Fundamental Relation Between Phase and Group Velocity, and Application to the Failure of Perfectly Matched Layers in Backward-Wave Structures

Fundamental relation between phase and group velocity, and application to the failure of perfectly matched layers in backward-wave structures

The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters.

Citation Loh, Po-Ru et al. “Fundamental relation between phase and group velocity, and application to the failure of perfectly matched layers in backward-wave structures.” Physical Review E 79.6 (2009): 065601. © 2009 The American Physical Society.

As Published http://dx.doi.org/10.1103/PhysRevE.79.065601

Publisher American Physical Society

Version Final published version

Citable link http://hdl.handle.net/1721.1/51780

Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. RAPID COMMUNICATIONS

PHYSICAL REVIEW E 79, 065601͑R͒͑2009͒

Fundamental relation between phase and group velocity, and application to the failure of perfectly matched layers in backward-wave structures

Po-Ru Loh,1,* Ardavan F. Oskooi,2 Mihai Ibanescu,3 Maksim Skorobogatiy,4 and Steven G. Johnson1 1Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 2Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 3OmniGuide Communications, One Kendall Square, Building 100, No. 3, Cambridge, Massachusetts 02139, USA 4Génie Physique, École Polytechnique de Montréal, CP 6079, Centre-Ville Montreal, Québec, Canada H3C3A7 ͑Received 4 March 2009; published 11 June 2009͒ We demonstrate that the ratio of group to phase velocity has a simple relationship to the orientation of the electromagnetic ﬁeld. In nondispersive materials, opposite group and phase velocity corresponds to ﬁelds that are mostly oriented in the propagation direction. More generally, this relationship ͑including the case of dispersive and negative-index materials͒ offers a perspective on the phenomena of backward waves and left-handed media. As an application of this relationship, we demonstrate and explain an irrecoverable failure of perfectly matched layer absorbing boundaries in computer simulations for constant cross-section waveguides with backward-wave modes and suggest an alternative in the form of adiabatic isotropic absorbers.

DOI: 10.1103/PhysRevE.79.065601 PACS number͑s͒: 41.20.Jb, 42.25.Bs, 42.79.Gn, 02.70.Ϫc

I. INTRODUCTION of adiabatic non-PML absorption tapers ͓12͔͑unrelated to the failure of PML in periodic media described in our previ- Recent years have seen a renewed interest in “left- ous work ͓12͔͒. handed” media in which the phase velocity ͑v ͒ and group p In what follows, we first review the fixed-frequency velocity ͑v ͒ of waves are antiparallel. Such media include g eigenproblem formulation of Maxwell’s equations ͓13,14͔ theoretical negative-index ͑␧,␮Ͻ0͒ media and their and then use this to derive Eq. ͑1͒͑an immediate precursor metamaterial realizations ͓1,2͔, wavelength-scale periodic of which can also be found in our book ͓14͔͒. We discuss the media that mimic some qualitative features of negative-index case of dispersive and negative-index media. Then, we re- media ͓3–5͔ while not having a strictly well defined v ͓6͔, p view important cases of positive-index nondispersive geom- and certain uniform cross-section positive-index waveguides etries with backward-wave modes and argue that PML irre- with “backward-wave” modes ͓7,8͔. In this Rapid Commu- deemably fails in such geometries. This failure can be nication, we derive a fundamental relationship between v , p physically understood in light of Eq. ͑1͒. We close by pro- v , and the orientation of the fields for any medium g posing a non-PML alternative for such cases. ͕␧͑x,y͒,␮͑x,y͖͒, where z invariance ensures that the phase velocity in the z direction is unambiguously defined. In par- II. RELATING GROUP AND PHASE VELOCITIES ticular, we prove that, for nondispersive materials, A. Notation and review of the generalized Hermitian ͑ ͒ ͑ ͒ vg = vp ft − fz , 1 eigenproblem ͑ ͒ where ft and fz are the fractions of the electromagnetic EM Employing Dirac notation, Maxwell’s equations with field energy in the transverse ͑xy͒ and longitudinal ͑z͒ direc- time-harmonic fields of frequency ␻ can be cast ͑exactly͒ in tions, respectively. Thus, the appearance of backward-wave the following form ͓13,14͔: ͑ Ͻ ͒ ץ modes vpvg 0 in nondispersive media coincides with the fields being mostly oriented in the longitudinal direction Aˆ ͉␺͘ =−i Bˆ ͉␺͘, ͑2͒ ץ ͒ Ͼ ͑ that is, fz ft . The situation of negative-index media in- z volves material dispersion and requires a modified equation where ͉␺͘ is the four-component state vector, discussed below. As an application of Eq. ͑1͒, we identify ͑ ͒ and explain a fundamental failure of perfectly matched layers Et x,y,z ͉␺͘ϵͩ ͪe−i␻t, ͑3͒ ͑PMLs͒, widely used as absorbing boundaries in simulating ͑ ͒ Ht x,y,z wave equations ͓9͔, for backward-wave structures. In particular, the stretched-coordinate derivation of PML suggests containing the transverse fields Ex ,Ey ,Hx ,Hy, and the opera- that such waves should be exponentially growing in the tors Aˆ and Bˆ are given by PML, and we explain this physically by pointing out that c 1 ϫ ١ ϫ ١ /PML is an anisotropic “absorber” with gain in the longitudi- ␻␧ c − t t 0 nal direction, which dominates for backward-wave modes ␻ ␮ Aˆ ϵ due to Eq. ͑1͒. In inhomogeneous backward-wave media, ΂ c 1 ΃ ϫ ١ ϫ ١ − ␻␮/c 0 ͔ ͓ unlike homogeneous negative-index media 10,11 ,weargue ␻ t ␧ t that the only recourse is to abandon PML completely in favor ͑4͒ ͑where ␧ and ␮ are the relative permittivity and permeabil- *[email protected] ity͒ and

