Theory of electromagnetic wave frequency upconversion in dynamic media
Frequency upconversion of an electromagnetic wave can occur in ionized plasma with decreasing electric permittivity and in split-ring resonator-structure metamaterials with decreasing magnetic permeability. We develop a general theory to describe the evolution of the wave frequency, amplitude, and energy density in homogeneous media with a temporally decreasing refractive index. We find that upconversion of the wave frequency is necessarily accompanied by partitioning of the wave energy into low-frequency modes, which sets an upper limit on the energy conversion efficiency. The efficiency limits are obtained for both varying permittivity and varying permeability.
The ability to coherently convert the frequency of an electromagnetic wave between arbitrary wavelengths has a broad range of applications, including expanding the wavelengths accessible by current laser technologies. Frequency conversion is typically achieved using nonlinear optical methods, which often require high light intensities. As an alternative linear method, frequency conversion by temporally changing the medium refractive index Morgenthaler (1958); Wilks et al. (1988); Mendonça (2001); Nerukh et al. (2012); Kalluri (2016) has attracted particular attention for its ability to create “tabletop” sources of coherent x rays out of near-infrared lasers at high efficiency and low cost Edwards et al. (2018); Froula et al. (2018). A rapid change in refractive index has also been proposed Yablonovitch (1989); Lähteenmäki et al. (2013) for observing the Unruh effect, which is crucial for testing fundamental quantum field theory. For these purposes, frequency upconversion has been experimentally demonstrated in the microwave Yablonovitch (1973); Joshi et al. (1990); Kuo (1990); Savage et al. (1993); Yugami et al. (2002), terahertz Nishida et al. (2012), and optical Geltner et al. (2002); Avitzour et al. (2005); Shcherbakov et al. (2017) regimes using dynamic plasmas and metamaterials Shcherbakov et al. (2017); Lähteenmäki et al. (2013).
Despite the wide spectra and disparate materials that can realize frequency upconversion using dynamic media, previous theories have generally focused on plasma with an increasing density Wilks et al. (1988); Stepanov (1993); Bakunov et al. (2000); Kalluri (2016). These theories have neglected the electromagnetic wave interaction with atomic resonances before the medium is ionized, and thus overestimated energy conversion efficiency. Previous theories have also ignored metamaterials as dynamic media capable of inducing large frequency shifts using engineered refractive indices.
In this paper, we formulate a general theory describing frequency upconversion of electromagnetic waves in dynamic homogeneous media. Compared to previous theories, our study obtains general constitutive relations when polarization and magnetization suddenly change in the frequency conversion process. Our theory describes how the electromagnetic field changes its frequency, amplitude, and energy density when the medium changes its refractive index. We show that the process of frequency upconversion necessarily leads to the division of wave energy into multiple modes with components at both higher and lower frequencies than the initial frequency.
This article is organized as follows: In Sec. II, we first introduce the theory of frequency upconversion of an electromagnetic wave starting from Maxwell’s equations. Then, we find the continuous quantities during the medium change by considering both polarization and magnetization and obtain the field amplitudes after the wave frequency is shifted. The obtained results are applied to media with decreasing permittivity in Sec. III to find the efficiency of energy conversion to the high-frequency mode. We specifically analyze ionized plasma media and supplement the known theory of energy conversion efficiency when the medium changes from gas or solid state to plasma. In Sec. IV, we study frequency conversion using magnetic metamaterials with split-ring resonator (SRR) structures Smith et al. (2000) which have variable magnetic permeability. We find that both variable permittivity and variable permeability yield energy conversion efficiencies reaching . In Sec. V, we summarize the key points of this paper.
Consider an electromagnetic plane wave with electric field and magnetic field propagating in a homogeneous medium. The field evolution is described by Maxwell’s macroscopic equations,
where and are the permittivity and permeability, respectively, of vacuum and . The total charge density and the total current density include components
where is the polarization charge density, is the polarization current, and is the magnetization current. Here, is the polarization density and is the magnetization density. Since the electric displacement is and the magnetic field is , the polarization and magnetization can be equivalently described by the permittivity and the permeability , respectively. The refractive index is defined as .
