## REFLECTION FROM A PLANAR INTERFACE

Plane-Wave Incidence

Radio waves are often reflected from smooth, flat surfaces, such as building walls or ground. When the reflecting surface is not a perfect conductor, part of the radio energy penetrates the surface, and part of the energy is reflected. Reflection of a plane wave from a uniform half space with a planar inter­face can be analyzed exactly by matching boundary conditions at the interface, and the derived reflection coefficients can then be used in other practical applications.

Consider the idealized geometry in Fig. 2. A free-space plane wave propagating at an angle в0 to the surface normal

 and

 (k/k0)2 cos90 — !(k/k0)2 — sin2 90 (k/k0)2 cos90 + V'(k/k0)2 — sin2 00

 (15)

 Tv =

 For the further simplification to a dielectric (a = 0) reflecting medium, Eqs. (14) and Eq. (15) reduce to

 — V’ V — SI

 sin2 90

 cos 9,

 (16)

 rh =

 cos 90 + л/er — sin2 90

 and

 — V’ V — SI

 sin2 9n

 er cos 9,

 (17)

 er COS 9n + л/Єr — si

 • 2 „ sin 90 Figure 2. Geometry for plane-wave reflection from a homogeneous half space.

 where er = e/e0. As incidence angle в0 varies from 0 (normal incidence) to пУ2 (grazing incidence), rh varies smoothly from (1 — Ver)/ (1 + Ver) to —1. However, for vertical polarization, the reflec­tion coefficient rv equals 0 at the Brewster angle, 0B = tan—1(Ver). At this angle, all of the incident energy is refracted into the dielectric. An examination of Eq. (15) reveals that the presence of nonzero conductivity a (which yields a complex k) prevents rv from going to zero. However, if the imaginary Figure 3. Far-field radiation from a vertical electric dipole over a homogeneous half space. The reflection coefficient Tv is a function of the incidence angle в.

part of k is small, there is nevertheless a pseudo-Brewster angle (14) where |rv| goes through a minimum.

Dipole Sources

Consider now a vertical electric dipole source located at a height h over a reflecting half space, as in Fig. 3. In the far field the electric field has only a в component Ee, which can be written as the sum of a direct and a reflected ray:

_ jeOflQIl sin в є_#оГ ^ A cos g + T^e-jk0hcoSe^

4n r

where Tv is given by Eq. (13). For the special case of a per­fectly conducting ground plane (a = oo), the reflection coeffi­cient equals 1, and Eq. (18) reduces to

I = jtoV’fyll sm6>e^or cos (k0h cos в) (19)

2n r

Equation (19) has a maximum at the interface, в = ет/2.

The dual case of a vertical magnetic dipole source is shown in Fig. 4. The source is a small loop of area A and current 1, and the loop axis is in the vertical direction. The electric field is horizontally polarized, and in the far field the ф component Eф is

_ f]0k0IA sin в ^_jk^r cos g _^^ jkQhcosg (20) ф 4n r h

where Гь is given by Eq. (12). For the special case of a per­fectly conducting ground plane (a = oo), the reflection coeffi- Figure 4. Far-field radiation from a vertical magnetic dipole over a homogeneous half space. The reflection coefficient rh applies to hori­zontal polarization.

cient equals — 1, and Eq. (20) reduces to „ jnklA sin в

Ефа=оо = ————– ~———– Є 3 0 Sin(&0/l COS 6>) (21)

2n r

Equation (21) has a null at the interface, в = пУ2.

For realistic (finite) ground parameters, the reflection coef­ficients for both vertical polarization Tv and horizontal polar­ization rh equal —1 at grazing incidence (в = пУ2). Hence the direct and reflected rays cancel in Eqs. (18) and (20), and the electric field is 0:

Eeв =n/2 = 0 and Ефв =п/2 = 0 (22)

In reality, only the space wave (the inverse-distance field that occurs for в > 0) is 0 at the interface. The ground wave is the dominant field component near the interface, and it will be discussed in detail later. It has a more rapid decay with dis­tance, but it does not equal 0 at the interface.