LOH et al. PHYSICAL REVIEW E 79, 065601͑R͒͑2009͒

1 ͵ ␧͉E ͉2 + ␮͉H ͉2 − ␧͉E ͉2 − ␮͉H ͉2 0 − zˆ ϫ −1 ␻ t t z z Bˆ ϵ ͩ ͪ = = Bˆ −1. ͑5͒ v = , ͑14͒ zˆ ϫ 0 ΂ −1 ΃ g ␤ ͵ ␧͉E ͉2 + ␮͉H ͉2 + ␧͉E ͉2 + ␮͉H ͉2 1 t t z z Moreover, under the inner product from which Eq. ͑1͒ follows. ͒ ͑ Ј ء Ј ء ͵␺͉␺Ј͘ϵ͗ Et · Et + Ht · Ht , 6 C. Subtleties with dispersive media and negative index both Aˆ and Bˆ are Hermitian. materials Specializing to the case of a z-uniform material allows us In deriving Eqs. ͑1͒ and ͑14͒, we assumed that the con- to choose modes with fixed propagation constant ␤, i.e., stituent materials were nondispersive. If ␧ and ␮ are ␻ de- ͉␺͘ = ei͑␤z−␻t͉͒␤͘, ͑7͒ pendent but we assume that absorption loss is negligible ͑so ͒ ͑ ␻ץ/ ˆ ץ ͒ where ͉␤͘ is z uniform ͓6͔. In this case Eq. ͑2͒ immediately that the problem is still Hermitian , then A in Eq. 9 ͑ ͒ reduces to has additional terms that change the denominator of Eq. 14 ˆ ˆ to d␻␧ d␻␮ A͉␤͘ = ␤B͉␤͘, ͑8͒ ͵ ͉E͉2 + ͉H͉2, ͑15͒ d␻ d␮ which is a generalized Hermitian eigenproblem in ␤. which is precisely the energy density of the EM field in a B. Case of continuous z-translational symmetry dispersive medium with negligible loss ͓15,16͔. Because the ͑ ͒ denominator has changed while the numerator is unchanged, The group velocity vg can be derived from Eq. 8 . View- one can no longer interpret the ratio as f − f . An interesting ing Aˆ as an operator parametrized by ␻, we know by the t z example of such a case is a negative-index metamaterial, in Hellmann-Feynman theorem ͓6͔ that the eigenvalues ␤ of which ␧ and ␮ are negative in some frequency range with Eq. ͑8͒ vary with ␻ according to low loss ͓1,2͔. In this case, the numerator of Eq. ͑14͒ flips ͔͒ ͑ ͓ Âץ Âץ ͗␤͉ ͉␤͘ ͗␤͉ ͉␤͘ sign, while the new denominator Eq. 15 is still positive -␻ ͑leading to a positive energy density ͓2,16͔͒. Hence in a disץ ␻ץ ␤ץ 1 = = = . ͑9͒ ␻ ͗␤͉ ˆ ͉␤͘ 1 persive negative-index medium a purely transverse fieldץ v g B ͗␤͉Aˆ ͉␤͘ ͑E H ͒ ␤ z = z =0 has opposite phase and group velocity. A nondispersive negative-index medium is not physical ͑it violates ͓ ͔͒ Plugging in Eqs. ͑4͒ and ͑6͒—assuming that the material is the Kramers-Kronig relations 15,16 ; the definition of vg in ͑ nondispersive so that Aˆ has only the explicit ␻ dependence such a case is subtle and