In our discussion, we focus on the polarization and magnetization effects of the media in response to an input electromagnetic wave. We neglect and since they do not contribute to polarization or magnetization. We count the electrons in metals and plasmas as polarization charges because their collective oscillation in response to electromagnetic fields induces a polarization field according to the Drude model Levi (2016). Then, we can write the wave equations as
where and denote time derivatives. The spatial operation and temporal operations in the wave equations (4) and (5) are separated. For homogeneous media, the right hand sides of Eqs. (4) and (5) are spatially invariant, so the spatial operators can be reduced . When the medium changes in time, Eqs. (4) and (5) describe damped harmonic oscillations of the fields and with a central frequency . The first-order time derivative terms of and describe damping, which causes a finite frequency spectrum when or is nonzero. The spectral width decreases as and approach zero. In the steady state , the wave equations (4) and (5) reveal a linear correspondence relation between the wave frequency and the wave number . Thus, an electromagnetic wave with a certain will change its frequency when the homogeneous refractive index changes in time. In the time scale that is defined, the wave frequency in the steady state is solely determined by but does not depend on the history of .
Assume an electromagnetic wave with initial frequency propagates in a medium whose refractive index is dynamic. The final wave frequency is determined by the equation , which may have multiple solutions in dispersive media where is frequency dependent. The initial wave energy can be projected into each of these frequency modes. To find out how the initial electric field amplitude and magnetic field amplitude are converted, we analyze the continuous quantities of the wave when the medium changes its refractive index, or specifically the polarization and magnetization.
We first write the general steady-state solution for the fields as a superposition of multiple frequencies
where runs through all the solutions to ; and represent the amplitudes of the electric and magnetic fields at frequency in the forward- and backward-propagating directions, respectively. An electrostatic or magnetostatic mode can be described with . It should be noted that the fields and oscillate in phase, and hence the amplitudes and are expressed as real values.
In the case of frequency upconversion, reducing the refractive index , which is proportional to , is accomplished by increasing , , and , which oscillate at the same frequency as the and fields. Examples of such processes include forming new molecules with larger polarizability or magnetizability, creating or increasing plasma density, and activating metamaterials that can interact with an electromagnetic wave. Wave frequency conversion arises from the coupling between the input electromagnetic wave and the oscillation of the new polarization and magnetization. For simplicity, we assume that the change in medium is triggered by an external field at which uniformly creates new charges and new currents in a time shorter than an optical cycle . The simultaneous creation ensures that the charges and currents begin to oscillate coherently under the drive of the input electromagnetic field, inducing polarization and magnetization and thereby reducing and . The initial values of the and oscillations are both zero at . We must emphasize that the thermal motion-induced and are not correlated with the input electromagnetic wave, hence their average contribution to polarization or magnetization is zero. With and both zero at , the values of and are continuous. Applying the continuity of and to Gauss’s law in Eqs. (1), we first find that the gradients of the and fields are continuous at . For a plane wave in a homogeneous medium, the continuity of the gradients can be generalized to the and fields themselves. Continuity of , and subsequently leads to the continuity of and at using Faraday’s law and Ampere’s law in Eqs. (1). Therefore, we obtain a set of continuity relations of the instantaneous fields,
We note that Eqs. (10) and (11) can only be satisfied simultaneously when the output wave has multiple frequency components. Therefore, they set an upper limit on the efficiency of energy transfer into the component with the highest frequency. Calculating the exact energy conversion efficiency requires additional initial conditions since Eqs. (9) and (12) are linearly dependent on each other.
The second set of continuity relations can be found by analyzing the polarization and magnetization fields. At , since and are continuous, the instantaneous fields , , and are continuous by definition. As a result, the fields and are also continuous. However, , whose curl is proportional to , is obviously not continuous. To find the continuity relation of , we treat as a number of charged particles in orbital motion. The magnetic momentum per atom or molecule is proportional to the product of particle charge and its angular momentum. The change in magnetic permeability can be a result of new charged particles joining the orbital motion or charged particles increasing their orbital momenta. In either scenario, the angular positions of the charged particles are continuous at , i.e. is continuous. Therefore, we obtain the second set of continuous quantities at the instant of medium change,
By using and , the continuity relation (13) can be written as
where and are the permittivity and permeability, respectively, of the medium at frequency .
Now, we have obtained eight continuity equations for the amplitudes of the electric and magnetic fields: Eqs. (9)-(12) and (14)-(17). These equations are, however, not all independent. We first note that Eqs. (9) and (12) are equivalent using relation considering is negative for the backward-propagating mode. Equation (16) can be constructed from Eqs. (11) and (15) by identifying . Thus, Eqs. (12) and (16) can be removed from the initial conditions. In Eq. (17), the terms and can be isolated and canceled by applying the identity to Eq. (14), which simplifies Eq. (17) to
Therefore, the electric and magnetic fields are determined by six independent relations: Eqs. (9)-(11), (14)-(15) and (18). Equations (9)-(10) and (14)-(15) explicitly describe the continuity of the fields , , , and . Equations (11) and (18), which describe the continuity of and , are the consequences of the continuity of the incremental values , , and .
