Average Linear and Angular Momentum and Power of Random Fields Near a Perfectly Conducting Boundary
11 Average Linear and Angular Momentum and Powerof Random Fields Near a PerfectlyConducting Boundary
Luk R. Arnaut and Gabriele Gradoni
Abstract —The effect of a perfectly conducting planar bound-ary on the average linear momentum (LM), angular (momentum(AM), and their power for a time-harmonic statistically isotropicrandom field is analyzed. These averages are purely imaginaryand their magnitude decreases in a damped oscillatory mannerwith distance from the boundary. At discrete quasi-periodicdistances and frequencies, the average LM and AM attain theirfree-space value. Implications for the optimal placement or tuningof power and field sensors are analyzed. Conservation of theflux of the mean LM and AM with respect to the difference ofthe average electric and magnetic energies and the radiationstresses via the Maxwell stress dyadic is demonstrated. Thesecond-order spatial derivatives of differential radiation stresscan be directly linked to the electromagnetic energy imbalance.Analytical results are supported by Monte Carlo simulationresults. As an application, performance based estimates for theworking volume of a reverberation chamber are obtained. Inthe context of multiphysics compatibility, mechanical self-stirredreverberation is proposed as an exploitation of electromagneticstress.
I. I
NTRODUCTION
Angular momentum (AM) in electromagnetics was firsttheorized by Poynting [1] and consists of spin (or intrinsic)and orbital contributions. Spin angular momentum (SAM)is represented by familiar left- and right-hand circular fieldpolarizations, as demonstrated experimentally by Beth [2].Recently, it was found that light waves with a helicoidalphase front exhibit another source of AM whose magnitudemay be several times larger than that of SAM [3]. Thisadditional part was named orbital angular momentum (OAM)and depends on the transverse spatial field variation. OAM canbe identified with a phase front rotating helicoidally aroundthe axial direction of energy propagation k that governs linearmomentum (LM).More recently, OAM has gained renewed interest in radioengineering because of its potential for increasing the numberof degrees of freedom and channel capacity of MIMO wirelesscommunications systems [4]. Its mechanism is the generationof orthogonal transmission modes with several different az-imuthal phase patterns (integer multiples of π ) and associatedAM power flux, while maintaining the overall point-to-pointLM power flux (Poynting vector).In the OAM literature, the focus has been almost universallyon unbounded propagation in free space. Real scenarios,however, involve antennas near a ground plane, scatterers,multipath reflection, etc. This raises the question to whatextent LM and (O)AM can then be preserved or detected. One motivation for studying OAM near a boundary is the useof reverberation chambers for evaluating (O)AM properties.Furthermore, OAM may arise unintentionally, e.g., from aparticular phase distribution of edge currents around aperturesor circuit loops. This may give rise to azimuthal AM powerin the near field in a manner that may not be detectable bywire or loop sensors for conventional LM power.LM and AM may also be instrumental in the conversion be-tween electromagnetic (EM) energy and other forms of phys-ical energy, in what may be termed multiphysics compatibility (MPC). In particular, MPC includes coupling phenomena be-tween EM fields and their induced or external forces on mate-rial bodies via radiation stress and shear, i.e., electromagnetic-mechanical compatibility (EMMC); cf. Sec. IV-B. Anothersubdomain of MPC is electromagnetic-thermodynamic com-patibility (EMTC), originating from ohmic losses that arealready incorporated in EM constitutive relations. EMMC orEMTC may be significant in e.g. vacuum chambers, aerospace,applications involving significant EM forces on small or lightobjects with large surface-to-volume ratios (“smart dust”),nanoscale components including VLSI circuits [5], etc., andin high-power applications where mechanical effects of EMforces and heating may lead to deformation and arcing inextreme cases. A basic tenet of MPC is that excitation inone subdomain (e.g., EM) should preserve functionality andnominal conditions of operation that remain within tolerancelevels also for other MPC subdomains. Here too reverberationchambers offer a primary test bed, because of their abilityto generate high field strengths and power densities, inducingextreme mechanical and thermal MPC effects.In this paper, we analyze the effect of a rigid infinite planarperfect electrically conducting (PEC) surface on the LM andAM power of an ideal isotropic random field incident froma half-space. The analysis extends earlier results for LM andAM in unbounded free space [6] and for energy density neara PEC plane [7], [8]. The approach differs from traditionalstudies of OAM, in which a helicoidal wavefront is consideredfrom the start. Instead, we analyze to what extent statisticalAM may be induced or reduced as a result of interaction witha PEC boundary. A previously developed methodology [9]based on local angular spectral plane-wave expansions nearan impedance boundary is applied and extended. Throughoutthis paper, general time-dependent EM quantities are shownin roman type; time-harmonic quantities have a suppressed exp(j ωt ) dependence and are denoted in italics. a r X i v : . [ phy s i c s . c l a ss - ph ] S e p II. L
INEAR AND A NGULAR M OMENTUM ,P OWER AND E NERGY
