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J. Opt. Soc. Am. B / Vol. 26, No. 2 / February 2009
P. P. Banerjee and M. R. Chatterjee
Negative index in the presence of chirality and material dispersion Partha P. Banerjee1,* and Monish R. Chatterjee2 1
Department of Electro-Optics and Electrical and Computer Engineering, University of Dayton, Dayton, Ohio 45469, USA 2 Department of Electrical and Computer Engineering, University of Dayton, Dayton, Ohio 45469, USA *Corresponding author:
[email protected] Received September 19, 2008; accepted November 6, 2008; posted November 11, 2008 (Doc. ID 101782); published January 7, 2009
By using a true phasor approach and slowly varying envelope approximations along with Maxwell’s equations and the constitutive relations in their respective domains (time and frequency, respectively), we derive expressions for the allowable propagation vector(s), electromagnetic fields, Poynting vectors, phase, energy, and group velocities in a medium where independent material parameters such as permittivity, permeability, and chirality are frequency dependent, including not only the carrier (e.g., optical) frequency but excursions around the carrier. One definition of negative index, viz., contradirection of the propagation and Poynting vector, demands a large value for the chirality parameter, which may not be physically attainable. We show that by incorporating dispersions in these (independent) material parameters, it may be possible to achieve negative index as defined through contradirected phase and group velocities for a range of carrier frequencies that are lower than the resonant frequency for the aforementioned material parameters as described through the Lorenz (for permittivity and permeability) and Condon (for chirality) models, and without violating the upper bound on the chirality. This has the added advantage that losses will be minimal, and further justifies our approach of using real functions for the material parameters. © 2009 Optical Society of America OCIS codes: 160.3918, 160.1585, 260.2030.
1. INTRODUCTION In recent years, the problem of negative index in a variety of materials, and its possible manifestation through a variety of mechanisms, has received considerable attention [1–6]. Traditionally, negative index behavior is identified in terms of specific physical interpretations of wave transportation in materials. An approach whereby the effective propagation vector is in direct opposition to the Poynting vector, in the presence of chirality, for instance, leads to a requirement for large magnitude of the chirality parameter [7,8]. Another established approach is to place the group and phase velocities in opposition (for instance, with the former being positive and the latter negative)— leading, under different analyses and assumptions, to the requirement of negative permittivity and permeability parameters simultaneously [1–3]. In the absence of chirality, the requirement of negative permittivity and permeability also satisfies the requirement of opposition of the propagation and Poynting vectors. By using the concept of simple (real and complex) dispersion relations, which show opposition of phase and group velocities, we have analyzed linear and nonlinear propagation in negative index materials [9–11]. Among the limitations of practical realizability of negative index through the traditional means, we note that large magnitude of chirality is not achievable [12,13]. In a recent approach to realizing negative index, fabricated Y structured metamaterials have been designed and tested for microwave and terahertz frequencies [14]. The possibility of realizing negative refractive index in gyrotropically magnetoelectric materials has been investigated by 0740-3224/09/020194-9/$15.00
Shen [15]. Yelin et al. [16] have recently shown that negative refraction with minimal absorption can be achieved through quantum interference effects and coupling between magnetic and electric dipole transitions, which leads to electromagnetically induced chirality. We remark here that there has not been enough attention given to the problem of possible variations between group and energy velocities as a function of frequency for sidebands in the neighborhood of a carrier. Commonly, the analysis centers on monochromatic behavior, in which case the concept of the group velocity loses any significance [17]. Calculations of these velocities have been performed for nonmonochromatic conditions for anomalous dispersion and for left-handed media [18–20]; however, the explicit effect of the sidebands on the velocity characteristics, in our opinion, have not been accurately established. To accurately establish electromagnetic (EM) behavior under signal modulation, one needs to use (slow) space- and time-dependent phasors for the EM fields, and apply frequency-dependent constitutive relations in conjunction with the Fourier transformed EM variables. It is found, as shown in this paper, that equivalence of energy and group velocities across the frequency spectrum is not uniformly valid, irrespective of the presence of chirality. A further problem to achieving negative index is often the high attenuation associated with the range of frequencies where the permittivity and permeability are negative [1,11]. This limitation can be alleviated by operating far from the resonant frequency. We show in this paper that, indeed, by operating in a frequency region far below the resonant frequency, where both permittivity © 2009 Optical Society of America
