電子技術(shù)專業(yè)外文翻譯4

1、<p><b>　　外文翻譯</b></p><p>　　外文原文： Considerations for modalBirefringence</p><p>　　There are two major aims of this chapter. Firstly, to provide a more explicit understanding o

2、f the meaning of polarization and modal birefringence in fibres such as microstructured fibres which have large refractive index contrasts. Secondly, in the instance of bound modes, to provide clear restrictions on the e

3、xpected behaviour of polarization in the modes of birefringent fibres.</p><p>　　Most conclusions that may be only strictly valid for bound modes, are in fact approximately valid for leaky modes with small co

4、nfinement losses. Note that even when confinement loss is high for practical purposes one usually finds that Im(neff ) <Re(neff ).Thus the proceeding analysis is approximately applicable to microstructured fibres supp

5、orting leaky modes.</p><p>　　3.1 Local polarization The concept of polarization in optical waveguides is subtle in its difference to the polarization of a freely propagating plane wave. Consider the reduced

6、 fields e, h which are obtained from the total fields</p><p><b>　　(3.1)</b></p><p>　　by eliminating the dependence on time, t, and the propagating direction, z. They satisfy the vect

7、or wave equations [Snyder and Love, 1983, pp. 591]</p><p><b>　　(3.2)</b></p><p>　　The refractive index distribution used to create modes in optical waveguides results in two key diff

8、erences. Firstly, the magnitude and direction of the reduced electric and magnetic field vectors may vary at different points in the waveguide cross-section.Secondly, the orthogonality between the transverse components o

9、f electric and magnetic fields are not preserved at all points in the cross-section. However, in the limit of small variation in refractive index the mode solutions are expected to</p><p>　　3.2 Restrictions

10、on the birefringence of translationally invariant waveguides</p><p>　　An open question is the extent to which waveguide geometry can be used to tailor the local polarization properties of bound modes in diel

11、ectric waveguides when the materials are assumed to be isotropic, lossless and non-magnetic. The analysis is completely original, although in retrospect some results may be inferred from statements regarding the transver

12、se and longitudinal field components which are given without proof in standard texts.</p><p>　　Take a bound mode solution, P, of some arbitrary waveguide such as that depicted in Fig. 3.1 and assume there ex

13、ists at least one point x0 in the infinite cross-sectionwhere the fields are not locally linearly polarized. There is some freedom in what</p><p>　　we mean by the polarization of the field at a point in a bo

14、und mode. Here we choose to consider the polarization of only the transverse field components, since they are the dominant components in all glass or polymer based optical fibres. Then this statement can be expressed exa

15、ctly by saying that and cannot both be made real valued at x0 by scaling with a complex constant. This is true since the phase difference between thetwo components of determines the ellipticity of the local polarization.

16、</p><p>　　A mode with label Q can then always be constructed by</p><p><b>　　(3.3)</b></p><p>　　which can easily be shown to also satisfy Eqs. (3.2) for the same refracti

17、ve index profile and equal propagation constant.Assuming that modes P and Q are not identical, then they are clearly linearly independent modes, but are generally not power orthogonal in the sense of Eq. (3.4). However,u

18、sing a suitable orthogonalisation procedure, provided in Eq. (3.5), an orthogonal and degenerate mode, , can be obtained from modes P and Q so that</p><p><b>　　(3.4)</b></p><p><b

19、>　　where</b></p><p><b>　　(3.5)</b></p><p><b>　　and</b></p><p><b>　　(3.6)</b></p><p>　　Hence we conclude that any mode featu

20、ring regions in the waveguide cross-sectionwhere the fields are not locally linearly polarized, must be degenerate and power orthogonal with at least one other mode. Furthermore, this mode can easily be constructed by th

21、e above procedure involving only complex conjugation and orthogonalisation.It has been shown [McIsaac, 1975a] and in section 2.1 that the maximum permissible occurrence of non-accidental modal degeneracy in such optical

22、waveguides is two. This</p><p>　　Only in the case where</p><p><b>　　(3.7)</b></p><p>　　are mode P and Q power orthogonal in the usual sense. The physical significance of

23、 this is highlighted in the special case where the fields are everywhere locally circularly polarized.Here modes P and Q are power orthogonal since the fields are locally orthogonally polarized everywhere. That is</p&

24、gt;<p><b>　　(3.8)</b></p><p>　　everywhere in </p><p>　　We can equivalently express our findings for bound or approximately bound modesas; only modes which are everywhere linea

