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Response to comments on boundary problems and electromagnetic constitutive parameters
ARTICLE IN PRESS
Optik
Optics
Optik 119 (2008) 250–252 www.elsevier.de/ijleo
Response to comments on boundary problems and electromagnetic constitutive parameters Akhlesh Lakhtakia,1 Computational & Theoretical Materials Sciences Group (CATMAS), Department of Engineering Science & Mechanics, Pennsylvania State University, University Park, PA 16802–6812, USA Received 26 September 2006; accepted 5 November 2006
Abstract The positive conclusion on the recognizable existence of the Tellegen parameter, which invalidates the Post constraint, arrived at in the Comment of Sihvola and Tretyakov through the solution of a boundary-value problem relies on a framework of macroscopic electromagnetism that lacks a microscopic basis. But the microscopic viewpoint underlying the framework of modern macroscopic electromagnetism invalidates the boundary conditions used therein, and instead validates the Post constraint. r 2006 Elsevier GmbH. All rights reserved. Keywords: Boundary conditions; Electromagnetic theories; Microscopic electromagnetism; Post constraint; Tellegen parameter
1. Introduction Since its rediscovery in 1994 in the context of linear electromagnetic materials, the Post constraint has been further developed and discussed [1–3]. This constraint reduces the number of complex-valued (frequencydomain) constitutive parameters of a linear material from a maximum of 36 to 35. In consequence, it negates the recognizable existence of the Tellegen constitutive parameter in macroscopic electromagnetism, and indicates that an isotropic material is described by at most 3, not 4, constitutive parameters. No credible experimental evidence against the Post constraint exists, as has been pointed out earlier in this journal [2,4].
Tel.: +1 814 863 4319; fax: +1 814 865 9974.
E-mail address: [email protected]. Also affiliated with Department of Physics, Imperial College, London SW7 2AZ, UK. 1
0030-4026/$ - see front matter r 2006 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2006.11.003
Solutions of boundary-value problems in macroscopic electromagnetism can be used to argue that the Tellegen parameter affects scattering, thereby leading to the twin conclusions that the Tellegen parameter has a recognizable existence and the Post constraint is invalid. This approach has been adopted in a variety of equivalent ways, most recently by Obukhov and Hehl [5] and Sihvola and Tretyakov [6]. Indeed, as I had solved boundary-value problems involving the Tellegen parameter 15 years ago [7,8], I have full confidence that Sihvola and Tretyakov have made the correct deductions from their solutions of their boundary-value problem. The actual issue is not the correct mathematical solution of that boundary-value problem, but the correct physical formulation thereof. Sihvola and Tretyakov have used the principles of a macroscopic electromagnetism for their formulation, without attending to the microscopic basis of modern electromagnetism. Now, electromagnetism has been a microscopic
ARTICLE IN PRESS A. Lakhtakia / Optik 119 (2008) 250–252
science for over a century, and any correctly formulated principles of macroscopic electromagnetism employed must emanate from a microscopic viewpoint. On that crucial issue, the formulation of Sihvola and Tretyakov fails – as shown in the following section – which renders their positive conclusion on the recognizable existence of the Tellegen parameter incorrect. Similarly incorrect is the implicit pessimism about the validity of the Post constraint.
2. Boundary-value problem Let all space V be divided into two distinct regions V 1 and V 2 , both filled with distinct linear, homogeneous materials labeled 1 and 2, respectively, and let S denote the boundary between the two regions. Following Ref. [4], let the (frequency-domain) constitutive tensors of the nth material, ðn ¼ 1; 2Þ, be denoted by n , nn , an , b , and n
Fn I, where Trace ½an b 0, I is the identity tensor, n
and Fn is the Tellegen parameter. As discussed elsewhere [2], the Post constraint cannot be derived in a macroscopic electromagnetism that takes E and H to be the primitive electric and magnetic fields and relegates D and B to the status of induction or displacement fields. Modern electromagnetism, however, is premised on E and B as being components of the primitive electromagnetic field and having microscopic counterparts, whereas D and H are merely convenient but inessential constructs without microscopic counterparts. Therefore, in all statements of the Maxwell postulates contained in this section, the only macroscopic fields retained are E and B.
