第4章集成光波导12

1、12344.1 4.1 电介质平板波导电介质平板波导54.1 4.1 电介质平板波导电介质平板波导1 . 4.sin12nnc2 . 4.sin13nnc64.1 4.1 电介质平板波导电介质平板波导74.2 4.2 对称平板波导中的模式对称平板波导中的模式84.2 4.2 对称平板波导中的模式对称平板波导中的模式3 . 4.sinsin10nkk94.2 4.2 对称平板波导中的模式对称平板波导中的模式aztyee4 . 4.sincosh1bztyee4 . 4.sinsinh1104.2 4.2 对称平板波导中的模式对称平板波导中的模式ggvv5 . 4.1 . 3.sin0kztee2

2、 . 3.vk11we find the phase velocity by setting the phase equal to a constant: phase = t - z = constanttake the derivative with respect to time:(10)0dzdtthe velocity is then:gdzvdt4.2 4.2 对称平板波导中的模式对称平板波导中的模式124.2 4.2 对称平板波导中的模式对称平板波导中的模式ggvv5 . 4.geffvcn134.2 4.2 对称平板波导中的模式对称平板波导中的模式5 . 4.gvgeffvcnc

3、vcngeffck06 . 4.0kneff7 . 4.sinsin10100nknkkneffevanescent waveezstanding waven1 n2envelopethe decaying wave carries no power in the z-direction.e- z154.2 4.2 对称平板波导中的模式对称平板波导中的模式2/8 . 4.sin2/2dyazteeedy2/8 . 4.sin2/2dybzteeedy16modes in the symmetric-slab waveguiden2n2n1n1 n2 range of for bound wav

4、es: 90 c17mode in the symmetric-slab waveguidefor an axial ray, = 90 and neff = n1for a critical angle ray = c and21211sinnnnnnnceff the range of neff is now (4.9)12nnneff define the effective index of refraction: sin 1nneff 184.2 4.2 对称平板波导中的模式对称平板波导中的模式9 . 4.12nnneff7 . 4.sin1nneff194.2 4.2 对称平板波导

5、中的模式对称平板波导中的模式4.2.1 4.2.1 模式条件模式条件204.2 4.2 对称平板波导中的模式对称平板波导中的模式4.2.1 4.2.1 模式条件模式条件10. 4.2m29. 3.sincossincos2212212222122122iiiipnnnnnnnn234.2 4.2 对称平板波导中的模式对称平板波导中的模式4.2.2 te4.2.2 te偏振和偏振和tmtm偏振偏振244.2 4.2 对称平板波导中的模式对称平板波导中的模式4.2.2 te4.2.2 te偏振和偏振和tmtm偏振偏振25keh磁场磁场h电场电场en2n1n3电场电场e磁场磁场hn2n1n3te（横

6、电波）tm（横磁波）dn1n2n2a光线1光线2光线向上传播时的相前光线向上传播时的相前c光线向下传播时的相前光线向下传播时的相前 dn1n2n2a光线1光线2光线向上传播时的相前cs s1 1s s2 2db be edn1n2n2a光线1光线2光线向上传播时的相前光线向上传播时的相前cs s1 1s s2 2db be e1023210122nkcdnkabdn1n2n2a光线1光线2cs s1 1s s2 2db be e1023210122nkcdnkabmcdabnk222321021（1）111111cos22sin22,cos2dintgdofoctgdefoftgefdoccd

7、abdcdab证明：mdnkmcdabnk32110321021cos222可化简为因此（2）mdcosnk32110（3）32 mhd（4）32 mhd（4）上下两个反射面发生两次相位突变上下两个反射面发生两次相位突变2 2 1 1、2 2 2 2，12131231212122cos/sinarctancos/sinarctannnnn344.2 4.2 对称平板波导中的模式对称平板波导中的模式4.2.3 te4.2.3 te模式图模式图10. 4.2m)11. 4.()sin(cos12tan22211nnndh112212212cossinarctannnncos2cos11. 4201

8、23232nkhhdmmhd354.2 4.2 对称平板波导中的模式对称平板波导中的模式4.2.3 te4.2.3 te模式图模式图)11. 4.()sin(cos12tan22211nnndhcos2cos1nkh364.2 4.2 对称平板波导中的模式对称平板波导中的模式4.2.3te4.2.3te模式图模式图consider an algaas laser diode55.36.321 nn 4.80sin121nnc 904.80 213.553.6effeffnnnn cos2cos211nhdddnhd37for the te0 mode the calculations yiel

