TM mode propagation

Fig. 9.7 The effective index change induced by the adsorption of a 2 nm thick molecular layer with index n 1.5, calculated as a function of core thickness for the TM mode propagating in glass, polymer, silicon nitride, and silicon waveguides. The right vertical axis shows the equiva lent modal sensitivity...

Figure 3. Normalized propagation constant as a function of normalized frequency for three low-order TE and TM modes n, 1.45, n2=l. 8, n3=l. 32.

As the refractive index within the slice depends on the vertical coordinate only, TE and TM modes can propagate independently in the slices, and Eqs (35) and (36) are fully applicable to this geometry without any change, too. Since we are interested in the propagation of waves in the azimuthal direction, it is advantageous to decompose the Hertz vectors into the eigenmodes of the slice as follows ... 

In anisotropic materials, the electronic bonds may have different polarizabilities for different directions (you may think of different, orientation-dependent spring constants for the electronic harmonic oscillator). Remembering that only the E-vector of the light interacts with the electrons, we may use polarized light to test the polarizability of the material in different directions, lno is one of the most important electro-optic materials and we use it as an example. The common notations are shown in Figure 4.7. If the E-vector is in plane with the surface of the crystal, the wave is called a te wave. In this example, the te wave would experience the ordinary index na of LiNbOs (nG 2.20). If we rotate the polarization by 90°, the E-ve ctor will be vertical to the surface and the wave is called tm. In lno, it will experience the extraordinary index ne 2.29. Therefore these two differently polarized waves will propagate with different phase velocities v c/n. In the example of Figure 4.7, the te mode is faster than the tm mode. 

The following equations define the requirements for the transverse-electric (TE) and the transverse-magnetic (TM) modes of light propagation, respectively ... 

Fig. 6 Sensitivity factor per unit length for absorbance measurement through propagating guided modes. Refractive index profile is given by 1.33 (water cover), 1.56 (waveguide film e.g. Coming glass 7059), and 1.46 (fused silica substrate). Solid lines TE modes dashed lines TM modes. Wavelength for calculations 1 = 550 nm...

The eigenvalue Eqs. 34 and 35 are transcendental equations for imknown modal propagation constants. After solving the eigenvalue equations, the field profiles can be determined by substituting the values of modal propagation constants fi into the boundary conditions and calculating the amplitudes and a i for TE modes and fcf and hj for TM modes (i = 1,2,3). 

In a rectangular cavity, electromagnetic waves are classified as transverse electric (TE) or transverse magnetic (TM) modes. AU the field combinations can be obtained by the superposition of TE and TM modes. TE modes are defined as the waves that have no electric field component in a defined propagation direction. In this discussion, the propagation direction is assumed to be the f-direction. Similarly, TM modes have no magnetic field component in the f - direction. By assuming a cavity with dimensions, a X b X d in the x-,y-, and z-directions, respectively, the frequencies at which nontrivial solutions of the Helmholtz Equation occur are... 

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