# Showing papers in "IEEE Transactions on Microwave Theory and Techniques in 1963"

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TL;DR: In this paper, the influence of the unevenness of the reflectors is studied theoretically, and the experimental results are in qualitative agreement with the theory of thin parallel wire gratings.

Abstract: Fabry-Perot Interferometers (FPI's) have been constructed for the far infrared and submillimeter wave region with metal grids as reflectors. Two-dimensional grids with square holes (metallic mesh) can be used for unpolarized radiation, too, and they are more convenient for the construction than one-dimensional (parallel wire) grids. The performance of several FPI's has been measured in the 100-600 /spl mu/ wavelength region. Q values in first-order range from 5 to 30, and peak transmissions up to 0.9 have been reached. The experimental results are in qualitative agreement with the theory of thin parallel wire gratings. The influence of the unevenness of the reflectors is studied theoretically. Applications of a submillimeter FPI include its use as the dispersion element in a spectrometer, as a narrow-band filter to check the radiation purity of a grating spectrometer, and as a separator of the harmonics from a crystal harmonic generator.

131 citations

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TL;DR: In this article, an exact method for the design of band-stop filters which adapts synthesis techniques due to Ozaki and Ishii is discussed, which places no theoretical limit on the width of the stopband, although, for practical reasons, different circuit configurations are used for stop-bands of different widths.

Abstract: An exact method for the design of band-stop filters which adapts synthesis techniques due to Ozaki and Ishii is discussed. This method places no theoretical limit on the width of the stop-band, although, for practical reasons, different (but equivalent) circuit configurations are used for stop-bands of different widths.These configurations include a form having open-circuited shunt stubs separated by lengths of line; a second form using resonators which are separate from the main line but parallel to it, so that coupling takes place by way of fringing fields; and a third form in which the resonators are attached directly to the main line, but are folded parallel to it so that coupling is both by direct connection and by fringing fields. Easy to use formulas are given for the exact design of band-stop filters from low-pass prototype filters and equations are given for converting from one form of filter structure to any of the other equivalent forms. The experimental results of trial designs are presented.

122 citations

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TL;DR: In this article, an exact synthesis procedure for a class of asymmetric multielement coupled-transmission-line directional couplers with any number of elements is derived based on the equivalence between the theory of the directional coupler and that of a stepped quarterwavelength filter.

Abstract: An exact synthesis procedure is derived for a class of asymmetric multielement coupled-transmission-line directional couplers with any number of elements. It is based on the equivalence between the theory of the directional coupler and that of a stepped quarter-wavelength filter. This can be treated using Richards' theorem for the synthesis of transmission-line distributed networks, as described previously by Riblet. The method is extended to give a general expression for the input reflection coefficient of the stepped filter, which corresponds to the voltage coupling of the directional coupler. Explicit formulas for the parameters of two, three, four and five couplers are derived and the extension to larger number of elements is straightforward. Two and three element couplers have been designed on this theoretical basis, and show excellent agreement with theory, for example a three element coupler of 20 db /spl plusmn/ 0.5 db over a 6:1 bandwidth, and a two element coupler of 3.2 db /spl plusmn/ 0.85 db over a 6.7:1 bandwidth. It is possible to design a 3-db /spl plusmn/ 0.43 db-coupler for decade bandwidths using only four elements. The 3 db-couplers may be used as 90/spl deg/ hybrids by careful choice of reference planes in the output parts.

115 citations

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TL;DR: In this article, a reentrant cross section was proposed for parallel-coupled TEM-mode cross sections and a series of modifications resulted in a final model having a coupling variation within 0.5 db, and a minimum directivity of about 22 db.

Abstract: A new type of parallel-coupled TEM-mode cross section is described and named the re-entrant cross section. An analysis of the even- and odd-mode characteristic impedances of the re-entrant cross section shows it to have advantages in the case of tight coupling when compared to previously used parallel-strip cross sections. Close tolerances are easily held, and considerable misalignment is permissible. Two single-section 3-db couplers were tested with coupling curves very close to theoretical, and with good directivity. Then a three-section coupler having a re-entrant center section was designed for the 400- to 2000-Mc band, yielding a coupling variation less than 0.4 db and a minimum directivity of 29 db. Next a three-section coupler was designed for the 1- to 5-Gc band. A series of modifications resulted in a final model having a coupling variation within 0.5 db, and a minimum directivity of about 22 db.

