A uni-directional optical pulse propagation equation for materials with both electric and magnetic responses

In the past few years, composite materials (“metamaterials”) that demonstrate both an electric and magnetic response have been the subject of both experimental and theoretical investigation. Often the motivation for this research is the potential for exotic applications: for example, superresolution Pendry (2000), or the possibility of “trapped rainbow” light storage Tsakmakidis et al. (2007). Despite these interesting possibilities, there is also the more basic need for efficient methods for propagating optical pulses in such metamaterials, in particular in one-dimensional (1D) waveguide geometries. Indeed, methods for doing so have already led to interesting predictions (see, e.g., Scalora et al. (2005); Wen et al. (2007); D’Aguanno et al. (2008)). However, these methods, and earlier ones, (e.g. Agrawal (2007); Brabec and Krausz (1997); Geissler et al. (1999); Kinsler and New (2003)) tend to rely on mechanisms such as the introduction of a co-moving frame, and assumptions that the pulse profile has negligible second-order temporal or spatial derivatives. Assuming second-order derivatives are small may well be reasonable, but it means that the pulse profile must remain well behaved. This approximation therefore might well become poorly controlled Kinsler and New (2003); Kinsler (2002), particularly for ultrashort or otherwise ultrawideband pulses, or exotic or extreme material parameters. Ideally we would prefer to make approximations based solely on the material parameters of our device, so as to avoid making assumptions about the state of an ever-changing propagating pulse.

Here I derive 1D wave equations for a waveguide with both electric and magnetic dispersion, and electric and magnetic nonlinearity. I use the directional fields approach Kinsler et al. (2005); Kinsler (2007a), which allows us to directly write down a first-order wave equation for pulse propagation without complicated derivation or approximation. We simply look at the coupled forward and backward wave equations that are a direct re-expression of Maxwell’s curl equations, and substitute in the appropriate dispersion and nonlinearity. I also show separate examples for second- and third-order nonlinearities in both electric and magnetic responses, although the effects can be combined if desired. Note that these directional fields are applicable to more than just pulse propagation, as they have been used to simplify Poynting-vector-based approaches to electromagnetic continuity equations Kinsler et al. (2009).

The derivation makes only a single, well-defined approximation to reduce the bi-directional forward-backward coupled model down to a single first-order wave equation – that of assuming small changes over the scale of a wavelength. This approximation is remarkably robust for all physically realistic parameter values – see Kinsler (2007b) for an analysis focused on nonlinear effects; more general considerations have been dealt with in terms of factorized wave equations Kinsler (2008). The resulting wave equation retains all the usual intuitive and analytical simplicity of ordinary wave propagation equations, unlike the computationally intensive approach of a direct numerical solution of Maxwell’s equations (see, e.g., Yee-1966tap; Flesch-PM-1996prl; Gilles-MV-1999pre; Brabec and Krausz (1997); Tarasishin-MZ-2001oc; Tyrrell-KN-2005jmo).

This paper is structured as follows: Directional fields and their re-expression of the Maxwell curl equations is outlined in Section II, followed by the reduction of the bi-directional wave equation into a uni-directional form in Section III. Section IV shows wave equations for a doubly-nonlinear third-order nonlinearity material, and Section V does the same for a second-order case. In Section VI propagation under the influence of typical metamaterial responses is discussed, and I conclude in Section VII.

The directional fields approach Kinsler et al. (2005) allows us to write down wave equations for hybrid electromagnetic fields $\textbf{G}^{\pm}$. Note that here I define $\textbf{G}^{\pm}$ with an alternate (and more sensible) sign convention than previously. Further, I also allow for more general types of polarization and magnetization in such a way as to provide a simpler presentation. For propagation along the unit vector u, the propagation (curl) equation for $\textbf{G}^{\pm}$ is written in the frequency domain as¹¹1See derivation in Appendix A.