To establish notions and notations, the definitions of LMand AM for general time-dependent and harmonic fields, andtheir connection to EM energy and power are briefly reviewed.For spatiotemporal fields E( r, t ) and B( r, t ) , the (real) localinstantaneous AM density with reference to a location r is[10, ch. 6], [11, ch. 1] M( r, r , t ) ∆ = ( r − r ) × P = µ (cid:15) ( r − r ) × S (1)where P( r, t ) ∆ = D( r, t ) × B( r, t ) = S( r, t ) / c (2)is the LM density and S( r, t ) = E( r, t ) × H( r, t ) is thelocal LM power flux density (Poynting vector). With ∇ =( ∂/∂r )1 r , ∇ × E = − ∂ B /∂t , and ∇ · D = ρ , (1) becomes M( r, r , t ) = (cid:15) ( r − r ) × (cid:90) t [ ∇ (E · E) − ( ∇ · E)E] d t = ( r − r ) × (cid:90) t ( ∇ U em − ρ E) d t (3)where U em ( r, t ) = U e + U m ∆ = E · D / · H / . The electricand magnetic energy densities U e and U m are quadraticfunctions of the EM field that are of purely electric andmagnetic type, whereas S is of mixed types.For time-harmonic fields E ( r, ω ) and B ( r, ω ) , the complexPoynting vector is S ( r, ω ) ∆ = ( E × H ∗ ) / , with Re ( S ) =E × H representing the time averaged LM power flux density.The corresponding (complex) AM density is M ( r, r , ω ) ∆ = ( r − r ) × P = µ (cid:15) ( r − r ) × S = ( r − r ) × j (cid:15) ω (cid:2) ∇| E | − ( ∇ · E ) E ∗ (cid:3) . (4)For a plane wave, (3) with ∇ = − j k ≡ − j( ω/ c)1 k yields M ( r, r , k ) = ( r − r ) × (cid:18) U em c 1 k − j ρ ω E ∗ (cid:19) (5)where U em ( r, k ) = U e + U m = (cid:15) E · E ∗ / µ H · H ∗ / [11].It is further assumed that r = 0 and ρ (cid:28) k || D || , so that thesecond term in (5) can be neglected. In Cartesian coordinates, P ( r, ω ) = µ (cid:15) (cid:2) ( E y H ∗ z − E z H ∗ y )1 x + ( E z H ∗ x − E x H ∗ z )1 y + ( E x H ∗ y − E y H ∗ x )1 z (cid:3) (6) M ( r, , ω ) = µ (cid:15) × (cid:8)(cid:2) y ( E x H ∗ y − E y H ∗ x ) − z ( E z H ∗ x − E x H ∗ z ) (cid:3) x + (cid:2) z ( E y H ∗ z − E z H ∗ y ) − x ( E x H ∗ y − E y H ∗ x ) (cid:3) y + (cid:2) x ( E z H ∗ x − E x H ∗ z ) − y ( E y H ∗ z − E z H ∗ y ) (cid:3) z (cid:9) . (7)The total AM can be decomposed into SAM and OAMcontributions [10, ch. 7] in dual-symmetrized form [12] as M ( r, r , ω ) = 14 [ (cid:15) ( E × A ∗ ) + µ ( H × F ∗ )]+( r − r ) ×
12 [ (cid:15) ( ∇ A ) · E ∗ + µ ( ∇ F ) · H ∗ ] ∆ = M s ( r, ω ) + M o ( r, r , ω ) (8) where ( ∇ A ) · E ∗ ≡ ∇ ( A · E ∗ ) , etc. This identification requiresknowledge of the magnetic and electric vector potentials A ( r | r (cid:48) ) and F ( r | r (cid:48) ) , and hence the spatial distributions ofelectric and magnetic source currents J ( r (cid:48) ) and K ( r (cid:48) ) needto be specified, respectively. In source-free regions, A and F in (8) are replaced with j E/ω and j H/ω , respectively.For unbounded plane waves, both M s and M o are purelyimaginary.III. A VERAGE LM AND AM OF R ANDOM F IELDS
A. Arbitrary Distance or Frequency
To calculate (6) and (7) explicitly, we employ the angularspectral plane-wave expansion of random fields [7], [9], [13] E ( r ) = 1Ω (cid:90) (cid:90) Ω E (Ω) exp( − j k · r )dΩ (9)with a similar expansion for H ( r ) , leading to E α ( r ) H ∗ β ( r ) = 1Ω (cid:18)(cid:90) (cid:90) Ω (cid:19) ( E (Ω ) · α )( H ∗ (Ω ) · β ) × exp[ − j( k − k ∗ ) · r ] δ (Ω ∆Ω )dΩ dΩ (10)for α, β ∈ { x, y, z } . Here, Ω = 2 π sr is the solid angle ofthe half space of incidence above the boundary ( z ≥ ); Ω ∆Ω = (Ω ∪ Ω ) \ (Ω ∩ Ω ) for Ω , and Ω as pointsets; δ ( · ) is Kronecker’s delta ( δ ( ∅ ) = 1 , and 0 otherwise); dΩ i = sin θ i d θ i d φ i with elevation angle θ i and azimuthangle φ i in standard spherical coordinates ( < θ i ≤ π/ , < φ i ≤ π ).For each plane-wave component {E i , H i , k i } , a TE/TM de-composition is performed with respect to its plane of incidence ok i z defining φ i = 0 [7], [9]. The incident plus reflected fieldis aggregated across the angular spectrum by integration across φ i , θ i and the uniformly distributed polarization angle − ψ i in the locally transverse plane. For example, for α = x and β = y , substituting [9, eqs. (10)–(15)] into (10) yields E x ( z ) H ∗ y ( z ) = j π (cid:90) π/ cos θ sin(2 k z cos θ ) sin θ d θ × (cid:90) π (cid:0) E θ H ∗ φ cos φ − E φ H ∗ θ sin φ (cid:1) d φ (11)with the locally transverse E i and H i ( i = 1 , ) given by [9] E iφ = E cos ψ i , E iθ = −E sin ψ i , E ik = 0 (12) H iφ = E η sin ψ i , H iθ = E η cos ψ i , H ik = 0 (13)where E ≡ E (cid:48) − j E (cid:48)(cid:48) is the complex amplitude of the circularelectric field of each plane-wave component, with (cid:104)|E | (cid:105) =2 (cid:104)E (cid:48) ( (cid:48) ) (cid:105) = (cid:104)| E | (cid:105) / and η = (cid:112) µ /(cid:15) . Integration of (11)followed by ensemble averaging (denoted as (cid:104)·(cid:105) ) over E and ψ yields (cid:104) E x H ∗ y (cid:105) = −(cid:104) E y H ∗ x (cid:105) = − j (cid:104)|E | (cid:105) η j (2 kz ) (14) where here and for later use j (2 kz ) ∆ = sinc(2 kz ) (15) j (2 kz ) ∆ = sin(2 kz )(2 kz ) − cos(2 kz )2 kz = − j (cid:48) (2 kz ) (16) j (2 kz ) ∆ = (cid:18) kz ) − kz (cid:19) sin(2 kz ) − kz )(2 kz ) (17) j (2 kz ) ∆ = (cid:18) kz ) − kz ) (cid:19) sin(2 kz ) − (cid:18) kz ) − kz (cid:19) cos(2 kz ) (18)are spherical Bessel functions of the first kind and order zeroto three, respectively. With an analogous calculation, (cid:104) E x H ∗ z (cid:105) = (cid:104) E z H ∗ x (cid:105) = (cid:104) E y H ∗ z (cid:105) = (cid:104) E z H ∗ y (cid:105) = 0 (19)because their kernel’s azimuthal dependence is of the form sin φ or cos φ , as opposed to their square in (11). Combining(7), (14) and (19) yields the ensemble averaged LM and AMas (cid:104) P ( r, ω ) (cid:105) ≡ (cid:104) S (cid:105) c = − j (cid:104)|E | (cid:105) c η j (2 kz ) 1 z (20) (cid:104) M ( r, , ω ) (cid:105) = − j (cid:104)|E | (cid:105) c η j (2 kz ) (cid:0) y x − x y (cid:1) (21)i.e., (cid:104) M (cid:105) = r × (cid:104) P (cid:105) = − r (cid:104) P (cid:105) φ . The result for (cid:104) M ( r, , ω ) (cid:105) in (21) extends to general (cid:104) M ( r, r , ω ) (cid:105) by subtracting r ×(cid:104) P ( r, ω ) (cid:105) from (cid:104) M ( r, , ω ) (cid:105) .The zero real parts of the power flux densities (cid:104) S (cid:105) = c (cid:104) P (cid:105) and c (cid:104) M (cid:105) indicate zero time-averaged energy flow in nor-mal (i.e., z -directed, longitudinal) and transverse azimuthaldirections, respectively. Thus, the ensemble averaged incidentpower and the propagating power after reflection off a PECboundary cancel. The combined LM and AM flux densitiesdepend on all three spatial coordinates via j (2 kz ) and theradial transverse distance || r × z || = (cid:112) x + y . At any height kz above the boundary, (cid:104) M (cid:105) is tangential ( (cid:104) M z (cid:105) = 0 ) andpurely solenoidal ( ∇ × (cid:104) M (cid:105) (cid:54) = 0 , ∇ · (cid:104) M (cid:105) = 0 ). Similar todeterministic fields in free space, (cid:104) M (cid:105) is complementary to theirrotational and normal (cid:104) P (cid:105) , i.e., ∇ × (cid:104) P (cid:105) = 0 , ∇ · (cid:104) P (cid:105) (cid:54) = 0 .In summary, for random fields near a planar PEC boundary,the combined linear (longitudinal) and angular (azimuthal)average power fluxes are reactive and characterized by a 3-D hybrid vector (cid:104) Π ( r, k ) (cid:105) ∆ = − j (cid:104)|E | (cid:105) η j (2 kz ) (cid:0) y x − x y + 1 z (cid:1) . (22)Its LM and AM components can be measured, e.g., using amodified magic T or a turnstile junction with L-shaped in-plane sections that contain inductive (diaphragm) shunt loads.Alternatively, helical and planar chiral power sensors withreactive loading may be used to measure the longitudinaland transverse progression of the phase for SAM and OAM,respectively.The results (20)–(21) are verified using a Monte Carlo (MC)simulation of the random plane-wave spectrum (9). We used370 values of kz ranging from 0.01 to 50 in logarithmic stepsof 0.01 and define x = y = 1 m. For each kz , a set of n θ × n φ × n ψ = 32 × × uniformly spaced angles of incidence andpolarization were generated across (0 , π/ × (0 , π ] × (0 , π ] with n = 30 complex random fields E for each k . This yields plane waves per kz . Fig. 1 shows the resulting averageLM power density (cid:104) S ( kz ) (cid:105) and, by extension, the transversecomponents of the AM power density c (cid:104) M ( kz ) (cid:105) normalizedby the values of x and y .Fig. 1: Amplitude of mean linear or angular power flux densitycomponents (cid:104) S z (cid:105) = c (cid:104) P z (cid:105) = c (cid:104) M x (cid:105) /y = − c (cid:104) M y (cid:105) /x [in unitsW/m ] as a function of height kz above a PEC boundary for (cid:104)E (cid:48) (cid:105) = (cid:104)E (cid:48)(cid:48) (cid:105) = 10 − (V/m) . Blue solid: theory [eq. (20)]; blue dashed:envelopes Ξ ± s for kz (cid:29) / [eq. (33)] and for kz (cid:28) √ [eq.(42)]; black dashed: MC simulation. B. Optimum Locations of Field vs. Power Sensors
On and at asymptotically large distances from the boundary ( kz → + ∞ ) , (cid:104) P (cid:105) and (cid:104) M (cid:105) vanish, which confirms the resultsfor unbounded random fields [6, eqs. (19)-(20)], [13, eq. (51)].They also vanish for j (2 kz ) = 0 , i.e., at frequencies anddistances that are related as tan(2 kz s ) = 2 kz s (23)viz., at z s = 0+ or z s (cid:39) (2 m + 3) π/ (4 k ) when kz s (cid:29) ( m = 0 , , , . . . ). Such measurement locations are preferentialwhen aiming to avoid the influence of a PEC boundary on theaverage LM and AM using a power sensor. Note that theselocations z s apply strictly to dot sensors and CW excitation,and vice versa. Inevitably, the optima become blurred for asensor of finite size or for nonzero bandwidths owing to localaveraging.These findings for (cid:104) P (cid:105) and (cid:104) M (cid:105) complement those forthe average energy densities (cid:104) U e (cid:105) and (cid:104) U m (cid:105) of the 3-Dtotal electric or magnetic vector field (no subscript), the 2-D tangential (subscript t ), and the 1-D Cartesian components( x, y, z ) [7], [8], written here in an alternative but equivalentform as (cid:104) U ( kz ) (cid:105) = (cid:15) (cid:104)|E | (cid:105) (cid:20) ∓ j (2 kz )3 ± j (2 kz )3 (cid:21) (24) (cid:104) U t ( kz ) (cid:105) = 2 (cid:15) (cid:104)|E | (cid:105) (cid:20) ∓ j (2 kz ) ± j (2 kz )2 (cid:21) (25) (cid:104) U ( x )( y ) ( kz ) (cid:105) = (cid:15) (cid:104)|E | (cid:105) (cid:20) ∓ j (2 kz ) ± j (2 kz )2 (cid:21) (26) (cid:104) U z ( kz ) (cid:105) = (cid:15) (cid:104)|E | (cid:105) ± j (2 kz ) ± j (2 kz )] (27)where upper and lower signs apply to electric ( U = U e ) andmagnetic ( U = U m ) densities, respectively. Unlike (23) for (cid:104) S (cid:105) , the asymptotic values of (cid:104) U (cid:105) for kz → ∞ are nowreached for j (2 kz ) = 2 j (2 kz ) , i.e., when tan(2 kz u ) = 2 kz u − kz u ) , ( kz u (cid:54) = 0) (28)viz., at z u (cid:39) ( m + 1) π/ (2 k ) for kz u (cid:29) . The solutions of(23) and (28) are separated by kz u − kz s (cid:39) π/ for m (cid:29) .Similarly, the frequencies and locations for reaching theasymptotic 2-D tangential energy (cid:104) U t ( kz → ∞ ) (cid:105) follow from(25) as solutions of j (2 kz ) = j (2 kz ) / , i.e., when tan(2 kz u,t ) = 2 kz u,t − (2 kz u,t ) , ( kz u,t (cid:54) = 0) (29)viz., at z u,t (cid:39) ( m + 1) π/ (2 k ) when kz u,t (cid:29) . For the 1-Dtangential components (cid:104) U x,y ( kz → ∞ ) (cid:105) , the solutions followfrom (26) as being identical to those for (cid:104) U t (cid:105) . Finally, for the1-D normal component (cid:104) U z ( kz → ∞ ) (cid:105) , the locations are thosefor (cid:104) S (cid:105) but exclude the boundary plane: from (27), j (2 kz ) = − j (2 kz ) is satisfied when tan(2 kz u,z ) = 2 kz u,z , ( kz u,z (cid:54) = 0) (30)viz., at z u,z (cid:39) (2 m + 3) π/ (4 k ) when kz u,z (cid:29) .The practical significance of these different optimum val-ues is that, depending on whether one measures either