25、rly polarized in the waveguide cross-section can be non-degenerate. Noting that the orientation of local polarization need not be uniform in the waveguide cross-section. This is the one special case in which the analysis

26、 given above fails. That is when andcan be made everywhere real with a complex scaling constant and would therefore be everywhere imaginary (evident from Eqs.</p><p>　　3.3 Derivation of local-mode coupling

27、 equations for a class of length-dependent perturbations</p><p>　　It is evident from the previous section that translation invariance in optical waveguides(isotropic and lossless) restricts the possib

28、le types of non-degenerate bound modes to those with linearly polarized transverse components. We are thus motivated to explorethe effects of z-dependent perturbations such as spinning of the local polarization. We use c

29、oupled mode theory to investigate this problem with an approach which is parallel to that presented by [Snyder and Love, 1983, Section 31-11] and </p><p>　　Take the modes of the unperturbed fibre</p>

30、<p><b>　　(3.9)</b></p><p>　　where x = (x; y; z) and p= 1,2 specifies the mode -considering only the coupling between a pair which may resemble a polarization pair with close propagation cons

31、tants. The corresponding local modes</p><p><b>　　(3.10)</b></p><p>　　are the solutions for the perturbed waveguide obtained by assuming the waveguide is nearly translationally invari

32、ant at each point z0. This result is an approximation to the exact solution of Maxwell's equations for the z-dependent problem but where coupling of power is expected between local modes [Snyder and Love, 1983, pp. 5

33、53-66 ].</p><p>　　Assume that the actual fields of the perturbed fibre are expressible as some supermode</p><p><b>　　(3.11)</b></p><p>　　over some length, L, of the wave

34、guide containing z0. We have used the term supermode,in a context different to its traditional meaning in translationally invariant waveguides.</p><p>　　Here it describes the construction of a z-dependent mo

35、dal distributionwith a constant-phase travelling wave component . Clearly this assumption is strongly dependent on the type of z-dependent perturbation being considered. In order for to be independent of z throughout L t

36、wo conditions are sufficient; that the propagation constants of the local modes , and the coupling coefficients between modes (see Eq. (3.17)) are constant throughout L. The types of perturbations that preserve this pro

37、perty </p><p>　　In order to evaluate the unknown coefficients, , , we use the reciprocity theorem to relate the perturbed and unperturbed waveguides in an in finitesimally thin slice centered at z0 where the

38、 two are identical, as depicted graphically in Fig. 3.2. In this vanishingly thin volume we can readily show [Snyder and Love, 1983, pp. 602-4] that</p><p><b>　　(3.12)</b></p><p><

39、;b>　　Where</b></p><p><b>　　(3.13)</b></p><p><b>　　(3.14)</b></p><p>　　The right hand side of Eq. (3.12) is exactly 0 since the difference in refract

40、ive index profiles is identically 0 at z0.Substituting Eq. (3.11) into Eq. (3.13) then Eq. (3.12) leads to the expression</p><p><b>　　(3.15)</b></p><p>　　where the normalization cons

41、tants are</p><p><b>　　(3.16)</b></p><p>　　and the coupling coefficients are</p><p><b>　　(3.17)</b></p><p>　　Note that in obtaining these express

42、ions we used the fact that and at z0 and made use of the power orthogonality of modes 1 and 2.</p><p>　　Other than the trivial solution, = =0, to Eq. (3.15) a non-trivial solution canbe found by expressing

43、this equation as an eigenvalue equation</p><p><b>　　(3.18)</b></p><p>　　where is the eigenvalue and the eigenvectors are given by = (，).As the matrix we have constructed is 2 by 2, w

44、e expect 2 supermodes to result from the orthogonal eigenvectors of Eq. (3.18). There are two cases in which valuable simplifications can be made and which provide some physical insight.</p><p><b>　　Ca

45、se 1</b></p><p>　　Where both local modes are locally linearly polarized in the waveguide cross-sectionand therefore can be scaled so that ande are purely real. Then it is straightforward to show that

46、 = = 0 since they are simply the derivative of unity. Furthermore it is possible to show that = = . By making the further assumption that the local modes are normalized and degenerate with propagation constant = = we

47、 find that the solutions to Eq. (3.18) are</p><p><b>　　(3.19)</b></p><p>　　Thus when any of the perturbations previously discussed are made to a waveguide with degenerate modes, the

48、resulting supermodes of the z-dependent waveguide are locally circularly polarized.</p><p><b>　　Case 2</b></p><p>　　Where local modes 1 and 2 are degenerate, orthogonal and satisfy ,

