Orthogonality radiation modes

In order to simplify the discussion we will here restrict ourselves to a quantum memory for a two-mode radiation field, realized for example in a weak-coupling resonator allowing for two orthogonal polarization modes of the same frequency described by annihilation and creation operators , [Lukin 2000... 

Representation of the modal fields 25-2 Propagation constant 25-3 Radiation-field expansion 25-4 Orthogonality and normalization 25-5 Power of the radiation field 25-6 Excitation of radiation modes... 

In Section 11-4, we showed that each bound mode is orthogonal to every other bound mode and to all radiation modes. The orthogonality properties of radiation modes are derived in Section 31-3. On a nonabsorbing waveguide, the th and kth forward-propagating radiation modes obey... 

By analogy with Section 11-4, the orthogonality, normalization and orthonormality relations satisfied by backward-propagating radiation modes are obtained from Eqs. (25-4) to (25-6) by applying the convention of Eq. (11-7). 

We use Eq. (30-9) for the transverse-field components and determine the constants from continuity of e j, h j, e j and h j at the interface. With the aid of the Wronskian of Eq. (37-77) this leads to the expressions in Table 25-3 for the ITE and ITM modes. The orthogonality and normalization of each radiation mode is identical to the corresponding free-space normalization of Table 25-2 for reasons given above. Alternatively, we can parallel the derivation of Nj(Q) in Section 25-7 using the radiation-mode fields. 

The combination of e j and eyj which satisfies Eq. (25-24a) is not unique. However, when Tj is constructed according to Eq. (25-21), the orthogonality of the free-space modes, like the normalization, automatically ensures orthogonality of the corresponding radiation modes [4]. This may be verified... 

Marcuse, D. (1974) Theory of Dielectric Optical Waveguides, Academic Press, New York. The radiation modes derived here are not orthogonal, and lead to errors. 

We showed how to determine the radiation modes of weakly guiding waveguides in Sections 25-9 and 25-10, starting with the transverse electric field e, which is constructed from solutions of the scalar wave equation. However, unlike bound modes, the corresponding magnetic field h, of Eq. (25-23b) does not satisfy the scalar wave equation. This means that the orthogonality and normalization of the radiation modes differ in form from that of the bound modes in Table 13-2, page 292, as we now show. 

Note that this implies orthogonality between identical radiation modes with different values of Q, i.e. when j = k. 

The radiation field of the scalar wave equation can be represented by the continuum of scalar radiation modes discussed above, or by a discrete summation of scalar leaky modes and a space wave. This is clear by analogy with the discussion of vector radiation and leaky modes for weakly guiding waveguides in Chapters 25 and 26. Scalar leaky modes have solutions P of Eq. (33-1) below their cutoff values when P becomes complex. Many of the properties of bound modes derived in this chapter also apply to leaky modes. For example, the orthogonality condition of Eq. (33-5a) applies to leaky modes, provided only that the cross-sectional area A. is replaced by the complex area A of Section 24-15 to ensure that the line integral of Eq. (33-4) vanishes. 

Then the moment induced by the electric vector of the incident light is parallel to that vector resulting in complete polarization of the scattered radiation. The A lg i>(CO) mode of the hexacarbonyls provides a pertinent example08. Suppose we have a set of coupled vibrators, equidistant from some origin. Then it must be possible to express the basis functions for the vibrations in terms of spherical harmonics, for the former are orthogonal and the latter comprise a complete set. The polarization of a totally symmetric vibration will be determined by its overlap with the spherically symmetrical term which may be taken as r2 = x2 + y1 + z2. Because of the orthogo-... 

In Section 31-3 we show that each bound mode is orthogonal to the radiation field in Eq. (11-2). Hence Eq. (31-15) gives... 

We recall from Section 11-4 that each mode is orthogonal to the radiation field and all other bound modes. Assuming that the fiber is nonabsorbing, we take the cross product of Eq. (20-la) with h, Eq. (20-lb) with ej and integrate over the infinite cross-section A. We deduce from Eq. (11-13) that... 

We are primarily interested in radiation from the fundamental modes of bent, single-mode fibers. Within the weak-guidance approximation, the power radiated is insensitive to polarization, since p. Thus we can conveniently assume that the transverse electric field is parallel to the Z-axis in Fig. 23-2(a), i.e. orthogonal to the plane of the bend. Close to and within the core, the magnitude of the electric field on the bend is given by aj Fo (R) exp (ifiz), using the local-mode approxinution, where is the modal amplitude, Fq (R) is the... 

The fundamental mode on a weakly guiding, bent fiber has been replaced by an antenna, or thin wire, carrying the current distribution of Eq. (23-4). Because the currents are orthogonal to the antenna, it may be helpful to think of them as a continuous distribution of Z-directed current dipoles. To calculate the power radiated, we assume, for simplicity, that the antenna is a closed loop of radius as shown in Fig. 23-2(a). If (s. O, < ) are spherical polar coordinates... 

The first application of the conjugated form of the reciprocity theorem demonstrates orthogonality of bound, radiation and leaky modes of nonabsorbing waveguides. Consider two modes propagating in the forward direction with propagation constants Pj nd The subscripts jand k may refer to two different bound modes, but, in the case... 

Radiation inodes of the scalar wave equation 33-7 Orthogonality and normalization 33-8 Leaky modes... 

In order to determine the polymer chain orientation, infrared measurements of transmittance or absorbance must be performed with polarized IR radiation, parallel or perpendicular to the mechanical draw direction, for a particular absorption band of the spectrum. Fig. 7.11 shows the FTIR spectra obtained for two orthogonal polarization directions, parallel and perpendicular to the draw direction of a poly(vinylidene fluoride) (PVDF) polymer sample. It is possible to observe that the overall spectra are similar with neither modes being totally suppressed nor new modes seeming to appear. Nevertheless, the amount of IR radiation absorbed by some particular modes are clearly different for IR spectra obtained with radiation polarized in the perpendicular or parallel to the polymer draw direction (Fig. 7.11). 

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