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Front. Astron. Space Sci., 09 November 2023
Sec. Space Physics
Volume 10 - 2023 |

Dispersive propagation of nuclear electromagnetic pulse in the ionosphere

www.frontiersin.orgYongli Wei1 www.frontiersin.orgDinghan Zhu1 www.frontiersin.orgZongxiang Li1 www.frontiersin.orgLihua Wang1,2 www.frontiersin.orgYuan Wang1 www.frontiersin.orgTianchi Zhang1 www.frontiersin.orgBin Xing1 www.frontiersin.orgBaofeng Cao1* www.frontiersin.orgPeng Li1*
  • 1State Key Laboratory of NBC Protection for Civilian, Beijing, China
  • 2College of Information and Communication Engineering, Harbin Engineering University, Harbin, China

Introduction: On the propagation path to the satellite, the ionosphere will distort the nuclear electromagnetic pulse (NEMP) and change its physical properties.

Methods: This paper proposes a method for calculating the propagation of NEMP to the satellite. The method decomposes NEMP into the superposition of simple harmonic waves, and each simple harmonic wave is calculated separately in the ionosphere. With the consideration of different time of arrival and critical frequency of the ionosphere, the NEMP after propagating in the ionosphere is obtained by superposition of simple harmonic waves in time domain rather than the inverse Fourier transform which will erase the time domain information.

Results: The results show that NEMP is dispersive in ionosphere with the pulse broadened, the speeds changed and the bandwidth narrowed. The time-frequency spectrum can provide the frequency band where the signal energy is located.

Discussion: Our proposed method provides a simple and effective way to calculate the NEMP propagation in the ionosphere, which should afford help to the design of NEMP receivers and the selection of satellite orbit altitude.

1 Introduction

Nuclear electromagnetic pulse (NEMP) is one of the important effects of nuclear explosions, with fast propagation speed, high peak value, wide frequency band, wide coverage and other characteristics. NEMP is appropriate for use as a detection target for space-based nuclear explosion monitoring systems. As a consequence of the Comprehensive Nuclear Test Ban Treaty, non-nuclear experimental and theoretical efforts are expanded to study and observe the NEMP phenomenology and to develop appropriate descriptive models. For the non-nuclear experimental efforts, especially in space-based NEMP explosion monitoring, in 1997, the United States launched a near Earth satellite, FORTE (Fast On orbit Recording of Transient Events), with an optical and radio frequency detection system. The satellite is equipped with a 90 MHz receiver and a pair of 20 MHz receivers, both of which can be tuned in the range of 30–300 MHz. The FORTE received the pulse signal sent by the ground electromagnetic pulse generator (Massey et al., 1998). After band-pass filtering and converting it into a digital signal, it was transmitted back to the ground for analysis, so as to obtain the ionospheric information of the propagation path (Roussel-Dupre et al., 2001; Huang and Roussel-Dupre, 2005; Minter et al., 2007) and carry out the ground signal source positioning algorithm (Jacobson and Shao, 2001). With the above mentioned techniques, FORTE has the ability to detected the NEMP signal.

Theoretical methods for studying NEMP propagation can be classified into two categories: time-domain methods and frequency-domain methods. For the time-domain methods, the Maxwell equations are the most essential parts. The electric field components and magnetic field components are calculated with appropriate current sources, medium parameters and boundary conditions. The widely used numerical calculation method is Finite Difference Time Domain (FDTD) method. With this method, the physical process of electromagnetic pulse excited by Compton electron currents is simulated (Meng et al., 2003; Meng, 2013). The discrete iterative equations for the electromagnetic field components are presented in the CPML (Convolutional Perfect Matched Layer) media under the two-dimension and three-dimension prolate-spheroidal coordinate system (Gao et al., 2005a; Gao et al., 2005b). Chen et al. derived the time-domain difference equations for the interaction between ionosphere and electromagnetic pulse (Chen et al., 2011). Those results show that the combination of ionospheric time-domain calculation method and NEMP calculation method is reasonable and effective. However, FDTD method is suitable for the electromagnetic field calculation of subtle structures, and it takes a very long time for the calculation of long distance electromagnetic pulse propagation.

In frequency domain methods, the ionosphere is considered as the transfer function, and the Fourier transform is used to obtain the spectrum of NEMP signal. The amplitude and phase of each frequency component after propagation in the ionosphere are calculated respectively. Then, the inverse Fourier transform is used to obtain the NEMP signal received at the satellite orbit (Meng et al., 2004; Yao et al., 2019). Nevertheless, the Fourier transform will erase the time domain information of NEMP signal, and the inverse Fourier transform ignores that the different frequency components have different speeds in dispersive material.