In media with dynamic and , the input electromagnetic field is coupled to the polarization and magnetization fields. They form a coupled three-mode system determining three eigenfrequencies with each containing two propagation directions. To find the wave amplitudes, we rewrite the continuity relations as six equations of using the relations and . Then the general solution for the wave amplitudes can be expressed in matrix form as
and . If either or remains constant when the medium changes, the coupled two-mode system determines four waves at two different eigenfrequencies whose amplitudes are given as
The general solutions expressed in Eqs. (19)-(23) describe the amplitudes of the electromagnetic waves after the sudden change in the refractive index of a homogeneous medium. Since the polarization and magnetization respond to the input electromagnetic wave linearly, our results can be extended to media with slowly varying by dividing the change in into multiple intermediate steps. At each step, the wave frequencies are solely determined by the equation , and the incremental changes in fields amplitudes and are related to their values at the previous step through Eqs. (19)-(23). Between the steps, the forward- and backward-propagating modes collect different phases . The phase difference between waves generated at different times causes constructive or destructive interference affecting the wave amplitudes. Thus, the field amplitudes and can be calculated by iterating the upconversion and propagation operations.
With the wave amplitudes, one can first find that the total momentum carried by the electromagnetic waves is generally not conserved. The result is not surprising because dividing a photon flow into multiple directions while maintaining its momentum would require an increase in total photon energy. We next find how wave energy is partitioned in each mode by specifying media with decreasing electric permittivity and with decreasing magnetic permeability .
Iii Frequency upconversion with decreasing permittivity
We first consider an electromagnetic wave propagating in a homogeneous medium whose permittivity suddenly decreases. In general, permittivity is caused by the polarization of the atom and molecule or by currents in metals and plasmas. When polarization charges or current are suddenly created, they couple to the input electromagnetic field, increasing the wave frequency. Evolution of the field and the field can be described by
where is the dipole resonance frequency ( for metals and plasmas), is the number density of the dipoles, is the natural charge, and is the electron mass. Absorption can be neglected when in the course of frequency upconversion the wave frequency is kept away from by at least the atomic relaxation rate. With the coupling strength , the wave eigenfrequencies are found by Fourier transforming Eqs. (24) and (25) to obtain
which on solving for yields
The same result can be obtained by solving the dispersion relation provided that is known. At a finite value of , Eq. (28) reveals two eigenmodes: an electromagnetic wavelike mode with frequency , and a dipole-oscillation-like mode with frequency . The values of are plotted in Fig. 1(a). The eigenfrequency increases at larger values of . This correlation relation enables upconverting the frequency of a propagating electromagnetic wave by suddenly increasing the medium dipole density . In the limit of large coupling strength , and .
As the coupling strength increases, we find that the group velocity decreases asymptotically to whereas approaches . The medium impedance is which equals in non-magnetized media, where is the impedance of vacuum. Thus we can obtain the energy density in each mode,
In Fig. 1(b), we plot the energy density of each mode using Eq. (30) as well as the total energy density. From the figure, we observe that decreases monotonically whereas increases monotonically as the coupling strength increases. The total energy density remains equal to which is the initial wave energy density in vacuum. At large coupling strengths, both and approach regardless of the frequency ratio .
The underlying physics can be explained by examining the coupling between the initial electromagnetic wave and the dipole oscillations. The dipole oscillation radiates in phase with the electromagnetic wave when and out of phase otherwise. Thus, destructive interference between the high-frequency mode and the dipole radiation causes a decrease in the wave amplitude. In the limit of large coupling strength, the wave energy is completely coupled into the dipole oscillation and is then radiated in both positive and negative directions. The radiation is also equally distributed in two modes with frequencies and . Since the amplitude ratio of the electric field and the magnetic field is , the energy in the higher- and lower-frequency modes is carried almost purely by electric fields and magnetic fields, respectively.