88 citations

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TL;DR: The results of detailed calculations based on Bosma's theory of 3-port stripline circulators are presented in this paper, where the analysis has been modified to enable the scattering matrix, and therefore the circulator bandwidth, to be found.

Abstract: The results of detailed calculations based on Bosma's theory of 3-port stripline circulators are presented. The analysis has been modified to enable the scattering matrix, and therefore the circulator bandwidth, to be found. The equations of the 4-port junction have been derived and an expression for the scattering matrix of the general m-port junction is given. A comparison of experimental and theoretical results for the 3-port junction shows good agreement for circulators designed in the frequency band 2-3 Gc. ,

82 citations

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TL;DR: In this paper, a new waveguide for low-loss transmission of millimeter-wave is presented, which consists of two parallel conducting walls with grooves in the central region of the guide cross section.

Abstract: A new waveguide for the low-loss transmission of millimeter waves is presented. The guide consists of two parallel conducting walls with grooves in the central region of the guide cross section. The grooves run along the guide in the direction of the wave propagation. It is shown that the waveguide, if excited in the TE-wave mode, has properties similar to those of the H guide, which contains a dielectric slab between the conducting walls in the center. The new guide is characterized by an exponential transverse decrease of the field distributions in direction from the center and by low attenuation. Theoretical considerations dealing with the field distribution and the data of the guide are presented.

79 citations

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Bell Labs

^{1}TL;DR: In this paper, a simplification of the diode by a simple equivalent circuit and the determination of the effective diode parameters from transmission measurements is presented, which enables one to make negligible the complicating effects of posts and shunting cartridge capacitance usually present in microwave diode circuits.

Abstract: A means has been found which enables one to make negligible the complicating effects of posts and shunting cartridge capacitance usually present in microwave diode circuits. This simplification permits the representation of the diode by a simple equivalent circuit and the determination of the effective diode parameters from transmission measurements. Parameters so obtained at X band (8.2 to 12.4 Gc) are compared with audio frequency bridge measurements. Measurements at M band (50-60 Gc) of a new 800-Gc cutoff frequency varactor are also described. This varactor has zero bias junction capacitance in the 0.016 pf range and spreading resistance on the order of 12 /Spl Omega/. It is expected to extend the useful range of parametric devices well into the millimeter region.

78 citations

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TL;DR: This paper tabulates the element values of negative-resistance terminated, prototype, low-pass, lumped-element ladder networks of normalized impedance and bandwidth and provides a technique for relating the characteristics of any actual narrow-band, negative-Resistance device to the value of the prototype susceptive element adjacent to the negative resistance.

Abstract: This paper tabulates, as functions of reflection gain and ripple, the element values of negative-resistance terminated, prototype, low-pass, lumped-element ladder networks of normalized impedance and bandwidth. (the values are calculated using known synthesis methods.) Next, it provides a technique for relating the characteristics of any actual narrow-band, negative-resistance device to the value of the prototype susceptive element adjacent to the negative resistance. When an actual negative-resistance device has been related to a prototype in this manner, the performance of the device with one, two or three additional cascaded resonators can be predicted from given graphs. This allows trade-offs among gain, ripple, and bandwidth, within limits. Finally, the predicted performance can be used with simple formulas and the table of prototype element values to design suitable resonators to broadband the actual amplifier. The tables and techniques of this paper are used successfully to broadband tunnel-diode, maser and parametric-amplifier circuits. This paper allows the practical engineer to estimate the broad-banding potential of any given negative-resistance device and provides him with the proper element values to do so with only a few very simple calculations required.

69 citations

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TL;DR: In this article, the design of direct-coupled resonator filters over wide (as well as narrow) frequency bands is presented using the quarter-wave transformer as a prototype circuit.

Abstract: Direct-coupled resonator filters in transmission line are discussed. The resonators consist of sections of transmission line coupled either by series capacitances or shunt inductances. Over narrow frequency bands, such filters show characteristics similar to those of lumped-constant filters and their design is straightforward. The design of direct-coupled resonator filters over wide (as well as narrow) frequency bands is presented here using the quarter-wave transformer as a prototype circuit. Several numerical examples (with fractional bandwidths between 10 per cent and 85 per cent) are worked out to illustrate the method. It is shown that the response can be improved by optimizing the line impedances.

65 citations

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TL;DR: In this paper, backward diodes suitable for small-signal detection applications in the millimeter-wave region have been fabricated from n-type germanium, and their initial performance at selected frequencies from 11 Gc to 300 Gc is discussed.