This paper proposes a method for NEMP propagation in the ionosphere. The method decomposes the NEMP signal into the superposition of simple harmonic waves. The propagation of each simple harmonic wave in the ionosphere is calculated separately. With the consideration of the time differences among the simple harmonic waves, the NEMP signal is obtained by superposition of simple harmonic waves after propagation in the ionosphere. This paper is organized as follows: In Section 2, the model of ionosphere, decomposition of NEMP and superposition of simple harmonic waves are presented. In Section 3, the waveforms chosen for simulation are introduced and the results and discussion are presented. In the last section, we conclude this paper and present the applications.

2 Methods

2.1 Model of ionosphere

2.1.1 Multilayer model

The multilayer model is used for calculation, as a result of the non-uniform distribution of the plasma density in the ionosphere. Each layer is considered to be uniform and has a thickness of Δh, as is shown in Figure 1. It is assumed that the p0 layer is vacuum due to the negligible plasma density.


FIGURE 1. Ionospheric electron density profile and multilayer model.

The plasma frequency of the ionosphere is fp=Nee2/4π2meε0, where e=1.602176634×1019C is the charge of electron, me=9.10956×1031kg is the mass of electron, ε0=8.854187817×1012F/m is the vacuum permittivity, and Ne is the electron density.

2.1.2 Critical frequency

Ne increases with the increase of altitude, and after reaching the peak Nem, Ne starts to decrease. The plasma frequency corresponding to Nem is the critical frequency, namely, fc=Neme2/4π2meε0. The frequency components below the critical frequency cannot continue to propagate upward.

2.1.3 Collision frequency

Particles in the ionosphere are in thermal motion and inevitably collide. Frequencies of electron-neutral and electron-ion collisions affect the ionosphere propagation of the signals (He and Zhao, 2009).

The collision frequencies of electron-neutral particle include.

(1) Collision frequency of electrons with Nitrogen molecules N2


(2) Collision frequency of electrons with Oxygen molecules O2


(3) Collision frequency of electrons with Oxygen atoms O


(4) Collision frequency of electrons with Helium atoms He


From Eqs 14, Te is the electron temperature (in eV, 1 eV = 11,600 K), nN2, nO2, nO, nHe are the density of neutral particles N2, O2, O, He, respectively.

The collision frequency of electron-ion is


where νe,i is the collision frequency of electrons with the ion i, which represents NO+, O2+, O+ in this paper. ni represents the density of the ion i.

The total electron collision frequency νe can be expressed as


2.1.4 Refraction index

The propagation of electromagnetic waves in the ionosphere follows the Appleton-Lassen equation. In the absence of geomagnetic field (Huang, 2000), the refraction index can be expressed by


where μ and η are real and imaginary parts, respectively; X=fp2/f2; Z=νe/2πf; f is the frequency of the simple harmonic wave.

In vacuum, the speed of a simple harmonic wave v is equal to the speed of light c, namely, v=c3×108m/s. In the ionosphere considering collisions, the phase speed vp can be expressed by (Liu et al., 2010)


where, σ=Nee2/meνe is the conductivity. εreal is the real part of ε, where ε=ε01+2πfp2/2πfive2πf. The item 2/1+σ/2πfεreal2+1 is always great than 1, resulting in vp<c. According to Eq. 8, as the frequency f increases, vp also increases, shown in Figure 2.


FIGURE 2. Speed of different frequencies of simple harmonic waves.

The propagation time in each layer of ionosphere is given by


In vacuum, the amplitude is a function of distance from the source point


where E0 is the initial amplitude, E1 is the amplitude at r distance from the source point.

In the ionosphere, the attenuation of amplitude with distance r can be expressed as


where E2 is the amplitude at r distance from the source point; β=2πfη/c is the attenuation coefficient with frequency f.

2.2 Dispersive propagation in the ionosphere

2.2.1 Decomposition of NEMP

In the light of Eqs 711, the influence of the ionosphere differ with the frequency f. Therefore, we decompose NEMP into K simple harmonic waves, and study the propagation of each simple harmonic wave in the ionosphere.

