Electromagnetic wave propagation in spatially homogeneous yet smoothly time-varying dielectric media
We explore the propagation and transformation of electromagnetic waves through spatially homogeneous yet smoothly time-dependent media within the framework of classical electrodynamics. By modelling the smooth transition, occurring during a finite period , as a phenomenologically realistic and sigmoidal change of the dielectric permittivity, an analytically exact solution to Maxwell’s equations is derived for the electric displacement in terms of hypergeometric functions. Using this solution, we show the possibility of amplification and attenuation of waves and associate this with the decrease and increase of the time-dependent permittivity. We demonstrate, moreover, that such an energy exchange between waves and non-stationary media leads to the transformation (or conversion) of frequencies. Our results may pave the way towards controllable light-matter interaction in time-varying structures.
keywords:wave propagation, time-dependent media, amplification and attenuation of waves, energy exchange, frequency conversion, exactly solvable systems
Email address: email@example.com (Armen G. Hayrapetyan)
Controlling the optical properties of photonic structures has been a topic of significant interest for last few decades both in fundamental (1); (2) and applied research (3); (4); (5). Recent advances in technology and instrumentation have made it possible to realize such control systems via ultrafast switching of the time-dependent dielectric permittivity (or the refractive index) (6); (7); (8); (9); (10); (11); (12). The two most relevant mechanisms of modifying the permittivity in the time domain are the excitation of the so-called free charge-carriers and the electronic Kerr effect. While the free-carriers are typically induced by strong pump pulses creating electron-hole pairs in semiconductors (13); (14); (15); (16), the Kerr effect arises from the non-linear (instantaneous) response of bound electrons to the applied field (1); (17); (18); (19). These mechanisms enable one to employ the optical switching of the refractive index for various purposes, such as quantum interference (20), information processing (21); (22); (23), material science (24); (25); (26), control of spontaneous emission (27), and several others (28); (29); (30); (31); (32); (33).
The investigation of dynamics of both optical and matter waves in instantaneously time-varying structures have also been in the focus of intense research throughout the last decades. Until now, several effects have already been proposed, for instance, to account for the stimulated electron-light interaction (34) and photon generation (35), to investigate irregular alternation of phases of electromagnetic waves (36) and the energy exchange between waves and non-stationary media (37). In other scenarios, modulation and conversion of frequencies as well as the transformation of waves have been explored in non-stationary waveguides, resonators and plasmas [cf. books (38); (39); (40) and references therein]. The idea of looking at such phenomena stems from the pioneering papers by Morgenthaler (41) and Ginzburg and Tsytovich (42); (43), who considered the velocity modulation of waves and the transition radiation of charged particles in time-dependent environments. Recently, the dynamics of sound waves have also been examined in non-stationary fluids with either a sudden (37) or smooth (44); (45) change of the medium, leading to the frequency conversion of waves that have already become accessible in an experiment (46).
In most of the studies related to the transformation of electromagnetic waves, the parameters describing the medium are assumed to vary sharply (or step-like) with time, i.e., when the time duration during which the medium experiences a change is much shorter than the propagation period of the wave, . Although such an assumption simplifies the theory and describes the relevant effects, it rarely describes the reality of switching processes and should be replaced with a smooth transition. This is especially important for experiments in which the switching duration is comparable to the period of light, so that .
In this paper, we re-visit the problem of transformation of electromagnetic waves in time-dependent media and provide a step forward towards understanding of the impact of non-stationary environments, continuously changing in time, on the dynamical properties of waves. To this end, we examine the propagation and transformations of waves in time-varying and, at the same time, spatially homogeneous (i.e., uniform) dielectric media. A particular emphasis is placed on studying how a smoothly time-dependent dielectric permittivity affects the energy (flux) and the frequency of transformed waves. In view of this, we derive a generalized wave equation for the electric displacement and obtain an analytically exact solution expressed via hypergeometric functions for a judiciously chosen sigmoidal change of the permittivity, explicitly accounting for the finite transition period . Using this solution, we show that an energy exchange occurs between electromagnetic waves and non-stationary media, which is further demonstrated to lead to either amplification or attenuation of waves depending on whether the refractive index decreases or increases as a function of time. For a monochromatic incident wave, moreover, such an energy non-conservation gives rise to the transformation (or conversion) of frequency due to the (more or less) abrupt change of the refractive index, quite similar to that of the sound wave frequency (44); (45).
