18.1.Reflection and Refraction at The Boundary of Two Nonconducting Media: Normal Incidence
Since only the field ratios are determined, it is convenient to introduce a special notation for them,
and are called Fresnel coefficients for normal incidence for reflection and transmission, respectively. The subscripts indicate that the wave is incident from medium 1 onto medium2. Thus the solution is given as
,
We define the reflectance Rn and the transmittance Tn for normal incidence by the ratios the intensities
,
,
18.2.Reflection and Refraction at The Boundary of Two Nonconducing Media: Oblique Incidence
A more general case than that discussed in the preceding section is that of reflection of obliquely incident plane waves by a plane dielectric interface. Consideration of this case leads to three well-known optical laws: Snell’s law, the law of reflection, and Brewster’s law governing polarization by reflection.
E1 =
E2 =
Where , , . The propagation vectors are k1 = k1u1, and so on, and the unit normal to the boundary is n = k. The plane defined by k1 and n is called the plane of incidence, and its normal is in the direction of n x k1. In fact, (n x k1)/│n x k1│= j, the unit vector in the y direction. The p-component of polarization has been chosen to lie parallel to the plane of incidence (p for”parallel”).
18.3.Brewster’s Angle: Critical Angle
Provides a means for determining the value of . Using in Snell’s law gives
or
18.4.Complex Fresnel Coefficients : Reflection From A Conducting Plane
The complication that arose in the last section for angels of incidence greater than the critical angle, namely , leads us to consider complex Fresnel coefficients. Since , a real value of sin greater than 1 implies a pure imaginary value of cos , so that in the Fresnel coefficient is imaginary and they are complex. They would also be complex if medium 2 were conducting, since in that case n2 is complex. Snell’s law
Since all the transmitted energy is eventually absorbed in a semi-infinite conducting medium, we define the absorptance as
A = 1 – R
For normal incidence,
An =
18.5.Reflection and Transmission by A Thin Layer: Interference
Note that for given n1 and n2, depends only on the angle of incidence ; it varies from 0 to as increases from to /2. The exponent y depends on the thickness d (as well as op through ). If y is not too small,
Note also that is proporsional to , so frustrated total reflection can be observed with microwaves on a larger scale.
18.6.Propagation Between Parallel Conducting Plates
Guided waves are another problem that can be treated by considering the interfrence between an incident and a reflected wave, or alternatively by starting with a new boundary-value problem that involves simultaneously satisfying the conditions at multiple boundaries. We again begin with the first approach. Now we are interested in the wave propagating in a dielectric medium, say air, which is bounded by conducting surfaces. Waveguides for microwaves are an application of this problem. As a simplification we idealize the conductivity of the metal to be infinite.
18.7.Waveguides
In addition to these wave equations, Maxwell’s equation must be satisfied. For the transverse electric (TE) case propagating in the z direction, Ez = o;
18.8.Cavity Resonators
Another type of device closely related to waveguides and of considerable practical importance is the cavity resonator. Cavity resonators display the properties typical of resonant circuits in that they can store energy in oscillating electric and magnetic fields. Furthermore, practical cavity resonators dissipate a fraction of the stored energy in each cycle of oscillation. In this letter respect, however, cavity resonators are usually superior to conventional L-C circuits by a factor of about twenty—that is, the fraction of the stored energy dissipated per cycle in an L-C circuit. An additional advantage is that cavity resonators of practical size have resonant frequencies which range upward from a few hundred megahertz—just the region where it is almost impossible to construct ordinary L-C circuits.
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