The theme of this paper is to establish a reliable ionospheric very low frequency (VLF) transmitter, which is also broad band. Two approaches are studied that generate VLF waves in the ionosphere. The first, classic approach employs a ground-based HF heater to directly modulate the high latitude ionospheric, or auroral electrojet. In the classic approach, the intensity-modulated HF heater induces an alternating current in the electrojet, which serves as a virtual antenna to transmit VLF waves. The spatial and temporal variations of the electrojet impact the reliability of the classic approach. The second, beat-wave approach also employs a ground-based HF heater; however, in this approach, the heater operates in a continuous wave mode at two HF frequencies separated by the desired VLF frequency. Theories for both approaches are formulated, calculations performed with numerical model simulations, and the calculations are compared to experimental results. Theory for the classic approach shows that an HF heater wave, intensity-modulated at VLF, modulates the electron temperature dependent electrical conductivity of the ionospheric electrojet, which, in turn, induces an ac electrojet current. Thus, the electrojet becomes a virtual VLF antenna. The numerical results show that the radiation intensity of the modulated electrojet decreases with an increase in VLF radiation frequency. Theory for the beat wave approach shows that the VLF radiation intensity depends upon the HF heater intensity rather than the electrojet strength, and yet this approach can also modulate the electrojet when present. HF heater experiments were conducted for both the intensity modulated and beat wave approaches. VLF radiations were generated and the experimental results confirm the numerical simulations. Theory and experimental results both show that in the absence of the electrojet, VLF radiation from the F-region is generated via the beat wave approach. Additionally, the beat wave approach generates VLF radiations over a larger frequency band than by the modulated electrojet.

## I. INTRODUCTION

Communication to submerged submarines dictates the very low frequency (VLF) band. This is because sea water skin depth is frequency dependent; the optimal frequency for an electromagnetic wave to penetrate to a depth H (in meters) into the sea water is given by f_{m} ∼ (16/H^{2}) kHz, which is in the VLF band. A ground-based VLF communications system requires a very large-size antenna for effective radiation. Moreover, VLF waves are ducted by the earth-ionosphere waveguide to polarize vertically and thus ought to be transmitted by a vertical antenna, which has a limitation on its physical height L. In other words, a practical vertical VLF antenna is electrically short, i.e., L/λ ≪ 1. Experiments and theory have shown an ionospheric current in combination with an HF heater can generate VLF waves with the ionospheric current becoming a large virtual antenna. The physical scale of an ionospheric virtual antenna suggests that there is potential in the development of a reliable broadband “ionospheric VLF transmitter.” This paper explores and compares two ionospheric VLF transmitter approaches through theory, numerical simulation, and experiment.

The electrojet appearing in the D and lower E regions of high latitude ionosphere has been considered as a potential virtual antenna, where the electron-neutral elastic collision frequency ν_{en} ∝ T_{e}^{5/6}. The Pederson current in the electrojet is driven by a dc electric field; this current can be modulated in time by an intensity modulated HF heater, which modulates the conductivity σ ≅ n_{0}e^{2}ν_{en}/mΩ_{e}^{2} of the Pederson current through modulated Ohmic heating; Ω_{e} is the electron gyrofrequency. The induced alternating current (AC) in the electrojet is then the antenna current for VLF radiation at the modulation frequency and its harmonics. Generation of VLF waves in the high latitude ionosphere via electrojet modulation has been studied experimentally^{1–13} and theoretically.^{12–22}

The heating and cooling rates as well as the available heating and cooling times in one modulation period affect the efficacy of conductivity modulation by an intensity modulated HF heater. Consequently, the efficacy of VLF generation via electrojet modulation is expected to decrease as the modulation frequency increases.

Moreover, such an ionospheric transmitter relies on the appearance of the electrojet; the signal quality and intensity of the radiation depend on the stability and strength of the electrojet, which often varies in time and does not appear all the time in the polar ionosphere. Contrary to the auroral electrojet, the equatorial electrojet that flows above the magnetic equator is rather predictable by time of day and season of the year. This suggests that the equatorial electrojet could be set up to become a reliable virtual ionospheric antenna. However, an operational facility on the magnetic equator needs to be possibly on a barge or on a platform similar to the platforms used for oil drilling and is not currently available.