## Wing Arrangements

Wing arrangements have a significant influence on the types of missile control to be used. Three types of wing arrange­ments are discussed here.

1. Cruciform. The most commonly used configuration in missile design is the cruciform, which possesses four wing surfaces and four tail surfaces. There are several major advantages in the use of this type of configura­tion: (i) fast response in producing lift in any direction, (ii) identical pitch and yaw characteristics, and (iii) sim­pler control system as the result of item (ii). One of the most important aspects associated with a cruciform de­sign is the orientation of the tail surface with respect to the wing planes. The significant conclusion from consid­erable experience and experimental data was that an in-line tail surface (i. e., all the four tail surfaces are in the same orientations as the four wing surfaces) pro­vides the best overall aerodynamic characteristics for most missile applications. The other possible wing-tail geometrical relation is called interdigitated configura­tion where there is a 45° separation between the wing and tail orientation. For a cruciform missile, the most difficult parameter to determine accurately is the in­duced rolling moment. The rolling moments arise when­ever the missile simultaneously executes pitch and yaw maneuvers that are unequal in magnitude. Such ma­neuvers result in unequal or asymmetric flow patterns over the aerodynamic lifting surface; consequently, roll­ing moments are induced on the airframe. Hence, roll stabilization or control is a critical issue for cruciform missiles.

2. Monowing. The monowing arrangements are generally used on cruise-type missile (i. e., missiles design to cruise for relatively a long range like crewed aircraft). This type of design is generally lighter and has less drag than the cruciform configuration. The wing area and span are, however, somewhat larger. Although the monowing missile must bank to orient its lift vector in the desired direction during maneuvering flights, the response time may be sufficiently fast and acceptable from a guidance-accuracy standpoint. The induced-roll problem for the monowing configuration is substantially less severe than that associated with the cruciform con­figuration. A separate set of lateral control surfaces, such as flaps, spoilers, and wing-tip ailerons, is gener­ally used in a monowing design. This stems from the fact that the canard or tail surfaces that are usually employed for pitch control on monowing design are gen­erally inadequate for lateral control.

3. Triform. This type of wing arrangement, which em­ploys three wings of equal area spaced 120° apart, is seldom used because no noticeable advantage can be re­alized. Results of a brief preliminary analysis indicate that the total wing area of the triform is equal to that used on a cruciform arrangement and that consequently no noticeable change in drag may be realized. In addi­tion, little or no weight saving will be realized, even though one less arrangement or fitting is required be­cause the total load remains the same.

## SEMICONDUCTOR TYPES

The nonlinear resistance of semiconductor junctions is used in the bridge-type fault current limiter, which combines power electronics and an SC coil (21). Figure 3(b) shows the circuit of a single phase. A bias power supply provides a cur­rent in all four semiconductors. As long as the load current is less than the bias current, all four semiconductors are for­ward biased, and the ac load current flows unimpeded by the bridge circuit, assuming negligible losses in the semiconduc­tors. In the quiescent condition, each thyristor conducts half the bias current superimposed with half of the line current. When a short circuit occurs, one pair of semiconductors is turned on, and the other pair is turned off in each half cycle, automatically inserting the bridge inductor into the circuit. The inductor limits the fault current. Because the inductor carries a bias current under normal condition, use of an SC coil reduces the overall system losses. The bridge-type fault current limiter has several attractive features, such as auto­matic insertion of the current-limiting reactor, reduction of the first half-cycle short-circuit current, precise control of the amplitude of the short-circuit current, complete current inter­ruption in less than a cycle if desired, operation with multiple fast reclosures, and high efficiency. In addition, it is conceiv­able to use the controlled thyristor bridge for other transmis­sion or distribution network functions such as var control. For distribution and transmission system semiconductor SFCLs, the equivalent thyristors indicated in Fig. 3(b) are series strings of thyristors, similar to those used in static var com­pensators or electronic transfer switches. A 2.4 kV, 2.2 kA single-phase fault current limiter was successfully tested in 1995 and a 15 kV, 800 A, three-phase unit is being designed to be tested in 1998 (2).