somewhat artificial naively, the ˆ energy density is negative͒ and is not discussed here. ͒ ͑ Aץ ␻ simplifies—the numerator of Eq. 9 evaluates to An interesting application of Eq. ͑1͒ is to backward-waveץ and 2 c 1 ء 1 ϫ E ͪ waveguides made of positive-index materials. For example, a ١ ϫ ١ + E · ͩ␧E ͵ c t t ␻2 t ␮ t t hollow metallic waveguide containing a concentric dielectric cylinder was shown to support backward-wave modes ͓7͔. 1 c2 1 ϫ ͪ ͑ ͒ More recently, the same phenomenon was demonstrated in ١ ϫ ١ ␮ͩ ء ͵ + Ht · Ht + t t Ht . 10 c ␻2 ␧ all-dielectric ͑positive-index nondispersive͒ photonic-crystal Bragg and holey fibers and in general can be explained as an Splitting the first integral into two summands, we obtain an ␤ ͐␧͉ ͉2 avoided eigenvalue crossing from a forced degeneracy at Et contribution from the first term; as for the second, ͓ ͔ integrating by parts gives =0 8 . An example of such a structure is shown in the inset 2 of Fig. 1, which shows the cross section of a Bragg fiber 2 1 ء c ϫ E ͒ = ͵ ␮͉H ͉ , ͑11͒ formed by alternating layers of refractive indices nhi=4.6 ١͑ · ͒ ϫ E ١͑ ͵ ␻2 t t ␮ t t z ͑ ͒ ͑ ͒ thickness 0.25a and nlo=1.4 thickness 0.75a with period ␻ a. The central high-index core has radius 0.45a and the first .ϫE =i ␮H zˆ in the last step ١ where we used the relation t t c z low-index ring has thickness 0.32a. For this geometry, one of Simplifying the second integral in an analogous manner and the guided modes ͑with angular dependence eim␾ and m=1͒ putting everything together, we find has the dispersion relation ␻͑␤͒ shown in the inset of Fig. 1: Aˆ 1 ␤ 2␻/ ␤2 Ͻץ ͗␤͉ ͉␤͘ ͵ ␧͉ ͉2 ␮͉ ͉2 ␧͉ ͉2 ␮͉ ͉2 ͑ ͒ at =0, d d 0, resulting in a downward-sloping = Et + Ht + Ez + Hz . 12 Ͻ ␻ c backward-wave region with vgvp 0. As the index of theץ core cylinder is varied, this curvature can be changed from By a similar computation, we may rewrite the denominator negative to positive in order to eliminate the backward-wave ͑ ͒ of Eq. 9 as region. Equation ͑1͒ tells us that the backward-wave region 1 ␻ coincides with fields that are mostly oriented in the z ͑axial͒ ͗␤͉ ˆ ͉␤͘ ͵ ␧͉ ͉2 ␮͉ ͉2 ␧͉ ͉2 ␮͉ ͉2 ͑ ͒ A = Et + Ht − Ez − Hz . 13 direction. ␤ c␤ A further consequence of Eq. ͑1͒ appears if we consider Thus the problem of terminating a waveguide like the one in Fig.