The paper is organized as follows. In the next section, we derive a generalized wave equation for the electric displacement when the time-varying dielectric permittivity remains uniform in space. While continuity conditions for the electric displacement and magnetic induction are used to account for the sudden transition (Subsection 2.1), rigorous exact solutions to the time-dependent wave equation are obtained in the case of the smooth transition (Subsection 2.2). These solutions are then discussed in Section 3 and a few effects are predicted, such as the energy exchange, amplification and attenuation as well as frequency conversion of waves. In Section 4, we conclude with future research directions.
2 Theory of time-dependent propagation and transformation of electromagnetic waves
In order to describe the temporal dynamics of light in spatially homogeneous and isotropic media, we start from the source-free Maxwell equations in Gaussian units
where is the vector differential operator, “cross” and “dot” mean vector and inner products, respectively. Maxwell’s equations self-consistently characterize the electromagnetic field state only in a vacuum. In a general medium, however, constitutive relations must be added to Eqs. (1-4) to provide a complete description of waves (47). For an isotropic medium, the electric field and displacement are related via the standard constitutive relation , with being the scalar dielectric permittivity. For non-magnetic media, moreover, the magnetic permeability as in most of the experiments on optical switching, so that the magnetic field equals to the induction . In this particular case, therefore, the dielectric permittivity and the refractive index are related by a simple formula .
For time-varying and space-independent media, i.e., when the dielectric permittivity is only a function of time, , an exact wave equation can be derived for the electric displacement from Eqs. (1)-(3)
Given that the medium is spatially uniform, we seek for the solution of this equation in the form
where is a unit vector along the displacement , while and are the wave and position vectors, respectively. The ansatz (6), which is necessary to assure time evolution of the system, characterizes the transformation of waves and accounts for a modification of frequencies of transformed waves. This is reminiscent of the counterpart scenario, when the space dependence of the medium implies for a monochromatic wave traveling with frequency (47).
where the permittivity appears in the denominator of the prefactor of , in contrast to the position dependent wave equation with the permittivity in the nominator [cf., e.g., Eqs. (88.3-4) of Ref. (47)]. Built upon the explicit form of , Eq. (7) allows solutions which characterize dynamics of waves in non-stationary media independent of the nature of switching or tuning of the permittivity. Moreover, an analogous generalized equation for sound waves can also be derived from Euler’s and continuity equations in non-stationary fluids [cf. Eq. (34) of Ref. (45)].
In the following, we shall investigate solutions of Eq. (7) for two distinct cases: when the time-dependent dielectric permittivity experiences either sudden or smooth change. On each of these scenarios, we shall (i) apply continuity conditions for electric displacement and magnetic induction or (ii) rigorously solve the differential equation with a phenomenologically realistic sigmoidal change of the permittivity, judiciously chosen to qualitatively coincide with experimentally determined behaviour. Our approach is based on the direct and exact integration of the time-dependent wave equation, in contrast to other theoretical methods, such as Green’s functions representation (48), the Wentzel-Kramers-Brillouin-Jeffreys approximation (49); (50) or the Volterra integral equation approach (51); (52) (see also Refs. (40); (53)).
2.1 Sudden change of the dielectric permittivity
Before we examine the dynamics of waves in spatially uniform media with a smoothly changing time-dependent permittivity, we re-visit the case of the sudden change, as reported in Ref. (37). In this scenario, it is assumed that the permittivity undergoes a discrete change at some time from (for ) to (for ) [cf. Fig 1]. For these constant values of the permittivity, the solution of the wave equation (7) can be expressed in terms of plane monochromatic waves. For a wave with the initial frequency and the constant amplitude
the sudden change of the permittivity gives rise to the superposition of two – reflected (“”) and transmitted (“”) – waves
propagating with opposite frequencies and and distinct amplitudes and . In accordance with our adopted assumption about the spatial homogeneity, the frequencies of the initial and transmitted waves are related via the conversion relation
where the constant numbers and represent the refractive indices of the medium before () and after () the change [cf. Ref. (45) for a detailed discussion].
The reflected wave in Eq. (9) can be interpreted as a propagation backward-in-time with positive frequency as , or else, propagation forward-in-time with negative frequency as . We can nevertheless demonstrate that the reflected wave describes an actual reflection in space by calculating the Poynting vector of the three waves (8)-(9)
As it can be readily seen, is anti-parallel to the corresponding Poynting vectors in the initial and transmitted waves. Moreover, the phase velocities
also indicate that the reflected wave propagates opposite to the propagation direction of the initial and transmitted waves. This theoretical construct for waves is still possible to interpret in terms of dynamical observables by taking into account the fundamental relation between the homogeneity of time and the conservation of energy, known as Noether’s theorem (54), as we show below.
