1、外文翻译 外文原文： Considerations for modalBirefringence There are two major aims of this chapter. Firstly, to provide a more explicit understanding of the meaning of polarization and modal birefringence in fibres such as microstructured fibres which have large refractive index contrasts. Secondly, in the i
2、nstance of bound modes, to provide clear restrictions on the expected behaviour of polarization in the modes of birefringent fibres. Most conclusions that may be only strictly valid for bound modes, are in fact approximately valid for leaky modes with small confinement losses. Note that even when co
3、nfinement loss is high for practical purposes one usually finds that Im(neff ) Re(neff ).Thus the proceeding analysis is approximately applicable to microstructured fibres supporting leaky modes. 3.1 Local polarizationThe concept of polarization in optical waveguides is subtle in its difference to t
4、he polarization of a freely propagating plane wave. Consider the reduced fields e, h which are obtained from the total fields ( ) ( )( ) ( )( , , , ) ( , ) ( ( , ) , ( , ) ) ,( , , , ) ( , ) ( ( , ) , ( , ) ) ,i z t i z ttzi z t i z ttzE x y z t e x y e e x y e x y eH x y z t h x y e h x y h x y e (
5、3.1) by eliminating the dependence on time, t, and the propagating direction, z. They satisfy the vector wave equations Snyder and Love, 1983, pp. 591 12201201 2 2201201( ) ,1( ) ,1( ) ( ( ) ,1( ) .ot t t zot t t zoz t t t t t toz t t t te z h i hknh z e i ekie i z h e e I n nknih i z e hk (3.2) The
6、 refractive index distribution used to create modes in optical waveguides results in two key differences. 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 transv
7、erse components of electric and magnetic fields are not preserved at all points in the cross-section. However, in the limit of small variation in refractive indexthe mode solutions are expected to locally resemble a plane travelling wave. Hence we expect to only deal with these two differences when
8、refractive index contrasts are large. In this latter regime it is the local polarization of the electromagnetic field in an arbitrary waveguide mode that has an unambiguous meaning. Birefringence defines the instance where two modes of different effective indices are found which also have nearly ort
9、hogonal local polarization everywhere in the waveguide cross-section. 3.2 Restrictions on the birefringence of translationally invariant waveguides An open question is the extent to which waveguide geometry can be used to tailor the local polarization properties of bound modes in dielectric waveguid
10、es 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 transverse and longitudinal field components which are given without proof in standard texts. Take a bo
11、und mode solution, P, of some arbitrary waveguide such as that depicted in Fig. 3.1 and assume there exists at least one point x0 in the infinite cross-sectionA where the fields are not locally linearly polarized. There is some freedom in what we mean by the polarization of the field at a point in a
12、 bound 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 exactly by saying that pte and pth cannot both be made real valued at x0 by scal
13、ing with a complex constant. This is true since the phase difference between thetwo components of pte determines the ellipticity of the local polarization. A mode with label Q can then always be constructed by ( , ) ( ( ) , ( ) ) ,( , ) ( ( ) , ( ) ) ,Q Q Q P Pt z t zQ Q Q P Pt z t ze e e e eh h h h
14、 h (3.3) which can easily be shown to also satisfy Eqs. (3.2) for the same refractive 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). Howeve
15、r,using a suitable orthogonalisation procedure, provided in Eq. (3.5), an orthogonal and degenerate mode, Q , can be obtained from modes P and Q so that ( ) ( ) 0 , ( ) ( ) 0P Q Q PAAP Q Q Pe h z d A e h z d Ae h z e h z (3.4) where ( , ) ( , ) ( , ) ,Q Q Q Q P Pe h e h e h (3.5) and ().()QPAPPAe h
16、z d Ae h z d A (3.6) Hence we conclude that any mode featuring 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 the above procedure invo
17、lving 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 waveguides is two. This property is derived purely from the rotational and reflectional symmetries of th
18、e waveguide and in this way degenerate modes often resemble conventional polarization pairs. This is also true of modes P and Q . Only in the case where 0,P P Q QAAe h z d A e h z d A (3.7) are mode P and Q power orthogonal in the usual sense. The physical significance of 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