By setting the atomic resonance frequency to zero, our results can describe the reported phenomenon of electromagnetic wave frequency upconversion via suddenly ionized plasma Wilks et al. (1988). During ionization, the liberated electrons begin to oscillate at the plasma frequency leading to an increased wave frequency . Here, denotes the number density of free electrons which can be larger than for fully ionized media. Note that plasma has no dipole frequency since the electrons are not bound to the atomic nucleus. The wave amplitudes can be found by reducing Eqs. (20)-(23) into
The field amplitudes are in agreement with Wilks et al. Wilks et al. (1988). The energy density of each mode can be found as
They include a forward propagating wave, a reflected wave, and a static magnetic field. Note that the energy density in each mode incorporates both the field energy and the electron mechanical energy. The energy partitioning in different modes can be verified by the numerical simulation results reported in Ref. Edwards et al. (2018). As the plasma frequency increases, the total energy density remains a constant . At high plasma frequencies, and approach , and approaches . They agree with the results in Sec. 3.4 of Ref. Kalluri (2016).
However, our theory also reveals that the total energy density equals the energy density of the initial wave only if the initial permittivity equals that of vacuum or plasma, i.e. . If the initial medium has a non-negligible , the wave energy density is
which is larger than the total energy density after the medium changes by the second term in the brackets. During frequency conversion, the energy stored in the dipole oscillation is lost from the electromagnetic wave.
In an experiment, one is often more interested in the propagation of a finite-duration pulse. When the pulse frequency is upconverted, its energy is partially reflected and lost. Since the amplitude of the reflection mode decreases proportionally to as increases, the reflection can be suppressed by gradually ionizing the plasma if the incremental plasma frequency in each step is much slower than , or . In this limit, the amplitude of the forward-propagation wave from step to step are related through Eq. (33) by where and are the wave amplitude and frequency at step , respectively. By recursively applying the relation, we can obtain the output electric field in the asymptotic limit of gradually ionized plasma
Here, we use and to denote the initial and final frequencies, respectively. In the last step, we used Euler’s formula for the function . The energy density is then
Therefore, the energy density of the wave decreases inversely with the wave frequency . Although the same result in a gradually ionized plasma has been shown in previous studies Bakunov et al. (2000); Dodin and Fisch (2010a, b) by treating the electromagnetic wave as the ponderomotive potential for the electron motion, our derivation has shown the connection between the sudden ionization limit and the adiabatic ionization limit.
Iv Frequency Upconversion with decreasing permeability
Although the electric permittivity can be controlled through coupling the wave to the atomic or molecular polarization, tunable magnetic permeability has been realized prevalently in artificial materials. In this section, we investigate the use of SRR structures Smith et al. (2000) for upconverting the frequency of an electromagnetic wave. We consider a specific type of SRR whose permeability is controlled by an external field. For simplicity, we neglect the change in permittivity.
Consider a microwave pulse propagating in a metamaterial with uniformly distributed SRR structures oriented perpendicular to the magnetic field of the microwave. We assume that the SRRs begin to interact with the microwave pulse only when activated by an external field at time . Once activated, a loop current is induced in the rings when the magnetic flux changes. For each ring, the split gap stops the current and helps accumulate positive and negative charges on two sides of the gap, thus forming a capacitor with capacitance . Variation of the loop current at the microwave frequency is also subject to self-inductance and a mutual inductance . Here, is the area filling fraction and denotes the coupling strength between the electromagnetic wave and the SRR structure. Thus, the SRR can be modeled as an LC circuit and the current satisfies
where is the lattice constant of the SRR structure. Using relation , it becomes
where is the resonance frequency of the LC circuit. Without , the effective resonance frequency of the magnetization is which increases when approaches . During the change in medium, the continuity relations are given by the same equations (8) and (13). Here, the continuity of is caused by the fact that the SRRs are not charged initially, i.e. .
The wave equation for the magnetic field can be obtained from Maxwell’s equations (1) using
The same result can be obtained by solving the dispersion relation where in such materials Smith et al. (2000). The frequencies as a function of filling fraction is plotted in Fig. 2(a). It is evident that an increased coupling strength with any finite SRR resonance frequency leads to an upshifted eigenfrequency and a downshifted . In the limit , they approach, respectively,
Compared with Eq. (28) where increases linearly with the coupling strength when is varied, we find that in Eq. (44) scales inversely with when is varied. The special scaling relation of arises from the existence of mutual inductance of SRRs which combines with the self inductance causing a near-zero effective inductance when . Then it elevates the effective SRR resonance frequency as well as the eigenfrequency .
