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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 17 — Aug. 17, 2009
  • pp: 15167–15169
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Comment on “Correct definition of the Poynting vector in electrically and magnetically polarizable medium reveals that negative refraction is impossible.”

Alberto Favaro, Paul Kinsler, and Martin W. McCall  »View Author Affiliations


Optics Express, Vol. 17, Issue 17, pp. 15167-15169 (2009)
http://dx.doi.org/10.1364/OE.17.015167


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Abstract

Physically valid electromagnetic continuity equations can be generated from either the usual form of the Poynting vector E⃗×H⃗ or the alternative E⃗×B⃗ form. However, the continuity equations are not identical, which means that quantities following from E⃗×H⃗ cannot always be compared directly to those from E⃗×B⃗. In particular, the work done on the bound current densities are attributed differently in the two representations. We also comment on the negative refraction condition used.

© 2009 Optical Society of America

Markel has computed the rate at which an electromagnetic field does work on currents in a medium in two different ways and noted an inconsistency between them [1

1. V. A. Markel, “Correct Definition of the Poynting Vector in Electrically and Magnetically Polarizable Medium Reveals that Negative Refraction is Impossible,” Opt. Express 16, 19,152 (2008). [CrossRef]

]. On the basis of this difference, he then claimed that E⃗×B⃗ correctly describes the electromagnetic flux (Poynting vector) in a magnetically polarizable medium, and the usual Abraham form E⃗×H⃗ does not. In fact, even though they differ, neither form is more or less “correct” than the other – the point is to ensure that the form used is compatible with the situation, as explained at length in the recent and comprehensive review by Pfeifer et al. [2

2. R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Momentum of an electromagnetic wave in dielectric media,” Rev. Mod. Phys. 79, 1197–1216 (2007). [CrossRef]

]. We clarify the issues by comparing the different electromagnetic continuity equations for E⃗×H⃗ and E⃗×B⃗; these are easily derived using the appropriate Maxwell’s equations and a standard vector identity. The two continuity equations describe energy conservation and the material response in different but complementary ways [3

3. P. Kinsler, A. Favaro, and M. McCall, “Four Poynting theorems,” Eur. J. Phys., to appear.

], and here we focus on how the electromagnetic work is represented in each.

Firstly, let us compare the two forms of electromagnetic continuity equation. We derive the E⃗×B⃗ form by starting with Maxwell’s equations in a medium in the absence of external currents and charges, i.e.

·E=1ε0ρb,
·B=0,
×E=tB,
×B=μ0Jb+μ0ε0tE,
(1)

where ρb and Jb are the bound charge density and bound current density respectively, and we use SI units. By utilizing the vector identity ∇⃗·(X⃗×Y⃗)=Y⃗·∇⃗×X⃗-X⃗·∇⃗×Y⃗, one readily shows that

t12[ε0E2+1μ0B2]+1μ0·(E×B)+E·Jb=0.
(2)

The E⃗×H⃗ electromagnetic continuity equation is derived by first defining

Jb=tP+×M,
(3)
ρb=·P,
D=ε0E+P,
(4)
H=1μ0BM,
(5)

where P⃗ and M⃗ are the polarization and magnetization. Hence Maxwell’s equations can be written

·D=0,
·B=0,
×E=tB,
×H=tD.
(6)

Using these, we can now derive an alternative continuity equation, in the same manner followed for Eq. (2). The result is

E·tD+H·tB+·(E×H)=0.
(7)

Already we see a difference between Eqs. (2) and (7), even though both relate an electromagnetic power flux out of a unit volume to a rate of change of an electromagnetic energy density, plus a rate at which the fields do work on the bound current density. In Eq. (2) this work on the bound current density is expressed explicitly as E⃗·Jb; but in Eq. (7) it is implicit, being included via P⃗ and M⃗ and the definitions of D⃗ and H⃗. The analogous terms between the two equations are, however, clearly distinct e.g. µ 0 -1∇⃗·(E⃗×B⃗)≠∇⃗·(E⃗×H⃗) in general; consequently, interpretations based on one form cannot be directly applied to the other. At this point, although it is certainly reasonable to prefer to use E⃗×B⃗ and its associated continuity equation in magnetic media, it is not required that we do so.

Secondly, in the regime of propagating plane waves, E⃗=Re{E0 exp[ι(k·r-ωt)]}, Eqs. (2) and (7) become

1μ0·E×B=|E·Jb,
(8)
·E×H=E·tD+H·tB,
(9)

where < ⋯ > denotes temporal averaging over many optical cycles. Respective calculations of the right hand sides (RHS) of Eqs. (8) and (9), using the frequency domain constitutive relations D⃗(ω)=ε (ω)E⃗(ω) and B⃗(ω)=µ(ω)H⃗(ω), show that

E·Jb=ωE02Im{μ(ω)ε(ω)}e2k·r,
(10)
E·tD+H·tB=ω[μ(ω)ε(ω)+ε(ω)μ(ω)]E0μ(ω)e2k·r.
(11)

Following this, Markel does an independent calculation of the left hand side (LHS) of Eq. (8), the result being consistent with the E⃗×B⃗-based Eq. (10). On this basis, he claims to have demonstrated that E⃗×B⃗ is the correct definition of the Poynting vector.