with

Here the electric response of the material is encoded in two parts: a spatially invariant linear response component $\alpha_{r}$, and the remaining contributions (of any type) in $\textbf{P}_{c}$. Similarly, the magnetic response is divided up in the same way between $\beta_{r}$ and $\mu_{0}\textbf{M}_{c}$. Generally we will put the entire non-lossy linear response of the medium (i.e. the dispersion) into the reference parameters $\alpha_{r}$ and $\beta_{r}$, although it may also be convenient to specify only that the product $\alpha_{r}\beta_{r}$ is real (cf. Kinsler-2009pra). All the nonlinear responses and other complications (“corrections”), such as spatial variations in the material parameters, remain in $\textbf{P}_{c}$ and $\textbf{M}_{c}$. As an example, in Kinsler et al. (2005) this approach was applied to second-harmonic generation in a periodically poled dielectric crystal. The time derivatives of these corrections $\textbf{P}_{c}$ and $\textbf{M}_{c}$ correspond to bound electric and magnetic currents respectively Kinsler et al. (2009). These $\textbf{P}_{c}$ and $\textbf{M}_{c}$ are functions of both fields E and H, i.e. $\textbf{P}_{c}\equiv\textbf{P}_{c}(\textbf{E},\textbf{H})$ and $\textbf{M}_{c}\equiv\textbf{M}_{c}(\textbf{E},\textbf{H})$. If we choose instead to have frequency-independent $\alpha_{r}$ and $\beta_{r}$, then the remaining linear response can simply be included in $\textbf{P}_{c}$ and $\textbf{M}_{c}$; in this case the “weak loss and nonlinearity” condition I use later to decouple forward and backward fields would then need to be broadened to include weak dispersion as well (also see Kinsler et al. (2005) for more discussion). However, neither version imposes any requirements on the pulse profile.

It is useful to give a simple example of the directional fields to provide some insight into their nature. In the pure transverse plane-polarized case, with fields propagating along the $z$ direction, and frequency-independent (material parameters) permitivitty $\epsilon_{r}$ and permeability $\mu_{r}$, we can write

where this simple $G_{x}^{\pm}$ definition matches the original proposal of Fleck Fleck (1970).

It is worth considering how reflections arise in this picture based on spatially invariant reference parameters augmented by corrections terms. Leaving aside for now the distinctions between spatially propagated fields and temporally propagated ones (see the discussion in Kinsler (2008)), transition to a new media can be handled in two ways. First, we could map the existing fields $\textbf{G}^{\pm}$ onto new ones $\textbf{G}_{n}^{\pm}$ based on new reference parameters $\alpha_{rn}$ and $\beta_{rn}$. Here a pure $\textbf{G}^{+}$ field would separate into two pieces, one a forward propagating $\textbf{G}_{n}^{+}$, and the other a “reflected” backward propagating $\textbf{G}_{n}^{-}$. Second, we might retain the existing reference parameters, and have modified corrections $\textbf{P}_{c}^{\prime}$ and $\textbf{M}_{c}^{\prime}$. These altered correction terms then couple the forward and backward directed fields, inducing the necessary reflection in $\textbf{G}^{-}$; although as a side-effect of our now no longer optimal $\alpha_{r}$ and $\beta_{r}$, the forward evolving field is made up of coupled $\textbf{G}^{+}$ and $\textbf{G}^{-}$ components Kinsler et al. (2005).

Starting with the vectorial curl equation (1), I first take the 1D, but bi-directional, limit, and describe the approximation necessary to produce a simpler uni-directional form. After this, I discuss how the common transformations used in optical wave equations can be applied in this context. All equations and field quantities are in the frequency ($\omega$) domain, unless explicitly noted otherwise.

Third-order nonlinearities are common in many materials, for example in the silica used to make optical fibres (see, e.g., Agrawal (2007)). There are many applications of significant scientific interest, for example, white light supercontinnua Alfano and Shapiro (1970a, b); Dudley et al. (2006), optical rogue waves Solli et al. (2007); or filamentation Braun et al. (1995); Berge and Skupin (2009).