In a medium with varying but constant , the material impedance becomes . The group velocity of each mode is
for . Assuming , the energy densities of each mode with different values of filling fraction and are plotted using Eq. (30) in Fig. 2(b). We observe that an increasing value of causes a decrease of energy density of the high-frequency mode and an increase in the energy density of the low-frequency mode . The total energy density is conserved.
As increases and decreases, the high-frequency mode becomes purely electric, i.e. and ; The low-frequency mode remains electromagnetic and . The wave energy partially resides in the current oscillations of the split rings. At the maximum coupling strength with , the energy density of each mode approaches
where is the initial wave energy density in vacuum. The result shows that the efficiency of energy conversion into the high-frequency mode is generally lower than for a finite value of although the efficiency asymptotically approaches as increases.
For frequency upconversion with simultaneously decreasing permittivity and permeability, it is straightforward to calculate the generated mode frequencies by solving the dispersion relation . We do not provide the explicit form of the energy density here as it is lengthy but not illuminative. We can expect a large frequency upshift resulting from the coupling between the upconverted oscillation frequencies of the electric field and the magnetic field. The energy density would be divided into three modes: an electromagnetic mode at upconverted frequency, a dipole-like mode at a low frequency, and a ring-shaped oscillating current mode at a low frequency.
To conclude, we have shown that upconverting the frequency of an electromagnetic wave using a temporally changing refractive index is accompanied by energy loss to low-frequency modes. We analytically obtained the field amplitudes and energy densities after frequency upconversion in a homogeneous medium when an external field causes sudden refractive index reduction. We have also outlined how our theory can be extended to media with gradually changing refractive index and how to calculate the modes amplitudes by iterating the sudden change processes. Our results show that media with decreasing permittivity and with decreasing permeability have comparable energy conversion efficiencies, but the upconverted frequencies scale differently with the coupling strengths. The difference is rooted in the special structure of SRRs that a larger filling fraction not only increases the coupling strength between the magnetic field and the magnetization current, but also elevates the effective resonance frequency of SRRs.
Experimental observations of frequency upconversion using dynamic media are subject to technological constraints. Using ionized plasma, the most prominent difficulty is to create high density plasma in few optical cycles. For this reason, the most successful demonstration achieved only about upconversion by ionizing semiconductors Nishida et al. (2012) using a moderate intensity () laser. Frequency upconversion in the optical range needs higher plasma density, which might be achieved using high intensity lasers. Using metamaterials, the foreseeable obstacle lies at the low resonance frequency limited by the fabrication technology. Even with the state-of-the-art technology, structures at the precision of micrometers Wong et al. (2017) can only be fabricated, which limits the resonance frequency to the range of terahertz. Frequency upconversion beyond the optical frequency will require either advances in the nanofabrication technology or using a cascaded sequence of sudden refractive index change.
Acknowledgements.This work was supported by NNSA Grant No. DE-NA0002948, and AFOSR Grant No. FA9550-15-1-0391.
- Morgenthaler (1958) F. R. Morgenthaler, “Velocity modulation of electromagnetic waves,” IEEE Trans. Microw. Theory Tech. 6, 167 (1958).
- Wilks et al. (1988) S. C. Wilks, J. M. Dawson, and W. B. Mori, “Frequency up-conversion of electromagnetic radiation with use of an overdense plasma,” Phys. Rev. Lett. 61, 337 (1988).
- Mendonça (2001) J. T. Mendonça, Theory of photon acceleration (Institute of Physics Publishing, Bristol, UK; Philadelphia, USA, 2001).
- Nerukh et al. (2012) A. Nerukh, N. Sakhnenko, T. Benson, and P. Sewell, Non-stationary electromagnetics (CRC, Boca Raton, FL, 2012).
- Kalluri (2016) D. K. Kalluri, Electromagnetics of time varying complex media: frequency and polarization transformer, Second Edition (CRC, Boca Raton, FL, 2016).
- Edwards et al. (2018) M. R. Edwards, K. Qu, Q. Jia, J. M. Mikhailova, and N. J. Fisch, “Cascaded chirped photon acceleration for efficient frequency conversion,” Phys. Plasmas 25, 053102 (2018).
- Froula et al. (2018) D. H. Froula, D. Turnbull, A. S. Davies, T. J. Kessler, D. Haberberger, J. P. Palastro, S.-W. Bahk, I. A. Begishev, R. Boni, S. Bucht, J. Katz, and J. L. Shaw, “Spatiotemporal control of laser intensity,” Nature Photonics 12, 262 (2018).