Abstract: Backward diodes (low peak current tunnel diodes) suitable for small-signal detection applications in the millimeter-wave region have been fabricated from n-type germanium. The diodes have the dimensions and geometry of point-contact diodes. For millimeter-wave signal levels below about - 20 dbm, the current sensitivity of these units is an order of magnitude greater than that of selected existing diodes for this frequency range. When employed as millimeter-wave frequency converters, the minimum conversion loss is comparable to that of conventional diodes, but the beating oscillator power requirements may be somewhat reduced. The diode noise factor at megacycle IF frequencies is comparable to that of conventional units, and in the low audio IF range it is expected to be markedly decreased. The fabrication of these diodes is described and their initial performance at selected frequencies from 11 Gc to 300 Gc is discussed.

57 citations

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TL;DR: A homogeneous coupled line configuration realizing the characteristics of an all-pass network in the distributed network sense is useful for delay equalization in the UHF range.

Abstract: A homogeneous coupled line configuration realizing the characteristics of an all-pass network in the distributed network sense is useful for delay equalization in the UHF range. All-pass networks of first and second order are presented, while nth-order networks may be realized directly or built out of first- and second-order networks, analogous to the lumped-constant element network technique.

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TL;DR: In this article, three different waffle-iron filters with staggered stop bands are connected in series to give a combined stop band that extends from 2.2 Gc to 13.7 Gc where the attenuation is 60 db or greater.

Abstract: This paper describes the design and measured performance of an L-band model of a high-power wide-band low-pass waffle-iron filter. Three different waffle-iron filters with staggered stop bands are connected in series to give a combined stop band that extends from 2.2 Gc to 13.7 Gc. where the attenuation is 60 db or greater. The waffle-iron filter attenuates all propagating waveguide modes which can propagate at frequencies in the above stop band. In the pass band the waffle-iron filters are matched to full size L-band waveguide using quarter-wavelength stepped transformers. The pulsed power-handling capacity without breakdown is measured to be over 1.4 Mw peak power with air at atmospheric pressure filling the filter.

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TL;DR: In this article, a non-resonant structure of conducting cylinders was used as an open surface-wave transmission line with a simple matching network, and it was shown that the line response is flat over a 20 per cent frequency range at X band for several combinations of cylinder lengths and spacings.

Abstract: Periodic structures of conducting cylinders have been used as radiators (Yagi antennas), and, more recently, as slow-wave lines in traveling-wave tubes and masers. In this report it is shown that a nonresonant structure may have interesting capabilities as an open surface-wave transmission line. By means of a relatively simple matching network, efficient excitation of a surface wave on the periodic line is obtained. Response is flat over a 20 per cent frequency range at X band for several combinations of cylinder lengths and spacings. Total insertion losses are less than 3 db and largely independent of length of transmission line. Conducting cylinders are embedded in styrofoam. The effects of bends and twists in the line have also been investigated. It is shown experimentally that a guided wave on this periodic structure can follow a circular path having 1.5/spl lambda/ radius of curvature with very little loss. The plane of polarization can be rotated 90/spl deg/ by inserting a short twisted section. By terminating the transmission line with short circuits at both ends, a discrete series of transmission maxima is observed. Since these resonant peaks of transmission are of high Q factor, the dispersion characteristic of the line is obtained with very good accuracy. This type of open transmission line may offer advantages over heavy-weight and bulky conventional waveguides for some specialized applications.

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TL;DR: In this paper, optical components such as directional couplers, attenuators, introduction and phase shifters have been developed using optical techniques in oversize rectangular waveguide for operation in the 300-to 350-Gc range.

Abstract: Components such as directional couplers, attenuators, introduction and phase shifters have been developed using optical techniques in oversize rectangular waveguide These components were designed for operation in the 300- to 350-Gc range They were scaled from a design that was successful at 27 Gc Preliminary data taken at 330 Gc indicates the feasibility of this technique The advantages of oversize waveguide as compared with conventional waveguide and freespace optical components are 1) lower attenuation and 2) simpler construction

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TL;DR: In this article, an experimental investigation of the Fabry-Perot interfometer (FPS) using spherical mirrors is reported, where the FPS was operated as a microwave resonant cavity at 60 to 70 Gc.