theCartesian or vector field energy density (or intensity) using afield sensor (wire or loop probe) or the reactive LM or AMpower flux using a power sensor (aperture antenna), thesedevices should be placed at different heights above a PECboundary in order to eliminate the effect of the boundary onthe measurement.The first few optimum locations for kz s , kz u , kz u,t ≡ kz u, ( x )( y ) and kz u,z are listed in Tbl. I and shown in Fig. 2. Ineach case, the optimal distances are spaced by asymptotically π/ , as follows from the asymptotic approximation [17] j (cid:96) (2 kz ) (cid:39) sin(2 kz − (cid:96)π/ kz for kz (cid:29) (cid:96) ( (cid:96) + 1)4 . (31)Excluding the boundary and DC regime ( kz (cid:54) = 0 ), the shortestoptimum distance ( m = 0 ) is attained for a 3-D isotropic fieldsensor ( u ), followed by a 1-D ( u x or u y ) or 2-D ( u t ) tangentialfield probe. A normal field sensor ( u z ) and power sensor ( s )must be placed farthest, at more than twice the distance forthe isotropic field probe. If the boundary plane is included,however, then power sensors exhibit the shortest (viz., zero)optimum distance. m kz s kz u kz u, ( t )( x )( y ) kz u,z TABLE I:
First six locations kz s and kz u for optimal placementof power or field sensors with various orientations. Note that for field intensity (energy) sensors, the optimal kz values are irrespective of the electric or magnetic typeof the sensor. On the other hand, for a combined electric-magnetic sensor or an (un)intentional receptor that is sensitive Fig. 2: First three optimum locations kz opt ( m = 0 , , for areactive power sensor (black) vs. various field sensors (red) near aPEC boundary. Symbols: actual optima (cf. Tbl. I), dashed lines:asymptotic laws. to different components in different measures and orientations,its optimal distance or frequency can be estimated on a caseby case basis by decomposing its orientation into normaland tangential components, which provide the weights for anappropriate superposition of the components of its averageresponse based on (20) and (24)–(26). C. Maximum Deviations of Mean Power and Energy1) Envelopes:
The results in Sec. III-B listed optimumlocations z opt for a specified frequency k , or vice versa. InEMC practice, signals are often wideband or being sweptacross frequency. In addition, surface imperfections, proximityof scatterers or other surfaces, etc., may perturb the field andaffect the optimality of these locations. Given these spatio-spectral uncertainty or aggregation effects, a more distantoptimum ( kz opt (cid:29) ) where the fluctuations of (cid:104) S ( kz ) (cid:105) are weaker may actually be preferential, in order to reducethe sensitivity on the optimum placement. The envelopesof (cid:104) S ( kz ) (cid:105) then represent the maximum deviation from theasymptotic value (cid:104) S ( kz → + ∞ ) (cid:105) that may be expected as asuboptimal alternative.The asymptotic analytic representation ˜ (cid:104) S (cid:105) ∆ = (cid:104) S (cid:105) +j H [ (cid:104) S (cid:105) ] follows with the Hilbert transformation H [ j (2 kz )] (cid:39) cos(2 kz − π/ / (2 kz ) for (31) and − j ≡ exp( − j π/ as (cid:104) (cid:101) S ( kz (cid:29) / (cid:105) (cid:39) (cid:104)|E | (cid:105) η kz exp [j(2 kz − π )] (32)whose signed magnitude defines the upper and lower envelopes Ξ ± s ( kz (cid:29) / (cid:39) ± (cid:104)|E | (cid:105) η kz . (33)These are indicated in Fig. 1. Their half separation ∆ Ξ s ( kz ) ∆ =[ Ξ + s ( kz ) − Ξ − s ( kz )] / measures the maximum absolute de-viation from the asymptotic mean power (cid:104) S ( kz → + ∞ ) (cid:105) =[ Ξ + s ( kz ) + Ξ − s ( kz )] / . For both electric and magnetic energy densities, similar definitions and calculations yield Ξ ± u ( kz (cid:29) /
2) = (cid:15) (cid:104)|E | (cid:105) (cid:18) ± kz (cid:19) (34) Ξ ± u, ( x )( y ) ( kz (cid:29) /
2) = (cid:15) (cid:104)|E | (cid:105) (cid:18) ± kz (cid:19) = 12 Ξ ± u,t ( kz (cid:29) / (35) Ξ ± u,z ( kz (cid:29) /
2) = (cid:15) (cid:104)|E | (cid:105) (cid:18) ± kz ) (cid:19) (36)where upper and lower signs now correspond to upper andlower envelopes. These are shown in Fig. 3 together with (24)–(27). (a)(b)Fig. 3: Theoretical averages [eqs. (24)–(27)] (light solid), their MCsimulations (dark dashed), and asymptotic envelopes Ξ ± [eqs. (34)–(36)] (light dotted), for (a) electric and (b) magnetic energy densities[in units J/m ] as a function of height kz above a PEC boundary for (cid:104)E (cid:48) (cid:105) = (cid:104)E (cid:48)(cid:48) (cid:105) = 10 − (V/m) . Black/grey: (cid:104) U (cid:105) ; blue/cyan: (cid:104) U t (cid:105) ;maroon/red: (cid:104) U z (cid:105) ; olive/green: (cid:104) U x (cid:105) . The associated maximum relative deviations are ∆ ξ u ( kz (cid:29) / ∆ = ∆ Ξ u ( kz (cid:29) / Ξ u ( kz → + ∞ ) = 12 kz (37) ∆ ξ u, ( x )( y ) ( kz (cid:29) /
2) = ∆ ξ u,t ( kz (cid:29) /
2) = 34 kz (38) ∆ ξ u,z ( kz (cid:29) /
2) = 34( kz ) (39)For example, at kz = 10 we find (cid:104) U x (10) (cid:105) = 0 . (cid:15) (cid:104)|E | (cid:105) ,whereas Ξ ± u,x (10) = [(1 ± . / (cid:15) (cid:104)|E | (cid:105) results in ∆ Ξ u,x (10) = 0 . (cid:15) (cid:104)|E | (cid:105) and ∆ ξ u,x (10) = 0 . . For (cid:104) S (cid:105) , a different normalization is adopted than for (cid:104) U (cid:105) , viz., ∆ ξ s ( kz (cid:29) / ∆ = ∆ Ξ s ( kz (cid:29) / (cid:104)|E | (cid:105) /η = 12 kz (40) In (36), the envelopes of (cid:104) U z (cid:105) are of leading second order ( kz ) − , be-cause of cancelling first-order terms. For the other energies, additional second-and higher-order terms in ( kz ) − provide corrections when kz (cid:54)(cid:29) / . in view of Ξ s ( kz → + ∞ ) = 0 . The motivation for this partic-ular choice will become clear in Sec. III-C3 and correspondsto a normalization of ∆ Ξ u and (cid:104) U (cid:105) by (cid:15) (cid:104)|E | (cid:105) .
