray 2,the two would destroy each other.Both rays initially start in phase;ray
17、1 at A just before reflection and ray 2 at A.Ray 1 at B,just after two reflections,has a phase k1 AB-2.Ray 2 at B has a phase k1(AB).The difference between the two phase must be m(2)and leads to the waveguide condition in Eq.(3).,上面的处理可以被认为是有一些人为假设的成分，原因为是小入射角的前提条件。等式（3）是一个基本的导波条件，无论我们使用小入射角还是大入射角，一
18、束光还是多束光。如果我们取两束任意的平行光，输入到如图3所示的波导中，同样可以得出导波条件。初始光线1和2是同相的，用平面波表示。光线1在A点和B点经过反射，传输又与光线2平行。光线1的波阵面在B点反射之后，除非光线2中的B同相，否则两束光将相互破坏。两光线初始同相位，光线1在A点反射之后，光线2到达A点。光线1在B点，经过两次反射后，相位为AB-2。光线2在B点的相位k1(AB)，两相位之间的差值应该是m(2)，可以倒出（3）式的波导条件。,To obtain the waveguide condition and solve the propagation modes for the sy
19、mmetric planar dielectric waveguides:(1)The wave optics approach Solve Maxwells equations.There is no approxima-tions and the results are rigorous.(2)The coefficient matrix approach Straightforward.Not suitable for multilayer problems.(3)The transmission matrix method Suitable for multilayer wavegui
20、des.(4)The modified ray model method It is simple,but provides less information.,k m:transverse propagation constant,The simplistic analysis as embedded in the wave-guide condition in Eq.(3)shows quite clearly that only certain reflection angles are allowed within the guide corresponding to m=0,1,2,
21、.We note that higher m values yield lower m.Each different m value leads to a different propagation constant along the guide given by Eq.(4).If we were to consider the interference of many rays,as in Figure 2.3,we would find that the resultant wave has stationary electric field pattern along the y-d
22、irection,and this field pattern travels along the guide,z-axis,with a propagation constant m.We can show this by considering the resultant of the two parallel rays in Figure 2.3 that have incidence angles m satisfying the waveguide condition.,Figure 2.5,The lowest mode(m=0)has a maximum intensity at
23、 the center and moves along z with a propagation constant of 0.There is a propagating evanescent wave in the cladding near the boundary.,Figure2.6 illustrates the field patterns for the first three modes,m=0 to 2.Notice that the rays have been shown to penetrate the cladding，and reflected from an ap
24、parent plane in the cladding.,We have seen that each m leads to an allowed m value that corresponds to a particular traveling wave in the z-direction as described by Eq.(8)with a particular wavevector m as defined by Eq.(4).Each of these traveling waves,with a distinct field pattern,Em(y),constitute
25、s a mode of propagation.The integer m identifies these modes and is called the mode number.The light energy can be transported only along the guide via one or more of these possible modes of propagation as depicted in Figure 2.7.Notice that the rays have been shown to penetrate the cladding，and refl
26、ected from an apparent plane in the cladding.Sincem is smaller for larger m,higher modes exhibit more reflections but they also penetrate much more into the cladding as schematically depicted in Figure 2.7.,我们已经看到，每一个m确定一个可能的值，在z方向上对应一个由E(y)式描写的特殊的传输波，传输波的m传输矢量由(4)式给出。每一个传输波，具有不同场的图样，组成一个传输模式。整数m并成为
27、模式的模数。光能只能通过可能存在于波导中的的传输模式来传输能量，如图2.7所示。注意到光线渗透到包层中，在包层中一个明显反射回来。因为大数值的m，m 非常小，较高的模式，更多的渗透到包层中，存在更多的反射，如图2.7所示。,For the lower mode,m=0,which leads to m being closest to 90 and the wave is said to travel axially.Light that is launched into the core of the waveguide can travel only along the guide in
28、the allowed modes specified by Eq.(3).These modes will travel down the guide at different group velocities.When they reach the end of the guide they constitute the emerging light beam.If we launch a short-duration light pulse into the dielectric waveguide,the light emerging from the other end will b
29、e a broadened light pulse because light energy would have been propagated at different group velocities along the guide as depicted in Figure2.7.The light pulse therefore spreads as it travels along the guide.,对于低阶模，m=0,使m接近于90度，光波是被认为是沿轴传输的。发射光进入波导的芯层后，只能以(3)式（导波条件）决定的模式沿着波导传输。这些模式传输在波导中，具有不同的群速度。当
30、传输到波导的另一端时，产生新的光束。如果我们发射一个短周期的光脉冲输入到介质波导中，在另一端产生的出现的光脉冲将发生加宽，原因是传输在波导中的光具有不同的群速度。因此，光脉冲沿着波导传输会被展宽。,The phase change at TIR depends on the polarization of the electric field,and it is different for E and E/.These two fields require different angles m to propagate along the waveguide.,Any other field
31、direction(perpendicular to the path of ray)can be resolved to have electric field components along E/and E.These two fields experience different phase changes,/and,and consequently require different anglesm to propagate along the guide.We therefore have a different set of modes for E/and E。The modes
32、 associated with E(or Ex)are termed transverse electric field modes,denoted by TEm,because E is actually perpendicular to the direction of propagation,z.,任何其他方向（垂直于光路经）的电场矢量都可以分解为E/和E两个电场分量。两个电场具有不同的相移/和，分别要求沿着波导传输有不同的m。因此，对于E/和E具有不同的传输模式。与E相关的称为横电场模式，用TEm表示，因为事实上E垂直于传播方向z。,The modes associated with
33、 the E field have a magnetic B,which is perpendicular to the direction of a propagation and are termed transverse magnetic field modes,denoted by TMm.It is interesting that E/has a field component parallel to the z-axis,shown as Ez,which is along the direction of propagation.It is apparent that Ez i