Abstract: An experimental investigation of the Fabry-Perot interfometer (FPS) using spherical mirrors is reported. The FPS was operated as a microwave resonant cavity at 60 to 70 Gc. Measurements were made of the loss and coupling as a function of mirror spacing. The electric field variation within the resonator was also measured. Other characteristics of the spherical Fabry-Perot resonator were observed and are discussed. A qualitative discussion of the behavior of a spheroidal cavity resonator is presented and its relation to the FPS and beam waveguide is demonstrated.

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TL;DR: In this article, the radiation characteristics of a line source of magnetic current embedded in a homogeneous electron plasma of infinite extent are investigated for the case in which a uniform magnetic field is impressed externally throughout the medium in the direction of the source.

Abstract: The radiation characteristics of a line source of magnetic current embedded in a homogeneous electron plasma of infinite extent are investigated for the case in which a uniform magnetic field is impressed externally throughout the medium in the direction of the source. The single-fluid theory of magnetohydrodynamics is employed. A very simple model is assumed for the plasma. Under this assumption, it is found that there are two modes of propagation of waves of small amplitude. By examining the behavior of these modes in the limiting cases of vanishing external magnetic field or infinite source frequency, they are identifiable as the modified forms of the usual plasma and optical modes which exist in an isotropic electron plasma. The dispersion relations for these two modes are discussed. The power radiated in each of the two modes is also evaluated. It is found that the power radiated in the optical mode is always lower than that due to the line source in free space, whereas the power radiated in the plasma mode is higher than that value for certain ranges of the source frequency.

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TL;DR: In this paper, the usual power loss method of evaluating the damping constant and Q of cavities and the attenuation constant of waveguides, as caused by finite wall conductivity, breaks down in the case of degenerate modes.

Abstract: The usual power loss method of evaluating the damping constant and Q of cavities and the attenuation constant of waveguides, as caused by finite wall conductivity, breaks down in the case of degenerate modes and fails to predict the coupling between degenerate modes. By means of variational formulations for the lossy case it is shown how the usual power loss method maybe generalized to treat the case when there are degenerate modes present. The generalized method turns out to be a particularly simple extension of the usual procedure.

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TL;DR: A concise and careful review of the Elliott analysis of how a pulsed carrier wave is distorted after propagating down a waveguide is given in this article, in conjunction with the more recent work of Haskell and Case, derives the time region for which the Elliott (corrected) solution is valid.

Abstract: A concise and careful review of the Elliott analysis of how a pulsed carrier wave is distorted after propagating down a waveguide is given. This review, in conjunction with the more recent work of Haskell and Case, derives the time region for which the Elliott (corrected) solution is valid. It is shown that if the pulse contains a sufficient number of carrier cycles, and if the guide is long enough compared to a free-space wavelength at the carrier frequency, then the Elliott solution is valid for practically all times.

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TL;DR: In this article, a precision coaxial to strip line transformer has been proposed to estimate the characteristic impedance of a square coaxial line (i.e., having inner and outer conductors of square cross section) which forms an intermediate part of the proposed transformer.

Abstract: Recent attempts to design a precision coaxial to strip line transformer have raised the need to know accurately the characteristic impedance of square coaxial line (i.e. a concentric line, as shown in Fig. 1, having inner and outer conductors of square cross section ) which forms an intermediate part of the proposed transformer.

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TL;DR: In this article, a class of integral equations with complex symmetric kernels which occur in laser theory are discussed. But they do not have some of the well-known properties of integral equation with Hermitian kernels.

Abstract: Mathematical comments are made on a class of integral equations with complex symmetric kernels which occur in laser theory. It is pointed out that integral equations of this class do not have some of the well-known properties of integral equations with Hermitian kernels. In particular, the usual extrernal principle by which the eigenvalues of a Hermitian kernel may be estimated using the Rayleigh-Ritz procedure does not apply to integral equations with complex symmetric kernels. It is suggested that the use of variational techniques to calculate diffraction losses in laser interferometers leads to results of doubtful accuracy.

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TL;DR: In this paper, a simple empirical formula for the design of band rejection radial-line coaxial filters is presented and the appropriateness of these filters for parametric work is discussed and a particular structure employing these filters to provide a high performance harmonic filter structure for rectangular waveguide is presented.

Abstract: Design techniques and a simple empirical formula for the design of band rejection radial-line coaxial filters are presented. The appropriateness of these filters for parametric work is discussed and a particular structure employing these filters to provide a high performance harmonic filter structure for rectangular waveguide is presented.