2.1. Macroscopic electromagnetism without a microscopic basis In an ab initio macroscopic formulation, the frequency-domain Maxwell postulates are represented as L½EðrÞ; BðrÞ; ðrÞ o I; nðrÞ m1 o I; aðrÞ; bðrÞ; FðrÞ ¼ 0, r 2 V,
ð1Þ
in the absence of externally imposed source charge and current densities, with L½ denoting a suitable operator, and o and mo being the permittivity and permeability of free space (i.e., vacuum). Thus V is taken to be filled with a nonhomogeneous continuum with a discontinuity across S; i.e., 8 8 8 8 < 1 < n1 < a1 < b1 ðrÞ ¼ ; nðrÞ ¼ ; aðrÞ ¼ ; bðrÞ ¼ , : 2 : n2 : a2 :b 2 ( ( F1 V1 ð2Þ FðrÞ ¼ ; r2 . F2 V2
251
Boundary conditions across S are derived from (1), and contain terms proportional to the difference F1 F2 ; ergo, the conclusions reached in Ref. [6].
2.2. Modern macroscopic electromagnetism (with a microscopic basis) But those conclusions are not acceptable in modern electromagnetism, which is not ab initio macroscopic but is actually a microscopic science. The frequency-domain Maxwell postulates everywhere are P½eðrÞ; bðrÞ; sðrÞ ¼ 0;
r 2 V,
(3)
in the absence of externally imposed source charge and current densities, with P½ denoting a suitable operator, e and b as the only two microscopic fields, and s representing an assembly of charges that represent matter. Eq. (3) leads to the macroscopic equations L½EðrÞ; BðrÞ; 1 o I; n1 m1 o I; a1 ; b ; 0 ¼ 0; 1
r 2 V1 (4)
and L½EðrÞ; BðrÞ; 2 o I; n2 m1 o I; a2 ; b ; 0 ¼ 0; 2
r 2 V2 (5)
for the two regions. These equations do not contain the microscopic sources s, which were homogenized to ultimately yield the constitutive tensors. These equations do not contain the microscopic fields e and b, which were homogenized simultaneously to yield the macroscopic fields E and B. These equations also do not contain F1 and F2 , the absence of which quantities underlies the conclusions reached in Ref. [4]. Eqs. (4) and (5) are indisputable, even in the framework of macroscopic electromagnetism without a microscopic basis. But there is more in those equations – which Sihvola and Tretyakov did not obtain because they did not adopt a microscopic viewpoint. The derivation of (4) from (3) at a specific point r 2 V 1 involves not only r but also a suitably small neighborhood thereof for homogenization. The neighborhood’s size must be finite and its shape must not introduce spurious effects. Furthermore, the neighborhoods of all points in V 1 must be chosen in an identical manner. If a point were to lie close enough to the edge of V 1 , its neighborhood will encompass a part of V 2 as well. Therefore, the only way that (4) can hold everywhere in V 1 is as if it were derived with the assumption that not only V 1 but also the neighborhood of V 1 is filled with material 1. 2 2 Of course, modern macroscopic electromagnetism is merely an approximation of true electromagnetism – which is microscopic and quantum–mechanical.
ARTICLE IN PRESS 252
A. Lakhtakia / Optik 119 (2008) 250–252
In consequence, any jump or boundary condition derived from (4) cannot contain any reference to F1 , since (4) does not contain F1 ; neither can that condition contain any reference to F2 . Likewise, any jump or boundary condition derived from (5) cannot contain any references to F2 and F1 . Therefore, the boundary conditions to be prescribed across S contain neither F1 nor F2 nor the difference F2 F1 . Thus, with the adoption of the microscopic viewpoint, the Tellegen parameter vanishes from all space V . In place of (1), we could have simply begun with L½EðrÞ; BðrÞ; ðrÞ o I; nðrÞ m1 o I; aðrÞ; bðrÞ; 0 ¼ 0, r 2 V,
ð6Þ
thereby quenching the specter of a constitutive parameter of a homogeneous material that manifests itself not throughout the material but only at its boundary! The discussion thus far in this subsection has been aimed towards the formulation of a macroscopic problem with a sharp boundary S between V 1 and V 2 . If the production of a sharp boundary is not required, both the regions V 1 and V 2 would shrink away from the boundary S which itself would swell into an intermediate region V S . For each r 2 V S , homogenization over a small neighborhood of r would result in macroscopic Maxwell postulates like (4) and (5). Thus, (6) would still hold but with 8 8 8 1 n1 a > > > > > > < < < 1 0 0 0 ðrÞ ¼ ðrÞ ; nðrÞ ¼ n ðrÞ ; aðrÞ ¼ a ðrÞ , > > > > > > : :n :a 2 2 2 8 8 b > > > 1 > V1 > > < 0 < ð7Þ bðrÞ ¼ b ðrÞ ; r 2 V S , > > > > > : > V2 :b 2
0
where ðrÞ, etc., are nonhomogeneous constitutive tensors.