2) Maximum Deviation:
The above relative deviations canbe used with a specified minimum distance of the sensor tothe PEC boundary to estimate the maximum local deviationfrom the asymptotic mean value, or vice versa. Tbl. II showsthese deviations for kz = π/ , π and π . The entries indicatethat a normally directed field sensor gives a smaller maximumdeviation than a power sensor, at sufficiently large distances.For example, ∆ ξ u,z = 7 . compared to ∆ ξ s = 15 . at z = λ/ with the chosen normalization, whereas at z = λ/ the maximum deviations of the normalized (cid:104) U z (cid:105) and (cid:104) S z (cid:105) aremore similar ( . vs. . , respectively). kz ∆ ξ s ∆ ξ u ∆ ξ u, ( t )( x )( y ) ∆ ξ u,z π/ π π TABLE II:
Maximum relative deviation for envelopes of (cid:104) S z (cid:105) / ( (cid:104)|E | (cid:105) /η ) and (cid:104) U α (cid:105) / ( (cid:15) (cid:104)|E | (cid:105) ) at quarter-, half-, and full-wavelength distances from a PEC boundary. At the other extreme of very low frequencies and/or smalldistances from the boundary, using the approximation [17] j (cid:96) (2 kz ) (cid:39) (2 kz ) (cid:96) (2 (cid:96) + 1)!! for kz (cid:28) (cid:114) (2 (cid:96) + 2)(2 (cid:96) + 3) (cid:96) + 1 (41)the asymptote for quasi-static deviations from (cid:104) S ( kz ) (cid:105) is Ξ − s ( kz (cid:28) √ (cid:39) − (cid:104)|E | (cid:105) kz η . (42)
3) Working Volume of a Reverberation Chamber:
The localdeviations from the asymptotic (cid:104) S (cid:105) or (cid:104) U (cid:105) may be used toestimate the working volume (WV) of a reverberation chamberbefore measurement. Consider a cubic cavity of side length L .Its WV is a symmetrically located interior cube of sides L − z W for some z W . On the boundary of this WV, a maximumtolerable relative deviation ∆ ξ of the mean energy or power isspecified with reference to its asymptotic (ideal) value when L → + ∞ , which yields kz W from (37)–(40). The relativeWV thus follows as V W (∆ ξ ) V = (cid:18) kL − kz W (∆ ξ ) kL (cid:19) . (43)This dependence of V W /V on ∆ ξ is shown in Fig. 4 foraverage power and energy components, for two cavity sizevalues of V /λ . It is seen that the total (vector) (cid:104) U e (cid:105) or (cid:104) U m (cid:105) and (cid:104) S (cid:105) ≡ (cid:104) S z (cid:105) yield the same WV. By contrast, (cid:104) U z (cid:105) givesa larger WV, whereas (cid:104) U t (cid:105) and (cid:104) U ( x )( y ) (cid:105) give a smaller WV.E.g., for ∆ ξ = 0 . , the value of V W /V based on (cid:104) U e (cid:105) , (cid:104) U m (cid:105) or (cid:104) S (cid:105) is . , whereas for (cid:104) U ( e )( m ) ,z (cid:105) it is . , whilstfor (cid:104) U ( e )( m ) ,x (cid:105) it is merely . . The larger value of WVfor (cid:104) U z (cid:105) can be traced to the more rapid decay of its envelope,viz., according to ( kz ) − , causing the specified uniformity tobe reached closer to the boundary.In practice, adjacent and opposite walls create additionalstanding waves, whence (43) serves merely as a first estimate, whose accuracy nevertheless increases with chamber volume V . The estimates and results serve as a guide to a more preciseevaluation to account for the effects of all boundaries, e.g.,based on a full-wave simulation or measurement.Note that, for a combined electric-magnetic field sensor, V W /V = 1 for any specified ∆ Ξ because the dependence of (cid:104) U e,α (cid:105) + (cid:104) U m,α (cid:105) on kz cancels for any component α .Fig. 4: Relative working volume V W as a percentage of cavityvolume V for specified relative deviation ∆ ξ of average power ( S )or average electric or magnetic energy ( U ). Blue solid: V = (5 λ ) ;red dashed: V = (50 λ ) . IV. C
ONSERVATION OF E NERGY AND M OMENTUM
A. Poynting Theorem for Average Power Flux and Energy
For deterministic fields in a lossless medium, the Poyntingtheorem states that ∇ · S + j ω ( U m − U e ) = 0 . (44)Application of (10) yields ∂∂z (cid:26) E x H ∗ y E y H ∗ x (cid:27) = ± j2 k δ (Ω ∆Ω ) (cid:0) E θ H ∗ φ − E φ H ∗ θ (cid:1) × [ j (2 kz ) − j (2 kz )] (45)where the upper and lower signs refer to E x H ∗ y and E y H ∗ x ,respectively. Substituting (12)–(13) herein followed by ensem-ble averaging yields ∂ (cid:104) S z (cid:105) /∂z , which is found to coincidewith the result obtained by direct differentiation of (20). Thus, (cid:104)∇ · S (cid:105) = ∇ · (cid:104) S (cid:105) . Moreover, by differentiation of (24) using d j n (2 kz )d z = 2 k n + 1 [ nj n − (2 kz ) − ( n + 1) j n +1 (2 kz )] (46)it is found that (44) also applies to (cid:104) S (cid:105) , viz., ∇ · (cid:104) S (cid:105) = − j ω (cid:15) (cid:104)|E | (cid:105) j (2 kz ) − j (2 kz )]= − j ω ( (cid:104) U m (cid:105) − (cid:104) U e (cid:105) ) . (47)The result (47) is confirmed by MC simulation shown in Fig.5. Thus, Poynting’s theorem for (physical) deterministic powerand energy extends to their (arithmetic) averages for randomfields, and also to the average power flux (divergence). Theextended theorem indicates that, for arbitrary kz , the spatiallocal flow of the reactive (nonradiating) linear and angularaverage power fluxes c (cid:104) P (cid:105) and c (cid:104) M (cid:105) are associated andin phase with temporal oscillations of the imbalance betweenelectric and magnetic average energies. For increasing kz , thesign of their difference (i.e., the dominance of local averagemagnetic over electric energy density, or vice versa) alternates,while the magnitude (i.e., strength of the imbalance) decreases. Fig. 5: Conservation of mean linear power flux and rate of changeof energy density ∇ · (cid:104) S (cid:105) ≡ j ∂ Im( (cid:104) S z (cid:105) ) /∂z = j ω ( (cid:104) U e (cid:105) − (cid:104) U m (cid:105) ) [in units W/m ] as a function of height kz above a PEC boundaryfor (cid:104)E (cid:48) (cid:105) = (cid:104)E (cid:48)(cid:48) (cid:105) = 10 − (V/m) at f = 100 MHz. Solid: MCsimulation; dashed: theory.