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Motorola

^{1}TL;DR: In this paper, an analysis of the operation of a 3-port Y-junction ferrite circulator was made, where it was assumed that the ferrite cylinder can support the propagation of a bound surface wave.

Abstract: An analysis is made of the operation of a 3 port Y-junction ferrite circulator wherein it is assumed that the ferrite cylinder can support the propagation of a bound surface wave The incident wave guided by the air-dielectric interface of the ferrite cylinder is taken to be linearly polarized parallel to the ferrite axis and after incidence on the cylinder is assumed to divide and propagate in opposite directions about the cylinder as two linearly polarized signals By appropriate selection of ferrite cylinder diameter, permeability, magnetizing field and saturation magnetization, constructive and destructive interference of the RF components are shown to occur at the two output ports The conditions for reinforcement and null at the outputs are discussed and compared with published experimental data It is possible to predict the occurrence of ferrite circulator diplexing reported by Brown and Clark although good quantitative comparison is lacking The theoretical dependence of ferrite diameter upon free space wavelength is computed and compares satisfactorily with recorded experimental data The experimental observations that greater circulator bandwidths are achievable when below resonance magnetizing fields are employed are shown to have a theoretical basis In addition, interrelationships among the designed variables: saturation magnetization, applied magnetic field, ferrite resonance linewidth and circulator insertion loss are noted in the theoretical results

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TL;DR: In this paper, the authors discuss the basic problems encountered in the general areas of generation, transmission, and detection of millimeter waves, and review representative examples of work in these three areas since 1959 with respect to the methods and techniques employed to circumvent present limitations.

Abstract: The aim of this survey is to discuss the basic problems encountered in the general areas of generation, transmission, and detection of millimeter waves. Representative examples of work in these three areas since 1959 are reviewed with respect to the methods and techniques employed to circumvent present limitations and extend the frontier into the submillimeter range. Subject classifications include classical and quantum electronics, harmonic generation optical frequency pumping and mixing, waveguide and optical transmission systems, resonators, and detectors. At the end of each section, a few critical evaluation remarks are made on the work in progress and the prospects of success in the near future. A fairly comprehensive list of some 157 references dating from 1959 is listed by year and subject title. While substantial progress has been made, especially in technology, in the last few years, the submillimeter wave problem appears as formidable as ever and no breakthrough idea has yet been recognized.

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TL;DR: In this paper, a general theory for analyzing scattering from loaded scatterers is available, and has been applied to small SCatterers suitable for electromagnetic field measurements, as long as the load is linear.

Abstract: A general theory for analyzing scattering from loaded scatterers is available, and has been applied to small scatterers suitable for electromagnetic field measurements. The theory is valid for both passive and active loads, as long as the load is linear. Ryerson has proposed the use of tunnel diodes to provide a negative resistance load, thereby enhancing the scattered signal. His predictions have been verified experimentally by measurements on dipoles and tunnel diodes at S band. The use of scatterers and tunnel diodes for field measurements is discussed in this communication.

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TL;DR: In this paper, a very compact E-plane waveguide circulator was used to put in the dewar and a compromise may be necessary in choosing between a compact stripline circulator and a comparatively bulky H-plane circulator.

Abstract: When a circulator is used with a parametric amplifier or maser, the noise contribution of the circulator may be reduced by cooling it in liquid nitrogen or liquid helium. Compact devices are required to put in the dewar and, depending on the microwave frequency, a compromise may be necessary in choosing between a compact stripline circulator and a comparatively bulky H-plane waveguide circulator, because waveguide feeds will have lower loss than coaxial line. This problem may be eased by using a very compact E-plane waveguide circulator, as shown in Fig. 1(a).

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TL;DR: In this paper, the diffraction loss of a new low-loss waveguide for millimeter and shorter wavelength, called the beam waveguide, was measured using a resonator technique, in the frequency range near 9 Gc.

Abstract: The diffraction loss of a new low-loss waveguide for millimeter and shorter wavelength, called the beam waveguide, was measured. The loss measurements were made using a resonator technique. The beam waveguide resonator derived from the beam waveguide consists of confocal paraboloids and is itself a very useful millimeter and sub-millimeter wave circuit component having already found application in some optical masers. Measurements made to determine the reflection loss of the resonator end plates also resulted in information on the loss of 90/spl deg/ bends in the beam waveguide. The results of the loss measurements made on the beam waveguide, in the frequency range near 9 Gc, are in good agreement with theoretical values given by Goubau.