P. P. Banerjee and M. R. Chatterjee
and permeability are positive, and by incorporating material dispersion in the presence of chirality, one can achieve opposing phase and group velocities, characteristic of a negative index medium, without violating the upper bound on the chirality. In Section 2 of this paper we consider a dispersive chiral material in which frequency-dependent constitutive relations are applied rigorously to the Maxwell equations, using specifically a fast spectrum corresponding to carrier frequency variations, and a slow spectrum corresponding to the sideband. Special care has been taken to distinguish fields and parameters in the space–time and their corresponding frequency domains. Slowly varying, complex time-dependent phasors are also derived in Section 2, and relationships between the time and frequency domains carefully formulated, retaining up to the first power in the sideband frequency. In Section 3 we determine the four possible (eigen)values of the propagation constant and the corresponding (eigen)solutions for the EM fields, both in the frequency domain as well as in the (slow) time domain. This is done to enable us to formulate the Poynting vector and the stored energy in Section 4, expressions for which are always formally expressed in the time domain. These are then transformed to the frequency domain and the energy velocity is derived as a function of the (sideband) excursions around the carrier. In Section 5 we derive expressions for the phase and group velocities, once again as a function of the carrier frequency and the sideband, and compare them with the energy velocity derived in Section 4. As shown in Section 6, there are some significant differences between energy and group velocities under dispersion, even when excursions around the carrier frequency are not present. It is evident from our calculations that while the carrier frequency term appears in the expression for group velocity under dispersion, both the energy and phase velocities are found to be independent of the carrier frequency. By examining the expressions for the velocities (or correspondingly the refractive indices), we suggest that if negative index in a dispersive material (with optional chirality) is defined via the opposition of the group and phase velocities (as is pursued frequently [3]), then the above results likely offer a more flexible route to realizing this goal. Since the group velocity contains derivatives of permittivity, permeability, and chirality terms, even around the carrier frequency, whereas the phase velocity does not, we show that one might effectively place the two velocities in opposition by appropriate choice of dispersion in the material parameters. We show this by assuming a Lorentz (or Drude) model for the permeability and the permittivity, and the Condon model for the chirality parameter [21].
2. ELECTROMAGNETIC PHASORS AND CONSTITUTIVE RELATIONS IN A DISPERSIVE CHIRAL MEDIUM ¯ , t兲 , B共r ¯ , t兲 , E共r ¯ , t兲 , H共r ¯ , t兲 in The four vector EM fields D共r Maxwell’s equations are time varying functions. However, as is true for all constitutive relations, the ones relating to ˜ 共r ˜ 共r ˜ 共r ˜ 共r ¯ , 兲 , B ¯ , 兲 , E ¯ , 兲 , H ¯ , 兲 in a reciprocal chiral meD dium are valid in the frequency domain:
Vol. 26, No. 2 / February 2009 / J. Opt. Soc. Am. B
195
˜ = ˜E ˜ − j˜冑 H ˜ D 0 0 ,
共1兲
˜ = j˜冑 E ˜ ˜H ˜, B 0 0 +
共2兲
where 0 and 0 are the permeability and permittivity of free space; ˜共兲 is the so-called frequency-dependent chirality parameter, which is dimensionless; and ˜共兲 and ˜ 共兲 are the frequency dependent electric permittivity and magnetic permeability parameters associated with the chiral material, often about a carrier frequency 0. To incorporate the constitutive relations into Maxwell’s equations, we need to express the vector fields D共t兲 , B共t兲 , E共t兲 , H共t兲 in terms of slowly time varying phasor fields as
冤 冥 冦冤 冥 冧 ¯ ,t兲 Dp共r
¯ ,t兲 D共r ¯ ,t兲 B共r
= Re
¯ ,t兲 E共r
¯ ,t兲 Bp共r
¯ ,t兲 Ep共r
e j0t ,
共3兲
¯ ,t兲 Hp共r
¯ ,t兲 H共r
where 0 is the carrier frequency. The Fourier transforms of the above variables, which are also space dependent, are
冤 冥 冦冤 ˜ 共兲 D ˜ 共兲 B
˜ 共兲 E
=
˜ 共 − 兲 + D ˜ * 共− − 兲 D p 0 0 p
1
˜ 共 − 兲 + B ˜ * 共− − 兲 B p 0 0 p
2
˜ 共 − 兲 + E ˜ * 共− − 兲 E p 0 0 p
˜ 共兲 H
˜ 共 − 兲 + H ˜ * 共− − 兲 H p 0 0 p
冥冧
.
共4兲
Now, the material parameters appearing in the constitutive relations are specified in the frequency domain, and similarly must possess a time domain counterpart. Accordingly, we express them as
冤 冥 冦冤 冥
冧
共t兲
p共t兲
共t兲 = 共t兲
p共t兲 ej0t + c.c. , p共t兲
共5兲
with their Fourier transforms
冤 冥 冦冤 ˜共兲
˜p共 − 0兲 + ˜p* 共− − 0兲
˜ 共兲 = ˜共兲
˜ p共 − 0兲 + ˜ p* 共− − 0兲 ˜p共 − 0兲 + ˜p* 共− − 0兲
冥冧
共6兲
.
