We present a simple model to estimate electromagnetic wave frequency up-conversion resulting from rapidly forming gaseous plasma slabs. Such a model aids in the interpretation or planning of realizable laboratory experiments, where the plasma is neither formed instantaneously nor infinite in spatial extent. The model uses, as a basis, the behavior of an unbounded plasma when the plasma forms over extended times and considers slab boundary conditions to estimate optimum transmitted sampling windows that capture the frequency spectra of the converted waves. The results of this model are compared to exact solutions using finite difference time domain calculations, confirming its effectiveness as a tool for understanding the fundamental nature of the wave-plasma slab interactions and for planning and interpreting experimental results.
I. INTRODUCTION
Recently, there has been a significant interest in the interaction of electromagnetic (EM) waves with linear, time-variant media.1 Unlike a time-invariant medium, in which a delay in an input to the system is reflected in its output, the delay of an input in a time-variant system can depend on the time of that input. In electromagnetic systems, a time-variant medium is characterized by how its constitutive parameters, such as the dielectric permittivity,2–10 permeability,11,12 and conductivity, vary over time either naturally or in some externally controlled manner. Despite this renewed interest, the theoretical exploration of this problem has a history spanning more than 50 years.13–18 The modulation of a medium over time opens up a diverse range of applications in engineering electromagnetics,19–21 such as non-reciprocity,22–25 wideband impedance matching,26 and frequency conversion,27–31 among others.
One potentially interesting material to study as a time-variant medium is a gaseous plasma. Experimentally, it is possible to vary the number density of electrons in time, hence varying the dielectric constant. In principal, one should be able to achieve frequency up-conversion32 with a time-switched non-magnetized plasma with a plasma density that increases in value with time.33 So far, there has been a significant body of theoretical research on the electromagnetics of time-variant plasmas.34–42 Experimental investigations, however, remain relatively limited.32,43–46
This second model is based on the assumption that the induced polarization is a superposition of dipole moments at different instants of time10,50 attributed to the collective response of plasma electrons, appropriate at typical conditions of laboratory plasmas. This model is also in line with that presented by Stepanov48 in 1976, for the case of an increase in plasma concentration due to ionization without taking recombination into account. In this paper, we adopt this model, which has also been derived and discussed by Kalluri.33
While in general, one can compute the problem using finite difference time domain (FDTD) simulations, we describe here, below, a useful approximate model which aids in the understanding of the physics of the problem of EM wave interaction with a time-varying plasma medium. In addition, it will simplify the design or interpretation of experiments that avoid having to carry out FDTD simulations, at least for simple one-dimensional configurations of an electromagnetic wave passing through a plasma slab with an increasing electron number density. Our discussion first starts with the problem of an electromagnetic wave traveling through a time-varying but spatially uniform and unbounded plasma. Then, we focus on a time-varying plasma slab with a spatially finite size.
II. TIME-VARYING BULK, UNBOUNDED PLASMA
We consider the case of a rapidly created plasma immersed in a monochromatic electromagnetic wave. We start our analysis by first considering the switching on, at a time t = 0, a spatially unbounded, and isotropic non-magnetized plasma.
A. Slow switching
Fairly routine pulsed power sources can have rise times of order of 20–2000 ns and when used to generate pulsed plasmas, can produce increases in plasma density via ionization on time scales of similar order. These times preclude the possibility of large frequency conversions in the GHz range and would be better suited for MHz scale frequency conversion. However, most practical plasma sources may have collisional damping rates that would considerably hinder the electron response at these lower frequencies. At GHz frequencies, this plasma density rise is therefore considered to be a slow one, i.e., what we would refer to as slow switching. Considering a range of switching speeds (leading to the square of the plasma frequency profiles shown in Fig. 2), we present the computed electric field for ns ns (wherein the plasma forms initially at ) in Fig. 3. The negative time refers to the presence of the original wave at GHz. It is clear from the behavior of the wave field at that a higher switching speed (lower characteristic time leads to a greater frequency component in the up-converted field. To better understand the spectrum generated, we consider the response within three sampling time windows following the onset of the plasmas.
First, we assume to have available to us a relatively long period of time to sample the converted fields, say and ns, respectively. The corresponding Fourier spectrum of the converted wave over these long sampling windows for the various switching speeds is shown in Figs. 4 and 5, respectively. The power values are normalized to the power of the reference signal, which is within the given time window ( and ns, respectively). It can be seen that, by increasing the switching speed (or, lowering the rise time, ), the power will be distributed over a wider range of frequencies. For example, for the case with ns, frequency conversion can be achieved up to almost 2.36, 6, and 10 GHz (below the limit GHz), for ns, ns, and ns, respectively.