B. Conservation of Average LM and AM1) Average EM Stress:
For stochastic fields, the (random)symmetrized Maxwell stress dyadic (cf. Appendix) T ( r, k ) = T e + T m ∆ = (cid:15) (cid:18) E E ∗ − | E | (cid:19) + µ (cid:18) H ∗ H − | H | I (cid:19) (48)characterizes the random radiation pressure T · z (LM flux)and shear T × z with reference to the surface normal z .The resultant exerted random EM force follows by integrating T across the oriented boundary surface (or ∇ · T across theenclosed volume). To evaluate (cid:104) T (cid:105) , note that (cid:104) E α E ∗ β (cid:105) = (cid:104) H α H ∗ β (cid:105) = 0 , ∀ α (cid:54) = β ∈ { x, y, z } (49)because the azimuthal dependence of their kernels are ofthe form sin φ , cos φ or sin(2 φ ) , or because (cid:104)E φ E ∗ θ (cid:105) ∝(cid:104) sin ψ cos ψ (cid:105) = 0 . From (48) and (49), with the aid of [8,eqs. (8)–(10)], it follows that (cid:104) T (cid:105) is diagonal, isotropic, andhomogeneous (i.e., nondispersive), viz., (cid:104) T ( kz ) (cid:105) = − (cid:104) U em,z (cid:105) I t + (cid:104) U em,z (cid:105) − (cid:104) U em,t (cid:105) z z = − (cid:15) (cid:104)|E | (cid:105) I (50)where I t ∆ = 1 x x + 1 y y is the transverse unit dyadic, with (cid:104) U em,z (cid:105) ∆ = (cid:104) U e,z (cid:105) + (cid:104) U m,z (cid:105) = 23 (cid:104) (cid:15) |E | (cid:105) , (cid:104) U em,t (cid:105) ∆ = (cid:104) U e,t (cid:105) + (cid:104) U m,t (cid:105) = 43 (cid:104) (cid:15) |E | (cid:105) , ∀ kz ≥ (51)and (cid:104) U ( e )( m ) ,t (cid:105) ∆ = (cid:104) U ( e )( m ) ,x (cid:105) + (cid:104) U ( e )( m ) ,y (cid:105) . The negativesign in (50) indicates that the average stress constitutes a pressure, as opposed to a tension. While all (cid:104) U ( e )( m ) , ( α )( t ) (cid:105) are individually dispersive, the sum of each matching pair (cid:104) U e,α (cid:105) + (cid:104) U m,α (cid:105) – and hence (cid:104) U em,t (cid:105) , (cid:104) U em,z (cid:105) and (cid:104) T (cid:105) –are real and dispersionless with respect to kz . Thus, (cid:104) T (cid:105) = (cid:104) T (cid:105) I = (cid:10) T (cid:11) (52)and ∇ · (cid:104) T (cid:105) = 0 , ∂ (cid:104) T αα (cid:105) ∂z = 0 , ∀ α ∈ { x, y, z } . (53)Fig. 6 shows MC results for (cid:104) T ( kz ) (cid:105) . The diagonal elementsare easily verified to correspond to their theoretical constantreal values (50). The residual off-diagonal (cid:104) T αβ (cid:105) originatefrom finite-precision errors in the numerical quadrature of (49),yet demonstrate that (cid:104) T (cid:105) is Hermitean.Unlike the overall (cid:104) T (cid:105) , the average individual electric (cid:104) T e (cid:105) and magnetic (cid:104) T m (cid:105) stress dyadics, i.e., (cid:104) T ( e )( m ) ( kz ) (cid:105) = − (cid:104) U ( e )( m ) ,z (cid:105) I t + (cid:104) U ( e )( m ) ,z (cid:105) − (cid:104) U ( e )( m ) ,t (cid:105) z z (54)are dispersive, as follows from (26)–(27). These are also shownin Fig. 6. From Fig. 3 and (54), it follows that the averagenormal electric stress (cid:104) T e,zz (cid:105) changes from tension ( > ) topressure ( < ) at kz (cid:39) . . By contrast, the average normalmagnetic stress always occurs as a pressure ( (cid:104) T m,zz (cid:105) < ).Fig. 6: MC simulated (cid:104) T (cid:105) and (cid:104) T ( e )( m ) (cid:105) [in units J/m or N/m ] asa function of height kz above a PEC boundary for (cid:104)E (cid:48) (cid:105) = (cid:104)E (cid:48)(cid:48) (cid:105) =10 − (V/m) . Blue: Re[ (cid:104) T αβ (cid:105) ]; red: Im[ (cid:104) T αβ (cid:105) ]; green: Re[ (cid:104) T e,αα (cid:105) ];cyan: Re[ (cid:104) T m,αα (cid:105) ]; black: Re[ (cid:104) T e,αα (cid:105) − (cid:104) T m,αα (cid:105) ]. As a practical EMMC application of T within the realmof mode-stirred reverberation, consider the concept of elec-tromechanical self-stirring . In this scenario, LM or AM arisesfrom the EM stress caused by a source field impingingonto suspended or free-flowing small scatterers, causing theirmotion or morphing if inertia is sufficiently small. In turn,this affects the cavity field distribution, hence T ( r ) via (48),and therefore P ( r ) , M ( r, r ) via (55)–(56), etc. The result is akin to that of chaff in radar, except that the mechanismfor its dynamics here is purely EM and equally feasiblein vacuum. The efficiency and control of self-stirring isgoverned by the field strength and mechanical properties ofeach scatterer. Contrary to conventional mechanical modestirring, self-stirring is most efficient for electrically small scatterers, as it relies on a net nonzero integrated T across theirsurface. As is well known, mechanical pressure exerted ontothe walls of an overmoded microwave cavity easily results insubstantial changes to mode degeneracy, coupling and spectra,even for geometric distortions smaller than λ/ [14, Fig.1]. A fortiori, corresponding effects of EM stress are relevantto macro- or mesoscale structures with small inertia. If leftuncontrolled, self-stirring produces noise additional to thermalnoise caused by ohmic dissipation.
2) Conservation of LM and AM:
For deterministic fields invacuum, the conservation of LM density states that [10], [11] ∇ · T − j ωP = 0 . (55)This follows from P in (4) upon adding ( ∇· H ) H ∗ ≡ , dual-symmetrizing (cid:15) ∇| E | , and recalling that ∇| E | = ∇· ( | E | I ) and ( ∇ · E ) E ∗ = ∇ · ( E E ∗ ) . The corresponding conservationof AM follows by pre-multiplying (55) by r × and using thedyadic identities r × ( ∇ · T ) = − r · ( ∇ × T ) = −∇ · (cid:0) T × r (cid:1) to yield ∇ · (cid:0) T × r (cid:1) + j ωM = 0 . (56)For arbitrary kz from the PEC boundary, (56) is in fact amanifestation of Noether’s theorem with generator z × r ≡ r φ , which holds because of rotational symmetry around z ,such that M z ( kz ) = 0 – as already found in (21) – and hence [ ∇ · (cid:0) T ( kz ) × r (cid:1) ] z = 0 follows from (56).Since P and M are proportional to linear and angular EMpower flux, eqs. (55) and (56) provide a conduit between EMand mechanical effects. On account of Gauss’s theorem, therate of change of the LM inside a finite volume can thus beobserved from the net flux of stress ∇· T through its boundarywhere EM forces act. For random fields, it appears that theaverage flux (cid:104)∇· T (cid:105) and the fluxes of the average ∇·(cid:104) T ( e )( m ) (cid:105) enable alternative physical interpretations of (cid:104) P (cid:105) in (20), aswill be shown next. a) Average Flux: To extend and