3. Concluding remarks Whereas without reference to a microscopic viewpoint, the Tellegen parameter has a ghostly presence in the equations of macroscopic electromagnetism without a microscopic basis and casts its shadow on boundaryvalue problems, I have shown here that recourse to a microscopic viewpoint negates the recognizable existence of the Tellegen parameter in boundary-value problems formulated for modern macroscopic electromagnetism. The difference between the two formulations is epistemic, and its possible resolution and the consequences thereof have been discussed elsewhere [2,4].
References [1] W.S. Weiglhofer, A. Lakhtakia, The Post constraint revisited, AEU¨. Int. J. Electron. Commun. 52 (1998) 276–279. [2] A. Lakhtakia, On the genesis of Post constraint in modern electromagnetism, Optik 115 (2004) 151–158. [3] R.E. Raab, O.L. de Lange, Multipole Theory in ElectroMagnetism, Clarendon Press, Oxford, UK, 2005 (Chapter 8). [4] A. Lakhtakia, Boundary-value problems and the validity of the Post constraint in modern electromagnetism, Optik 117 (2006) 188–192. [5] Yu.N. Obukhov, F.W. Hehl, Measuring a piecewise constant axion field in classical electrodynamics, Phys. Lett. A 314 (2005) 357–365. [6] A. Sihvola, S. Tretyakov, Comments on boundary problems and electromagnetic constitutive parameters, Optik 118 (2007). [7] A. Lakhtakia, J.R. Diamond, Reciprocity and the concept of the Brewster wavenumber, Int. J. Infrared Millim. Waves 12 (1991) 1167–1174. [8] A. Lakhtakia, On the Huygens’s principles and the Ewald–Oseen extinction theorems for, and the scattering of, Beltrami fields, Optik 91 (1992) 35–40.
Optik
Optics
Optik 119 (2008) 250–252 www.elsevier.de/ijleo
Response to comments on boundary problems and electromagnetic constitutive parameters Akhlesh Lakhtakia,1 Computational & Theoretical Materials Sciences Group (CATMAS), Department of Engineering Science & Mechanics, Pennsylvania State University, University Park, PA 16802–6812, USA Received 26 September 2006; accepted 5 November 2006
Abstract The positive conclusion on the recognizable existence of the Tellegen parameter, which invalidates the Post constraint, arrived at in the Comment of Sihvola and Tretyakov through the solution of a boundary-value problem relies on a framework of macroscopic electromagnetism that lacks a microscopic basis. But the microscopic viewpoint underlying the framework of modern macroscopic electromagnetism invalidates the boundary conditions used therein, and instead validates the Post constraint. r 2006 Elsevier GmbH. All rights reserved. Keywords: Boundary conditions; Electromagnetic theories; Microscopic electromagnetism; Post constraint; Tellegen parameter
1. Introduction Since its rediscovery in 1994 in the context of linear electromagnetic materials, the Post constraint has been further developed and discussed [1–3]. This constraint reduces the number of complex-valued (frequencydomain) constitutive parameters of a linear material from a maximum of 36 to 35. In consequence, it negates the recognizable existence of the Tellegen constitutive parameter in macroscopic electromagnetism, and indicates that an isotropic material is described by at most 3, not 4, constitutive parameters. No credible experimental evidence against the Post constraint exists, as has been pointed out earlier in this journal [2,4].
Tel.: +1 814 863 4319; fax: +1 814 865 9974.