Here we study propagation through such a material, with non-reference linear responses $\kappa_{\epsilon},\kappa_{\mu}$ (describing e.g. loss and dispersion), and instantaneous magnetic third-order nonlinearity $\chi_{\mu}$ Klein-WFL-2007oe along with the more common electric type $\chi_{\epsilon}$. For such a system, and for plane-polarized fields, the propagation equation is

This is a generalized nonlinear Schrödinger (NLS) equation, and it retains both the full field (i.e. uses no envelope description) and the full nonlinearity (i.e. includes third-harmonic generation). The only assumptions made are that of transverse fields and weak dispersive and nonlinear responses; these latter assumptions allow us to decouple the forward and backward wave equations. This decoupling allows us, without any extra approximation, to reduce our description to one of forward-only pulse propagation. The specific example chosen here is for a cubic nonlinearity, but it is easily generalized to the noninstantaneous case or even other scalar nonlinearities.

We can transform eqn. (13) into one closer to the ordinary NLS equation by representing the field in terms of an envelope and carrier

where we choose the carrier wave vector to be $k_{0}=k_{r}(\omega_{0})=\omega_{0}\alpha_{r}(\omega_{0})\beta_{r}(\omega_{0})$. After separating into a pair of complex-conjugate equations (one for $A$ and one for $A^{*}$), and ignoring the off-resonant third-harmonic generation term, this gives us the expected NLS equation without diffraction. The chosen carrier effectively moves us into a frame that freezes the carrier oscillations, but this phase velocity ($v_{p}=\omega/k$) frame differs from one that is co-moving with the pulse envelope (i.e., one moving at the group velocity $v_{g}=\partial\omega/\partial k$). After we transform into a frame co-moving with the group velocity at $\omega_{0}$, where $\Delta_{g}=\omega_{0}[v_{g}^{-1}(\omega_{0})-v_{p}^{-1}(\omega_{0})]$, the wave equation for ${A}(\omega)$ is

where $K(\omega)=k_{r}[\kappa_{\epsilon}(\omega)+\kappa_{\mu}(\omega)]/2+\Delta_{g}$; and $\chi=\chi_{\epsilon}-(\mu_{0}\epsilon_{r}^{2}/\epsilon_{0}\mu_{r}^{2})\chi_{\mu}$. All that has been assumed to derive this standard envelope NLS equation is uni-directional propagation and negligible third-harmonic generation. The self-steepening term, often seen in (or added to) NLS equations arises from the frequency dependence of $k_{r}$. This self-steepening has both electric and magnetic contributions, which can be adjusted independently, as has been pointed out by Wen et al. Wen et al. (2007) for the case of the SVEA limit. In Section VI, I discuss how the importance of each contribution varies with frequency for both a double-plasmon model (as in Wen et al. (2007)), and a wire-array and split-ring model more typical of practical metamaterials.

It is worth comparing this eqn. (15) to D’Aguanno et al.’s D’Aguanno et al. (2008) eqn. (5) [hereafter eqn. (DMB5)]. Although in many respects they appear to be the same, mine is far more general and can be applied (at least in principle) to an arbitrarily wide pulse bandwidth, whereas theirs is subject to the rather restrictive SVEA. For example, my eqn. (15) results from only one “slow evolution” approximation, as opposed to the numerous steps, substitutions, and approximations in Section 2 of D’Aguanno et al. (2008). I also retain the possibility of arbitrary dispersion $K(\omega)$, whereas theirs retains only the second-order part (i.e. as $\propto\partial_{t}^{2}$, which in the frequency domain would be $\propto\omega^{2}$). Indeed, with the dispersion and nonlinear factors in my eqn. (13) combined, that full-spectrum wave propagation equation is scarecly more complicated than eqn. (DMB5). Similar remarks also hold when comparing eqn. (15) to Wen et al.’s Wen et al. (2007): but although Wen et al.’s result is also restricted by the SVEA, it does at least allow for diffraction. Both, however, along with Scalora et al.’s form Scalora et al. (2005), cannot model the full non-envelope field, nor revert to an exact and explicitly bi-directional form, as in my eqn. (9) or (10).

Treating a second-order nonlinearity is more complicated than the third-order case, since it typically couples the two possible polarization states of the field together. Such interactions occur in materials used for optical parametric amplification, and have long been used for a wide variety of applications (see, e.g., Boyd (2003); Various (1993); Danielius et al. (1993)). To model the cross-coupling between the orthogonally polarised fields, it is necessary to solve for both field polarizations; and to allow for the birefringence we need two pairs of (non-reference) linear responses, i.e. $\kappa_{\epsilon x},\kappa_{\epsilon y}$ and $\kappa_{\mu x},\kappa_{\mu y}$.