065601-2 RAPID COMMUNICATIONS

FUNDAMENTAL RELATION BETWEEN PHASE AND GROUP… PHYSICAL REVIEW E 79, 065601͑R͒͑2009͒

1 ͑inset͒ in a computer simulation. A standard approach is to 1 1 terminate the waveguide with a perfectly matched absorbing i␴␧ i␴␮ ⌬␧ Ϸ 1 , ⌬␮ Ϸ 1 . ͑17͒ layer, which is an artificial absorbing material constructed so ␻ ΂ ΃ ␻ ΂ ΃ as to be theoretically reflectionless ͓9͔. PML is reflectionless −1 −1 because it corresponds merely to a complex coordinate Note the −1 in the z axis, which flips the sign of the imagi- →͑ i␴ ͒ i␤z stretching z 1+ ␻ z so that propagating waves e are nary part and hence corresponds to gain if the xy axes are i␤z−␴z/v transformed into exponentially decaying waves e p for loss. From standard perturbation theory, the first-order some PML strength ␴. From this perspective, an obvious change in ␻ resulting from perturbations ⌬␧ and ⌬␮ is given Ͻ Ͼ problem occurs for backward waves: if vp 0 for vg 0, by ͓18͔ ͑ Ͼ ͒ then a +z-propagating wave vg 0 will undergo exponen- ␮H⌬ء␧E + H⌬ءtial growth for ␴Ͼ0. ͑This is entirely distinct from the fail- ͵ E ⌬␻ ure of PML in a medium periodic in the z direction, which in Ϸ − . ͑18͒ that case is due to the nonanalyticity of Maxwell’s equations ␻ ␧͉ ͉2 ␮͉ ͉2 ͓12͔ and leads to reflections but not instability.͒ In a homo- ͵ E + H geneous backward-wave medium, this problem can be solved merely by making ␴Ͻ0 in the negative-index fre- Substituting Eq. ͑17͒, we find that quency ranges ͓10,11͔. This solution is impossible in the case ͵ ␧͉ ͉2 ␮͉ ͉2 ␧͉ ͉2 ␮͉ ͉2 of Fig. 1 ͑inset͒, however, because at the same ␻ one has Et + Ht − Ez − Hz ⌬␻ Ϸ ␴ ␴͑ ͒ both forward and backward waves—no matter what sign is − i =−i ft − fz ␴ chosen for , one of these waves will experience exponential ͵ ␧͉E ͉2 + ␮͉H ͉2 + ␧͉E ͉2 + ␮͉H ͉2 growth in the PML. t t z z Precisely this exponential growth is observed in Fig. 1. v We simulated the backward-wave structure of Fig. 1 ͑inset͒ =−i␴ g ͑19͒ with a finite-difference time-domain ͑FDTD͒ simulation in vp ͓ ͔ cylindrical coordinates 9,17 , terminated in the z direction upon applying our identity Eq. ͑1͒. with PML layers. Both the forward- and backward-wave The ramifications for PML behavior are now immediately modes were excited with a short-pulse current source, and apparent: turning on a small ␴ leads to an additional time- the fields in the PML region after a long time were fit to an dependent exponential factor exponential in order to determine the decay rate. Figure 1 v plots this decay rate as a function of the curvature expͩ− g · ␴tͪ. ͑20͒ v ͉ 2␤ץ/2␻ץ ␤=0 as the core-cylinder index is varied from 2.6 to p 5.0. The appearance of negative curvature, which indicates In the usual case in which group and phase velocities are the appearance of a backward-wave region, precisely coin- oriented in the same direction, the overall rate constant is cides with the decay rate changing sign to exponential negative and this causes absorptive loss in the PML. In the growth. / case of backward waves, however, the ratio vg vp is nega- This exponential growth is fully explained by the tive, and thus the overall rate constant is positive, i.e., PML coordinate-stretching viewpoint of PML, but physically it produces gain. Physically, we can now understand the ratio may still seem somewhat mysterious: PML can also be v /v as determining whether the fields are mostly transverse ͓ ͔ g p viewed as an artificial anisotropic absorbing medium 9 ,so or mostly longitudinal and hence whether the PML loss ͑xy how can an absorbing medium lead to gain? The answer lies axes͒ or gain ͑z axis͒ dominates. Forward waves are mostly in the anisotropy of PML: as we review below, PML is ac- transverse and hence ordinary PML is lossy. ͑Waveguides ͑ ͒ tually a gain medium in the z direction, and in fact Eq. 1 with backward waves also possess complex-␤ evanescent precisely explains how backward waves cause the z principal waves ͓8͔, and we believe that these may also have gain in axis of the PML to dominate and produce a net gain. In the PML; but as their appearance always coincides with the ␴ particular, we look at the case where is small so that we appearance of backward-wave modes we focus on the insta- can analyze the effect of the PML with perturbation theory. bilities due to the latter.͒ Using the e−i␻t time convention as above, a z-absorbing PML for an isotropic material is obtained by multiplying ␧ D. Numerical results, modifications, and alternatives to PML and ␮ by the ͑anistropic͒ tensor ͓9͔ With the understanding that the standard formulation of PML fails for backward waves, we now turn to a discussion i␴ 1+ of what can be done instead. As pointed out above, previous ␻ corrections for left-handed media ͓10,11͔ are inapplicable i␴ here because one has forward and backward waves at the 1+ ͑ ͒ ␻ ␻ . 16 same . Since the reflectionless property of PML fundamen- tally arises from the coordinate-stretching viewpoint and ΂ −1 ΃ i␴ gain is predicted by coordinate-stretching above, we are led ͩ1+ ͪ ␻ to the conclusion that PML must be abandoned entirely for such backward-wave structures. The alternative is to use a To first order in ␴, this changes ␧ and ␮ by scalar absorbing material, e.g., a scalar conductivity ␴, which