E-mail address: [email protected]. Also affiliated with Department of Physics, Imperial College, London SW7 2AZ, UK. 1
0030-4026/$ - see front matter r 2006 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2006.11.003
Solutions of boundary-value problems in macroscopic electromagnetism can be used to argue that the Tellegen parameter affects scattering, thereby leading to the twin conclusions that the Tellegen parameter has a recognizable existence and the Post constraint is invalid. This approach has been adopted in a variety of equivalent ways, most recently by Obukhov and Hehl [5] and Sihvola and Tretyakov [6]. Indeed, as I had solved boundary-value problems involving the Tellegen parameter 15 years ago [7,8], I have full confidence that Sihvola and Tretyakov have made the correct deductions from their solutions of their boundary-value problem. The actual issue is not the correct mathematical solution of that boundary-value problem, but the correct physical formulation thereof. Sihvola and Tretyakov have used the principles of a macroscopic electromagnetism for their formulation, without attending to the microscopic basis of modern electromagnetism. Now, electromagnetism has been a microscopic
ARTICLE IN PRESS A. Lakhtakia / Optik 119 (2008) 250–252
science for over a century, and any correctly formulated principles of macroscopic electromagnetism employed must emanate from a microscopic viewpoint. On that crucial issue, the formulation of Sihvola and Tretyakov fails – as shown in the following section – which renders their positive conclusion on the recognizable existence of the Tellegen parameter incorrect. Similarly incorrect is the implicit pessimism about the validity of the Post constraint.
2. Boundary-value problem Let all space V be divided into two distinct regions V 1 and V 2 , both filled with distinct linear, homogeneous materials labeled 1 and 2, respectively, and let S denote the boundary between the two regions. Following Ref. [4], let the (frequency-domain) constitutive tensors of the nth material, ðn ¼ 1; 2Þ, be denoted by n , nn , an , b , and n
Fn I, where Trace ½an b 0, I is the identity tensor, n
and Fn is the Tellegen parameter. As discussed elsewhere [2], the Post constraint cannot be derived in a macroscopic electromagnetism that takes E and H to be the primitive electric and magnetic fields and relegates D and B to the status of induction or displacement fields. Modern electromagnetism, however, is premised on E and B as being components of the primitive electromagnetic field and having microscopic counterparts, whereas D and H are merely convenient but inessential constructs without microscopic counterparts. Therefore, in all statements of the Maxwell postulates contained in this section, the only macroscopic fields retained are E and B.
2.1. Macroscopic electromagnetism without a microscopic basis In an ab initio macroscopic formulation, the frequency-domain Maxwell postulates are represented as L½EðrÞ; BðrÞ; ðrÞ o I; nðrÞ m1 o I; aðrÞ; bðrÞ; FðrÞ ¼ 0, r 2 V,
ð1Þ
in the absence of externally imposed source charge and current densities, with L½ denoting a suitable operator, and o and mo being the permittivity and permeability of free space (i.e., vacuum). Thus V is taken to be filled with a nonhomogeneous continuum with a discontinuity across S; i.e., 8 8 8 8 < 1 < n1 < a1 < b1 ðrÞ ¼ ; nðrÞ ¼ ; aðrÞ ¼ ; bðrÞ ¼ , : 2 : n2 : a2 :b 2 ( ( F1 V1 ð2Þ FðrÞ ¼ ; r2 . F2 V2
251
Boundary conditions across S are derived from (1), and contain terms proportional to the difference F1 F2 ; ergo, the conclusions reached in Ref. [6].
2.2. Modern macroscopic electromagnetism (with a microscopic basis) But those conclusions are not acceptable in modern electromagnetism, which is not ab initio macroscopic but is actually a microscopic science. The frequency-domain Maxwell postulates everywhere are P½eðrÞ; bðrÞ; sðrÞ ¼ 0;
r 2 V,
(3)
in the absence of externally imposed source charge and current densities, with P½ denoting a suitable operator, e and b as the only two microscopic fields, and s representing an assembly of charges that represent matter. Eq. (3) leads to the macroscopic equations L½EðrÞ; BðrÞ; 1 o I; n1 m1 o I; a1 ; b ; 0 ¼ 0; 1
r 2 V1 (4)
and L½EðrÞ; BðrÞ; 2 o I; n2 m1 o I; a2 ; b ; 0 ¼ 0; 2
r 2 V2 (5)
for the two regions. These equations do not contain the microscopic sources s, which were homogenized to ultimately yield the constitutive tensors. These equations do not contain the microscopic fields e and b, which were homogenized simultaneously to yield the macroscopic fields E and B. These equations also do not contain F1 and F2 , the absence of which quantities underlies the conclusions reached in Ref. [4]. Eqs. (4) and (5) are indisputable, even in the framework of macroscopic electromagnetism without a microscopic basis. But there is more in those equations – which Sihvola and Tretyakov did not obtain because they did not adopt a microscopic viewpoint. The derivation of (4) from (3) at a specific point r 2 V 1 involves not only r but also a suitably small neighborhood thereof for homogenization. The neighborhood’s size must be finite and its shape must not introduce spurious effects. Furthermore, the neighborhoods of all points in V 1 must be chosen in an identical manner. If a point were to lie close enough to the edge of V 1 , its neighborhood will encompass a part of V 2 as well. Therefore, the only way that (4) can hold everywhere in V 1 is as if it were derived with the assumption that not only V 1 but also the neighborhood of V 1 is filled with material 1. 2 2 Of course, modern macroscopic electromagnetism is merely an approximation of true electromagnetism – which is microscopic and quantum–mechanical.