34、s a propagating longitudinal electric field.In free space,it is impossible for such a longitudinal field to exist but within an optical guide,due to the interference phenomena,it is indeed possible to have a longitudinal field.Similarly,those with B/,have a magnetic field along z that propagate alon
35、g this direction as a longitudinal wave.,E有一个磁场分量B，与它相关的垂直于传输方向的模式，称为横磁场模式，用TMm表示。E/有一个分量平行与z轴，沿着传输方向,表示为Ez。Ez明显是一个纵向传输的电场。在自由的空间中，像这样的纵向电场是不可能存在的。但由于干涉现象，存在于波导中的场，确实可能是一个纵向场。类似的，对于B/同样有一个纵向的磁场沿着z轴方向传输。,Questions and Problems,2.3 TE and TM Modes in dielectric slab waveguide(p.98)Consider a planar di
36、electric guide with a core thickness 20 m,n1=1.455,n2=1.440,light wavelength of 1.3 m.Given the waveguide condition and the expressions for phase changes and in TIR for the TE and TM mode respectively,Using a graphical solution find the angle for the fundamental TE and TM modes and compare their pro
37、pagation constants along the guide.,Questions and Problems,2.5 Dielectric slab waveguide(p.98)Consider a dielectric slab guide that has a thin GaAs layer of thickness 0.2 m between two AlGaAs layers.The refractive index of GaAs is 3.66 and that of AlGaAs the layers is 3.40.What is the cut-off wavele
38、ngth beyond which only a single mode can propagate in the waveguide,assuming that the refractive index does not vary greatly with the wavelength?If a radiation of wavelength 870 nm(corresponding to bandgap radiation)is propagating in the GaAs layer,what is the penetration of the evanescent wave into
39、 the AlGaAs layers?What is the mode field distance of this radiation?,Chapter 2 Dielectric Waveguides and Optical Fiber,2.1 Symmetric planar dielectric Slab waveguide2.2 dispersion in the planar waveguide 2.3 Step index fiber2.4 Numerical aperture2.5 Dispersion in single mode fibers2.6 Dispersion,el
40、ectrical,and optical bandwidth2.7 The graded index optical fiber2.8 Light absorption and scattering2.9 Attenuation in optical fibers2.10 Fiber manufacture,专业词汇选编,Ray Path:光程Propagation Constant:传输常数 Propagation Mode：传输模式Group Velocity vg:群速 vgDispersion Diagram:色散图Modal Dispersion(Intermodal Dispers
41、ion):模式色散（模间色散）Intermode Coupling:模式耦合Waveguide Dispersion：波导色散Material Dispersion：材料色散Intramode Dispersion：模内色散Normalized Index Difference:归一化折射率差,专业词汇选编,Meridional Ray：子午光线Skew Ray:斜光线Helical Path:螺旋路径Weakly Guiding Fibers：弱导光纤Linearly Polarized：线偏振Fundamental mode：基模Normalized Propagation Constan
42、t：归一化传输常数Numerical Aperture：数值孔径Maximum Acceptance Angle：最大受光角Total Acceptance Angle：总受光角Monochromatic:单色的（单频的）,The propagating modes that exist in a slab waveguide are determined by the waveguide condition.Each choice of m from 0 to its maximum value results in one distinct solution and one possibl
43、e propagation constant m.We see that each mode propagates with a different propa-gation constant even if illumination is by monochromatic radiation.Examination of Figure 2.7 gives the impression that the axial ray has the least reflections and therefore seems to arrive more quickly than a higher-mod
44、e ray.,Higher modes zigzag more along the guide and appear to have longer ray paths.However,there are two important wrong impressions with this view.First,what is important is the group velocity vg along the guide,the velocity at which the energy or information is the transported.Second is that the
45、higher modes penetrate more into the cladding where refractive index is smaller and the waves travel faster.,(2),Equation(2)considers only two extreme modes,the lowest and the highest,and does not consider whether some intermediate modes can have group velocities falling outside the range c/n1 to c/
46、n2(consider the slope d/d in Figure 2.10).Neither does it consider how the light energy is proportioned between various modes.Taking n1=1.48(core)and n2=1.46(cladding),we find/L6.710-11sm-1 or 67nskm-1.In general,intermodal dispersion is not as high as indicated by this estimate due to an“intermode
47、coupling”the discussion of which is beyond the scope of this book.The spread in Eq.(2)is between extremes of the broadened output light pulse.In optoelectronics we are frequently interested in the spread 1/2 between the half intensity points that is smaller than the full width.The determination of 1
48、/2 depends on the temporal shape of the output light pulse,but as a first order approximation,when many modes are present,we can take 1/2.,Thus,a greater portion of the light energy is carried by the cladding in which the phase velocity is higher.Longer wavelength propagate faster,even though by the
49、 same mode.This is called waveguide dispersion inasmuch as it results from the guiding properties of the dielectric structure and it has nothing to do with the frequency(or wavelength)dependence of the refractive of the guide,the disper-sion can also be state as due to the wavelength dependence of t
50、he V-number.,The higher the wavelength(lower the frequency),the greater the penetration of the field into the cladding,as depicted in Figure.,The refractive index of the guide material will also depend on the wavelength and thus modify the-m behavior in Figure.The change in the group velocity of a g
