Quantum Features of Microwave Propagation in a Rectangular Waveguide

Nicolae Marinescu Department of Physics, University of Bucharest, Romania

Rudolf Nistor Department of Physics, Politechnic Institute Bucharest, Romania

Z. Naturforsch. 45a, 953-957 (1990); received September 7, 1990

The formal analogy between the distribution of the electromagnetic field in waveguides and microwave cavities and quantum mechanical probability distributions is put into evidence. A wave- guide of a cut-off frequency a)c acts on an electromagnetic wave as a quantum potential barrier Ug = hcoc. A non-habitual time independent Schrödinger equation, describing guided wave propaga- tion, is established.

Introduction length of the photon, we can state that the quantum potential associated to a body exerts an action up to The association of a quantum potential barrier to a a distance one order of magnitude larger than the given body with respect to the quantum particle pen- wave length of the incident photon. We are led to the etrating it or passing in its vicinity has been described idea that the distance of the interaction photon-object in several papers. The notion of quantum potential has a dispersive character. has first been mentioned by de Broglie [1], Böhm [2] and Philippidis [3]. In these works, the authors use a quantum potential depending on the wave function of 2. The Waveguide as a Quantum Potential Barrier the quantum particle. The concept of a waveguide as a quantum potential The dual character of microparticles and electro- barrier is not new. It has been introduced for the first magnetic fields is well known. Moreover, in an equal time by Blatt and Weisskopf [4] to explain the mecha- number of cases the physical interaction between the nism of nuclear reactions. A quantum potential has electromagnetic field and the medium is described by also been used to describe the interaction of electro- wave or corpuscular concept. Usually, for short-wave magnetic waves with the medium; thus in [5] a Fermi length electromagnetic fields (y-rays) we use the corpus- pseudopotential has been used in the Schrödinger cular concept, and for long wavelength electromag- equation for solving Maxwell's equations for anten- netic fields (radio waves) the wave concept. In mi- nas. crowave circuits all the authors use the wave concept. In this work the potential for photons is written as To test the corpuscular concept in microwave electro- a function of the frequency of the associated waves, magnetic field theory, we start with some very com- and not of their amplitude as in [2] and [3]. We also mon remarks. The waveguide wall is impenetrable for give an explicit expression for the total energy of the the electromagnetic field. For a rectangular, wave- photon in the medium. Thus we have shown that the guide, the photons are placed in an infinite height same object can be described by different quantum bidimensional well. A quantum particle in such a po- potentials for different frequencies of the wave. In this tential well cannot be at rest, and also the photons are way we obtain a dispersion relation of the potential. permanently reflected by the wall of the waveguide. Moreover, if we take into account that the guiding Taking the above remarks for granted, an incident effect on the wave disappears for dimension of the photon normal to the cross section of a rectangular guide by an order of magnitude larger than the wave waveguide must change its propagation direction. The energy of the photon must remain unchanged after Reprint requests to Prof. A. A. Raduta, Institut für Theoreti- penetration inside the waveguide (we consider that the sche Physik, Universität Tübingen, Auf der Morgenstelle 14, 7400 Tübingen. waveguide has an infinite mass and that it has per-

0932-0784 / 90 / 0700-967 $ 01.30/0. - Please order a reprint rather than making your own copy. 954 N. Marinescu and R. Nistor • Microwave Propagation in a Rectangular Waveguide fectly reflecting walls) If we consider now a free space photon of momentum p0 denoting the momentum Ug components inside the waveguide by px, py, p,, we can incident wave write transmitted wave reflected wave 2 Pl=P x+Py+Pzy (1) Z = l Z or, dividing by h2, we obtain Fig. 1. The waveguide along its axis is analogous to an quan- 2 2 2 2 tum potential barrier of height U . k = k x + k + k . (2)

The allowed values of wave numbers kx and ky are fixed by the transversal dimensions of the rectan- gular waveguide (a bidimensional (a, b) well of infinite mn nn length): kx= and kv=— (m and n are positive a b integers), k, is given by

,2 2 2 m n n n K.k ,-k —- K. o (3) a b~

In the case of electromagnetic field propagation along Fig. 2. The cross-section of a rectangular waveguide. the waveguide the axial wavenumber k, must be real, , ,, fm 7z\2 fn n\2 and consequently k^> I— 1 +1 ^ | . The critical Fig. 2, for normal incidence with respect to the wave- guide cross section, an equal number of photons is deviated to the left and to the right, up and down, so case , =k2 marks the transition be- a plane electromagnetic wave is decomposed inside a ) \ b the waveguide in four (or two) plane waves. tween a progressive and an evanescent wave, respec- We shall try further to provide a description tively, and (3) can be written 2 2 2 analogous to that used in nonrelativistic quantum me- k z=k 0-k . (4) chanics of a one-dimensional stationary particle. The The equation (4) is obtained in a more complicated quantum mechanical counterpart of (4) is way in microwave electromagnetic field theory by E2=p2c2 + U2, (7) solving the Helmholtz equation with boundary condi- tions for the electromagnetic field. where we used the notation