49、and. This case describes the situation when the two modes chosen for the expansion are everywhere locally circular polarized in opposite directions. Then it is possible to show that, while is real and equivalent to that

50、 of the case 1.When the modes are normalized we have the solutions</p><p><b>　　(3.20)</b></p><p>　　This result is formally equivalent to that of case 1 as the supermode propagation c

51、onstants are equal. Furthermore, the insight we gain is that when the unperturbed modes are approximately locally circularly polarized then the perturbed fields remain so.</p><p>　　3.4 Coupling coefficients

52、for spun fibre</p><p>　　When the pitch length, , of a spun fibre is long enough to assume that the perturbation is adiabatic, we can apply the analysis in the previous section to investigate its polarization

53、 properties. This proceeding analysis initially will follow closely that provided in [Bassett,1988], which uses coupled local modes in the same formulation as that provided in [Snyder and Love, 1983, section 31-14]. The

54、analysis in this reference is formally equivalent to that presented here but the representation i</p><p>　　For this type of perturbation we can relate the local modes of the spun fibre with those of the unpe

55、rturbed fibre using</p><p><b>　　(3.21)</b></p><p>　　where the anti-clockwise rotation matrix is given by</p><p><b>　　(3.22)</b></p><p>　　Substit

56、uting Eq. (3.21) into Eq. (3.17) gives a reduced expression for the coupling coeffficients in terms of the fields of the unperturbed fibre at z = 0;</p><p><b>　　(3.23)</b></p><p>　　w

57、here a anti-clockwise rotation matrix is defined by</p><p><b>　　(3.24)</b></p><p>　　and the operatoris simply . The resulting coupling coefficients of Eq. (3.23) are identical to th

58、ose found in [Bassett, 1988].If we consider case 1, that is where the modes and therefore are locally linearly polarized and degenerate, then our supermode solutions from Eq. (3.19) are</p><p><b>　　(3

59、.25)</b></p><p><b>　　whereand</b></p><p><b>　　(3.26)</b></p><p>　　In order to understand the behavior of the local polarization we use a simplfied, but

60、 equivalent, expression for the turning fields in complex notation</p><p><b>　　(3.27)</b></p><p>　　Substituting Eq. (3.27) into Eq. (3.25) gives the following transverse components&l

61、t;/p><p><b>　　(3.28)</b></p><p><b>　　where</b></p><p>　　We can gain insight into the meaning of this expression by applying the approximations; which states th

62、at the transverse field components of the two modes are approximately orthogonal, which is only strictly true for waveguides with small refractive index contrasts. The resulting supermodes are</p><p><b&g

63、t;　　(3.29)</b></p><p>　　where.By considering only the transverse components we find that propagation along the spun fibre demonstrates a circular birefringence of magnitude</p><p><b>

64、;　　(3.30)</b></p><p>　　中文翻譯： 在本章有兩個(gè)主要目的。第一，提供光纖中偏振和模式雙折射率含義的一個(gè)更明確的理解，例如具有較大折射率差異的微光纖結(jié)構(gòu)。第二，在耦合模式的實(shí)例中，提供在雙折射率模式下最佳偏振態(tài)下的明確限制。</p><p>　　許多可能只對(duì)耦合模式有效結(jié)論事實(shí)上對(duì)有少量限制損耗的泄露膜是近似有效的。當(dāng)模有效折射率的虛部小于它的實(shí)部時(shí)，在實(shí)

65、際的試驗(yàn)中限制損耗才會(huì)高。因此進(jìn)行的分析近似的可應(yīng)用于具有泄露膜的微結(jié)構(gòu)的光纖。</p><p><b>　?。?）部分偏振</b></p><p>　　在光波導(dǎo)中偏振的解釋的跟在自由傳播的平面波中有細(xì)微的區(qū)別?？紤]從總場(chǎng)獲得的簡(jiǎn)化的電場(chǎng)和磁場(chǎng) (3.1)</p><p>　　消去關(guān)于時(shí)間的量和z傳播方向。他們滿足下邊的

66、矢量波方程：</p><p><b>　　(3.2)</b></p><p>　　常常用來(lái)創(chuàng)建光波導(dǎo)的模式的折射率的分布導(dǎo)致了兩個(gè)主要的差異。第一、部分電場(chǎng)和磁場(chǎng)的矢量的大小和方向在波導(dǎo)的橫截面的不同點(diǎn)處可能不同。第二、不是在橫截面的所有點(diǎn)都具有電場(chǎng)和磁場(chǎng)橫向要素的正交性。但是，在極小的范圍內(nèi)變化的折射率，模式的解與平面?zhèn)鬏數(shù)牟ǖ南嗨?。所以，我們只要在折射率差別比較大