## Laser Doppler Velocimetry

Basic Principle. Laser Doppler velocimetry (LDV) is a rela­tively new clinical method for assessing cutaneous blood flow. This real-time measurement technique is based on the Dopp­ler shift of light backscattered from moving red blood cells and is used to provide a continuous measurement of blood flow through the microcirculation in the skin. Although LDV provides a relative rather than an absolute measure of blood flow, empirical observations have shown good correlation be­tween this technique and other independent methods to mea­sure skin blood flow.

According to the fundamental Doppler principle, the fre­quency of sound, or any other type of monochromatic and co­herent electromagnetic radiation such as laser light, that is emitted by a moving object is shifted in proportion to the ve­locity of the moving object relative to a stationary observer. Accordingly, when the object is moving away from an ob­server, the observer will detect a lower wave frequency. Like­wise, when the object moves toward the observer, the fre­quency of the wave will appear higher. By knowing the difference between the frequencies of both the emitted and the detected waves, the Doppler shift, it is possible to calcu­late the velocity of the moving object according to the follow­ing equation:

f = 2vjf cos в/е (1)

where f is the Doppler shift frequency, в is the mean angle that the incident light makes with the moving red blood cells, е is the speed of light, f0 is the frequency of the incident light, and v is the average velocity of the moving red blood cells. Because the red blood cells do not move through the microcir­culation at a constant velocity and light scattering leads to a wide distribution of angles в, the Doppler-shifted light con­tains a spectrum of different frequency components.

The Doppler shift of laser light caused by the average blood velocity in the capillaries (around 103 m/s) is very small and difficult to measure directly. Therefore, the frequency shifted and unshifted backscattered light components from the skin are mixed on the surface of a nonlinear photodiode. The out­put from the photodiode, an average dc offset voltage and a small superimposed ac component, is amplified and band pass filtered to eliminate low-frequency components in the range between 10 and 50 Hz. These frequencies are attributed to noise resulting from motion artifact and high-frequency noise components (typically in the kilohertz range) resulting from nonbiological noise. As the average red blood cells (RBC) ve­locity is increased, the frequency content of the ac signal changes proportionally.

Assuming a constant blood flow geometry, Stern (24) pro­posed the following empirical relationship between the ampli­tude of the Doppler-shifted spectrum and the velocity of the blood flow:

F =(o2P((o)dco (2)

where F is the root-mean-square (rms) bandwidth of the Doppler power spectrum signal, w is the angular frequency, and P(w) is the power spectral density of the Doppler signal. To compensate for laser light intensity, skin pigmentation, and numerous other factors that affect the total amount of light backscattered from the skin, the flow parameter is usu­ally calculated by multiplying the percentage of light reflected from the moving RBCs by the mean photodiode current, which is a function of the average backscattered light in­tensity.

Instrumentation. The original light source used in LDV was a HeNe laser (25,26). Newer systems use a much smaller and less expensive single-mode semiconductor laser diode in the near-infrared region around 750 to 850 nm as a light source. These wavelengths are near the isosbestic wavelength of oxy and deoxyhemoglobin (i. e., 810 nm) so that changes in blood oxygenation have no effect on the measurement. Some LDV systems are equipped with different light sources (e. g., green, red, or near-infrared), which allow measurement from differ­ent tissue layer depths because light penetration depth is wavelength-dependent. Typical output powers used in LDV range from 1 to 15 mW.

In most LDV systems, the laser output is coupled through a small focusing lens into the polished end of a flexible plastic or silica optical fiber (25 to 1000 ^m core diameter), which illuminates the blood directly in invasive measurements or the surface of the skin in noninvasive applications. Light backscattered from the biological media is collected either by the same optical fiber used for illumination or by a separate receiving fiber mounted in close proximity to the illuminating fiber tip. A rigid probe helps to maintain the two optical fiber tips parallel to each other and also perpendicular to the sur­face of the illuminated sample. Depending on the application, a wide selection of probe geometries and sizes are available commercially. In invasive applications, the optical fibers can also be inserted through a catheter for measurement of flow inside a blood vessel. In most noninvasive applications, the flow probes are attached to the surface of the skin by a dou­ble-sided adhesive ring. Because blood perfusion is strongly dependent on skin temperature, some LDV systems also have probes with built-in heaters to control and monitor skin tem­perature. Absolute calibration of an LDV instrument is inherently difficult to obtain because blood flow in the skin is highly com­plex and variable. Because accurate calibration standards or suitable physical models of blood flow through the skin do not exist, instrument calibration is usually accomplished empiri­cally either from an artificial tissue phantom, which is often made out of a colloidal suspension of latex particles, or by comparing the relative output from the laser Doppler instru­ment with other independent methods for measuring blood flow.