By introducing the photon energy E = hco0 and the E = hk c, p =—ihV,, U = hco , oo = c k . waveguide potential 0 z g c c c The operators in (7) thus lead us to an equation U=hcoc, (5) characterizing the propagation of the wave along the the propagation-evanescent condition k0^kc is seen z direction: to correspond to + (8) (6)

Thus the constraint on the motion of the photons in Hence the term denoted by Ug will be designated as the cross section of the waveguide gives rise to a po- the quantum potential of the medium interacting with tential barrier as shown in Figure 1. We note this is the electromagnetic wave [2]. We remark that (8) rep- true under the condition that the wavelength is com- resents a stationary form of the equation for the waves parable to the linear dimensions "a" and "6" [6-8]. propagating through a guide having properties of ho- If we accept, statistically speaking, that a free space mogeneity in the plane perpendicular to the direction incident electromagnetic plane wave is composed of a of propagation. large number of photons, and accepting the symmetry Using (7), the de Broglie relation as well as the of the physical system relative to the AA' and BB' axis, relation between the photon energy E and the wave- 955 N. Marinescu and R. Nistor • Microwave Propagation iRectangulan a r Waveguide length in free space we obtain for the refraction index of a medium of quantum potential C7g, the expression

2 n = s/\-{UJE) . (9)

3. The Waveguide Filled with Lossless Dielectric Medium

It is interesting to apply (8) to the description of several physical situations. For example, in the case of Fig. 3. Rectangular cavity with dielectric cylinder. a lossless plasma of frequency cop, by multiplying with h the wave propagation condition co > co one obtains The solutions of (8) for the potential U given by (16) E>Up. (10) are the unperturbed normalized eigenfunctions From (9) we obtain the well known relation [9] 8 /sin\m7c /sin\nn /sin\ p n wo _ z mnp W —x hrn )-T W n„ = (11) abd\cosJ a \cos/ b \cos/ a

even In the same way we obtain the waveguide refraction with m, n, p I 1 and the eigenvalues of energy index odd 1/2 n = i-i^ (12) Wl=hcn + + (18) co "a) U

One may now add two or more potentials (for ex- If we consider that the perturbating term W^np W®np, ample a waveguide filled with a plasma or an usual we obtain from (15) the perturbed eigenvalues dielectric). It is known that the propagation condition 2 2 2 1 W' in a waveguide filled with plasma is [9] co0 > >/co + co . mnp ~ +1 - Wmnp— W°mnp ^ yyQ (19) By multiplying this relation with h one obtains

2 E>y/Up + U< (13) In a first order approximation, the term W„np results [10] from We can say that the photons interact with a potential \y>2 _ f ip0* JJ'2 ipO dy barrier of height "mnp J mnp 1 mnp u ' • (20) ^cylinder U2 = TJ2 4- U2 (14) U% + P UP + UG ' According to (9), the cylinder potential or more generally we can write U' = E-J l-e/e0 (21) (15) i is constant inside the cylinder, and if we note that the integral is independent of p, we obtain

2 2 4. Microwave Perturbed Cavity Wm np = E (\-s/z0)Qmn, (22) (Nondegeneratetd Case) where Qmn is a constant depending on the cavity di- Let us examine a parallelepipedical microwave cav- mensions, dielectric cylinder dimensions and cavity ity of dimensions, a, b, c, and a dielectric cylinder of mode. radius R(R<£a,b, d) as perturbing element a dielectric For a cavity resonant, W°np and (19) becomes cylinder of radius R(R<£a, b, d), as perturbing ele- W mnp ~ W mnp z i-- Q (23) ment, as shown in Figure 3. The unperturbed poten- nnp\_ 2\ tial is where the constants Qmn are summarized in Table 1. JJ_\0 inside the cavity, Here % is the relative dielectric volume/cavity vol- j oo on the walls and outside cavity. ume, i.e. = n R2/a b. 956 N. Marinescu and R. Nistor • Microwave Propagation in a Rectangular Waveguide