However, the discrepancy between the RHS’s of Eqs. (8) and (9), as revealed through Eqs. (10) and (11), clearly cannot be used to establish the correctness of E⃗×B⃗ over E⃗×H⃗ (or vice-versa) for the Poynting vector. Indeed, an independent calculation of the LHS of Eq. (9) shows compatibility with the RHS of Eq. (11), as expected. Eqs. (8) and (9) are distinct formulations of energy conservation – both are physically valid, but their components have different meanings [3

3. P. Kinsler, A. Favaro, and M. McCall, “Four Poynting theorems,” Eur. J. Phys., to appear.

].

Thirdly, the condition for Negative Phase Velocity (NPV) propagation [4

4. M. McCall, A. Lakhtakia, and W. S. Weiglhofer, “The negative index of refraction demystified,” Eur. J. Phys. 23, 353–359 (2002). [CrossRef]

]

Im[μ(ω)ε(ω)]<0,
(12)

whose LHS appears on the RHS of Eq. (10), is applicable only to doubly passive media (i.e. with both electric and magnetic losses, Im[ε]<0, Im[µ] <0) [5

5. P. Kinsler and M. W. McCall, “Criteria for negative refraction in active and passive media,” Microwave Opt. Tech. Lett. 50, 1804 (2008). [CrossRef]

], or even to doubly active media, if we know in advance, and reverse the sense of the inequality. This is just one of several NPV criteria relating ε and µ to whether (E⃗×H⃗) · k⃗<0, but it is not the most general, since it does not apply to partly active media [5

5. P. Kinsler and M. W. McCall, “Criteria for negative refraction in active and passive media,” Microwave Opt. Tech. Lett. 50, 1804 (2008). [CrossRef]

] – note that gain in NPV media is not prohibited by causality [6

6. P. Kinsler and M. W. McCall, “Causality-based conditions for negative refraction must be used with care,” Phys. Rev. Lett. 101, 167,401 (2008). [CrossRef]

]. Also, by convention, NPV is determined with respect to E⃗×H⃗ – but for any chosen medium with Re(µ)<0, the sign of Markel’s preferred Poynting vector E⃗×B⃗ will oppose that of E⃗×H⃗!

In summary, we have shown that whilst E⃗×B⃗ can be a useful choice of Poynting vector for some (see e.g. the recent [7

7. F. Richter, M. Florian, and K. Henneberger, “Poynting’s theorem and energy conservation in the propagation of light in bounded media,” Europhys. Lett. 13, 117–121 (2008).

]), it is no more or less correct than E⃗×H⃗. In addition, claiming that NPV is impossible while using an E⃗×B⃗ based argument is dubious.

References and links

1.

V. A. Markel, “Correct Definition of the Poynting Vector in Electrically and Magnetically Polarizable Medium Reveals that Negative Refraction is Impossible,” Opt. Express 16, 19,152 (2008). [CrossRef]

2.

R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Momentum of an electromagnetic wave in dielectric media,” Rev. Mod. Phys. 79, 1197–1216 (2007). [CrossRef]

3.

P. Kinsler, A. Favaro, and M. McCall, “Four Poynting theorems,” Eur. J. Phys., to appear.

4.

M. McCall, A. Lakhtakia, and W. S. Weiglhofer, “The negative index of refraction demystified,” Eur. J. Phys. 23, 353–359 (2002). [CrossRef]

5.

P. Kinsler and M. W. McCall, “Criteria for negative refraction in active and passive media,” Microwave Opt. Tech. Lett. 50, 1804 (2008). [CrossRef]

6.

P. Kinsler and M. W. McCall, “Causality-based conditions for negative refraction must be used with care,” Phys. Rev. Lett. 101, 167,401 (2008). [CrossRef]

7.

F. Richter, M. Florian, and K. Henneberger, “Poynting’s theorem and energy conservation in the propagation of light in bounded media,” Europhys. Lett. 13, 117–121 (2008).

OCIS Codes
(160.1245) Materials : Artificially engineered materials
(350.3618) Other areas of optics : Left-handed materials

ToC Category:
Metamaterials

History
Original Manuscript: March 24, 2009
Revised Manuscript: June 12, 2009
Manuscript Accepted: June 15, 2009
Published: August 11, 2009

Citation
Alberto Favaro, Paul Kinsler, and Martin W. McCall, "Comment on “Correct definition of the Poynting vector in electrically and magnetically polarizable medium reveals that negative refraction is impossible.”," Opt. Express 17, 15167-15169 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-17-15167


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References

  1. V. A. Markel, "Correct Definition of the Poynting Vector in Electrically and Magnetically Polarizable Medium Reveals that Negative Refraction is Impossible," Opt. Express 16, 19,152 (2008). [CrossRef]
  2. R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Momentum of an electromagnetic wave in dielectric media," Rev. Mod. Phys. 79, 1197-1216 (2007). [CrossRef]
  3. P. Kinsler, A. Favaro, and M. McCall, "Four Poynting theorems," Eur. J. Phys., to appear.
  4. M. McCall, A. Lakhtakia, andW. S.Weiglhofer, "The negative index of refraction demystified," Eur. J. Phys. 23, 353-359 (2002). [CrossRef]
  5. P. Kinsler and M. W. McCall, "Criteria for negative refraction in active and passive media," Microwave Opt. Tech. Lett. 50, 1804 (2008). [CrossRef]
  6. P. Kinsler and M. W. McCall, "Causality-based conditions for negative refraction must be used with care," Phys. Rev. Lett. 101, 167,401 (2008). [CrossRef]
  7. F. Richter, M. Florian, and K. Henneberger, "Poynting’s theorem and energy conservation in the propagation of light in bounded media," Europhys. Lett. 13, 117-121 (2008).

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