Energy Flow in a Waveguide below Cutoff Marco Moriconi Instituto de F´ısica, Universidade Federal Fluminense, 24210-340, Niter´oi, RJ, Brasil Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 (May 20, 2009; updated July 2, 2013)

1Problem

Electromagnetic waves of frequency below a cutoff cannot propagate down a hollow wave- guide. One view of why this is so has been given in sec. 24-8 of [1], where it is argued that constructive interference between the physical source of the waves and the infinite set of related image sources can only occur if the free-space wavelength is short enough. In this problem, discuss the flow of energy (as described by the Poynting vector [2]) supposing that source currents in the plane z = 0 launch fields according to the TE10 mode pattern of a rectangular, vacuum waveguide of inner dimensions a in x and b in y.

2Solution

The electromagnetic fields of the TE10 mode with angular frequency ω in a guide whose interior is 0

πx ( − ) E E ei kgz ωt , y = 0 sin a (3) i ∂Ey kg πx ( − ) B − E ei kgz ωt , x = ω ∂z = ω 0 sin a (4) i ∂Ey iπ πx ( − ) B − − E ei kgz ωt , z = ω ∂x = ωa 0 cos a (5) in SI units, and the guide wave number kg is given by   ω2 π2 π2 ω2 k = − = i − ≡ iα, (6) g c2 a2 a2 c2 where c is the speed of light in vacuum, such that eqs. (3)-(5) satisfy the wave equation ∇2ψ = ∂2ψ/∂(ct)2.

1 The TE10 fields (3)-(5) can also be thought of as the sum of a pair of free-space waves that zig-zag down the guide at angle θ (see, for example, sec. 9.9 of [3]), E   ω  0 i(k+·r−ωt) i(k−·r−ωt) i(k z−ωt) E = E+ + E− = e − e yˆ = E0 sin cos θx e g yˆ, (1) 2i c

k+ × E+ k− × E− ω ω ω B , k± ± θ xˆ θ ˆz ± θ xˆ k ˆz. = ω + ω = c cos + c sin = c sin + g (2)

The requirement that Ey(x = a) = 0 implies that (ω/c)cosθ = π/a,sothatkg =(ω/c)sinθ is given by eq. (6).

1 2.1 Above Cutoff

For frequencies ω>πc/athe guide wave number kg is real, and the fields (3)-(5) can be written πx E E k z − ωt , y = 0 sin a cos( g ) (7) k πx B − g E k z − ωt , x = ω 0 sin a cos( g ) (8) π πx B E k z − ωt . z = ωa 0 cos a sin( g ) (9) The (instantaneous) Poynting vector in this case is × c E B E B − E B S = μ = Z ( y z xˆ y x zˆ) 0 0  2 cE0 π 2πx kg 2 πx 2 = sin sin[2(kgz − ωt)] xˆ + sin cos (kgz − ωt) zˆ , (10) Z0 4ωa a ω a  where Z0 = μ0/0 = 377 Ω, and 0 and μ0 are the permittivity and the permability of the vacuum, respectively. The time-average Poynting vector is k cE2 πx c E2 πx S = g 0 sin2 zˆ = 0 sin2 zˆ, (11) ω 2Z0 a vp 2Z0 a corresponding to flow of energy down the guide with phase velocity ω c v = =  >c, (12) p k  2 g − πc 1 ωa and with group velocity  dω πc 2 c2 vg = = c 1 − =

2.2 Below Cutoff

For frequencies ω<πc/athe guide wave number kg is imaginary, and the fields (3)-(5) can be written in terms of the real parameter α (defined in eq. (6)) as πx E E e−αz ωt, y = 0 sin a cos (14) α πx B − E e−αz ωt, x = ω 0 sin a sin (15) π πx B − E e−αz ωt. z = ωa 0 cos a sin (16)

2 The Poynting vector in this case is

cE2 π 2πx α πx S = 0 − sin xˆ + sin2 zˆ e−2αz sin 2ωt. (17) 2Z0 2ωa a ω a The time-average Poynting vector is zero, and there is no net flow of energy away from the sources in the plane z =0. The Poynting vector (17) does describe an energy flow in the z-direction, but this energy propagates only characteristic distance z ≈ 2/α (≈ a for low ω) before returning back to the sources. Again, there is also a flow of energy in the x-direction that oscillates in time, indicating an oscillatory rearrangement of energy stored within the guide and close to the sources. In more detail, the instantaneous density u of electromagnetic energy in the guide is  E2 B2 u 0 = + μ 2  2 0  E2 πx α2 πx π2 πx 0 2 −2αz 2 2 −2αz 2 2 −2αz 2 = 0 sin e cos ωt + sin e sin ωt + cos e sin ωt 2 a μ ω2 a μ ω2a2 a  0  0 E2 πx π2 0 2 −2αz −2αz 2 = 0 sin e cos 2ωt + e sin ωt 2 a μ ω2a2  0  cE2 1 πx π2 0 2 e−2αz ωt e−2αz 2 ωt , = 2 sin cos 2 + 2 2 sin (18) 2Z0 c a ω a recalling eq. (6). Then,   ∂u cE2 π2 2ω πx 0 − 2 e−2αz ωt = 2 2 sin sin 2 ∂t 2Z0 ωa c a   cE2 π2 2πx 2α2 πx 0 2 e−2αz ωt −∇ · , = 2 sin + sin sin 2 = S (19) 2Z0 ωa a ω a again using eq. (6) for α, which verifies that electromagnetic energy is conserved under the flow (17). To use the language of antennas, we can say that the electromagnetic fields in a waveguide operated below its cutoff frequency include only near field terms; there are no far fields in this case.2

References

[1] R.P. Feynman, R.B. Leighton and M. Sands, The Feynman Lectures on Physics,Vol.II (Addison-Wesley, Reading, MA, 1964), http://www.feynmanlectures.caltech.edu/II_24.html#Ch24-S8

2Another term associated with waves is “evanescent,” meaning waves that are ”tied” to some matter and do not transport energy to “infinity.” All waves in waveguides are “evanescent.” See, for example, [4] for a general discussion.

3 [2] J.H. Poynting, On the Transfer of Energy in the Electromagnetic Field, Phil. Trans. Roy. Soc. London 175, 343 (1884), http://physics.princeton.edu/~mcdonald/examples/EM/poynting_ptrsl_175_343_84.pdf

[3] S.J. Orfanidis, Waveguides, http://www.ece.rutgers.edu/~orfanidi/ewa/ch09.pdf

[4] K.T. McDonald, Decomposition of Electromagnetic Fields into Electromagnetic Plane Waves (July 11, 2010), http://physics.princeton.edu/~mcdonald/examples/virtual.pdf

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