Table 1. potential barrier for the electromagnetic wave. The main advantage of a quantum formalism consists in m n Qmn the ability of performing the guided wave analysis onto a single particle, i.e. a photon instead of using A 2 b 2 2 ü 2 odd odd 4x — m ii - % —n it - % Maxwell's field equation. By this method we per- a b 2 2 formed the analysis of directional couplers, a periodic b m n Q2 odd even m n x structure in the waveguide, a microwave interaction a b with anistropic medium a.s.o. [12]. . a m2 n2 iz2 We note that (1) is generated by the wave equation even odd n n x X b b by supposing a plane structure for its solution. The m 2 n 27 T2 vectorial properties of the e.m. field against Lorentz even even x transformation is preserved by our formalism. Indeed, b by means of (8) we determine the energy E character- izing the propagation along the z direction. The en- Using only the first term in the expression of ergy of this mode is influenced by the transversal the relative frequency variation of the fundamental modes through the potential term Ug involved in (8). cavity mode is Therefore the reflection on the guide walls yields a coupling between the transversal modes on one hand Aco 2nR2f e\ - = -_-! . (24) and the longitudinal one, on the other hand. Further, SQJ co0 ab \ (7) defines the momentum pz. In this way the 4 compo- nents p , p , p , E of the fourdimensional pM are given The Eq. (24) is in agreement with the usual electro- x y : in a quantized form, and consequently the plane solu- magnetic theory [11]. It can be seen from Table 1 that tion of the wave equation is discretized. there is a dependence of the perturbation on the parity of the integers m and n. Our procedure describes microwave propagation by replacing the problem of solving the wave equation with specific boundary conditions by an eigenvalue 5. Conclusions problem of the Schrödinger type. The advantage of the latter procedure over the former one consists in its The electrodynamc characterization of wave pro- simplicity. The transfer of a formalism from one cesses in transmission lines in terms of the vector func- branch of physics to another one is a frequent practice. tions E(ux, u2) • exp( + y z) and H(ul, u2) • exp( + y z) Indeed the specific methods of e.m. field theory ae

(where u1,u2 are the coordinates in the cross-section) often used for solving elegantly some quantum me- carries much more information than may be necessary chanical problems. Among many examples are the for the design of a microwave system and also in path integral method [13], the formalism developed by microwave measurements. Indeed, we are interested in Albeverio et al. [14] in connection with some solvable the power to be transmitted, the relative magnitude of models in quantum mechanics, the treatment of elec- the incident and reflected waves and the phase shift tron propagation in microelectronic devices [15]. and attenuation over a definite length of the transmis- Various aspects of microwave propagation have sion line. The power, phase shift and reflection coeffi- previously been investigated by standard methods of cient can be found by experiment, whereas the mea- e.m. field theory. Thus the reflection by an obstacle surement of the field components and their distribu- was treated by Boström in [16, 17], while the theory tion function entails appreciable difficulties. of resonant cavities is given in [18]. The major result reported in this paper is that a The extension of this formalism to the description of propagation medium, a microwave circuit, a dielectric the microwave propagation in a guide having a more or a plasma medium can be considered as a quantum complex profile, is in progress. 957 N. Marinescu and R. Nistor • Microwave Propagation in a Rectangular Waveguide

[1] L. De Broglie, Found. Phys. 1, 5 (1970). [10] A. Mesiah, Mecanique Quantique, Dunod, Paris 1985. [2] D. Böhm, Phys. Rev. 15, 166 (1952). [11] R. F. Harrington, Time Electromagnetic Fields, McGraw [3] C. Philippidis, et al. Nuovo Cim. 52 B, (N. 1) 11 (1979). Hill, London 1961, pp 373. [4] J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear [12] N. Marinescu and R. Nistor, Quantum Theory of Mi- Physics, John Wiley, New York 1952, pp. 364. crowave Filters, CNETAC Bucharest, Romania, Nov. [5] Tai Tsun Wu, Lecture Notes in Physics 211 Resonances 1988. - Models and Phenomena - Proceedings Bielefeld 1984, [13] R. P. Feynam and A. R. Hibbs, Quantum Mechanics Edited by S. Albeverio, L. S. Ferreira, and L. Streit, and Path Integrals, McGraw-Hill, London 1965. Springer-Verlag, Berlin 1900. [14] S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, and H. [6] N. Marinescu and R. Nistor, Rev. Roum. Phys. 29, Holden, Solvable Models in Quantum Mechanics, No. 2 (1984). Springer, Berlin 1988. [7] N. Marinescu and R. Nistor, Proceedings of the 4th [15] A. M. Kriman and P. P. Ruden, Phys. Rev. B321, 8013 Romanian Microwave Conference, Bucharest, Romania (1985). Nov. 1984. [16] A. Bostöm and P. Olson, J. Appl. Phys. 52, 1184 (1981). [8] N. Marinescu and R. Nistor, Second International Con- [17] A. Boström, Wave Motion 2, 167 (1980). ference Trends in Quantum Electronics Bucharest, Ro- [18] D. S. Jones, The Theory of Electromagnetism, 64, Perg- mania, September 1985. amon Press, Oxford 1964. [9] D. Quemada, Ondes dans les plasmas, Herman, Paris 1968, pp 142.