67、的時(shí)候考慮這兩個(gè)差異就可以了。在后一種結(jié)構(gòu)中，在任意波導(dǎo)模式中的電場(chǎng)的部分極化有有一個(gè)明確的的意義。雙折射定義了兩種模式的不同指標(biāo)在波導(dǎo)橫截面上的正交局部極化無(wú)處不在。</p><p>　?。?）平行變量固定的光波導(dǎo)雙折射的限制 一個(gè)未解決的問(wèn)題，當(dāng)介質(zhì)材料被假設(shè)各向同性，無(wú)損，非磁性時(shí)，波導(dǎo)幾何適合在介質(zhì)波導(dǎo)中約束模式的部分偏振特性。在此，我們將說(shuō)明在這樣的平行變量固定的光波導(dǎo)中，只有橫向成分部分線性

68、極化的模式具有雙折射。這個(gè)分析是原創(chuàng)的，盡管回顧的一些結(jié)果是參考一些在標(biāo)準(zhǔn)文本中沒(méi)有給出證明的關(guān)于橫向和縱向場(chǎng)的分量的的報(bào)告，例如Snyder and Love [1983] and Vassallo [1991].</p><p>　　求一個(gè)任意模式波導(dǎo)的約束模式的解，例如在圖3.1中描繪的，假設(shè)在無(wú)限大的沒(méi)有部分線性極化的場(chǎng)分布的橫截面存在至少存在一個(gè)x0。在束縛模式中的一個(gè)點(diǎn)的附近存在一些自由模式。在此我們

69、只考慮橫向場(chǎng)分量的極化，因?yàn)樗鼈兪撬胁ＡЩ蚓酆衔锏墓饫w的主導(dǎo)分量。也可以這樣準(zhǔn)確的表述，和可以通過(guò)用一個(gè)復(fù)數(shù)來(lái)估算的x0來(lái)得到真值。因?yàn)閮蓚€(gè)分量的相位差異確定了部分極化的橢圓極化率。</p><p>　　用Q標(biāo)注的模式可以這樣建立：</p><p><b>　　(3.3)</b></p><p>　　由于有相同的折射率分布和相同的傳播常數(shù)，則

70、這個(gè)式子同樣滿足等式（3.2）。</p><p>　　假設(shè)，P和Q模式不同，它們顯然是線性無(wú)關(guān)的模式，但是從3.4來(lái)看它們通常不正交。用一個(gè)恰當(dāng)?shù)恼贿^(guò)程，3.5所示，一個(gè)正交簡(jiǎn)并的模式，可以從P和Q模式中獲得，由此：</p><p><b>　　(3.4)</b></p><p>　　其中

71、 (3.5)</p><p><b>　　(3.6)</b></p><p>　　因此我們得出結(jié)論，在波導(dǎo)界面的任何區(qū)域模式，它的場(chǎng)分布不是部分線性極化的，必定至少有一種其它模式是簡(jiǎn)并的和正交的。此外，這種模式可以通過(guò)以上的只包含復(fù)數(shù)的共軛和極化步驟很容易的建立。</p><p>　　它已被[Mclsaac,1975a]證明，

72、在第2.1部分，在這樣的光波導(dǎo)最大允許發(fā)生的主模式簡(jiǎn)并有兩個(gè)。</p><p>　　這個(gè)性質(zhì)是純粹是從旋轉(zhuǎn)和反射的對(duì)稱波導(dǎo)推導(dǎo)出來(lái)的，用這樣的方法簡(jiǎn)并的模式與常規(guī)的極化對(duì)相似,對(duì)于P和模式來(lái)說(shuō)也是正確的。</p><p><b>　　只有在這種情況下</b></p><p><b>　　(3.7)</b></p>

73、;<p>　　P和Q一般是正交的。在這個(gè)到處部分圓極化的場(chǎng)的特例中物理意義很明顯的體現(xiàn)出來(lái)，由于是任何地方都是部分正交極化的，P和Q是正交的。在處都滿足</p><p><b>　　(3.8)</b></p><p>　　我們可以同樣表達(dá)我們的研究結(jié)果為束縛或近似束縛模式；只有在波導(dǎo)截面處到處線性極化的模式是非簡(jiǎn)并的。在波導(dǎo)截面處得部分極化的方向不需要是