Dispersion of the above material parameters can be expressed as an expansion around the carrier frequency as
冤
˜p共 − 0兲
冥 冦冤 冦冤
˜ p共 − 0兲 ⬇ ˜p共 − 0兲
⬅
˜p共0兲 + 共 − 0兲共˜p/兲 0 ˜ p共0兲 + 共 − 0兲共 ˜ p/兲 0 ˜p共0兲 + 共 − 0兲共˜p/兲 0 ˜p0 + ⍀ ˜ p0 ⬘
˜ p0 + ⍀ ˜ p0 ⬘ ˜p0 + ⍀˜p0 ⬘
冥冧
.
冥冧 共7兲
˜ in the presence of Using the constitutive relation for D chirality, the displacement field may be expressed in terms of the phasor electric and magnetic fields as
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P. P. Banerjee and M. R. Chatterjee
1 ˜ = 关E ˜ 共 − 兲 ˜* ˜ p* 共− − 0兲兴 D p 0 ˜ p共 − 0兲 + Ep共− − 0兲 2
˜ + ˜k H ˜ ˜ = 0, ˜pE ␣ pH py z px − j˜ py
1
˜ 共 − 兲˜ 共 − 兲 + H ˜ * 共− − 兲 − j冑00关H p 0 p 0 0 p 2 ⫻˜p* 共− − 0兲兴,
共8兲
where it has been assumed that all fields and material parameters are band limited, thus eliminating cross coupling between positive and negative frequencies. From Eqs. (4), (7), and (8), ˜ 共⍀兲 = 关E ˜ 共⍀兲 ˜ 共⍀兲˜ 共⍀兲兴 ˜ p共⍀兲 − j冑00H D p p p p ˜ 共⍀兲共 ˜ 共⍀兲共˜ + ⍀˜⬘ 兲兴. ˜ p0 + ⍀ ˜ p0 ⬘ 兲 − j冑00H = 关E p p p0 p0 共9兲 The corresponding slowly time varying displacement is obtained from Eq. (9) as ˜ p0 ˜ p0 − j ⬘ /t兲Ep共t兲 − j冑00共˜p0 − j˜p0 ⬘ /t兲Hp共t兲兴. Dp共t兲 = 关共 共10兲 The corresponding relation for the flux density Bp共t兲 may be similarly found. Note that in the absence of dispersion and chirality, the above result reduces to the expected constitutive relationship between the electric displacement and field vectors.
共13d兲
where the parameter ˜␣p = ˜p冑00, which has the dimension rad m−1, is the chiral wavenumber. In Eqs. (13a)–(13d), all quantities with tildes are functions of ⍀. Finding nontrivial solutions for the homogeneous Eqs. (13a)–(13d) requires that the determinant of the coefficient matrix must vanish. This leads to the well-known expressions for the wavenumbers in the chiral medium: ˜k = + ˜ 冑 + 冑 ˜ p˜p , z1 p 0 0
共14a兲
˜k = + ˜ 冑 − 冑 ˜ p˜p , z2 p 0 0
共14b兲
˜k = − ˜ 冑 + 冑 ˜ p˜p , z3 p 0 0
共14c兲
˜k = − ˜ 冑 − 冑 ˜ p˜p , z4 p 0 0
共14d兲
which indicates a set of four possible values of the wavenumber that satisfy the nontrivial field solutions. We may note that the ˜kz values depend on the chirality parameter ˜. ˜ is known. We next assume that the field component E px From the homogeneous set of equations (13a)–(13d), as˜ value, the field solutions are obsuming the known E px tained after some algebra as follows: ˜ arbitrary, E px
3. ELECTROMAGNETIC FIELD SOLUTIONS IN THE PRESENCE OF DISPERSION To determine possible propagation constants and the corresponding field solutions, we start from Maxwell’s curl equations. For instance, from ⵜ ⫻ E = −B / t, Fourier transforming and using the constitutive relations, we get for the phasor fields 共 ⬎ 0兲 the relation
˜ = E py
˜ = H px
¯k ⫻ E ˜ 共⍀兲 = 关j˜ 共⍀兲冑 E ˜ ˜ 共⍀兲兴. 共11兲 ˜ p共⍀兲H p p 0 0 p共⍀兲 + p Similarly, from the other Maxwell curl equation, ¯k ⫻ H ˜ 共⍀兲 = 关 ˜ 共⍀兲 − j˜ 共⍀兲冑 H ˜ ˜ p共⍀兲E p p p 0 0 p共⍀兲兴. 共12兲
˜ = H py
˜ p˜p + ˜␣p2 + ˜kz2 − 2 2j˜␣p˜kz ˜ p˜p + ˜␣p2 − ˜kz2 + 2 ˜p 2j˜␣p ˜ p˜p − ˜␣p2 + ˜kz2 + 2 ˜ p˜kz 2
˜ , E px
˜ , E px
˜ . E px
共15兲