9、d the table:(1)(2)(3)(4)(5)(6) nefftan(hd/2)hd2 n1cos d/ 80.43.550003.7570823.5650.6511.1553.1480.367843.5801.2611.7802.3640.753863.5912.1612.2751.5781.442883.5984.6532.7180.7893.445903.6003.1420 4.2 4.2 对称平板波导中的模式对称平板波导中的模式4.2.3 te4.2.3 te模式图模式图for the table on the previous slide, the values were c

10、alculated as follows:(1) the chosen angle (2) neff = n1sin = 3.6 sin (3) 222211sincos12tannnnhd (4) 2tan-1(3), in radians (5) 2 n1cos (6)5()4(cos2cos211 nhddndhd39the plotted result is the mode chart:te0te1te2te3te4te5d/ neff 4.2 4.2 对称平板波导中的模式对称平板波导中的模式4.2.3te4.2.3te模式图模式图404.2 4.2 对称平板波导中的模式对称平板波导

11、中的模式4.2.3 te4.2.3 te模式图模式图4.2.4 4.2.4 高阶模高阶模)11. 4.()sin(cos12tan22211nnndhmhd22112212212cossinarctan*nnncos22cos2/2, 0) 1 (12102nnhddhdmcos2/cos2cos22cos2/2, 0)2(1011212nmdnmnnhddmhdmm)12. 4()12. 4(.cos2/10nmddmmhd222222, 0) 1 (hdhdm112212212cossinarctan*nnnthis applies for even mode patterns given

12、 by:)4 . 4(.)cos(ee)(1aehyztj)11. 4.(sincos12tan222211annnhdmhd22112212212cossinarctan*nnn22222, 1)2(hdhdmin this case, the resonance condition is:)11. 4.(sincos122tan222211bnnnhdfor odd mode patterns, the electric field can be written as:)4 . 4.(.)sin(ee)(1behyztj)13. 4.(.cos21101dnddcos21101nddfor

13、 the te1 mode, we then havethe mode chart for the te1 mode can be plotted from this result.)12. 4(.cos2/10nmddm46higher order modesthe next mode is the te2 mode. it is an even mode. (d/ )2 is found from (4.11a) by taking the second lowest solution as illustrated on the following plot of the tangent

14、function.)11. 4.(sincos12tan222211annnhd47higher order modes(hd/2)te0 te2 te425 vertical scale: tan (hd/2)horizontal scale: hd/248higher order modes cos210nmddm where m is the mode order. m is even for even modes and m is odd for odd modes.in general(4.12)49higher order modesthus, we increment d/ by

15、 an amount11( / )2cosdnbetween successive modes. the table on the next slide was obtained from the te0 mode table in this way. 50 neff (d/ )(d/2 280.43.5500.8360.8361.672823.5650.9981.3652.363843.5801.3292.0823.410863.5911.9913.4335.424883.5983.9807.42511.40903.600 (d/0 000.3670.7531.4423.445 (d/1 1

16、 51example: let d = 1.64 m, = 0.82 m. 2 then dfind , neff, and the number of allowed modes. from the mode chart (next slide): 1.81557.3 7.83578.3 7.86594.3 210 effeffeffntentente111sinsinnnnneffeff te mode chart d/ neff only three modes propagated in the example where d/ = 2. in general, the m-th mo

17、de cuts off (no longer propagates) when d/ satisfies:2221,2nnmdcm (4.14)joseph c. palais4.254higher order modesthe order of the highest ordered propagating mode is found by solving (4.14) for m. the result is: 22212nndm use the integer part of m. since the lowest ordered mode has m = 0, the number o

18、f modes allowed is the integer part of 222121nndn (4.16)55higher order modesas seen from this last equation, we can minimize the number of modes by:(1) making n1 close to n2(2) making d/ smallthese are general conclusions for all waveguides. 222121nndn (4.16)suppose we wish to allow only one mode. s

19、olve (4.14) for m = 1.2221, 121nndc to insure only the te0 mode will propagate, make222121nnd this produces a single-mode waveguide.(4.17)2221,2nnmdcm (4.14)in a multimode waveguide:at a fixed thickness, lower-ordered rays travel with larger angles (ray travel more axial) than higher-ordered rays. lower higherhigher-ordered modes travel with smaller angles (rays travel