Time-harmonic fields

The vector Fc is a complex representation of the real field F. If all our operations on time-harmonic fields are linear (e.g., addition, differentiation, integration), it is more convenient to work with the complex representation. The reason this may be done is as follows. Let be any linear operator we can operate on the field (2.10) by operating on the complex representation (2.11) and then take the real part of the result  

Note that there is a degree of arbitrariness associated with the complex representation of a real field F could just as easily have been written F - Re(F, where F = C exp(/to/) and the asterisk denotes the complex conjugate. Thus, there are two possible choices for the time-dependent factor in u complex representation of a time-harmonic field exp(/co/) and exp( — iat). It mukcs no difference which choice is made the quantities of physical interest arc ttlwuys real. But once a sign convention has been chosen it must be used conniMcntly in all analysis. We shall take the time-dependent factor to be exp(-/u /) this is the convention found in standard books on optics (Born Mini Wolf, 1965) and electromagnetic theory (Stratton, 1941 Jackson, 1975) as 

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To obtain the frequency-dependent susceptibility x(<°), we need the polarization in response to a time-harmonic field E0e ... 

Besides its appearance in the FFMF equation in plasma physics, as well as associated with time-harmonic fields in chiral media, the chiral Beltrami vector field reveals itself in theoretical models for classical transverse electromagnetic (TEM) waves. Specifically, the existence of a general class of TEM waves has been advanced in which the electric and magnetic field vectors are parallel [59]. Interestingly, it was found that for one representation of this wave type, the magnetic vector potential (A) satisfies a Beltrami equation ... 

Let us now assume that all fields and sources are time harmonic (or monochromatic), which means that their time dependence can be fully described by expressing them as sums of terms proportional to either cos cot or sin where co is the angular frequency. It is standard practice to represent real monochromatic fields as real parts of the respective complex time-harmonic fields, e.g.,... 

It is straightforward to verify that the choice of the exp(itime dependence in the complex representation of time-harmonic fields in Eq. (4.1) would have led to m = - iirii with a non-negative mi. 

R. F. Harrington, Time-Harmonic Fields, McGraw-Hill, New York, 1961, pp. 163-168. 

The source of radiation, in the classical electromagnetic theory, is an accelerated charge. For time-harmonic fields, electrical current serves as the source. There is a considerable amount of literature on the radiation properties of apertures and antennas at radio and microwave frequencies. At these frequencies, the penetration of the fields into a metal is small. Thus, it is frequently quite acceptable to model these structures by assuming the metals are PECs. At optical frequencies, a significant portion of the incident energy can be dissipated in the metal. In addition, typical metals exhibit surface plasmon resonances at optical frequencies. Associated with a surface plasmon is an oscillating charge distribution on the surface of the structure, localized within the skin depth of the metal. 

The electromagnetic fields (x,r) and H(x, t) associated with scattering from a microsphere satisfy Maxwell s equations. For a homogeneous, isotropic linear material the time-harmonic electrical held E and the magnetic held H satisfy vector wave equations, which in SI units are (Bohren and Huffman, 1983)... 

Let us now assume that the fields are time harmonic with angular frequency... 

Thus, the perhaps unfamiliar constitutive relations (2.23)-(2.25) yield familiar results when the fields are time harmonic moreover, because of (2.26) and (2.27), physical meaning can now be attached to the phenomenological coefficients even for arbitrarily time-dependent fields. 

Consider an electromagnetic field (E, H), which is not necessarily time harmonic. The Poynting vector S = E X H specifies the magnitude and direction of the rate of transfer of electromagnetic energy at all points of space it is... 

In addition to irradiance and frequency, a monochromatic (i.e., time-harmonic) electromagnetic wave has a property called its state of polarization, a property that was briefly touched on in Section 2.7, where it was shown that the reflectance of obliquely incident light depends on the polarization of the electric field. In fact, polarization would be an uninteresting property were it not for the fact that two waves with identical frequency and irradiance, but different polarization, can behave quite differently. Before we leave the subject of plane waves it is desirable to present polarization in a systematic way, which will prove to be useful when we discuss the polarization of scattered light. 

We showed in Chapter 3 that a physically realizable time-harmonic electromagnetic field (E, H) in a linear, isotropic, homogeneous medium must satisfy the wave equation... 

The electric field is taken to be time harmonic with frequency co. As in previous chapters, we shall deal with the complex representations of the real... 

Given the response of a single oscillator to a time-harmonic electric field, the optical constants appropriate to a collection of such oscillators readily follow. The induced dipole moment p of an oscillator is ex. If 91 is the number of oscillators per unit volume, the polarization P (dipole moment per unit... 

For time derivative operators, the time derivative of A is known to have a direchon different from A. However, in some cases, and as observed in spacelike derivations, we may expect that second-order derivations of A with respect to time lead to a vector having the same direction as A. This, obviously, is the case of A vectors, which are time-harmonic functions. These remarks can be extended, and therefore applied, to the case of the electromagnetic field, whose calculation is based on the following set of relations ... 

INTRODUCTION. A standard and universal description of various nonlinear spectroscopic techniques can be given in terms of the optical response functions (RFs) [1], These functions allow one to perturbatively calculate the nonlinear response of a material system to external time-dependent fields. Normally, one assumes that the Born-Oppenheimer approximation is adequate and it is sufficient to consider the ground and a certain excited electronic state of the system, which are coupled via the laser fields. One then can model the ground and excited state Hamiltonians via a collection of vibrational modes, which are usually assumed to be harmonic. The conventional damped oscillator is thus the standard model in this case [1]. 

XVII. Beltrami Field Relations from Time-Harmonic EM in Chiral Media... 

A complete set of standard time-harmonic solutions to Maxwell s equations usually involve the plane wave decomposition of the field into transverse electric... 

TE) and transverse magnetic (TM) parts. However, Rumsey [53] detailed a secondary method of solving the same equations that effected a decomposition of the field into left-handed and right-handed circularly polarized parts. For such unique field solutions to the time-harmonic Maxwell equations (e = electric permittivity, p = magnetic permeability) ... 

XVII. BELTRAMI FIELD RELATIONS FROM TIME-HARMONIC EM IN CHIRAL MEDIA... 

When we consider time-harmonic electrodynamics in more general media (chiral-biisotropic), A. Lakhtakia also underscored the importance of the Beltrami field condition [54], In particular, he found that time-harmonic EM fields in a homogeneous reciprocal biisotropic medium are circularly polarized, and must be described by Beltrami vector fields. 

A. Lakhtakia et al., Time-Harmonic Electromagnetic Fields in Chiral Media, Springer-Verlag, Berlin, 1989. 

It was indicated numerous times that the SHG intensity is dependent on both the magnitude of x<2) tensor elements as well as the phase relationships between fundamental and harmonic fields in the crystal. Under certain circumstances, it is possible to achieve phase matched propagation of the fundamental and harmonic beams. Under these conditions, power is continually transferred from the fundamental to harmonic beam over a path length, which is only limited by the ability... 

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