065601-3 RAPID COMMUNICATIONS

LOH et al. PHYSICAL REVIEW E 79, 065601͑R͒͑2009͒

) 10 0 0.22 2

c loss -0.01 π 10-5 a/2 0.21 v > 0 ω g units of-0.02 a/c a ( 1/L4 linear -0.03 0.2 ~ reflection / L -10 10 1/L6 2

| quadratic 8 (L) L 1/L -0.04 frequency vg<0 L 0.19 a cubic b 0 0.1 0.2 0.3 0.4 0.5 -H -15 s o 10

10 a r 1/L b b

-0.05 s i n

o propagation constant βa/2π g

gain r quartic

b i (L+1) n 12

g 1/L -0.06 v < 0 |H quintic g -20 10 PML field decay rate 1 2 -7 -6 -5 -4 -3 -2 -1 0 10 10 absorber length L/a band curvature 2ω/ 2 at β=0 (units of ca/2π) ͑ ͒ ͑ϳ / 2͒ FIG. 1. ͑Color͒ Field decay rate within the PML vs curvature of FIG. 2. Color Field convergence reflection L vs absorber ␴ ͕͓␴͑ / ͒ϳ͑ / ͔͖͒ the dispersion relation at ␤=0, showing onset of gain for v Ͻ0 and length for various ranging from linear z L z L to quin- g tic ͓␴͑z/L͒ϳ͑z/L͒5͔. For reference, the corresponding asymptotic loss for vg Ͼ0. Inset: dispersion relation ͑of the first TE band͒ with v Ͻ0 region at ␤=0 and cross section of the Bragg fiber. power laws are shown as dashed lines. Inset: Bragg-fiber structure g in cylindrical computational cell with absorbing regions used in is absorbing for all field orientations and therefore cannot simulation. lead to gain ͑unlike our previous work where an anisotropic “pseudo-PML” could still be employed ͓12͔͒. At the inter- absorber ͑albeit thicker than a PML for purely forward-wave face of such a material, however, there will be reflections. modes͒ and displays the expected scaling 1/L2d+2 for ␴ Such reflections can be made arbitrarily small, however, by ϳ͑z/L͒d ͓12͔. turning on the absorption by a sufficiently gradual taper tran- Colloquially, the term “left-handed medium” is some- sition, similar to our approach for an unrelated failure of times applied to photonic-crystal structures with periodicity PML ͓12͔. Even for PML, numerical reflections due to dis- on the same scale as the wavelength that mimic some quali- cretization require a similar gradual ␴ taper. In both cases, tative feature of negative-index media, such as negative re- ͑ ͒ ͓ ͔ the reflection R L goes to zero as the absorber thickness L is fraction 3,4 . In such media, however, the phase velocity vp made longer ͑and more gradual͒, and the impact of PML is not uniquely defined ͑any reciprocal lattice vector can be ͑when it works͒ is merely to multiply R͑L͒ by a smaller added to ␤͓͒6͔,soEq.͑1͒ is not directly applicable. It may ͓ ͔ ͑ ͒ constant coefficient 12 . Even without PML, the rate at be interesting, however, to use Eq. 1 as the definition of vp which R͑L͒ goes to zero can be made more rapid by reducing in periodic media, in which case it turns out that one obtains ␴ ␴ϳ͑ / ͒2 ͑ Ͼ ͒ the discontinuity in : for example, if z L for z 0 a result similar in spirit to previous work that defined vp as a then its second derivative is discontinuous at the transition weighted average of the phase velocity of each Fourier com- z=0 and R͑L͒ consequently scales as 1/L4, while if ␴ ponent ͓5͔. ϳ͑ / ͒3 ͑ ͒ϳ / 6 z L then R L 1 L . Figure 2 shows how a scalar con- ACKNOWLEDGMENTS ductivity ␴ can be used as a last-resort replacement for PML in the backward-wave structure of Fig. 1 ͑inset͒. The plot This work was supported in part by the National Defense shows the difference squared of the magnetic field at a test Science and Engineering Council, the MRSEC Program of point for absorber lengths L and L+1 ͓which scales as the National Science Foundation under Award No. DMR- R͑L͒/L2͔͓12͔ versus L for various conductivity profiles ␴. 0819762, and the Army Research Office through the Institute Even with both forward and backward waves excited, the for Soldier Nanotechnologies under Contract No. DAAD-19- reflection can indeed be made small for a sufficiently thick 02-D0002.