- Yablonovitch (1989) E. Yablonovitch, “Accelerating reference frame for electromagnetic waves in a rapidly growing plasma: Unruh-Davies-Fulling-Dewitt radiation and the nonadiabatic Casimir effect,” Phys. Rev. Lett. 62, 1742 (1989).
- Lähteenmäki et al. (2013) P. Lähteenmäki, G. S. Paraoanu, J. Hassel, and P. J. Hakonen, ‘‘Dynamical Casimir effect in a Josephson metamaterial,” Proc. Natl. Acad. Sci. 110, 4234 (2013).
- Yablonovitch (1973) E. Yablonovitch, “Spectral broadening in the light transmitted through a rapidly growing plasma,” Phys. Rev. Lett. 31, 877 (1973).
- Joshi et al. (1990) C. J. Joshi, C. E. Clayton, K. Marsh, D. B. Hopkins, A. Sessler, and D. Whittum, “Demonstration of the frequency upshifting of microwave radiation by rapid plasma creation,” IEEE Trans. Plasma Sci. 18, 814 (1990).
- Kuo (1990) S. P. Kuo, “Frequency up-conversion of microwave pulse in a rapidly growing plasma,” Phys. Rev. Lett. 65, 1000 (1990).
- Savage et al. (1993) R. L. Savage, R. P. Brogle, W. B. Mori, and C. Joshi, “Frequency upshifting and pulse compression via underdense relativistic ionization fronts,” IEEE Trans. Plasma Sci. 21, 5 (1993).
- Yugami et al. (2002) N. Yugami, T. Niiyama, T. Higashiguchi, H. Gao, S. Sasaki, H. Ito, and Y. Nishida, “Experimental observation of short-pulse upshifted frequency microwaves from a laser-created overdense plasma,” Phys. Rev. E 65, 036505 (2002).
- Nishida et al. (2012) A. Nishida, N. Yugami, T. Higashiguchi, T. Otsuka, F. Suzuki, M. Nakata, Y. Sentoku, and R. Kodama, “Experimental observation of frequency up-conversion by flash ionization,” Appl. Phys. Lett. 101, 161118 (2012).
- Geltner et al. (2002) I. Geltner, Y. Avitzour, and S. Suckewer, “Picosecond pulse frequency upshifting by rapid free-carrier creation in znse,” Appl. Phys. Lett. 81, 226 (2002).
- Avitzour et al. (2005) Y. Avitzour, I. Geltner, and S. Suckewer, “Laser pulse frequency shifting by ionization and recombination fronts in semiconductor plasma,” âJ. Phys. B 38, 779 (2005).
- Shcherbakov et al. (2017) M. R. Shcherbakov, K. Werner, Z. Fan, N. Talisa, E. Chowdhury, and G. Shvets, “Non-integer harmonics generation by photon acceleration in rapidly evolving nonlinear semiconductor metasurfaces,” (2017), arXiv:1710.06966 .
- Stepanov (1993) N. S. Stepanov, “Waves in nonstationary media,” Radiophys. Quantum Electron 36, 401 (1993).
- Bakunov et al. (2000) M. I. Bakunov, V. B. Gildenburg, S. N. Zhukov, and N. A. Zharova, “Adiabatic frequency shifting of a surface wave guided by a time-varying plasma structure,” Phys. Plasmas 7, 1035 (2000).
- Smith et al. (2000) D. R. Smith, Willie J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184 (2000).
- Levi (2016) A. F. J. Levi, “The Drude model,” in Essential Classical Mechanics for Device Physics, 2053 (Morgan & Claypool Publishers, 2016) pp. 6–1.
- Dodin and Fisch (2010a) I. Y. Dodin and N. J. Fisch, “On generalizing the K- theorem,” Phys. Lett. A 374, 3472 (2010a).
- Dodin and Fisch (2010b) I. Y. Dodin and N. J. Fisch, “Damping of linear waves via ionization and recombination in homogeneous plasmas,” Phys. Plasmas 17, 112113 (2010b).
- Wong et al. (2017) Z. J. Wong, Y. Wang, K. O’Brien, J. Rho, X. Yin, S. Zhang, N. Fang, T.-J. Yen, and X. Zhang, “Optical and acoustic metamaterials: superlens, negative refractive index and invisibility cloak,” Journal of Optics 19, 084007 (2017).