The solutions for the above fields are obtained more readily by assuming a wave vector ¯k共⍀兲 = ˜kz共⍀兲aˆz pointed in the Z direction (which is arbitrary, and hence general, in an unbounded medium). This approach simplifies the resulting characteristic matrix to a 4 ⫻ 4 instead of a 6 ⫻ 6. Using such a wave vector, it is simple to show that the longitudinal components of the fields vanish, resulting in a purely transverse propagation in the bulk chiral material. The following set of homogeneous equations for the field components is then obtained from Eqs. (11) and (12):
Based on the above algebraic equations for the field components, where excursions of the material parameters around a carrier frequency 0 are assumed, we may anticipate that the field components will assume the following forms on the basis of a linear expansion (up to ⍀1) in the sideband frequency:
˜ + ˜k E ˜ ˜ = 0, ˜ pH j˜␣pE px z py + px
共13a兲
˜ 共⍀兲 = j共A + B ⍀兲E ˜ 共⍀兲, H px 2 2 px
˜k E ˜ ˜ − ˜ = 0, ˜ pH ␣ pE z px − j˜ py py
共13b兲
˜ 共⍀兲 = 共A + B ⍀兲E ˜ 共⍀兲. H py 3 3 px
˜ − j˜␣ H ˜ ˜ ˜ ˜pE px p px − kzHpy = 0,
共13c兲
For future use, the corresponding inverse Fourier transforms are given below:
˜ 共⍀兲 arbitrary, E px ˜ 共⍀兲 = j共A + B ⍀兲E ˜ 共⍀兲, E py 1 1 px
共16a兲
P. P. Banerjee and M. R. Chatterjee
Vol. 26, No. 2 / February 2009 / J. Opt. Soc. Am. B
Epx共t兲 arbitrary,
A11 = − A13,
B11 = B13 ,
Epy共t兲 = jA1Epx共t兲 + B1Epx共t兲/t,
A21 = − A23,
B21 = − B23 ,
˜ 共t兲 = jA E 共t兲 + B E 共t兲/t, H px 2 px 2 px
A31 = A33,
˜ 共t兲 = A E 共t兲 − jB E 共t兲/t, H py 3 px 3 px
共16b兲
where t is the inverse transform variable corresponding to ⍀, and hence denotes “slow time.” The coefficients A1–3 and B1–3 will now be evaluated. We will demonstrate the derivation for ˜kz3 = −˜p冑00 ˜ p˜p = −˜␣p + 冑 ˜ p˜p, and for the pair A13, B13, where + 冑 the second subscripts refer to ˜kz3, and write the results for the other pairs. We also can expand the material parameters in first-order Taylor expansions around ⍀ = 0 as
冤 冥冤 ˜p共⍀兲
˜p共⍀兲
冥 冦冤
˜ p共⍀兲 ˜ p共⍀兲 = = ˜␣p共⍀兲 冑00˜p共⍀兲
˜p0 + ⍀ ˜ p0 ⬘ ˜ p0 + ⍀ ˜ p0 ⬘
⬘ 兲 冑00共˜p0 + ⍀˜p0
冥冧
,
共17兲 From the second of the relations in Eq. (15), ˜ = E py
˜ p˜p + ˜␣p2 + ˜kz2 − 2 2j˜␣p˜kz
˜ ⬅ E px
˜ p˜p 2˜␣p2 − 2˜␣p冑 2j˜␣p˜kz
˜ = jE ˜ . E px px 共18兲
Upon comparision with Eq. (16), it follows that A13 = 1,
共19兲
B13 = 0.
This shows that in the presence of chirality, the electric field is indeed circularly polarized. It is important to note that in Eq. (18), all terms involving cancel out, and therefore do not require expansion of in terms of ⍀. This is also true during the computation of the pairs A23, B23 and A33, B33. Proceeding similarly from the third relation in Eq. (15), ˜ =−j H px
冑
˜p ˜p
so that using Eq. (17), A23 = −
冑
˜p0 ˜ p0
1 ,
B23 = −
2
˜ , E px
冑 冉
共20兲
⬘ ˜p0 ˜p0
˜ p0 ˜p0
−
˜ p0 ⬘ ˜ p0
冊
Finally, from the fourth relation in Eq. (15), it follows after straightforward algebra, that A33 = − A23 =
冑
˜p0 ˜ p0
1 ,
B33 = − B23 =
2
冑 冉
⬘ ˜p0 ˜p0
˜ p0 ˜p0
−
˜ p0 ⬘ ˜ p0
冊
共23兲
The opposite circular polarization can be observed for ˜k . Note also that the EM fields do not involve any exz2,4 plicit dependence on the chirality; the chirality is, however, responsible for the circularly polarized eigenstates, and only manifests itself explicitly in the eigenvalues or propagation constants.