The Fourier transform is applied to Et and the signal is converted from the time domain Et to the frequency domain Ef. The complex value corresponding to frequency fk is Efk=ak+ibk. The simple harmonic wave decomposed from Et can be described as


where fk is the frequency, Ek=ak2+bk2 and φk=arctan (bk/ak) are the initial amplitude and phase, respectively. The superposition of all simple harmonics can be expressed as


where e1t is the simple harmonic wave with highest frequency f1, and eKt is the simple harmonic wave with the lowest frequency fK.

2.2.2 A simple harmonic wave propagation in the ionosphere Propagation from h0 to h1

For a simple harmonic wave with frequency fk, when propagating vertically upward from h0 to the bottom layer h1 of the ionosphere, where there is no plasma, the propagation time is calculated by


and the amplitude is calculated by


The expression is derived from Eq. 12 as

ekt=Ekh1cos2πfkt+tk0+φk(16) Propagation from h0 to h2

When propagating from h1 to h2, i.e., in p1 layer, the refraction index of p1 layer is expressed by


The propagation time is calculated by


The absorption loss can be derived from Eqs 10, 11


where βk1=2πfkηk1/c is the attenuation coefficient in p1 layer, Eq. 16 is then expressed as

ekt=Ekh2eβk1Δhcos2πfkt+tk0+tk1+φk(20) Propagation from h0 to hM

By analogy, when the simple harmonic wave ekt propagates to the height hM, the expression is


where βkm=2πfkηkm/c is the attenuation coefficient in pm layer, tkm=Δh/vkm is the propagation time in pm layer, and m=0M1tkm is the propagation time from h0 to hM. Definition domain

Each simple harmonic wave has the same duration T. The time difference Δtk between e1t and ekt, propagating from h0 to hM, is given by


Equation 22 represents that when e1t reaches hM, ekt needs Δtk to reach it. Simple harmonic waves of different frequencies propagation in the ionosphere are shown in Figure 3.


FIGURE 3. Simple harmonic waves of different frequencies propagation in the ionosphere.

Taking the moment when e1t reaches hM as the zero moment, then the time domain of ekt is defined as


In particular, the time domain of e1t is defined as


2.2.3 Superposition of simple harmonic waves

The ionosphere will reflect the simple harmonic waves with frequencies below the critical frequency fc. The simple harmonic waves that can penetrate the ionosphere are e1t, e2t, …, ekt, …, ect. ect is the last simple harmonic wave that can penetrate the ionosphere. ec+1t, …, eKt will be reflected by the ionospheric electron frequency peak (foF2).

With the consideration of the critical frequency and the different definition domain, the simple harmonic waves e1t, e2t, …, ect are superimposed, so NEMP signal Et that propagates from h0 vertically upward to hM is finally described as


3 Simulation

The NEMP is emitted at the ground and is received at an altitude of 800 km and let Δh=1 km. The data of electron density and ions density are downloaded from the International Reference Ionosphere (International Reference Ionosphere - IRI (2016) | CCMC ( The data of neutral particle density are downloaded from the NRLMSISE-00 Atm Model (NRLMSISE-00 Atm Model | CCMC ( The electron density of the ionosphere as a function of altitude is plotted in Figure 1. The parameters of ionosphere are shown in Table 1.


TABLE 1. Parameters of ionosphere.

3.1 Simulation waveform

We choose Bell Laboratory pulse as the simulation waveform. The first reason is that the method proposed in this paper is to study the dispersion of NEMP in the ionosphere, rather than the generation of NEMP. Therefore, the simulation waveform can be adjusted and replaced to meet the needs of the simulation. The second reason is that it is easier to be simulated, which facilitates post-calculation analysis.

Bell Laboratory pulse is represented by the double exponential function


where E0 is the maximum amplitude, k=eαtpeβtp1 is the modifying factor, α and β are the parameters characterizing both the rise time and fall time simultaneously, tp=lnβlnα/βα is the peak time.

Parameters of Bell Laboratory pulse are shown in Table 2. The original waveform Et and superposition waveform Et which is obtained from Eq. 13, are shown in Figure 4. The relative error is less than 1 × 10−11%. Compared to the order of original waveform, the error is negligible.


TABLE 2. Parameters of Bell Laboratory pulse.


FIGURE 4. Bell Laboratory pulse, superposition waveform and the relative error.

3.2 Results and discussion

Let EBt represents the Bell Laboratory pulse after propagation in the ionosphere. EBt becomes an oscillatory waveform with a long tail after propagation in the ionosphere, and is stretched from 1 to 30 μs. The rising edge of the envelope becomes slower, and the duration of the rising edge and the falling edge are almost the same. In addition, there is a peak time delay of 10 μs for EBt, shown in Figure 5.