͓1͔ D. R. Smith et al., Science 305, 788 ͑2004͒. ͓10͔ S. Cummer, IEEE Antennas Wireless Propag. Lett. 3, 172 ͓2͔ R. Marqués et al., Metamaterials with Negative Parameters: ͑2004͒. Theory, Design, and Microwave Applications ͑Wiley, New ͓11͔ X. Dong et al., IEEE Microw. Wirel. Compon. Lett. 14, 301 ͒ York, 2008 . ͑2004͒. ͓ ͔ ͑ ͒ 3 M. Notomi, Phys. Rev. B 62, 10696 2000 . ͓12͔ A. F. Oskooi et al., Opt. Express 16, 11376 ͑2008͒. ͓4͔ C. Luo et al., Phys. Rev. B 65, 201104͑R͒͑2002͒. ͓13͔ S. G. Johnson et al., Phys. Rev. E 66, 066608 ͑2002͒. ͓5͔ M. Ghebrebrhan et al., Phys. Rev. A 76, 063810 ͑2007͒. ͓14͔ M. Skorobogatiy and J. Yang, Fundamentals of Photonic Crys- ͓6͔ J. D. Joannopoulos et al., Photonic Crystals: Molding the Flow ͑ of Light, 2nd ed. ͑Princeton University Press, Princeton, NJ, tal Guiding Cambridge University Press, Cambridge, En- ͒ 2008͒. gland, 2009 . ͓7͔ P. J. B. Clarricoats and R. A. Waldron, Int. J. Electron. 8, 455 ͓15͔ J. D. Jackson, Classical Electrodynamics, 3rd ed. ͑Wiley, New ͑1960͒. York, 1999͒. ͓8͔ M. Ibanescu et al., Phys. Rev. Lett. 92, 063903 ͑2004͒. ͓16͔ L. D. Landau and E. M. Lifshitz, Electrodynamics of Continu- ͓9͔ A. Taﬂove and S. C. Hagness, Computational Electrodynam- ous Media ͑Pergamon, Oxford, 1960͒. ics: The Finite-difference Time-domain Method ͑Artech House, ͓17͔ A. Farjadpour et al., Opt. Lett. 31, 2972 ͑2006͒. Boston, 2000͒. ͓18͔ C. Kottke et al., Phys. Rev. E 77, 036611 ͑2008͒.

065601-4