74、一致的。這是在屢次失敗之上經(jīng)過(guò)分析得出的特殊情況。只有當(dāng)和在任何地方都能用一個(gè)固定復(fù)數(shù)的實(shí)部得到，并且因此是虛部。用3.3等式來(lái)建立Q模式與模式P是線性相關(guān)的，允許它是非簡(jiǎn)并的。</p><p>　?。?）一系列的跟長(zhǎng)度有關(guān)的波動(dòng)部分模式的耦合方程的推到</p><p>　　從前一部分可以明顯得知，在光波導(dǎo)中的平移不變性把非簡(jiǎn)并耦合模式的可能的種類限制到那些具有線性極化的部分。因此我們?nèi)パ?/p>

75、究Z方向的波動(dòng)的影響例如部分極化的旋轉(zhuǎn)。我們用耦合模理論來(lái)研究這個(gè)問(wèn)題，用與[Snyder and Love, 1983, Section 31-11] and Bassett [1988].提出的類似的方法</p><p>　　獲得非攝動(dòng)光纖的模式</p><p><b>　　(3.9)</b></p><p>　　在此X=（x,y,z）,P

76、=1,2決定考慮到模式，只有當(dāng)類似于具有密切傳輸常數(shù)的一個(gè)極化對(duì)之間的耦合。對(duì)應(yīng)的部分模式</p><p><b>　　(3.10)</b></p><p>　　它是通過(guò)假設(shè)在波導(dǎo)中的每一個(gè)點(diǎn)z0幾乎都是平移不變的情況下波動(dòng)波導(dǎo)的解。這個(gè)結(jié)果是Maxwell方程的Z方向的解的近似，但是它只有在部分模式之間才能耦合。</p><p>　　假設(shè)在某

77、一種模式下攝動(dòng)光纖的實(shí)際場(chǎng)可以用含有Z0的以下式子來(lái)描述：</p><p><b>　　(3.11)</b></p><p>　　在上下文中，我們用這個(gè)術(shù)語(yǔ)，超級(jí)模式，與它在平移不變的波導(dǎo)中的傳統(tǒng)意義不同。在此，它用一個(gè)行波的特定的式子描述Z方向的的分布。顯然這個(gè)假設(shè)依賴于我們所考慮的Z方向攝動(dòng)的形式。為了使與Z的長(zhǎng)度無(wú)關(guān)，有兩種情況；部分極化模式的，的傳輸常數(shù)和模式

78、間的耦合系數(shù)是不隨L變化的。保持這種性質(zhì)的擾動(dòng)的形式包括以固定速率旋轉(zhuǎn)和重復(fù)線性轉(zhuǎn)換。事實(shí)上這兩種攝動(dòng)的結(jié)合可以同時(shí)使用，這對(duì)纖維的和非纖維的光纖損傷是很有必要的。纖維光纖和非擾動(dòng)光纖部分模式之間的關(guān)系會(huì)在下一個(gè)部分講述，這個(gè)部分是為了保持分析進(jìn)程的正確性。</p><p>　　為了估算, ,這兩個(gè)未知的系數(shù)，我們用互易定理在一個(gè)極其小的以z0為中心的薄片分析這兩種波導(dǎo)，這兩種波導(dǎo)在z0處是相同的，如上圖圖解的描

79、述。在這個(gè)極其薄的體積里我們可以得出</p><p><b>　　(3.12)</b></p><p>　　在此 (3.13)</p><p><b>　　(3.14)</b></p><p>　　由于在z0處折射率剖面

80、上的差異同為0，則等式3.2的右端為0。</p><p>　　將等式3.11帶入3.13和3.12得出</p><p><b>　　(3.15)</b></p><p><b>　　式中歸一化常數(shù)</b></p><p><b>　　(3.16)</b></p>

81、<p><b>　　耦合系數(shù)為</b></p><p><b>　　(3.17)</b></p><p>　　為了獲得這些表達(dá)式，在z0處, ,，并應(yīng)用1和2 中模式的正交性。等式3.15 除了零以外的解（= =0），還可以通過(guò)用特征方程表達(dá)這個(gè)等式來(lái)獲得有非零解</p><p><b>　　(3.18

82、)</b></p><p>　　式中是特征值，特征向量由= (，)給出。</p><p>　　我們建了二維矩陣，我們希望兩個(gè)模式由3.18式的正交向量得出。下面的簡(jiǎn)化例子可以提供一些物理上的理解。</p><p><b>　　例子1</b></p><p>　　在這個(gè)例子中，在波導(dǎo)的橫截面的兩種局域模式的是線