As an example, I choose a magnetic nonlinearity that couples $H_{y}$ and $H_{x}$ in the same way as the electric nonlinearity couples $E_{x}$ and $E_{y}$, although other configurations are possible. This means that the $\beta_{r}\textbf{u}\times\textbf{P}$ term in eqn. (1), which represents the non-reference part of the electric response, needs to include those for the standard second-order nonlinear terms (here $P_{x}\propto E_{x}E_{y}$ and $P_{y}\propto E_{x}^{2}$). Similarly, the $\alpha_{r}\mu_{0}\textbf{M}$ term has ones for the complementary second-order nonlinear magnetic response. Note that second-order nonlinear magnetic effects have been measured in split ring resonators by Klein et al. Klein-EWL-2006s; Klein-WFL-2007oe.

Since it is convenient, I split the vector form of the $\textbf{G}^{\pm}$ wave equation up into its transverse $x$ and $y$ components. By noting that the definition of $G^{\pm}_{y}$ means that $H_{x}^{+}=-F_{y}^{+}\alpha_{r}/\beta_{r}$, the 1D wave equations can be written as

These wave equations for the field are strikingly similar to the usual SVEA equations used to propagate narrowband pulses; the main differences are the addition of terms for magnetic dispersion ($\kappa_{\mu x},\kappa_{\mu y}$) and nonlinearity ($\chi_{\mu}$), and the lack of a co-moving frame.

We can transform eqns. (16) and (17) into a form close to the usual equations for a parametric amplifier by representing the $x$ and $y$ polarized fields in terms of three envelope and carrier pairs:

where $\omega_{3}=\omega_{1}+\omega_{2}$. After separating into pairs of complex-conjugate equations (one each for all $A_{i}$ and $A_{i}^{*}$), and ignoring the off-resonant polarization terms, we transform into a frame co-moving with the group velocity, although here we select the group velocity of a preferred frequency component, with $\Delta_{g}=\omega(v_{g}^{-1}-v_{p}^{-1})$. The wave equations for the ${A}_{i}(\omega)$ are then

Here $K_{1,2}(\omega)=k_{1,2}[\kappa_{\epsilon x}(\omega)+\kappa_{\mu y}(\omega)]/2+\Delta_{g}$ and $K_{3}(\omega)=k_{3}[\kappa_{\epsilon y}(\omega)+\kappa_{\mu x}(\omega)]/2+\Delta_{g}$; we choose $k_{r}$ for each equation differently (i.e., with $k_{r}\in\{k_{1},k_{2},k_{3}\}$); also the phase mismatch term is $\Delta k=k_{3}-k_{2}-k_{1}$. The combined nonlinear coefficient is $\chi^{\pm}=\chi_{\epsilon}\pm(\mu_{0}/\epsilon_{0})(\epsilon_{r}/\mu_{r})^{3/2}\chi_{\mu}$.

Examining the respective roles of the reference permittivity $\epsilon_{r}$ and permeability $\mu_{r}$ in eqns. (13) and (16), (17), we see that as far as dispersion and other linear effects are concerned, the two components simply add. In contrast, their effect on nonlinear terms is more dramatic: with the ratio $Y^{2}=\epsilon_{r}/\mu_{r}$ scaling the nonlinear corrections to the magnetization into the electric field units of $\textbf{F}^{\pm}$. This is because $Y^{2}$ determines how much of a given directional field $\textbf{F}^{\pm}$ is electric field and how much magnetic field; large values of $Y^{2}$ correspond to cases where the magnetic field is most prominent. Indeed, $Y$ is just the reciprocal of the electromagnetic impedance of our chosen reference medium, and only if $Y$ is real-valued do propagating fields exist, since otherwise the fields become evanescent.