Now, we focus on the case of a short sampling time period, i.e., ns. In other words, we are interested in frequency up-conversion within a short period of time after the initial plasma creation. The analysis here would be informative for the case of a finite plasma slab that will be discussed later. Figure 6 shows the frequency up-conversion of the system for various values of the switching speeds. For the very slow case of ns, the up-converted frequency is only slightly higher than the original frequency, GHz, but with a power that is of a similar level compared to that of the original frequency. For ns and ns, the range of converted frequency is shifted toward lower values, and the respective power is increased for the up-converted frequencies, compared to the results for the long sampling time period discussed above. While in any practical sense, we see that restricting the sampling time of the converted waves greatly restricts the range of frequency up-shift in this unbounded plasma. We shall show later that this restriction in sampling time occurs naturally, in a bounded plasma slab.
B. Moderate and fast switching
In this section, we focus on the response associated with faster switching speeds, i.e., equal to the wave period ( ns), and shorter, with ns and ns. Figure 7 shows the computed electric field for ns ns (where again, the onset of the plasma rise is at ).
If a long enough sampling period of time is considered (i.e., ns), we find that these moderate and fast switching speeds lead to frequency up-conversion very close to the limiting frequency ( GHz), as shown in Fig. 8. The up-converted power at 10.05 GHz for the ns is almost 3 and 5 dB larger than that of the cases with ns and ns, respectively. Also, moderate switching ( ns) leads to a sideband in the up-converted frequency at GHz, almost 27 dB below its converted power at GHz. As expected, the power will be distributed within a wider frequency range when lower switching speeds are used.
In the case of a shorter time window (i.e., ns), Fig. 9 illustrates that when ns, the frequency response is similar to that of the original wave, with the peak power up-converted to the desired frequency of GHz. One can also notice the spreading of the frequency response compared to the longer time window for ns and ns.
III. SWITCHED PLASMA SLAB
When the plasma slab is initially created at , the already present wave within the slab at the initial frequency ( ) converts to other frequencies with the same wavelength throughout time (i.e., the wavenumber within the slab is preserved). Theoretically, if the plasma slab considered is made increasingly larger, one would approach the problem of an unbounded plasma medium, for which one would have access to an opportunity for a longer sampling time window, as described below. We have already shown how the available sample time window can affect the generated frequencies and their respective powers within an unbounded plasma.
Ostensibly, one might not find any obvious relationship between the problems of the time-varying spatially unbounded and bounded plasmas (such as the finite slab shown in Fig. 10). However, here we show how one can analyze the problem of transmission through a switched finite plasma slab using the wave equation [i.e., Eq. (7)] for field solutions within a spatially unbounded plasma. Before moving into the details of the analysis and the proposed model for a finite slab, first, let's review what defines a time varying, spatially unbounded 1-D plasma. The spatially unbounded plasma is: (a) not bounded spatially by any boundaries, and, as a result, for a wave traveling in the direction (with a spatial dependency of ), there will be no spatial reflections within the plasma with a spatial dependency of . Also, (b) there exists an available sampling time window that adequately captures the transient behavior of the problem. Since the medium is unbounded, this time window is not dependent on the size of the plasma but on the capability of the sampler in time. To be able to link the problem of a time varying, finite slab to the unbounded one, two conditions should be satisfied: (a) for a finite slab, although reflections within plasma are unavoidable, it should be possible to neglect their effects in analyzing the transmission in the case of minimal transmission of multiple-reflected wave components through the medium (we will investigate this aspect here first); and (b) there exists an effective time window (i.e., ) that captures the transient behavior of the problem within a finite slab. Unlike an unbounded medium, here we show that such an effective time window is proportional to the slab size, and if transmission of the converted waves is greatly limited by the scattering at the interface, this effective time is proportional to .
Now, we observe that waves with higher frequency components, despite possessing faster phase velocities (as shown in Fig. 1) and passing through the slab within a shorter time window, have minimum transmission through the slab due to the filtering effect forced by the impedance discontinuity at the spatial boundary, . As a result, the slab behaves, in some ways, like a cavity for these frequency components. They will experience multiple reflections within the plasma medium, and their transmission decrease as time progresses. Their main contribution to the sampled transmitted fields will then be their first transmission (spatially) through the slab. In contrast, lower frequency components have high transmission from the plasma slab to free space but with lower phase velocities. The waves that are reflected at the boundary strike the boundary at and are reflected back into the plasma. Since the reflection is much smaller than their transmission, their second and following transmissions from the plasma slab to free space have much smaller magnitudes and can then be neglected. Accordingly, similar to the higher frequencies, their main contribution to the sampled transmitted fields will also be their first transmission (spatially) through the slab. To summarize, both low and high frequency components contribute to the transmission process mainly through their first transit through the slab. The problem of transmitting through a finite plasma slab, therefore, approximately mimics the problem of an unbounded medium with waves traveling in the direction when sampled over a time window comparable to this transit time through the slab.