In the present work, a beat wave modulation mechanism^{23,24} is explored for possibly establishing a reliable virtual ionospheric transmitter for VLF communications. It employs transmissions with slightly different frequencies from the two sub arrays of the HF transmitter divided equally. Theory shows that overlapping HF heater waves can modulate the electrojet in a similar way as that by an intensity modulated HF heater wave. Moreover, the beating of two HF heater waves in the F region of the ionosphere can also induce an AC for VLF radiation, which provides for the operation of an ionospheric VLF transmitter in the absence of a background electrojet.

In Sec. II, theory of VLF wave generation by modulated electrojet is formulated and numerical simulation results are presented. A beat wave mechanism for VLF wave generation in the F region of the ionosphere is presented in Sec. III. Experiments exploring these generation processes are reported in Sec. IV. A discussion of the work is given in Sec. V.

## II. VLF RADIATION OF MODULATED IONOSPHERIC ELECTROJET

### A. Formulation of electrojet modulation

The polar electrojet, consisting of Pederson and Hall current, is driven by a dc electric field, set to be **E**_{0} = $x\u0302$ E_{0}, in the D and lower E regions of the ionosphere, which is embedded in the geomagnetic field with the magnetic induction **B**_{o} = −$z\u0302$ B_{0}. The Pederson current is in the x-direction (i.e., in the same direction as the driving field) and the Hall current in the y-direction. The conductivities of the electron plasma in both current components vary with the collision frequency, which is electron temperature dependent. Thus, the electrojet current can be modulated through a time varying heating of the electron plasma; only the electron electrojet current modulation is considered because the electron plasma interacts with the HF heater much stronger and has much higher collision frequency than the ion plasma. The electron drift velocity is given to be

Then, the electron electrojet current density is obtained to be

where the conductivity tensor $\sigma \xaf\xaf$ = σ [$x\u0302$$x\u0302$ + $y\u0302$$y\u0302$ + (Ω_{e}/ν_{en})($x\u0302$$y\u0302$ – $y\u0302$$x\u0302$)] and the conductivity σ = n_{0}e^{2}ν_{en}/m(ν_{en}^{2 }+ Ω_{e}^{2}) is electron temperature dependent^{23} through the dependency ν_{en} ∝ T_{e}^{5/6}. Although the conductivity σ can be modulated through the electron density n_{0}, its modulation through ν_{en} by modulated electron heating is much more feasible and considered in the following.

We now model the modulated electrojet current as a localized transverse current source positioned at **r**′ = 0, and thus Eq. (2) becomes

where *V* is the volume of the current source.

The vector potential **A**(**r**, t) satisfies the inhomogeneous wave equation

under the transverse gauge condition ∇ **· A = **0; **J** is the current density induced by the radiation field **E** = −∂_{t}**A** in background plasma, which is derived to be **J** ≅ iε_{0}(ω_{p}^{2}/Ω_{e}) ∂_{t}**A**; thus, Eq. (4) becomes

which has the solution

where **p**(**r**, t) = (μ_{0}*V*n_{0}e^{2}E_{0}/4πm){[$x\u0302$ν_{en}(t − r/v_{A}) − $y\u0302$Ω_{e}]/[ν_{en}(t − r/v_{A})^{2 }+ Ω_{e}^{2}] = (μ_{0}*V*/4π) **J**_{e}(t − r/v_{A}); v_{A} ≅ 〈(ωΩ_{e}/ω_{pe}^{2})^{1/2}〉c is the average phase velocity of the radiation over the propagation path from the source to the receiver. The wave magnetic induction is obtained from the vector potential by

On the right hand side of Eq. (7), the first term represents the near field and the second term is the radiation (far) field. In the region right beneath the electrojet, the near field is dominant as the modulation frequency f_{1} < 50 Hz and can be neglected in the VLF region.

In Eq. (7), the near field is proportional to **J**_{e}(t − r/v_{A}) given by Eq. (2) and the radiation field is proportional to ∂_{t} **J**_{e}(t − r/v_{A}) given by

Set $r\u0302$** ****=**** **$x\u0302$ sin θ cos φ** **+** **$y\u0302$ sin θ sin φ −$z\u0302$ cos θ in Eq. (9), where θ and φ are the poloidal and azimuthal angles with respect to the downward direction, the magnitude of **B** is given by

The near field is included in the numerical analyses only for comparison with the extremely low frequency (ELF) experimental results which are measured near the heating site. In the VLF applications, only the radiation field needs to be considered and Eq. (10) is reduced to