ARTICLE IN PRESS 252
A. Lakhtakia / Optik 119 (2008) 250–252
In consequence, any jump or boundary condition derived from (4) cannot contain any reference to F1 , since (4) does not contain F1 ; neither can that condition contain any reference to F2 . Likewise, any jump or boundary condition derived from (5) cannot contain any references to F2 and F1 . Therefore, the boundary conditions to be prescribed across S contain neither F1 nor F2 nor the difference F2 F1 . Thus, with the adoption of the microscopic viewpoint, the Tellegen parameter vanishes from all space V . In place of (1), we could have simply begun with L½EðrÞ; BðrÞ; ðrÞ o I; nðrÞ m1 o I; aðrÞ; bðrÞ; 0 ¼ 0, r 2 V,
ð6Þ
thereby quenching the specter of a constitutive parameter of a homogeneous material that manifests itself not throughout the material but only at its boundary! The discussion thus far in this subsection has been aimed towards the formulation of a macroscopic problem with a sharp boundary S between V 1 and V 2 . If the production of a sharp boundary is not required, both the regions V 1 and V 2 would shrink away from the boundary S which itself would swell into an intermediate region V S . For each r 2 V S , homogenization over a small neighborhood of r would result in macroscopic Maxwell postulates like (4) and (5). Thus, (6) would still hold but with 8 8 8 1 n1 a > > > > > > < < < 1 0 0 0 ðrÞ ¼ ðrÞ ; nðrÞ ¼ n ðrÞ ; aðrÞ ¼ a ðrÞ , > > > > > > : :n :a 2 2 2 8 8 b > > > 1 > V1 > > < 0 < ð7Þ bðrÞ ¼ b ðrÞ ; r 2 V S , > > > > > : > V2 :b 2
0
where ðrÞ, etc., are nonhomogeneous constitutive tensors.
3. Concluding remarks Whereas without reference to a microscopic viewpoint, the Tellegen parameter has a ghostly presence in the equations of macroscopic electromagnetism without a microscopic basis and casts its shadow on boundaryvalue problems, I have shown here that recourse to a microscopic viewpoint negates the recognizable existence of the Tellegen parameter in boundary-value problems formulated for modern macroscopic electromagnetism. The difference between the two formulations is epistemic, and its possible resolution and the consequences thereof have been discussed elsewhere [2,4].
References [1] W.S. Weiglhofer, A. Lakhtakia, The Post constraint revisited, AEU¨. Int. J. Electron. Commun. 52 (1998) 276–279. [2] A. Lakhtakia, On the genesis of Post constraint in modern electromagnetism, Optik 115 (2004) 151–158. [3] R.E. Raab, O.L. de Lange, Multipole Theory in ElectroMagnetism, Clarendon Press, Oxford, UK, 2005 (Chapter 8). [4] A. Lakhtakia, Boundary-value problems and the validity of the Post constraint in modern electromagnetism, Optik 117 (2006) 188–192. [5] Yu.N. Obukhov, F.W. Hehl, Measuring a piecewise constant axion field in classical electrodynamics, Phys. Lett. A 314 (2005) 357–365. [6] A. Sihvola, S. Tretyakov, Comments on boundary problems and electromagnetic constitutive parameters, Optik 118 (2007). [7] A. Lakhtakia, J.R. Diamond, Reciprocity and the concept of the Brewster wavenumber, Int. J. Infrared Millim. Waves 12 (1991) 1167–1174. [8] A. Lakhtakia, On the Huygens’s principles and the Ewald–Oseen extinction theorems for, and the scattering of, Beltrami fields, Optik 91 (1992) 35–40.