Considering Fig. 1 again, the minimum phase velocity is equal to the speed of light in free space c and is attributed to the wave with a frequency of (initial frequency). For a finite slab with a size d, the effective time window should be selected in such a way that both low and frequency components pass through the slab at least once. The time it takes for the lowest frequency component (the slowest wave), located at , to reach for the first time is . As a result, we have . For our proposed simple model, we consider to have an effective time window wherein this condition is valid. In addition, we choose the effective time to be the time that it takes for the lowest frequency component (the slowest wave), located right next to , to approach the boundary at again (just before striking it for the second time), i.e., . This ensures that the first transmission out of the plasma occurs for the lowest frequencies.
As a result, it can be shown that one can link the problem of transmission through a time varying plasma slab to the wave solutions within an unbounded plasma slab by selecting an effective time window that almost captures the physics of the problem. In summary, our model can be described in two steps.
- Step 1: For a 1-D, finite plasma slab with the size d, one solves the following equation within the effective time window :
- Step 2: By taking the Fourier transform of , i.e., , the magnitude of the transmitted wave in the Fourier domain can then be found using
As follows, we present the results that show the effectiveness of the presented model, and also, to analyze the effect of a plasma slab with a finite size on the creation of new frequencies and their transmission through plasma to free space. Without loss of the generality of the problem, here we consider two cases with plasma slab widths of and . A slab width of leads to an effective time window of ns. The resulting normalized transmitted power is shown in Fig. 12. An interesting observation is that similar to the unbounded scenario shown in Figs. 6 and 9, a faster switching speed leads to higher converted frequencies. However, due to the filtering behavior of impedance discontinuity, the accessible transmitted power to free space is reduced drastically for frequencies far from the initial frequency ( ). On the other hand, the frequencies generated by slow switching, though being lower in terms of frequency conversion, will transmit similar power levels (compared to the unbounded case) from plasma to free space. Also, to complete the study and confirm the model, we have slightly modified the FDTD code presented in Lee's dissertation52 to solve for the finite slab, but with the square of plasma frequency mimicking the exponential profile presented in this paper. For the sake of brevity, we have not presented the details of the FDTD technique here. We refer the reader to Lee's dissertation52 for further details and the provided computational script. The FDTD results are shown in Fig. 13. It can be clearly seen that the presented model can predict the frequency response of the systems reasonably well. However, we believe that the presented approximate model conveys a better understanding of the physics of the problem.
For the case of a slab with , very slow switching speeds (i.e., ns and ns) lead to either no frequency conversion or a very slight frequency change as shown in Fig. 14. On the other hand, moderate and moderately fast speeds (i.e., ns and ns) lead to wideband frequency conversions. As a result, a smaller slab size requires faster switching speeds to be able to compensate for its shorter effective time window. Here, again, we have presented the FDTD results in Fig. 15. It can be seen how the results from such a simple presented model agree with the FDTD results.
IV. CONCLUSIONS
We have presented a simple model for analyzing frequency up-conversion in linear, time-varying plasma slabs. The model links the problem of transmission through a time-varying plasma slab to the wave solutions within an unbounded plasma slab by selecting an effective time window that captures the most important contributions to the physics of the problem. We have considered various slab widths and plasma switching speeds, and the associated results of this model have been compared to the FDTD calculations, confirming its effectiveness as a useful tool paving the way for understanding the interactions of EM waves with time-varying plasma slabs, and also, for planning and interpreting experimental results.
ACKNOWLEDGMENTS
The authors would like to thank Professor Kalluri for useful discussions and for directing us to the FDTD simulation in the thesis of J. H. Lee. This work was supported by an Air Force Office of Scientific Research Multi-University Research Initiative (MURI) (Grant No. FA9550-21-1-0244), with Dr. Arje Nachman as the Program Officer. The authors acknowledge the original support of Dr. Mitat Birkan, who was the AFOSR Program Officer prior to his retirement.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Hossein Mehrpour Bernety: Conceptualization (equal); Data curation (lead); Formal analysis (lead); Investigation (equal); Methodology (equal); Project administration (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Mark A. Cappelli: Conceptualization (lead); Data curation (lead); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (equal); Resources (lead); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).
DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.