In practice, most commercial systems express and display the Doppler-shifted quantity measured by the instrument ei­ther in terms of blood flow (in units of milliliters per minute per 100 g of tissue), blood volume (in milliliters of blood per 100 g of tissue), or blood velocity (in centimeters per second).

The clinical and medical research applications of LDV range from cutaneous studies of ischemia in the legs (27) to general subcutaneous physiological investigations related to the response of various organs to physical (temperature, pres­sure) and chemical (pharmacological agents) perturbations that can alter local blood perfusion. LDV has been used exten­sively in dermatology to assess cutaneous microvascular dis­ease (28,29), arteriosclerosis, or diabetic microangiopathy; in plastic surgery to determine the postoperative survival of skin grafts; in ophthalmology to evaluate retinal blood flow (30,31); and in evaluating skeletal muscles (32). To date, LDV re­mains mainly an experimental method. Although it has been widely used as a research and clinical tool since the mid 1970s, LDV has not reached the stage of routine clinical ap­plication.

Fluorescence Spectroscopy

Many dyes that absorb energy can reemit some of this energy as fluorescence. Laser-induced fluorescence emission is cur­rently being investigated for the early detection, localization, and imaging of normal and abnormal tissues, determining whether a tumor is malignant or benign and identifying ex­cessive areas of atherosclerotic plaque. One of the future goals is to incorporate this technique into special fiber optic based guidance systems used during ablation or laser angioplasty particularly inside the coronary arteries.

Diffuse Reflectance and Transillumination Spectroscopy

Several methods are being developed to measure the absorp­tion spectra of tissues illuminated by laser light. In a rela­tively new technique known as photon time-of-flight spectros­copy, researchers are trying to measure the temporal spreading of very short pulses of laser light as photons un­dergo multiple scattering in the tissue. By measuring the time it takes the light to travel through the tissue it is possi­ble to estimate how much light scattering and absorption oc­curs. Some of these time-resolved or ‘‘photon migration’’ meth­ods are being evaluated clinically as a potential alternative to ionizing radiation used in X-ray mammography for early noninvasive diagnosis of breast cancer.

## Radiation from a Hertzian Dipole

This article covers only steady-state, time-harmonic sources and fields (12) with time variation exp( jwt), where the angu­lar frequency w = 2nf and f is the radio frequency. The time dependence is suppressed in the equations. The basic source is a current 1 extending over an incremental length l. This elementary dipole source is called a Hertzian dipole and has a moment 1l. If the dipole is directed along the z axis, as shown in Fig. 1, the radiated electric field has two compo­

 z Equation (4) gives the electric field components of charges ± 1/jw separated by a distance l. The magnetic field of an elec­tric dipole has no quasi-static term. At intermediate dis­tances (k0r « 1), the inverse-square terms dominate Eqs. (1)- (3), and the field is called the induction field.

For most practical applications, such as communications or radar, the far fields are of interest. At large distances (k0r > 1), the inverse-distance terms dominate Eqs. (1) and Eq. (3), and the fields are approximated by

where the free-space wave number k0 = wV^0e0, the free – space impedance = V^0/e0, is the magnetic permeability of free space, and e0 is the dielectric permittivity of free space. The equations for radiation by the dual source (a magnetic dipole or small loop) are given in Ref. 12.

