## Comment on “Correct definition of the Poynting vector in electrically and magnetically polarizable medium reveals that negative refraction is impossible.”

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 *E*⃗×*H*⃗ or the alternative *E*⃗×*B*⃗ form. However, the continuity equations are not identical, which means that quantities following from *E*⃗×*H*⃗ cannot always be compared directly to those from *E*⃗×*B*⃗. 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

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]

*E*⃗×

*B*⃗ correctly describes the electromagnetic flux (Poynting vector) in a magnetically polarizable medium, and the usual Abraham form

*E*⃗×

*H*⃗ 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]

*E*⃗×

*H*⃗ and

*E*⃗×

*B*⃗; 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], 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

*E*⃗×

*B*⃗ form by starting with Maxwell’s equations in a medium in the absence of external currents and charges, i.e.

*ρ*and

_{b}*J*⃗

_{b}are the bound charge density and bound current density respectively, and we use SI units. By utilizing the vector identity ∇⃗·(

*X*⃗×

*Y*⃗)=

*Y*⃗·∇⃗×

*X*⃗-

*X*⃗·∇⃗×

*Y*⃗, one readily shows that

*E*⃗×

*H*⃗ electromagnetic continuity equation is derived by first defining

*P*⃗ and

*M*⃗ are the polarization and magnetization. Hence Maxwell’s equations can be written

*explicitly*as

*E*⃗·

*J*⃗

*; but in Eq. (7) it is*

_{b}*implicit*, being included via

*P*⃗ and

*M*⃗ and the definitions of

*D*⃗ and

*H*⃗. The analogous terms between the two equations are, however, clearly distinct e.g.

*µ*

_{0}

^{-1}∇⃗·(

*E*⃗×

*B*⃗)≠∇⃗·(

*E*⃗×

*H*⃗) 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

*E*⃗×

*B*⃗ and its associated continuity equation in magnetic media, it is not required that we do so.

*Secondly*, in the regime of propagating plane waves,

*E*⃗=Re{

*E*⃗

_{0}exp[

*ι*(

**k**·

**r**-

*ωt*)]}, Eqs. (2) and (7) become

*D*⃗(

*ω*)=

*ε*(

*ω*)

*E*⃗(

*ω*) and

*B*⃗(

*ω*)=

*µ*(

*ω*)

*H*⃗(

*ω*), show that

*E*⃗×

*B*⃗-based Eq. (10). On this basis, he claims to have demonstrated that

*E*⃗×

*B*⃗ is the correct definition of the Poynting vector.

*E*⃗×

*B*⃗ over

*E*⃗×

*H*⃗ (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].

*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]

*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]

*ε*and

*µ*to whether (

*E*⃗×

*H*⃗) ·

*k*⃗<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]

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]

*E*⃗×

*H*⃗ – but for any chosen medium with Re(

*µ*)<0, the sign of Markel’s preferred Poynting vector

*E*⃗×

*B*⃗ will

*oppose*that of

*E*⃗×

*H*⃗!

*In summary*, we have shown that whilst

*E*⃗×

*B*⃗ can be a useful choice of Poynting vector for some (see e.g. the recent [7]), it is no more or less correct than

*E*⃗×

*H*⃗. In addition, claiming that NPV is impossible while using an

*E*⃗×

*B*⃗ 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 |

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. |

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. |

5. | P. Kinsler and M. W. McCall, “Criteria for negative refraction in active and passive media,” Microwave Opt. Tech. Lett. |

6. | P. Kinsler and M. W. McCall, “Causality-based conditions for negative refraction must be used with care,” Phys. Rev. Lett. |

7. | F. Richter, M. Florian, and K. Henneberger, “Poynting’s theorem and energy conservation in the propagation of light in bounded media,” Europhys. Lett. |

**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

- 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]
- 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]
- P. Kinsler, A. Favaro, and M. McCall, "Four Poynting theorems," Eur. J. Phys., to appear.
- M. McCall, A. Lakhtakia, andW. S.Weiglhofer, "The negative index of refraction demystified," Eur. J. Phys. 23, 353-359 (2002). [CrossRef]
- P. Kinsler and M. W. McCall, "Criteria for negative refraction in active and passive media," Microwave Opt. Tech. Lett. 50, 1804 (2008). [CrossRef]
- 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]
- 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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