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TL;DR: In this paper, four different millimeter wave equipments, which have been made for plasma diagnostic work, are described and the minimum temperature changes which can be detected, at short millimeter wavelengths, with various types of superheterodyne radiometers, including the Golay cell, the barretter, the crystal video radiometer, the carbon bolometer and the 1.5/spl deg/K InSb photoconductive detector.

Abstract: In this paper four different millimeter wave equipments, which have been made for plasma diagnostic work, are described. They are: 1) A straightforward 70-Gc superheterodyne radiometer with an over-all noise factor of 13 db; 2) An early 140-Gc radiometer, with second harmonic mixing, which has an over-all noise factor of about 25 db; 3) A later and more sensitive 140-Gc radiometer which contains a fundamental local oscillator, VX 3352 mixer crystals and a 408-Mc IF amplifier commencing with an Adler tube; 4) A very simple 140-Gc transmission measuring equipment containing a 1-watt source and a crystal video receiver which has a tangential sensitivity of ---42 dbm. The last part of this paper discusses the minimum temperature changes which can be detected, at short millimeter wavelengths, with various types of superheterodyne radiometers, the Golay cell, the barretter, the crystal video radiometer, the 1.5/spl deg/K carbon bolometer and the 1.5/spl deg/K InSb photoconductive detector. The performances expected from straight traveling-wave tube radiometers and traveling-wave masers at short millimeter wavelengths are also considered. The Appendices are devoted to mixer crystal performance in the millimeter and submillimeter regions, a theory of second harmonic mixing and the voltage sensitivity of a forward biased detector crystal.

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TL;DR: In this article, the authors deal with two basic problems in the measurement of the Q factor of low-loss transmission lines: 1) long lines and 2) short lines with appreciable direct coupling between the driving and pickup probes.

Abstract: This paper deals with two basic problems in the measurement of the Q factor of low-loss transmission lines: 1) long lines and 2) short lines with appreciable direct coupling between the driving and pickup probes. The results are given as relative ordinates on the (distorted) resonance curve which correspond to the correct values of /spl plusmn//spl 1/2//spl delta//spl iota/ or /spl plusmn//spl 1/2//spl delta//spl conint/ defined by Q =/spl conint/ / /spl delta//spl conint/=/spl iota/ / /spl delta//spl iota/.

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TL;DR: In this article, the electromagnetic wave propagation in arbitrarily magnetized homogeneous ferrites between two perfectly conducting parallel planes has been investigated by using the operational calculus method and the discrete propagation constants and the eigenvalues are to be determined from an algebraic equation of the fourth order and a determinantal equation derived from the boundary conditions.

Abstract: The electromagnetic waves propagating in arbitrarily magnetized homogeneous ferrites between two perfectly conducting parallel planes have been investigated by using the operational calculus method. The discrete propagation constants and the eigenvalues are to be determined from an algebraic equation of the fourth order and a determinantal equation derived from the boundary conditions. The hybrid modes thus found degenerate to the solutions already found for the particular cases of longitudinally and transversely (parallel and perpendicular to boundaries) magnetized ferrite cases.

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TL;DR: Yeh derived a general relation between Q and /spl alpha/, namely, /splAlpha/ = /spl upsi//sub p/ / / spl upsi-sub g/ /spl beta/ /2Q where /splUpsi// Sub p/ and /Spl upsi/ are the phase velocity and group velocity of the wave respectively.

Abstract: The relation between the quality factor Q and the attenuation constant /spl alpha/ of a transmission line has been known as follows: /spl alpha/ = /spl beta/ / /2Q where /spl beta/ is the phase constant. Recently from the following relation of propagation constant at resonance /spl Gamma/(/spl omega//sub 0/) + /spl part//spl Gamma/ / /spl part//spl omega/ /spl Delta//spl omega//spl cong/ i/spl beta/(/spl omega//sub 0/), where /spl Gamma/(/spl omega//sub 0/) = /spl alpha/(/spl omega//sub 0/) + i/spl beta/(/spl omega//sub 0/). Yeh derived a general relation between Q and /spl alpha/, namely, /spl alpha/ = /spl upsi//sub p/ / /spl upsi//sub g/ /spl beta/ / /2Q where /spl upsi//sub p/, and /spl upsi//sub g/ are the phase velocity and group velocity of the wave respectively. This general relation can be derived very simply from the generally accepted definition of /spl alpha/ and Q.