Figures 1 and 2 show how $Y$ varies with frequency for two different metamaterial types, with the dispersions encoded on $\epsilon_{r}$ and $\mu_{r}$ and scaled by $\omega^{2}$ to moderate the low-frequency singularity of the Drude response. The extreme limits of large $Y$ occur when $|\mu_{r}|\ll|\epsilon_{r}|$, that is, usually just at an edge of a non-propagating band, where $\mu_{r}$ is about to change sign. In such a region, it would be better to revert to the $\textbf{G}^{\pm}$ fields, or to rescale the propagation equations into units of magnetic field (e.g., with some $\textbf{K}^{\pm}=\textbf{G}^{\pm}/2\beta_{r}$).

In previous work Scalora et al. (2005); Wen et al. (2007); D’Aguanno et al. (2008), a Drude type response for both $\epsilon$ and $\mu$ was assumed, where $\epsilon_{r},\mu_{r}\propto 1-\omega_{\epsilon,\mu}^{2}/(\omega^{2}-\imath\gamma_{\epsilon,\mu}\omega)$; and this situation is shown on Fig. 1. However, although the dielectric response in metamaterials (e.g., a wire grid array Pendry-HSY-1996prl; Pendry-HRS-1998jpcm) often has this behaviour, the magnetic response of split ring resonators (SRRs) differs. SRR magnetization is best described by a pseudo-Lorentz model³³3Also known as the “F-model” Pendry-HRS-1999ieeemtt with $\mu_{r}\propto 1+F_{\mu}\omega^{2}/(\omega^{2}-\omega_{\mu}^{2}-\imath\gamma_{\mu}\omega)$, although sometimes a true Lorentz response is used instead, $\mu_{r}\propto 1+F_{\mu}\omega_{\mu}^{2}/(\omega^{2}-\omega_{\mu}^{2}-\imath\gamma_{\mu}\omega)$. Note the difference between the numerators in these latter two expressions: a frequency-dependent $\omega^{2}$ versus a constant material parameter $\omega_{\mu}^{2}$. The pseudo-Lorentz model has an incorrect high-frequency behaviour, and so it is incompatible with the Kramers-Kronig relations that enforce causality. However, at low and medium frequency it is a better match to the physical response of SRRs, and so I use it for Figs. 2 and 3.

There are frequency ranges over which the linear material responses vary dramatically, and in particular on Fig. 3 (which uses the same model as Fig. 2) this is evident for both the refractive index $n$ and group velocity $v_{g}$. If we aim to operate in such regions, this leads to two potential complications. First, if we have chosen reference parameters that do not match the linear material responses exactly, then the correction terms will become large, meaning that our wavelength-scale “slow evolution” approximation may come under threat. Second, even if our reference parameters do match the linear material responses exactly, our wavelength-scale will have become frequency dependent, and so again our “slow evolution” approximation may be threatened. In either case the solution is simple – we just need to revert to the bi-directional wave equations [i.e., eqns. (9) or (10)]. However, this does not necessarily mean that any backward evolving fields are generated (as explained in Kinsler et al. (2005), and following a different approach in Kinsler (2008)), so that in principle one could optimize the propagation by reinstating it only over those frequency ranges where it becomes necessary.

I have derived a uni-directional optical pulse propagation equation for media with both electric and magnetic responses, based on the directional fields approach Kinsler et al. (2005). This involved a re-expression of Maxwell’s equations, and required only a single approximation to reduce a one dimensional bi-directional model, to a uni-directional first-order wave equation. The simplicity of this approach makes it very convenient in waveguides, optical fibres, or other collinear situations. The important approximation is that the pulse evolves only slowly on the scale of a wavelength; and indeed this is a valid assumption in a wide variety of cases – note in particular that nonlinear effects have to be unrealizably strong to violate it Kinsler (2007b). The result has no intrinsic bandwidth restrictions, makes no demands on the pulse profile, and does not require a co-moving frame – unlike other common types of derivation Agrawal (2007); Brabec and Krausz (1997); Geissler et al. (1999); Kinsler and New (2003).

The resulting equations have the advantage that they are straightforward to write down, despite containing the complications of both electric and magnetic responses, and that a carrier-envelope representation or co-moving frames are easy to apply if desired, requiring no further approximation. In this, they match the clarity and flexibility of factorized second-order wave equations Ferrando et al. (2005); Genty et al. (2007); Kinsler (2008), but they can more easily incorporate the effects of magnetic material responses – albiet at the cost of being restricted to one dimensional propagation.