Being a solution to the wave equation (5), Eqs. (8)-(9) allow us to reveal the behaviour of waves in non-stationary media. To discuss this, we shall construct the reflectivity and transmittivity which are correspondingly defined as ratios of (space) averaged energy fluxes of reflected and transmitted waves to the averaged flux of the initial wave (37)
A similar definition, leading to the famous Fresnel’s formulae, is provided in Ref. (47) to describe the transformation of waves in spatially inhomogeneous and time-independent media. Moreover, Refs. (55) and (56) are dedicated to some modified Fresnel’s formulae constituting the transformation of resonant light and polarized matter waves, respectively. Finally, another type of Fresnel’s formulae accounting for a time-dependent spatial reflection of waves from non-stationary interfaces are derived in Refs. (49); (57).
Using relations (13), we can establish the energy balance between the transformed waves (9) and the medium if we, without loss of generality, assume that the dielectric permittivity (or the refractive index) suffer an abrupt change at the time . For such an assumption, the continuity conditions for the electric displacement and magnetic induction
lead to the Fresnel-type formulae in the time domain (37)
that shows non-conservation of energy for the wave. As the energy is conserved for the whole “wave + medium” system, the expression (17) can be nevertheless interpreted as an energy (flux) exchange between the wave and the time-dependent medium. Depending on whether the wave propagates to optically denser () or rarer () medium, it is either attenuated () or amplified (), as also illustrated on Fig 2. This behaviour is quite in contrast to the propagation of sound waves in non-stationary fluids, when the wave is only amplified despite the increase or decrease of the appropriate quantities, such as distributions of the mass density and the sound velocity (37); (44); (45).
It is important to realize that in the absence of the time-dependent change of the permittivity, i.e., when the temporal inhomogeneity is “switched off” ( or ), the reflectivity vanishes, while , the energy of the wave is conserved (, the black line on Fig 2) and no frequency conversion occurs (), as one would expect.
|Spatial homogeneity and||Spatial inhomogeneity and|
|temporal inhomogeneity (37)||temporal homogeneity (47)|
For a better understanding, a comparative analysis of the transformation of waves in the time domain and of its spatial counterpart is summarized in Table 1. For the sake of simplicity, we compare the Fresnel-type formulae (15) and (16) with the conventional Fresnel’s formulae for the case of the normal incidence of a wave on the interface between two different spatially homogeneous media with refractive indices and . While the wave does not conserve energy throughout the propagation in the non-stationary medium, the energy of the wave is conserved in a stationary medium, even if it is spatially inhomogeneous. These energy-related effects are a direct manifestation of either violation or observance of Noether’s conservation theorems (54).
After this overview, in the next subsection, we extend our studies of transformation of waves in abruptly-varying media to the case of the adiabatic change of the dielectric permittivity. In view of this, we shall assume that the permittivity varies smoothly during some finite transition period , which also plays the role of the switching duration. Such a smoothness is then modelled by a sigmoidal function, and an analytically exact treatment to the transformation of waves is developed based on the time-dependent wave equation (7).
2.2 Smooth change of the dielectric permittivity
In a more realistic case, the dielectric permittivity, instead of an abrupt variation, often experiences a change during a finite transition period . In order to account for such a finiteness, it is no longer sufficient to consider continuity conditions (14): we need to solve the time-dependent wave equation (7) where the sudden change of the permittivity is replaced by a judiciously chosen and smoothly time-varying function, as depicted in Fig. 3. We may therefore model the switching of the permittivity by a phenomenological sigmoidal function
to assure that the asymptotic values (for ) and (for ) are recovered when .
Choosing this shape of the permittivity, we derive an exact second order linear differential equation from Eq. (7)
where a new variable is introduced, , which converges with . The constant parameters
expressed also by means of the initial and transmitted (transformed) frequencies, carry information about the wave and the impact of the medium upon it.
Equation (19) has a singularity at , which can be removed by making the replacement
where is a constant and, in general, complex number, while represents an analytical function of and describes the time-dependent dynamics of the wave. The ansatz (21) we advocate here amounts to reducing Eq. (19) to the conventional form
where the constant parameters
are introduced for the sake of brevity. Equation (22) has an exact solution expressed in terms of the hypergeometric function, with being a constant, provided that to ensure the convergence of the solution when (58). This means that the solution is valid for negative values of since . Furthermore, we exploit Eqs. (6) and (21) in order to construct the explicit form of the electric displacement in the interval
The asymptotic behaviour of this function defines the constant number and the sign of . Given that , the anticipated initial wave (8) can be gained if
such that the electric displacement itself takes the final form
This holds for all times for the given ‘rate of change’ of the dielectric permittivity (18).