83、性極化的，因此可以得到，所以和都是實(shí)數(shù)。由于它們只是整體的導(dǎo)數(shù)，所以直接 = = 0，此外可以正陽(yáng)表示= = 。通過(guò)進(jìn)一步假設(shè)局域模式由傳播常數(shù)= = 歸一化和簡(jiǎn)并，我們得出3.18的解是：</p><p><b>　　(3.19)</b></p><p>　　因此當(dāng)任何前邊提到的擾動(dòng)光纖是簡(jiǎn)并模式的波導(dǎo)，z方向的波導(dǎo)的超級(jí)模式的解是部分圓極化的</p&g

84、t;<p><b>　　例子2</b></p><p>　　例子中的局部模式1和2 是簡(jiǎn)并的，正交的，且滿足和，這個(gè)例子描述了這樣的情況，被用來(lái)擴(kuò)充的兩種模式在相反的方向到處是部分圓極化的。由于是實(shí)數(shù)且和例1中的值是等價(jià)的，則可以得出，當(dāng)模式是歸一化的時(shí)，我們得以下解：</p><p><b>　　(3.20)</b></p&

85、gt;<p>　　由于超級(jí)模式的傳播常數(shù)是相等的，這個(gè)解通常和例1中的解是等價(jià)的。因此，我們可以這樣理解，當(dāng)非擾動(dòng)模式近似局域圓極化時(shí)，擾動(dòng)區(qū)域仍然是這樣。</p><p>　　3.4在纖維光纖中的耦合系數(shù)</p><p>　　當(dāng)纖維光纖中的極限長(zhǎng)度大到可以假設(shè)擾動(dòng)部分是絕熱的,我們可以把這個(gè)分析應(yīng)用到前面的情況里去探究它的極化特性。這種過(guò)程分析起初要緊緊跟隨在[Basset

86、t,1988]中提供的分析，它像[Snyder and Love, 1983, section 31-14]中提供的分析一樣，在相同的情況下應(yīng)用局部模式耦合，在這個(gè)參考里，這個(gè)分析與在此的分析在形式上是等價(jià)的，但在表示形式上有微妙的不同。此外，當(dāng)前的分析把[Bassett, 1988] 的結(jié)果擴(kuò)展到了全矢量。</p><p>　　對(duì)于這種形式的擾動(dòng)光纖，我們可以把纖維光纖的局部模式與那些非擾動(dòng)光纖用以下式子聯(lián)系起

87、來(lái)：</p><p><b>　　(3.21)</b></p><p>　　式中逆矩陣由下式給出</p><p><b>　　(3.22)</b></p><p>　　將等式3.21帶入等式3.17，在非擾動(dòng)光纖z=0的場(chǎng)分布情況下，我們得到耦合系數(shù)的簡(jiǎn)化表達(dá)式。</p><p&

88、gt;<b>　　(3.23)</b></p><p><b>　　式子中</b></p><p><b>　　(3.24)</b></p><p>　　運(yùn)算符為。等式（3.23）耦合系數(shù)與[Bassett, 1988]中的是相同的。</p><p>　　如果我們考慮例1，其中

89、和是局部極化和簡(jiǎn)并的，因此等式（3.19）的超級(jí)模式解為：</p><p><b>　　(3.25)</b></p><p><b>　　其中</b></p><p><b>　　(3.26)</b></p><p>　　為了理解局部極化的性質(zhì)我們?cè)诰哂袕?fù)雜符號(hào)的領(lǐng)域里用一個(gè)簡(jiǎn)

90、化、但是等價(jià)的的表達(dá)式：</p><p><b>　　(3.27)</b></p><p>　　將等式（3.27）帶入到等式（3.25）得到下邊的橫向分量：</p><p><b>　　(3.28)</b></p><p><b>　　其中式子中 </b></p>

91、<p>　　我們可以通過(guò)應(yīng)用近似來(lái)來(lái)理解這個(gè)表達(dá)式的意思；它表明這兩個(gè)模式的橫向場(chǎng)分量是是近似正交的，在相對(duì)比較小的折射率情況下的波導(dǎo)中才嚴(yán)格成立。得出的超級(jí)模式是：</p><p><b>　　(3.29)</b></p><p>　　其中，通過(guò)只考慮橫向分量，我們得出纖維光纖中的傳播的除了圓形雙折射率的大?。?lt;/p><p>&