Complex amplitude

Assuming kinematical diffraction theory to be applicable to the weakly scattering CNTs, the diffraction space of SWCNT can be obtained in closed analytical form by the direct stepwise summation of the complex amplitudes of the scattered waves extended to all seattering centres, taking the phase differenees due to position into aeeount. 

The diffraction space of MWCNTs can be computed by summing the complex amplitudes due to each of the constituent coaxial tubes. Taking into account possible differences in Hamada indices (Lj, Mj) as well as the relative stacking (described by zQ.j, can formally write... 

The diffraction space of ropes of parallel SWCNT can similarly be computed by summing the complex amplitudes of the individual SWCNTs taking into account the relative phase shifts resulting from the lattice arrangement at... 

In astronomy, we are interested in the optical effects of the turbulence. A wave with complex amplitude U(x) = exp[ irefractive index, resulting in a random phase structure by the time it reaches the telescope pupil. If the turbulence is weak enough, the effect of the aberrations can be approximated by summing their phase along a path (the weak phase screen approximation), then the covariance of the complex amplitude at the telescope can be shown to be... 

The total (complex) amplitude of the waves at a point r, V(r) is the superposition or sum of the amplitudes due to the waves from each pinhole,... 

At distances that are very large compared to either the source dimensions, or the aperture of our optical system the complex amplitude is just the Fourier transform of the object complex amphtude. Furthermore, it can be shown that the complex amplitude in the image plane of an optical system is just the Fourier transform of the complex amplitude at the aperture plane, (for a complete derivation of this see Goodman, 1996). 

The complex amplitude of the signal field at the output of the crystal is amplified by a factor g with respect to the input field, g is always larger than one so that there is always coherent amplification of the signal field one speaks of parametric amplification. Nonlinear optics provides an amplification different from the amplification occurring in a medium which present population inversion. For a t)q)ical value of the nonlinear coefficient of 1 pm/V, a pump of... 

A complete description of the method requires a procedure for selecting the initial conditions. At t 0, initial values for the complex basis set coefficients and the parameters that define the nuclear basis set (position, momentum, and nuclear phase) must be provided. Typically at the beginning of the simulation only one electronic state is populated, and the wavefunction on this state is modeled as a sum over discrete trajectories. The size of initial basis set (N/it = 0)) is clearly important, and this point will be discussed later. Once the initial basis set size is chosen, the parameters of each nuclear basis function must be chosen. In most of our calculations, these parameters were drawn randomly from the appropriate Wigner distribution [65], but the earliest work used a quasi-classical procedure [39,66,67], At this point, the complex amplitudes are determined by projection of the AIMS wavefunction on the target initial state (T 1)... 

We have used various integrators (e.g., Runga-Kutta, velocity verlet, midpoint) to propagate the coupled set of first-order differential equations Eqs. (2.8) and (2.9) for the parameters of the Gaussian basis functions and Eq. (2.11) for the complex amplitudes. The specific choice is guided by the complexity of the problem and/or the stiffness of the differential equations. 

Here complex amplitude of the pulse envelope E x, z, t) is also a slowly varying function of z and t. Spatiotemporal distribution of the electric field is described by E x,z,t) = E x,z,t)ex-p[iujQt— i/Sz), f3 being the longitudinal wavenumber of the waveguide mode at the pulse peak. 

According to (2.18), an arbitrary time-dependent function can be expressed as a superposition of time-harmonic functions exp( — o/), where the complex amplitude %(u>) depends on the frequency to. The condition that F(t) be real is that <3r (w) = <3r( — w) therefore, F(t) can be expressed as a superposition of time-harmonic functions with positive frequency ... 

The limit cycle is an attractor. A slightly different kind occurs in the theory of the laser Consider the electric field in the laser cavity interacting with the atoms, and select a single mode near resonance, having a complex amplitude E. One then derives from a macroscopic description laced with approximations the evolution equation... 

RF frequency is 10.6 GHz and bandwidths are 20 MHz. Phase and Amplitude are automatically controlled by the digital complex amplitude equalizers. Design of two dimensional systems in progress of 8x8 and 64x64 are discussed. 

Even more interesting sounds can be made by more complex usage of the FM formulas. With frequency modulation one might select more than one modulating waveform, or perhaps different waveforms than sinusoids. In addition, a complex amplitude modulation can be imposed. For example, one possibility is revealed in the trigonometric relation... 

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 ... 

The drawback of Eq. (14c) is that the integrands comprise a dynamic quantity p (f) and an induced distribution F( y). Both are perturbed by radiation field and therefore depend on its complex amplitude E. Because of that further calculations become cumbersome [18]. It is possible to overcome this drawback on the basis of a linear-response approximation. The field-induced difference 8p of the law of motion p (f) from the steady-state law p (f) is proportional to the field amplitude E. The same is supposed with respect to the difference F(y) — 1 of induced and homogeneous distributions (for the latter F = 1). A steady-state dipole s trajectory does not depend on phase y and therefore does not contribute (at F = 1) to the integral (14c). Then in a linear approximation we may represent the average7 of p (f)F(y) over y as a sum... 

We express the left-hand part of this formula through the complex amplitudes and then equate expressions by the same exponential factors exp( / y). In view of Eq. (29) we find... 

To relate the complex permittivity s of a polar medium with the complex susceptibility % provided by motions of the dipoles, we suggest that a polar medium under study is influenced by the external macroscopic time-varying electric field Ee(f) = Re[Em exp(imf)], where Em is the complex amplitude. This field induces some local field EM(f) = Re[ ) exp(icof)] in a cavity surrounding each polar molecule. A given molecule directly experiences the latter field. 

If we neglect the difference between two complex amplitudes, Em and E , then the complex permittivity s of a polar medium and the complex susceptibility x provided by motions of the dipoles would be related as follows ... 

A more rigorous theory [40, 41] accounting for an internal field correction yields the following ratio of two field complex amplitudes ... 

Consider the response of this nonlinear medium to a harmonic electric field of angular frequency to (wavelength Ao = 2nco/u>) and complex amplitude E uj) ... 

The source S(t) = -pod2PNL(t)/dt2 corresponding to (4.22) has a component at frequency 2lo and complex amplitude S(2lo) = 4pb0u)2dE(uj)E u), which radiates an optical field at frequency 2co (wavelength Ao/2). Thus the scattered optical field has a component at the second harmonic of the incident optical field. Since the amplitude of the emitted second-harmonic light is proportional to S(2lo), its intensity is proportional to S 2uj) 2au)id I2, where I = E(uj) 2/2r/ is the intensity of the incident wave. The intensity of the second-harmonic wave is therefore proportional to d2, to 1/Ag, and to I2. Consequently, the efficiency of second-harmonic generation is proportional to / = P/A, where P is the incident power and A is the cross-sectional area. It is therefore essential that the incident wave have the... 

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