The solution (25) does also contain the transformed waves (9) that in turn incorporates the two – reflected and transmitted – waves. For , however, the hypergeometric function diverges because of the argument . We therefore need to employ its symmetry properties to circumvent this divergence. By building a new convergent variable at , in fact, one can re-write the solution (25) as (58)
that represents a superposition of two waves with time-varying complex ‘amplitudes’
expressed in terms of the hypergeometric functions.
In Eq. (26), the presence of the two exponentials indicates, generally, the frequency conversion
owing to the terms
and [cf. Eq. (23)], which
mean that the non-stationary medium serves as a frequency transformer for waves
which shows explicitly the occurrence of two counter-propagating waves with modified amplitudes determined by the -functions. As we expect the solution (27) to take the form (9) for , further comparison of Eqs. (9) and (27) gives
These expressions define the -dependent amplitudes of the reflected and transmitted waves normalized to
the amplitude of the initial wave (see also Eqs. (20) and (23)).
In the limiting case , moreover, our general treatment
confirms the results of Refs. (34); (37); (41); (61)
as for , but disagrees with
Ref. (62) where the authors derive incorrect coefficients
despite using the same continuity conditions as Eq. (14)
Solutions (25)-(27) show how the smooth change of the dielectric permittivity (18) manifests itself in the electric displacement, and therefore, affects the dynamical properties of electromagnetic waves in a non-stationary medium. Since these solutions – established already for the entire time axis – depend explicitly on the switching duration (but not the mechanism!) of the refractive index, the transformation of waves and their energy exchange with the non-stationary medium will also depend on the switching duration and the conversed frequency . To demonstrate this, we employ the recurrence relation and Euler’s reflection formula for the -function
This is one of the main results of this work. By generalizing the Fresnel-type formulae (15) and (16) from sudden () to smooth () transition of the permittivity, the expressions (29)-(30) describe the energy transport of an electromagnetic wave when propagating through a spatially homogeneous yet smoothly time-varying medium for the specific time dependence (18). We will utilize formulae (29)-(30) in the next section in order to discuss the energy balance between waves and non-stationary media, which depends on the switching duration and reveals the amplification and attenuation of waves.
3 Results and discussion
|Electromagnetic waves||Sound waves (45)|
|General, -dependent||✳ Eqs. (29)-(30)||✳ Eqs. (53)-(54) of (45)|
|Energy balance||✳||✳ ,|
|[Eq. (31)]||[Eq. (56) of (45)]|
|Amplification of waves||✳||✳ ,|
|Attenuation of waves||✳||✷|
|[Eq. (32)]||[Eq. (57) of (45)]|
For both the sudden and smooth changes of the permittivity, the wave exchanges energy with the medium [cf. Eq. (17)] and, as a result, is either amplified or attenuated. For a smooth change, that occurs during the finite period , the sum of the reflectivity (29) and the transmittivity (30) takes the form
As the expression in the parentheses is always larger than unity, we immediately see that the wave is either amplified or attenuated depending on whether the conversed frequency is increased or decreased (as compared to the initial one), or else, the refractive index is decreased or increased [cf. Eq. (10)]. A similar situation also holds for the sudden change of the medium, as described by Eq. (17). Being one of the main results of this paper, the expression (31) generalizes Eq. (17) to explicitly include the switching duration and shows the universal nature of the amplification and attenuation of electromagnetic waves also in the case of smooth change of the medium. This is in contrast with sound waves which due to their inherent structure are only amplified, irrespective, whether the mass density and sound velocity distributions increase or decrease as functions of time. In Table 2, a comparison is made in terms of generalized Fresnel-type formulae for the electromagnetic and sound waves.
which is independent of and maintains the same form as that for the sudden change (37). While the wave travels from optically rarer to denser medium (), the transmitted wave carries an energy flux smaller than the sum of the energy fluxes in the reflected and initial waves, and vice versa for : the energy flux of the transmitted wave surpasses that of the two other waves. This is again in contrast to sound waves where the transmitted wave carries an energy flux exactly equal to the sum of the fluxes of the reflected and initial waves (37); (45) [cf. Table 2].
