Time-averaged Poynting vector

Figure 17. z-component of time-averaged Poynting vector in the focal plane. 

When it is clear from the context that it is the time-averaged Poynting vector with which we are dealing, the brackets enclosing S will be omitted. 

Once we have obtained the electromagnetic fields inside and scattered by the particle, we can determine the Poynting vector at any point. However, we are usually interested only in the Poynting vector at points outside the particle. The time-averaged Poynting vector S at any point in the medium surrounding the particle can be written as the sum of three terms ... 

The intensity is the time-averaged Poynting vector multiplied by c/in (see Eq. (32)) ... 

The quantities observable spectroscopically are the reflectance Rn and the transmittance T12 of the interface. They are defined as the ratios of the z-components (normal to the interface) of the time-averaged Poynting vectors (1.1.15°) for the reflected and transmitted waves to that of the incident wave [8] ... 

Ideally, we would like to impose and then solve for E / and E ut- However, it is not possible to do this since it would not assign a boundary conditirai to the system (11.82) and (11,95). We can get this around in the following way. From the time-average Poynting vector (11.92), it follows that... 

Reflectivity The ratio of the reflected to the incident time-averaged Poynting vector component normal to the interface. 

The power dPj lost from the modal fields due to absorption over a differential length dz of the waveguide follows from the time-averaged Poynting vector theorem in Section 30-9, and is given by Eq. (30-28) with E replaced by the modal field OjEj. If denotes the volume between cross-sectional planes at z and z + dz, then... 

The modal power flow along the fiber per unit cross-sectional area, or intensity, is given by the time-averaged Poynting vector S in Table 14-3, where a is the modal amplitude. To describe the change in this distribution with changes in V, we keep the total modal power P fixed, and define a normalized intensity S = S/P = S/ a N. Hence... 

The intensity, or power density, S has magnitude and direction determined by the time-averaged Poynting vector. Thus we deduce from Eq. (35-lb) that... 

Substituting for the time-averaged Poynting vector from equation (2.46) and expanding the triple vector product, we obtain L in terms of the fields ... 

Thus, the time average of the dot or cross product of two time-harmonic complex quantities is equal to half of the real part of the respective product of one quantity and the complex conjugate of the other. In this regard, the time-averaged Poynting vector is given by... 

We now consider extinction from a computational point of view. In order to simplify the notations we will use the conventional expressions of the Poynting vectors and the electromagnetic powers in terms of the transformed fields introduced in Sect. 1.1 (we will omit the multiplicative factor 1/ eoMo)- The time-averaged Poynting vector S) can be written as [17]... 

Henceforth, Poynting vector refers to the time-averaged Poynting vector. 

The average energy flux in the evanescent wave is given by the real part of the Poynting vector S = (c/47t)ExH. However, the probability of absorption of energy per unit time from the evanescent wave by an electric dipole-allowed transition of moment pa in a fluorophore is proportional to lnfl - El2. Note that Re S and pa E 2 are not proportional to each other they have a different dependence on 0. 

The instantaneous Poynting vector (2.38) is a rapidly varying function of time for frequencies that are usually of interest. Most instruments are not capable of following the rapid oscillations of the instantaneous Poynting vector, but respond to some time average (S) ... 

The intensity I of the radiation along z is the energy passing per unit of time through a unit area perpendicular to the z-axis and is given by the time-averaged value of the Poynting vector S. 

The intensity of the wave, I, is the energy, time averaged over one period of oscillation, transferred by the wave across a unit area perpendicular to the direction of the Poynting vector per unit time. For a damped homogeneous wave of the type (1.14) propagating in the z-direction [6-13],... 

The flux density of an electromagnetic wave is described by the Poynting vector. For the case of the plane wave field one obtains for the time average in the direction of propagation... 

Thus, S3) = f e E H2 ) is constant throughout the domain, both inside and outside the NLC cell. Note that it is only the time-average of the z-component of the Poynting vector that is constant throughout the domain, not the overall Poynting vector, its magnitude, or S3 itself. We define the intensity 7 as this constant so that... 

The reflectivity, R, is defined as the ratio of the time averaged z-component of the Poynting vector of the reflected wave to that of the incident wave. The transmissivity, T, is defined as a similar quantity for the transmitted wave. The Poynting vector component of the corresponding wave normal to the boimd-... 

Poynting s vector theorem can be derived from Maxwell s equations as a consequence of power conservation. It shows that the time-averaged power produced by a distribution of currents with density J within volume is given by[l]... 

Using the properties of the plane-wave solutions discussed in section 2.3, we find that the time-averaged energy flux is given by the real part of the complex Poynting vector... 

The emitted radiation is linearly polarized in the plane containing tlie z-axis and the point of observation, P. The time-averaged intensity at P is given by the Poynting vector, equation (2.46) ... 

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