Very close (k0r < 1) to the Hertzian dipole, the electric field is dominated by the r~3 inverse cube term, and Eqs. (1) and (2) can be approximated by

Even though Eq. (5) applies to a Hertzian dipole, it illustrates the more general far-field properties that the electric and magnetic fields are related by the free-space impedance and they are orthogonal to each other and to the radial direction of propagation. Hence the radial electric field Er in Eq. (2) has no inverse-distance term.

The power density of the electromagnetic field is called the Poynting vector S and can be written

where r is the unit vector in the radial direction. The sin20 factor in Eq. (7) is specific to the radiation pattern of a Hert­zian dipole, but the inverse-square dependence applies to the far field of any radiator. The total radiated power P can be obtained by integrating Eq. (7) over a far-field sphere (12):

 and Нф = e-Jkor (J^ + 4 ) sin0

 e-‘Vsin0 and EB

 k0Il sin0 4 nr

 Il cos в 2n joe0r3

 S=ExH*

 jKr3

 jk0r3

 sin в

 cos в

 4n r

 1

 2

nents, Ee and Er, and the radiated magnetic field has only a single component Иф. The expressions for these field compo­nents can be derived from scalar and vector potentials (12):

where boldface denotes vectors and * denotes complex conju­gate. The far-field expression for S can be obtained by substi­tuting Eq. (5) into Eq. (6):

 щ(к0т2 6ж

 d9 r2 sin 9r ■ S =

 P =

 (8)

 (12)

 rh =

 r ptGtG^0

 (9)

 (4n D)2

 The result for the total radiated power in Eq. (8) is indepen­dent of the radius r at which the integration is evaluated, but the evaluation is simplest in the far field, where S can be approximated by Eq. (7). Free-Space Transmission Loss Now consider free-space transmission between a pair of an­tennas in the far fields of each other. The received power Pr can be written (13)

 p PtGfi r (2 k0D)2

 р2ж рл / йф / J0 J0

 is incident on a half space with permittivity e, electrical con­ductivity a, and permeability x. The angle of reflection 6r equals the angle of incidence, 6r = 60. The reflection coefficient depends on the polarization of the incident field. For horizontal polarization (electric field perpendicular to the plane of incidence), the reflection coefficient rh is given by (1)

 where k = wV^(e — ja/ш). For vertical polarization (electric field parallel to the plane of incidence), the reflection coeffi-

 likо cos 90 — /x0Vk2 — k2 sin2 90 jikо cos 90 + jif^k2 — k2 sin2 90

 cient rv is
 0
 (13)
 Гу =

where Pt is the transmitted power, Gt is the gain of the trans­mitting antenna, Gr is the gain of the receiving antenna, D is the separation distance between the antennas, and the free – space wavelength A0 = 2w/k0. The D—2 factor represents the inverse-square dependence of the radiated power density available at the receiving antenna.

The antenna and propagation effects can be separated by writing Eq. (9) in the following form:

(10)

4n D

The squared factor on the right side of Eq. (10) is dimen – sionless and does not involve the antenna gains. The recipro­cal is called the free-space transmission loss L0 and is usually expressed in decibels: jiffe2 cos 90 — jikf^k2 — k2 sin2 90 jiffk2 cos 90 + iiko’Jk2 — k2 sin2 90

The reflection coefficients in Eq. (12) and (13) apply to both the reflected electric and magnetic fields. The power reflection coefficients are obtained by taking the squares of the magni­tudes of the field reflection coefficients, |Tv|2 and |ГЬ|2. In gen­eral, the reflection coefficients in Eqs. (12) and (13) are com­plex because k is complex. Thus the reflected field undergoes phase shift as well as reduction in amplitude. In the limiting case of grazing incidence (00 = w/2), both rh and rv equal —1.

For the common case where the reflecting medium is non­magnetic (x = juo), the reflection coefficients in Eqs. (12) and (13) simplify to

 cos 00 — V’ (k/k0)2 — sin2 00 cos 90 + V(k/k0)2 — sin2 90

 (14)

 rh =

L0 = 101og10 dB = 201og10 dB

 (11)

Normally the antenna gains and the power ratio in Eq. (10) are also expressed in decibels.