Here, represents the dimensionless transition period, while the parameter shows the ratio between the refractive indices before and after the change, so that equals to unity when the tuning of the refractive index is “switched off”. Figure 4 demonstrates the energy balance between the wave and the medium as a function of for different values of as well as for both the sudden (dashed lines) and smooth (solid curves) changes of the dielectric permittivity. The fact that the sum of the reflectivity and transmittivity is either greater (Fig. 4(a)) or less (Fig. 4(b)) than the unity is a signature of the wave amplification and attenuation, respectively. This change in the sum of the energy fluxes of transformed waves is quantified by means of the ratio between the refractive indices. As seen, the larger the ratio or , the stronger is the amplification or attenuation of the wave. The inclination of curves for the smooth change is more pronounced as increases (decreases) from unity due to the variation for small . Particularly, the increase of % in corresponding to the decrease of the refractive index () gives rise to the change of % in the amplification () [cf. black curves in Fig. 4a]. Whereas the decrease of % in results in the decrease of % in the sum of the reflectivity and transmittivity and leads to an attenuation of waves () [cf. grey curves in Fig. 4b].
Such an asymmetry between the increase and decrease of energy fluxes is due to the fact that Eqs. (33) and (34) do not remain symmetric under the interchange (or ). As expected, moreover, when the transition period is much less than the period of the initial wave () both curves for sudden and smooth changes at given merge to each other as for . Figure 5, in turn, illustrates the dependence of the energy balance on the ratio between the refractive indices for selected values of the dimensionless time . As the dimensionless time approaches its asymptotic value (), representing the sudden change of the permittivity, the difference between curves decreases. The dependence on is markedly pronounced in the domain of amplification, which is again due to the asymmetry of the energy balance (33).
Thus, apart from being a frequency transformer, the non-stationary medium acts as a ‘source or sink of energy’ for electromagnetic waves. The prediction of an increase and decrease of normalized energy flux for the wave, which can correspondingly be interpreted as an amplification and attenuation due to an energy exchange with a medium, can be tested experimentally, if it is possible to disentangle the energy transport through the dielectric from its internal energy. A microscopic theory should be developed in order to explicitly reveal the source of energy, as in our phenomenological approach the time-dependent permittivity characterizes only a ‘net’ structure, which can be generated, for instance, by laser pulses in photonic systems (63); (64).
We have derived an analytically exact theory to describe the propagation and transformation of electromagnetic waves in spatially homogeneous yet smoothly time-varying dielectric structures. The emphasis has been put on exploring how the finite transition period for the dielectric permittivity influences the dynamical properties of waves, such as the energy (flux) exchange between waves and non-stationary media and the conversion of frequencies of transformed waves. The exchange is shown to lead to the -dependent amplification or attenuation of waves correspondingly linked to the wave propagation from optically denser to rarer medium or vice versa. We have provided a detailed comparison between predictions of our generalized theory and those of the sudden change approximation. The peculiar differences in transformations of electromagnetic and sound waves in smoothly-varying media are also pointed out. Being manifestations of the temporal inhomogeneity, both the energy exchange and the transformation of frequencies can be tested experimentally should the switched dielectric permittivity follow the sigmoidal shape, as shown in Fig. 3. However, a rigorous study of ubiquitous processes of relaxation following the switching of the refractive index would constitute modification of the sigmoidal change, and therefore correction to the reflectivity (29) and transmittivity (30).
Although our results are valid for a wide range of frequencies (from radio frequency up to ultraviolet) and for various types of dielectric media, a generalized approach is needed to simultaneously account for smoothly time-dependent dielectric permittivity and magnetic permeability, especially relevant for studying transformation of waves in non-stationary plasmas (65); (66); (67); (68) and magnetoelectric systems (69). Such a general study would enable one to resolve the debate about various ways of deriving the reflection and transmission coefficients in suddenly changing media (70); (71). In recent years, moreover, controlling waves in both space and time domains has raised considerable interest (72); (73). Simultaneous investigation of space- and time-dependent transformation of waves will lead to an intriguing ‘interplay’ of energy and momentum exchange between waves and spatially inhomogeneous and non-stationary media.
AGH acknowledges useful discussions with Jörg Evers at early stage of this work and thanks Willem Vos and Georgios Ctistis for their insightful comments on the experimental realization of time-varying refractive index.
- Similar effects related to frequency conversion, modulation and shift attracted continuously growing interest since the past decades (40). See also Refs. (59); (60) for the frequency modulation in parity-time symmetric non-stationary structures and for the frequency conversion in the quantum regime, respectively.
- This mistake is later corrected in the subsequent paper (61).
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