4. POYNTING VECTOR, STORED ENERGY, AND ENERGY VELOCITY Based on the above derivations of the various components of the electric and magnetic fields, one can write down the expression for the complex Poynting vector, averaged over fast time or the carrier period. This average Poynting vector may still be a slowly varying function of time, or slow time as defined earlier, and is given by 1 1 * 共t兲 − E 共t兲H* 共t兲兴a ˆ z. Sav共t兲 = 关Ep共t兲 ⫻ Hp* 共t兲兴 = 关Epx共t兲Hpy py px 2 2 共24兲 The slowly time varying phasors in Eq. (24) can be found by using Eq. (16b) leading to (for kz3, for instance), 1 * 共t兲 − jB E* 共t兲/t其 Sav3共t兲 = 关Epx共t兲兵− A23Epx 23 px 2 * 共t兲 + B E* 共t兲/t其兴a ˆz − jEpx共t兲兵− jA23Epx 23 px 2 2 = 兵− A23Epx 共t兲 − j共B23/2兲Epx 共t兲/t其aˆz ,
assuming Epx共t兲 to be real. Now Fourier transforming back to the frequency domain, ˜ 共⍀兲 = 兵− A + B ⍀/2其I 关E2 共t兲兴aˆ , S av3 23 23 t z px
共25兲
where A23, B23 have been derived in Eq. (21). We will now compute the stored electric and magnetic energies given by the scalar product definitions 1 we共t兲 = Dp* 共t兲 · Ep共t兲, 4
. 共21兲
B31 = B33 .
197
1 wm共t兲 = Bp共t兲 · Hp* 共t兲. 4
共26兲
Using Eqs. (10) and (16b), 1 * E + E* E 兲 + j冑 共˜ ˜ p0 − j ˜ p0 ⬘ /t兲共Epx we3共t兲 = 关共 px 0 0 p0 py py 4 * E + H* E 兲兴 ⬘ /t兲共Hpx − j˜p0 px py py
.
共22兲 ˜ = jH ˜ , indicating circular polarIt is readily seen that H py px ization, as stated above. Similar circular polarization can be observed for ˜kz1. Incidentally, for this case, the A and B coefficients take the form
1 2 ˜ p0 − j ˜ p0 ⬘ /t兲Epx ⬘ /t兲 + 冑00共˜p0 − j˜p0 = 关共 2 2 2 + j共B23/2兲Epx /t兲兴, ⫻共A23Epx
assuming Epx共t兲 to be real once again. Now, simplifying, taking the Fourier transform, and upon retaining only up to ⍀1, we obtain
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P. P. Banerjee and M. R. Chatterjee
˜ t3共⍀兲 = w ˜ e3共⍀兲 + w ˜ m3共⍀兲 w
1 ˜ e3共⍀兲 = 关共 ˜ p0 + 冑00A23˜p0兲 − 兵冑00共B23/2兲˜p0 w 2 2 ˜ p0 ⬘ + 冑00A23˜p0 ⬘ 兲其⍀兴It关Epx − 共 共t兲兴.
冋
˜ p0 + 冑00A23˜p0兲 − = 共
共27兲
再
冑00共B23/2兲˜p0 −
冎册
2 2 ˜ p0 ⬘ + A23 ⬘ 兲 ⍀ It关Epx + 2冑00A23˜p0 共t兲兴.
Similarly,
共28兲
The total stored energy can then be written as
˜ 共⍀兲/w ˜ve3共⍀兲 ⬅ S ˜ t3共⍀兲 = av3
冋
˜ p0 + 冑00A23˜p0兲 − 共
再
˜ p0 ⬘ 共 共29兲
In the absence of chirality and dispersion, the above reduces to the expected stored energy, furthermore, in the presence of chirality without dispersion, stored energy has a sideband dependence through the chirality parameter. We are now in a position to compute the energy velocity, including its frequency dependence. This reads
1 2 ˜ m3共⍀兲 = 关共 ˜ p0A23 + 冑00A23˜p0兲 − 兵冑00共B23/2兲˜p0 w 2 2 2 ˜ p0 ⬘ A23 ⬘ 兲其⍀兴It关Epx − 共 + 冑00A23˜p0 共t兲兴.