FIGURE 5. Bell Laboratory pulse after propagation in the ionosphere.

The delay can be explained with the simple harmonic waves’ arrival time differences and amplitudes, shown in Table 3. From 0 to 6 μs, the frequencies of the simple harmonic waves are all above 50 MHz, but their amplitudes are small, so the envelope of EBt is flat. The envelope begins to rise at 6 μs, corresponding to the arrival time of a 50 MHz harmonic wave. Simple harmonic waves with lower frequencies but larger amplitudes then arrive and superimposed one after another. The energy of the pulse accumulates and reaches the peak at 10 μs. After 10 μs, although the amplitude of the arriving simple harmonic waves is larger than those that have arrived before 10 μs, the time difference of arrival is also increasing, so that the overlap between adjacent simple harmonic waves is constantly reduced. The energy of the pulse fails to accumulate, and the envelope begins to fall.


TABLE 3. Amplitudes and time differences of simple harmonic waves.

Because of the critical frequency, the simple harmonic waves below it will be reflected. The critical frequency in the ionosphere is 6.24 MHz, and there are no amplitudes below the critical frequency, which are shown in Figure 6. It should be noted that Figure 6 is the one-to-one correspondence between simple harmonic waves and their amplitudes after propagation in the ionosphere. The frequency spectrum has the same shape as the original waveform.


FIGURE 6. Frequency spectrum of Bell Laboratory pulse after propagation in the ionosphere.

The spectrum of the time-frequency of EBt shows that the simple harmonic waves with higher frequencies arrive earlier, which is consistent with the time-frequency spectrum of the signal observed by the satellite (Massey et al., 1998). The brightest area in Figure 7 represents the frequency band with the highest concentration of energy and corresponds to the peak of EBt, too.


FIGURE 7. Time–frequency spectrum of Bell Laboratory pulse after propagation in the ionosphere.

Comparing the above results with other scholars’ great work (Liang and Zheng, 2009; Chen et al., 2011), our proposed method can explain the changes of waveforms after propagation in the ionosphere, such as the delay of the peak time, in light of the arrival time differences and amplitudes of simple harmonic waves. Besides, the application of inverse Fourier transform requires that the frequency components have the same wave front, in other words, the frequency components have the same propagation speed. Therefore, inverse Fourier transform is not suitable for the dispersive materials (Yao et al., 2019). Our proposed method is to superimpose the simple harmonic waves directly, which coincides with realistic physical process.

We have chosen different values of Δh to calculate the peak value of EBt. The results are shown in Table 4. The origin peak value of the pulse is 50,000 V/m, and peak value of the pulse after propagation in the ionosphere is in the order of mV/m. The order of amplitude of the calculation is what we care about most.


TABLE 4. Peak value and thickness of each layer.

4 Conclusion

This paper proposes a method for calculating the propagation of NEMP in the ionosphere. Our method surpasses the weakness of the inverse Fourier transform that erases the propagation time difference for different frequencies. The results show that the NEMP signal is broadened as 30 times as its original length in the ionosphere. The spectrum of NEMP signal has the shape of the original signal envelope. The obtained dispersed spectrum is consistent with the satellite observations. The method can be used for the design of the space-based NEMP detectors and satellite-ground docking experiments. The propagation model may be optimized upon receipt of downlink data from the satellite.

Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding authors.

Author contributions

YLW proposed the model and the method, and wrote the paper. DHZ and ZXL ran the simulation. LHW and YW analyzed the results. TCZ and BX wrote the paper. BFC and PL did the literature review and checked the paper. All authors contributed to the article and approved the submitted version.


This paper is supported by the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDA17040503).

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Keywords: nuclear electromagnetic pulse, ionosphere, closed-form expression, electromagnetic pulse propagation, dispersive propagation

Citation: Wei Y, Zhu D, Li Z, Wang L, Wang Y, Zhang T, Xing B, Cao B and Li P (2023) Dispersive propagation of nuclear electromagnetic pulse in the ionosphere. Front. Astron. Space Sci. 10:1201921. doi: 10.3389/fspas.2023.1201921

Received: 07 April 2023; Accepted: 23 October 2023;
Published: 09 November 2023.

Edited by:

Chen Zhou, Wuhan University, China

Reviewed by:

Sampad Kumar Panda, K L University, India
Xiang Xu, Wuhan University, China

Copyright © 2023 Wei, Zhu, Li, Wang, Wang, Zhang, Xing, Cao and Li. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Peng Li,; Baofeng Cao,