evaluate (55)–(56) forrandom fields near a PEC plane, note that spatial variations of E α E ∗ β and H ∗ α H β are in the normal direction ( ∇ = 1 z ( ∂/∂z ) )for all α, β ∈ { x, y, z } , whence ∇ · T = (cid:88) α = z (cid:88) β = x,y,z ∂T αβ ∂α β . (57)Correspondingly, the dyadic skew product T × r may becalculated termwise, i.e., ( E E ∗ ) × r = E ( E ∗ × r ) , etc.,followed by differentiation, resulting in r × ( ∇ · T ) = (cid:88) α = z (cid:88) β = x,y,z ( r × β ) ∂T αβ ∂α . (58)It can be easily shown from (10) that (cid:104) T zx (cid:105) = (cid:104) T zy (cid:105) = 0 = (cid:104) T ( e )( m ) ,zx (cid:105) = (cid:104) T ( e )( m ) ,zy (cid:105) and that (cid:28) E α ∂E ∗ β ∂z (cid:29) = (cid:28) H α ∂H ∗ β ∂z (cid:29) = 0 , ∀ α (cid:54) = β ∈ { x, y, z } (59) for the same reasons as those given for (49). Therefore, (cid:104) ∂T zx /∂z (cid:105) = (cid:104) ∂T zy /∂z (cid:105) = 0 . Ensemble averaging of (57)and (58) yields (cid:104)∇ · T (cid:105) = (cid:104) (cid:0) ∇ · T (cid:1) z (cid:105) = (cid:28) ∂T zz ∂z (cid:29) z (60) (cid:104) r × ( ∇ · T ) (cid:105) = y (cid:28) ∂T zz ∂z (cid:29) x − x (cid:28) ∂T zz ∂z (cid:29) y . (61)Thus, whereas ∇· T and r × ( ∇· T ) depend on the z -derivativeof both the fluctuating EM stress ( T zz ) and shear ( T zx , T zy )components as generated by each plane wave component, onlythe dependence on the EM stress survives after averaging.Figs. 7 and 8 show (cid:104)∇· T (cid:105) and (cid:104)∇· ( T × r ) (cid:105) as a function of kz , respectively, for a MC simulation based on T calculatedfrom (10) and after finite differencing of ∇ z for discrete kz . Itis seen that conservation of LM and AM also holds betweentheir statistical averages and the average flux of EM stressesand their moments (or torque, in case of SAM), respectively,i.e., (cid:104)∇ · T (cid:105) − j ω (cid:104) P (cid:105) = 0 (62) (cid:104)∇ · (cid:0) T × r (cid:1) (cid:105) + j ω (cid:104) M (cid:105) = 0 . (63)While (62) and (63) follow of course trivially from (55)and (56) from a purely mathematical perspective, the demon-stration of their validity here through an independent MCcalculation of T , (cid:104) T (cid:105) , (cid:104)∇ · T (cid:105) and (cid:104)∇ · ( T × r ) (cid:105) starting fromthe plane-wave expansion serves to validate (55) and (56) asa starting point based on first principles.Fig. 7: Shear ( x, y ) and normal ( z ) components of average spatialgradient (cid:104)∇ · T (cid:105) [in units J/m or N/m ] as a function of height kz above a PEC boundary for (cid:104)E (cid:48) (cid:105) = (cid:104)E (cid:48)(cid:48) (cid:105) = 10 − (V/m) at f = 100 MHz. Blue solid: MC real part; red solid: MC imaginarypart; blue dashed, theoretical j ω (cid:104) P z (cid:105) . b) Flux of Average Electric and Magnetic Stresses: From(20), (53) and (55), it follows that in general ∇ · (cid:104) T (cid:105) − j ω (cid:104) P (cid:105) (cid:54) = 0 (64) ∇ · (cid:104) T × r (cid:105) + j ω (cid:104) M (cid:105) (cid:54) = 0 (65) Fig. 8: Angular shear ( x, y ) and stress ( z ) components of averagespatial gradient (cid:104)∇· ( T × r ) (cid:105) [in units J/m or N/m ] as a function ofheight kz above a PEC boundary for (cid:104)E (cid:48) (cid:105) = (cid:104)E (cid:48)(cid:48) (cid:105) = 10 − (V/m) at f = 100 MHz. Blue solid: MC real part; red solid: MC imaginarypart; blue dashed: theoretical j ω (cid:104) M ( x )( y ) (cid:105) . except at the optimum locations kz s where (cid:104) P (cid:105) or (cid:104) M (cid:105) vanish(cf. Sec. III-B). Generally, spatial differentiation and statisticalaveraging operations commute (in particular (cid:104)∇· T (cid:105) = ∇·(cid:104) T (cid:105) )provided that the probability distribution (of T ) is spatiallyhomogeneous [15, Sec. 7.2]. This condition has previouslybeen shown to be violated for U e near an EM boundary inits normal direction [16, Figs. 5-7]. Hence (64) and (65) areconsistent with this result. On the other hand, in Sec. IV-Ait was found that (cid:104)∇ · S (cid:105) = ∇ · (cid:104) S (cid:105) even though S and (cid:104) S (cid:105) (as well as U m − U e and (cid:104) U m (cid:105) − (cid:104) U e (cid:105) ) are dispersivenear the boundary. This result is again not inconsistent withcommutativity, because the boundary zone field is statisticallyinhomogeneous. Note, however, that the average total energydensity is nondispersive, viz., (cid:104) U em (cid:105) ≡ (cid:104) U e (cid:105) + (cid:104) U m (cid:105) =2 (cid:15) (cid:104)|E | (cid:105) .Expressions for ∇·(cid:104) T ( e )( m ) (cid:105) and ∇·(cid:104) T ( e )( m ) × r (cid:105) are readilyobtained as follows. Application of (10) to the calculation of ∂E α E ∗ β /∂z and ∂H α H ∗ β /∂z produces zero when α (cid:54) = β , and ∂∂z (cid:26) | E ( x )( y ) | | H ( x )( y ) | (cid:27) = ± k δ (Ω ∆Ω ) (cid:20)(cid:26) E θ E ∗ θ H θ H ∗ θ (cid:27) × [3 j (2 kz ) − j (2 kz )] + (cid:26) E φ E ∗ φ H φ H ∗ φ (cid:27) j (2 kz ) (cid:21) (66) ∂∂z (cid:26) | E z | | H z | (cid:27) = ∓ k δ (Ω ∆Ω ) (cid:26) E θ E ∗ θ H θ H ∗ θ (cid:27) × [ j (2 kz ) + j (2 kz )] (67)otherwise. Substituting (12)–(13) followed by ensemble aver- aging leads to ∂ (cid:104) U ( x )( y ) (cid:105) ∂z = ± k(cid:15) (cid:104)|E | (cid:105) j (2 kz ) − j (2 kz )] (68) ∂ (cid:104) U z (cid:105) ∂z = ∓ k(cid:15) (cid:104)|E | (cid:105) j (2 kz ) + j (2 kz )] (69) ∂ (cid:104) T ( e )( m ) (cid:105) ∂z = ± k(cid:15) (cid:104)|E | (cid:105) (cid:110) [ j (2 kz ) + j (2 kz )] I t − j (2 kz )1 z z } (70)where upper and lower signs again refer to electric ( U e , T e )and magnetic ( U m , T m ) quantities, respectively. Eqs. (68)–(70) also follow from (26)–(27) with the aid of (46). Eq. (70)confirms (53) for the sum (cid:104) T (cid:105) ≡ (cid:104) T e (cid:105) + (cid:104) T m (cid:105) and enables(64)–(65) to be refined to ∇ · (cid:104) T ( e )( m ) (cid:105) = ∓ j ω (cid:104) P (cid:105) = ∓ k(cid:15) (cid:104)|E | (cid:105) j (2 kz )1 z (71) ∇ · (cid:104) T ( e )( m ) × r (cid:105) = ± j ω (cid:104) M (cid:105) . (72)Furthermore, (71) and (72) reveal a conservation law betweenthe flux of the differential average EM stress ∆ (cid:104) T (cid:105) = (cid:104) T e (cid:105) −(cid:104) T m (cid:105) , and the rate of oscillation of the average LM or AM,viz., ∇ · ( (cid:104) T e (cid:105) − (cid:104) T m (cid:105) ) + j2 ω (cid:104) P (cid:105) = 0 (73) ∇ · (cid:104) ( (cid:104) T e (cid:105) − (cid:104) T m (cid:105) ) × r (cid:105) − j2 ω (cid:104) M (cid:105) = 0 . (74)Eq. (73) also follows more directly by substituting (25) and(27) into (54), which yields ∆ T zz = j (2 kz ) , and applying(46). To illustrate and validate these results, Figs. 6 and 9 show (cid:104) ∆ T αα ( kz ) (cid:105) ∆ = (cid:104) T e,αα ( kz ) (cid:105) − (cid:104) T m,αα ( kz ) (cid:105) and theirdivergence, respectively. The latter Figure confirms (73) for α = z and (70) for arbitrary α . C. Relation Between EM Stress and Energy