1 2
− A23 + 共B23/2兲⍀
冑00共B23/2兲˜p0 −
1
2 ˜ p0 ˜ p0 ⬘ + 2冑00A23˜p0 ⬘ + A23 ⬘ 兲 共
2
冎册
aˆz . 共30a兲
⍀
For later comparison with the group velocity ˜vg3共⍀兲, we express the above, after substituting for A23 and B23 from Eqs. (21) and (22) in the approximate form 1 ˜ve3共⍀兲 ⬇ =
冤冑
˜ p0 − 冑00˜p0其 + 兵 ˜p0
冦冑 冢 1
˜p0 ˜ p0
4
˜p0 ⬘
˜ p0 ⬘ −
˜p0
˜ p0
冣
1 +
冑˜p0˜ p0
2
冢
˜p0 ⬘
˜ p0 ⬘ +
˜p0
˜ p0
冣
冧冥
aˆz .
共30b兲
⬘ ⍀ − 冑00˜p0
It is interesting to observe that the velocity does not have any frequency dependence through the chirality parameter in the absence of dispersion. Also, the frequency (sideband) dependence of the velocity is asymmetric with respect to the dispersion in permittivity and permeability. Furthermore, in the nonchiral nondispersive limit, the above result converges to the standard solution for the energy velocity. We also note that the energy velocity does not depend explicitly on the carrier frequency. A similar calculation for ˜kz1 shows that while the Poynting vector is unchanged, the total energy exhibits a change of sign with respect to the ˜-dependent terms (originating from the similar change of sign with respect to ˜ in the expression for ˜kz1 relative to ˜kz3), which also manifests itself in the expression for ˜ve1共⍀兲: 1 ˜ve1共⍀兲 ⬇
冤冑
˜ p0 + 冑00˜p0其 + 兵 ˜p0
冦冑 冢 1
˜p0 ˜ p0
4
˜p0 ⬘
˜ p0 ⬘ −
˜p0
˜ p0
冣
1 + 2
冑˜p0˜ p0
冢
˜p0 ⬘
˜ p0 ⬘ +
˜p0
˜ p0
冣
冧冥
aˆz .
共31兲
⬘ ⍀ + 冑00˜p0
Note that the above result shows a similar asymmetry with respect to the dispersion in permittivity and permeability, as with the ˜kz3 case. Calculations for ˜kz2,4 (not shown here) would reveal that the Poynting vectors are in directions op˜ ˜ e3,1. ve2,4 = −v posite to that for ˜kz1,3 (note that ˜kz2,4 = −k z3,1), and consequently, ˜
5. PHASE AND GROUP VELOCITIES ˜ p˜p. Defining vp = 1 / 共kz / 兲, we now proceed to determine the phase velocity corresponding to ˜kz3 = −˜p冑00 + 冑 Straightforward computation, retaining up to ⍀1, yields
P. P. Banerjee and M. R. Chatterjee
Vol. 26, No. 2 / February 2009 / J. Opt. Soc. Am. B
199
1 ˜vp3共⍀兲 ⬇
冤冑
˜ p0 − 冑00˜p0其 + 兵 ˜p0
冦冑 冢 1
˜p0 ˜ p0
2
˜p0 ⬘
˜ p0 ⬘ + ˜ p0
˜p0
冣
冧冥
共32兲
aˆz .
⬘ ⍀ − 冑00˜p0
Next, we define 1 vg3 =
1
˜kz3
=
˜kz3
共33兲
,
⍀
where we express ˜kz3 explicitly in terms of the sideband ⍀: ˜k = − 共 + ⍀兲˜ 冑 + 共 + ⍀兲冑 ˜ p˜p , z3 0 p 0 0 0
共34兲
and differentiate with respect to ⍀ to obtain ˜vg3共⍀兲 ⬇
1
˜p
⍀
⍀
冑˜p˜ p + 共0 + ⍀兲 冑˜p˜ p − ˜p冑00 − 共0 + ⍀兲冑00
共35兲
aˆz .
After considerable algebra and retaining up to ⍀1 yields: 1 ˜vg3共⍀兲 ⬇ =
冤
冦冑
˜p0 ˜ p0 − 冑00˜p0 +
+
冑˜p0˜ p0
2
冦冑 冢 ˜p0 ˜ p0
0
˜p0 ⬘
˜ p0 ⬘ +
˜p0
˜ p0
冣
冢
˜p0 ⬘
˜ p0 ⬘ +
˜p0
⬘ − − 2冑00˜p0
0
˜ p0
冣
⬘ − 0冑00˜p0
冑˜p0˜ p0
4
冢
˜p0 ⬘
˜ p0 ⬘ − ˜ p0
˜p0
冣冧 2
Finally, the phase velocity for the ˜kz3 case can be written as
˜vp3共⍀兲 =
1 aˆz ⬇
kz3
冑˜p˜ p − ˜p冑00
冧 ⍀
冥
共36兲
aˆz .
1 aˆz ⬇ ˜ p0 − ˜p0冑00兲 + 共冑˜p0
Expressions for group and phase velocities for ˜kz1 can be obtained from Eqs. (36) and (37), respectively, by changing the sign of the ˜-dependent terms.
