Complex amplitude vector

Figure A6.1 Representation of an atomic scattering factor as a complex amplitude vector...

The complex amplitude vectors and u are conjugates and so the solution can be simplified in the following way ... 

In the frequency domain, the complex amplitude vector E q defined as... 

Let the wave vector k be normal to electric field (k LE), E and J are the corresponding complex amplitudes, is a complex-conjugation symbol, i = %/—1, and co is angular frequency of radiation. Since in the case of transverse wave div E = 0, Eq. (1) in representation (2) reduces to the following equation for the complex amplitudes ... 

Attenuation of a wave can be accounted with the help of either complex frequency 0) = 0)1 + ia>2 or complex wave vector k = Rek + /Imk. We will consider frequency as a real value, while the wave vector as a complex one and use notation k = (tt)/v — ia)sk, where is the unit vector of k-direction and attenuation coefficient a describes the reduction of amplitude by the factor exp(— r). Thus the expression ... 

The global energy conservation condition, Eq. (4.11), is explicitly demonstrated for resonant processes up to third order (5 3), particularly for resonant passive processes, at exact resonance, where population change can be achieved at a nonquadrature level by the fields. The phase matching condition is assumed, Ak = 0. As before, the material resonance at the one-photon level is taken into account by the complex wave vector, k, = k -I-ik", whose imaginary part is absorbed into the amplitude, ,( ) = sxp[-k" r]. The electrical susceptibility is expressed in terms of the scalar Cartesian component, as given in Eq. (2.17). 

For a spectrum of N lines we consider a vector x, of N components xf, where x, is the complex amplitude of the transition /- /,/= 1,. .., JV. We write xf as a vector since as a complex number it itself has two components. By a distribution of amplitudes we mean a probability density function P(x) (107), such that the observed value of the transition intensity is ( x, 2) ... 

In order to investigate the effect caused by finite dimensions of a transmitter coil let us first derive formulae for the vector potential of the electrical type corresponding to a current element. As is well known, complex amplitudes of the field are described by Maxwell s equations ... 

Here Eo and EH are the complex amplitudes associated with the incident and diffracted X-ray plane-waves, K0 and Kh are the respective complex wave vectors inside the crystal, and v is the X-ray frequency. The two wave vectors are coupled according to the Laue condition, with... 

Equations (12.4)-(12.9) describe an outgoing transverse spherical wave propagating radially with the phase velocity v = cojk and having mutually perpendicular complex electric and magnetic field vectors. The wave is homogeneous in that the real and imaginary parts of the complex wave vector kx are parallel. The surfaces of constant phase coincide with the surfaces of constant amplitude and are spherical. Obviously,... 

The reflectivity is calculated using the Fresnel reflection coefficient r, which is defined as 7 rp/ ip, where rp and Eip are the complex amplitudes of the electric vectors of the reflected and incident p-polarized light respectively. Values may also be calculated for = Ers/Eis, the reflection... 

Eq and Bq are constant complex amplitudes, and e- and 2 shall be constant unit vectors. As a consequence the electric field vector always points in the direction of Cy and such an electromagnetic wave is said to be linearly polarized. By superpositions of two such linearly polarized waves with different phases and amplitudes so-called elliptically polarized waves may be constructed. However, we do not need to further discuss this possibility here. 

Neglecting reflection and absorption, the incident beam of light with a polarizahon state can be conveniently described by the Jones vector, which is expressed in terms of complex amplitudes as a column vector ... 

Figure 9.7 Definition of the plane of incidence (p plane) and the incidence angle 4>o through the wave vectors of the incident and emerging (reflection set up) plane waves. Ap, A, Bp, and 6, denote the complex amplitudes of the p and s modes...

ML estimation of the (s, v) parameters. The GLRT in Eq. 7 allows also demonstrating the effectiveness of an imaging approach, as in SAR tomography, with respect to the classical interferometric processing. The PSI technique, in fact, exploits a similar test but for the use of only the phase information of each element of the vector g (Ferretti et al. 2000, 2001) the exploitation of the whole complex, amplitude plus phase, data as in Eq. 7 provides a significant increase, with respect to PSI, of the detection probability for a fixed false alarm rate. 

Here o. and j3 are complex amplitudes which are arbitrary as long as f2) is a proper state vector with norm 1. In particular, if ) and ) are orthogonal states this requires lop + /3 = 1. 

When 8j and 82 have different phases, the tip of the electric vector traces out an ellipse in any fixed plane perpendicular to k and the wave is said to be elliptically polarized. This general case is roost easily discussed if we choose a coordinate system in which the z-axis coincides with the direction of propagation k and 2 te in the directions Ox, Oy respectively. If the complex amplitudes are... 

The reflectivity of an interface is calculated from Fresnel s equations, which are derived from the boundary conditions that the tangential (t) component of the electric field vector, and the normal (n) component of the electric displacement vector, D = e - E , are continuous across the inter-face. We define the so-called Fresnel reflection coefficient r as the ratio of the complex amplitudes of the electric field vectors of the reflected and the incident waves ... 

A exp(i(p) defines the length and direction of the radius vector (A) at the initial instant of time and is referred to as complex amplitude (which we shall designate a)... 

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