For deterministic fields, substitution of (55) into (44) en-ables the second-order spatial derivatives (curvatures) of theradiation stress components to be related to the EM energyimbalance U m − U e (or the second-order time derivative ofthe total energy, for general time-dependent fields) withoutrecourse to the LM or power, viz., j ω ∇ · P = k ( U m − U e ) = ∇ · ( ∇ · T ) (75)where ∇ · ( ∇ · T ) = ∂ T zz /∂z in the present configuration.For an averaged random field, the corresponding identityfollows from (47) and (62) as ∇ · (cid:104)∇ · T (cid:105) − k ( (cid:104) U m (cid:105) − (cid:104) U e (cid:105) ) = 0 . (76)Hence (cid:104)∇·∇· T (cid:105) = ∇·(cid:104)∇· T (cid:105) but ∇·(cid:104)∇· T (cid:105) (cid:54) = ∇·∇·(cid:104) T (cid:105) = 0 in view of the foregoing analysis, except asymptotically for kz → + ∞ or at discrete locations and frequencies kz s atwhich the dispersion vanishes. In terms of the individual The transverse diagonal elements ( α = x, y ; cf. top two plots in Fig. 9)are only relevant in configurations for which ∇ x and/or ∇ y are also nonzero,e.g., when one or more additional adjacent boundaries are perpendicular to x or y . Fig. 9:
Spatial gradients of average differential EM stress com-ponents ∇ · (cid:104) ∆ T (cid:105) [in units J/m or N/m ] as a function of height kz above a PEC boundary and (cid:104)E (cid:48) (cid:105) = (cid:104)E (cid:48)(cid:48) (cid:105) = 10 − (V/m) at f = 100 MHz. Blue solid: MC real part, red solid: MC imaginarypart, black dashed: theoretical ∂ (cid:104) T e,αα − T m,αα (cid:105) /∂z for α = x, y, z [eq. (70)]. electric and magnetic average stresses and energies, (76) with(73) degenerates into separate equations, viz., ∇ · ∇ · (cid:104) T ( e )( m ) (cid:105) ± k (cid:104) U ( e )( m ) (cid:105) = 0 (77)where the upper and lower signs refer to ( (cid:104) T e (cid:105) , (cid:104) U e (cid:105) ) and ( (cid:104) T m (cid:105) , (cid:104) U m (cid:105) ) , respectively.V. C ONCLUSION
The average LM (20) and AM (21) (and the correspondinglinear and angular power flux densities) of a random field neara PEC boundary exhibit a dependence on the electric distance kz that contains similarities as well as differences comparedto the dependence of the average electric or magnetic energydensities (24)–(27). Similar to the energy, the strength ofthe average momentum and power decays with increasingdistance in a damped oscillatory manner and vanishes far awayfrom the boundary. By contrast, the average momentum andpower flux vanish on the boundary itself, their rate of decaywith kz is different, and their asymptotic deep values areattained at values of kz in (23) that are interleaved with (i.e.,spaced by approximately π/ from) those for the total energydensity (28). As an application to reverberation chambers, itwas shown that by specifying a maximum tolerable deviation(nonuniformity) of the average boundary power or energy fromits asymptotic free-space value, a performance based metric forthe size of the working volume of a chamber can be definedand calculated via (43).The Poynting theorem for conservation between energyimbalance and power flux in the absence of ohmic losseswas found to remain satisfied for their (arithmetic) statisticalaverages in the case of random fields near a PEC plane, extending its validity beyond deterministic fields to (47).Conservation of the average LM and AM can be expressedeither for the average flux of the full EM stresses (62) and itsmoment (63), or in terms of the flux of electric or magneticstress (71) or their difference (73), and their moments (72)and (74), respectively. The average EM energy imbalance andstress are directly related as (76) or individually as (77).As a final comment, LM and AM of random fields werefound to exhibit a partial and statistical behaviour, rather thanbeing an all-or-nothing property in the case of deterministicfields. This is similar to other wave characteristics of in-homogeneous random fields, in particular for the degree ofpolarization [18], [19]. This is not surprising in view of thefact that polarization content is already comprised as SAMwithin AM (8). R EFERENCES[1] J. H. Poynting, “The wave motion of a revolving shaft, and a suggestionas to the angular momentum in a beam of circularly polarised light,”
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AXWELL S TRESS D YADIC FOR T IME -H ARMONIC R ANDOM F IELDS
The Maxwell stress dyadic is usually formulated in the timedomain as (e.g., [10, Sec. 6.8]) T( r, t ) ∆ = (cid:15) E E + µ H H − (cid:0) (cid:15) E + µ H (cid:1) I. (78)With an assumed exp(j ωt ) dependence, the correspondingexpression for the complex T ( r, ω ) for a time harmonic ran-dom field ( E, H, ω ) can be derived as follows. First, considerexcitation by a single plane wave ( E , H , k ) from the angularspectrum of ( E, H ) . Being a sum of products of complex fieldquantities, the complex Lorentz force can be written as F ( r, k ) = 12 ( (cid:37) E ∗ + J × B ∗ ) . (79)Applying Gauss’s and Amp`ere’s laws to express (cid:37) and J interms of their source fields gives F = 12 [( ∇ · D ) E ∗ + ( ∇ × H ) × B ∗ − j ω D × B ∗ ] . (80)The last term in (80) equals − j ω P . Using dyadic algebra andthe vector identity H ∗ × ( ∇×H ) = ∇ ( H ∗ ·H ) − ( ∇·H ∗ ) H = ∇ |H| − ∇ · ( H ∗ H ) , this yields in vacuum F = 12 [ (cid:15) ∇ · ( E E ∗ ) + µ ∇ · ( H ∗ H ) − µ ∇ · ( |H| I ) (cid:3) − j ω P = 0 . (81)Expressing j ω P as ∇ · T in view of (55) gives T ( r, k ) = 12 (cid:0) (cid:15) E E ∗ + µ H ∗ H − µ |H| I (cid:1) . (82)This expression for T is dual-asymmetric because (79) is assuch. Formally applying H = E /η enables the last term in(82) to be dual-symmetrized [20, sec. 4.2] as µ |H| I = (cid:0) (cid:15) |E| + µ |H| (cid:1) I/ . Spherical integration of each dyad E α E ∗ β and H ∗ α H β according to (10) finally results in the stressdyadic for the time-harmonic random field as T ( r, ω ) = 12 [ (cid:15) E E ∗ + µ H ∗ H − (cid:0) (cid:15) | E | + µ | H | (cid:1) I (cid:21) ..