6. COMPARISON OF ENERGY, GROUP, AND PHASE VELOCITIES UNDER DISPERSIVE PERMEABILITY, PERMITTIVITY, AND CHIRALITY In the nondispersive limit (primed quantities are zero), we immediately observe that all three velocities are equal, and the modification due to chirality is selfevident. We next note some significant differences between energy and group velocities under dispersion, even when excursions 共⍀兲 around the carrier frequency 共0兲 are not present. Between the three velocities, we note that
冦
冑˜p0˜ p0 2
冢
˜p0 ⬘
˜ p0 ⬘ +
˜p0
˜ p0
冣
冧
aˆz .
共37兲
⬘ ⍀ − 冑00˜p0
the group velocity has additional terms dependent on both 0 and ⍀, while the energy and phase velocities contain only the sideband 共⍀兲 dependent terms. Note that in deriving ˜kzs in Eq. (14), we have incorporated the general frequency 共=0 + ⍀兲 instead of simply the carrier 0 because taking the derivative of ˜kz with respect to en route to finding the group velocity using only 0 leads to a result that in the nondispersive limit does not match the expected classical result. On the other hand, restoring the term in ˜kz nondispersively leads to the correct result. An interesting outcome of this approach is the persistence of 0 in the expression for group velocity under dispersion, as may be readily verified. On the other hand, both the energy and phase velocities are found to be independent of the carrier frequency 0. For the first case, this is due to the fact that the electric and magnetic field compo-
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P. P. Banerjee and M. R. Chatterjee
˜ , are independent of . In nents, expressed in terms of E px 0 the second case, the (total) frequency variable , which appears both in the numerator and the denominator, simply drops out. We now suggest that if negative index in a dispersive material (with optional chirality) is defined via the opposition of the group and phase velocities (as is pursued frequently [3]), then the above results likely offer a more flexible route to realizing this goal. Since it is well-known that achieving negative index in a nondispersive chiral material requires large magnitudes of the chirality parameter ˜ that are not practically realizable [12,13], it is meaningful to search for alternative and more attainable routes to obtaining the same. Since the group velocity ˜vg3共⍀兲 contains derivatives of permittivity, permeability, and chirality terms, even around the carrier frequency, whereas the phase velocity does not, it is conceivable that one might effectively place the two velocities in opposition by an appropriate choice of dispersion in the material parameters. To track the behavior of chiral dispersive materials in terms of the energy, group, and phase velocities derived earlier for propagating signals consisting of pulsed or modulated carriers, we assume the Lorentzian (Drude) dispersion models for relative permittivity and permeability and the Condon dispersion model for chirality, as follows [21]:
˜pr共兲 = 1 +
˜ pr共兲 = 1 +
˜ 共兲 = ␣ pr c
p2 2c
−
2
2 m
2c − 2
2c
− 2
,
,
共38a兲
,
共38b兲
共38c兲
˜ pr共兲, and ˜pr共兲 are, respectively, the relawhere ˜pr共兲, tive (spectral) permittivity, permeability, and chirality admittance of the material under consideration. The frequencies p and m arise from electric polarization and magnetization, respectively, while c represents a (single) resonance in the neighborhood of the applied signal. Likewise, ␣c represents a chiral frequency parameter. It can be shown that our chirality parameter ˜pr is related to the ˜ pr˜pr. relative chirality admittance above through ˜pr = − Let = 0 be the carrier frequency of operation, and (for ˜ pr). Next, we numerical modeling) p = m = ⬘c (i.e., ˜pr = introduce the following normalizations: n = 0 / c, ␣ = ␣c / c, and  = ⬘c / c. Note that although typical values for the normalized frequency 共n兲 may be less than unity, the frequency of operation of the carrier frequency 0 can certainly be in the optical range. Then we obtain the following expressions for normalized phase, energy, and group velocities (in terms of a monochromatic carrier, i.e., with ⍀ = 0) from Eqs. (30a), (30b), (36), and (37) as
c np = ne =
c
v p3
=
v e3
c ng =
c
v g3
=
v p3
˜ pr0 − ˜pr0 , = 冑˜pr0
共39a兲
˜ pr0 ⬘ − ˜pr0 ⬘ 兴, + 0关
共39b兲
where the primed terms refer to derivatives with respect to , evaluated at the carrier frequency, and c is the free space velocity. Equations (39) are indeed the phase, energy, and group indices for the dispersive material. Now, in terms of the stated normalizations, ˜pr0 = ˜ pr0 = 1 +
˜pr0 = − ␣
˜pr0 ⬘ =
1 − n2
共1 − n2 兲2
共1 − n2 兲2
共40a兲
,
关共1 + 2兲n − n3 兴
2 n 2
˜pr0 ⬘ =−␣
2
共40b兲
,
共40c兲
,
关共1 + 2兲 + 32n2 − n4 兴 共1 − n2 兲3
.
