# Dispersive propagation of nuclear electromagnetic pulse in the ionosphere

^{1}State Key Laboratory of NBC Protection for Civilian, Beijing, China^{2}College 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

The plasma frequency of the ionosphere is

#### 2.1.2 Critical frequency

#### 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

(2) Collision frequency of electrons with Oxygen molecules

(3) Collision frequency of electrons with Oxygen atoms

(4) Collision frequency of electrons with Helium atoms

From Eqs 1–4,

The collision frequency of electron-ion is

where *i*, which represents *i*.

The total electron collision frequency

#### 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

In vacuum, the speed of a simple harmonic wave

where,

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

In the ionosphere, the attenuation of amplitude with distance

where

### 2.2 Dispersive propagation in the ionosphere

#### 2.2.1 Decomposition of NEMP

In the light of Eqs 7–11, the influence of the ionosphere differ with the frequency

The Fourier transform is applied to

where

where

#### 2.2.2 A simple harmonic wave propagation in the ionosphere

##### 2.2.2.1 Propagation from ${\mathit{h}}_{\mathbf{0}}$ to ${\mathit{h}}_{\mathbf{1}}$

For a simple harmonic wave with frequency

and the amplitude is calculated by

The expression is derived from Eq. 12 as

##### 2.2.2.2 Propagation from ${\mathit{h}}_{\mathbf{0}}$ to ${\mathit{h}}_{\mathbf{2}}$

When propagating from

The propagation time is calculated by

The absorption loss can be derived from Eqs 10, 11

where

##### 2.2.2.3 Propagation from ${\mathit{h}}_{\mathbf{0}}$ to ${\mathit{h}}_{\mathit{M}}$

By analogy, when the simple harmonic wave

where

##### 2.2.2.4 Definition domain

Each simple harmonic wave has the same duration

Equation 22 represents that when

Taking the moment when

In particular, the time domain of

#### 2.2.3 Superposition of simple harmonic waves

The ionosphere will reflect the simple harmonic waves with frequencies below the critical frequency

With the consideration of the critical frequency and the different definition domain, the simple harmonic waves

## 3 Simulation

The NEMP is emitted at the ground and is received at an altitude of 800 km and let

### 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

Parameters of Bell Laboratory pulse are shown in Table 2. The original waveform ^{−11}%. Compared to the order of original waveform, the error is negligible.

### 3.2 Results and discussion

Let

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

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.

The spectrum of the time-frequency of

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

## 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.

## Funding

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.

## Publisher’s note

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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, ChinaCopyright © 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, lipeng@sklnbcpc.cn; Baofeng Cao, caobaofeng@sklnbcpc.cn