共40d兲
For variations in n, ␣, and , we are interested to find a range of parameters that satisfy the condition that vp and vg are in opposition (in order to ensure negative index behavior), while not requiring the following: (i) Large magnitude of chirality, , which is not realiz˜ pr0 = ˜pr0, since able (i.e., still maintaining 兩˜pr0兩 ⬍ 冑˜pr0 r = r); (ii) Negative permittivity and permeability, which are commonly used to realize negative index (our intention is to show that negative index may be induced by primarily invoking dispersion). A preliminary check with ␣ = 1 and  = 1 shows that conditions (i) and (ii) cannot be satisfied for any n. Therefore, it is necessary to arrive at the desired negative index condition by carrying out a more generalized analysis based on all three quantities treated as variables. It may be shown that conditions (i) and (ii) cannot be satisfied for large values of n (higher than the resonance value of 1). Hence, we begin by assuming n Ⰶ 1. Based on this, the following approximations apply: ˜pr0 ⬇ 1 + 2 ,
共41a兲
˜pr0 ⬇ − ␣n共1 + 2兲,
共41b兲
˜pr0 ⬘ ⬇ 0,
共41c兲
˜pr0 ⬘ ⬇ − ␣共1 + 2兲.
共41d兲
Now, we require (i) ˜pr0 − ˜pr0 ⬎ 0 (this makes vp ⬎ 0), which leads to ␣n ⬎ −1;
P. P. Banerjee and M. R. Chatterjee
Fig. 1. (Color online) Variation of phase refractive index np with normalized frequency for various values of .
Fig. 2. (Color online) Variation of group refractive index ng with normalized frequency for various values of .
Vol. 26, No. 2 / February 2009 / J. Opt. Soc. Am. B
201
Fig. 4. (Color online) Variation of difference between relative permittivity (which equals relative permeability) and the modulus of the chirality parameter with normalized frequency for various values of .
(ii) vg ⬍ 0, which using Eq. (39b) leads to ␣n ⬍ −0.5. Therefore, combining the two conditions, the overall range for expected negative index becomes approximately −1 ⬍ ␣n ⬍ −0.5. Figures 1 and 2 are drawn over the interval 0.05艋 n 艋 0.2 for the normalized frequency, and with ␣ = −7.5. Indeed as predicted from a calculation with n = 0.1 and  = 1, the phase velocity is positive and the group velocity is negative for kz3. In fact, we find that by using the exact formulas for the two velocities, they continue to be in similar opposition over a range of the normalized frequency n given by 0.065艋 n 艋 0.13 and over values of  ranging from 0.1 to 10. Opposition of phase and group velocities with the former negative and the latter positive can be realized with other values of kz. We have also checked that the permittivity and permeability are positive in this range (see Fig. 3) and, further, they are larger than the magnitude of the chirality parameter (see Fig. 4). Finally, it is clear that since the range of validity of frequency is smaller than the resonant frequency, one can safely neglect losses coming from the imaginary part(s) of the permittivity and permeability.
7. CONCLUSIONS
Fig. 3. (Color online) Variation of relative permittivity with normalized frequency for various values of .
For propagation in a material with arbitrary frequencydependent permittivity, permeability, and chirality, we have, from first principles, and using a rigorous phasor approach, carefully derived detailed expressions for the allowable (frequency-dependent) propagation vector(s); EM fields; Poynting vectors; and phase, energy, and group velocities, including not only the carrier frequency but also excursions around the carrier. For the first time to the best of our knowledge, we have shown that by incorporating dispersions in the material parameters, it may be possible to achieve negative index as defined through contradirected phase and group velocities for a range of carrier frequencies that are lower than the resonant frequency for the material parameters (as described through the Lorenz and Condon dispersion models), and without
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violating the upper bound on the chirality. This has the added advantage that losses will be minimal, and further justifies our approach of using real functions for the material parameters. Derivation of the dispersion relation depicting the variation of wavenumber with angular frequency by starting out from the dispersions in permittivity, permeability, and chirality is under development, and will be used to analyze propagation of modulated pulses in the future. Finally, although we have illustrated realization of negative index only assuming a carrier frequency (without sidebands) in our example, one can readily generalize this to cases where sideband frequencies are present over and above the carrier frequency (as expected in all modulated pulses), as demonstrated in the general theory developed above.
P. P. Banerjee and M. R. Chatterjee 7. 8.
9. 10.
11.
12.
ACKNOWLEDGMENTS This work has been partially funded by a Defense Advanced Research Projects Agency (DARPA) Small Business Initiative Research (SBIR) contract W31P4Q-08-C0154. The authors thank Mohammed al-Saedi for his assistance in plotting the figures.
13. 14.
15.
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