In the presence of HF heating, the electron temperature is governed by the electron thermal energy equation^{25,26}

where **v**_{e} is the electron fluid velocity, δ(T_{e}) is the average relative energy fraction lost in each collision, ν_{e}(T_{e}) is the effective collision frequency of electrons with neutral particles (it accounts for both elastic and inelastic collisions), T_{n} is the temperature of the background neutral particles; the ionization loss becomes significant as electrons are heated up to high temperature; Q is the total Ohmic heating power density in the background plasma and contributed by the electrojet current and the HF heater wave, $K\xaf\xaf$_{e} = ($x\u0302$$x\u0302$ + $y\u0302$$y\u0302$) K_{⊥} + $z\u0302$$z\u0302$ K_{z} is the thermal conduction tensor, K_{z} = 3n_{0}T_{e}/mν_{e}, K_{⊥} = (ν_{e}/Ω_{e})^{2}K_{z}, and m is the electron mass; the ionization loss term on the left hand side (LHS) of Eq. (12) will be neglected, though in the strong heating cases, the electron temperature can become high enough to contribute thermal ionization. Let T_{eb} be the background electron temperature in the absence of the heating wave and substitute it into Eq. (12), the solar source power = δ(T_{eb})ν_{e}(T_{eb})(T_{eb} – T_{n}) – (2/3)(Q_{E0}/n_{0}) is determined, here Q_{E0} = ν_{en}(T_{eb})n_{0}mu^{2}_{e}(T_{eb}) is the Ohmic heating power density contributed by the unmodulated electrojet current. The total Ohmic heating power density is given by

where **J**_{et} = −en(**u**_{e} + **v**_{pe}) is the total electron current density in the background plasma carrying an electrojet and interacting with the HF heater, angle brackets indicate the time average over the HF wave period, and σ_{0} = n_{0}e^{2}/mν_{en} is the conductivity of the plasma responsible for the Ohmic loss; **u**_{e} is given by Eq. (1) and **v**_{pe} is the quivering velocity of electrons imposed by the HF wave electric field; Q_{E} = ν_{en}n_{0}mu_{e}^{2} and Q_{H} = ν_{en}n_{0}m〈|v_{pe}|^{2}〉, where u_{e} = eE_{0}/mΩ_{e} and 〈|v_{pe}|^{2}〉 varies with the modulation form of the HF heater. An intensity modulated HF heater can be radiated directly by the HF transmitter. It can also be set up through a beat wave approach by overlapping two CW HF heaters in the electrojet,^{27} where the two HF heaters either have a frequency difference in the VLF band or introduce a spatially periodic heating that converts to temporal modulation with the aid of wind.^{28} We now determine the Ohmic heating power density Q_{H} in Eq. (13) delivered by an intensity modulated HF heater.

#### 1. Which is radiated directly by the HF transmitter

Consider electrojet heating by an intensity modulated HF heater of left-hand (LH) circular polarization (X-mode heater), which is transmitted along the magnetic zenith, the wave electric field is given by **E**_{p} = ($x\u0302$ − i $y\u0302$)(ε_{p}/2)exp[i(k_{0}z–ω_{0}t)] + c.c., where ω_{0} and k_{0} are the heater radian frequency and wavenumber, respectively. The quivering velocity **v**_{pe} of electrons is obtained to be **v**_{pe} ≅ −i($x\u0302$ − i $y\u0302$)[eε_{p}/2m(ω_{0} − Ω_{e})]exp[i(k_{0}z − ω_{0}t)] + c.c. The power of the HF heater is modulated periodically by a modulation function M(t), i.e., ε_{p}^{2 }= ε_{p0}^{2}M(t), where M(t) has a general form, M(t) = ∑_{k=0} M_{k} cos k(ω_{1}t + φ_{k}); ω_{1}/2π = *f*_{1} = 1/T_{1} is the modulation frequency. For example, in the three modulation schemes: (1) rectangular wave of 50% duty cycle, M_{R}(t) = [1/2 + ∑_{k=1} sinc(kπ/2)cosk(ω_{1}t + π/2)]; (2) sine wave, M_{S}(t) = [3/8 + ½ cos(ω_{1}t–π/2) − (1/8)cos2ω_{1}t]; and (3) half-wave rectified wave, M_{H}(t) = {1/4 – (4/π)∑_{k=1} [(sin kπ/2)/k(k^{2}–4)]cos k(ω_{1}t–π/2)}. Thus, in Q_{H}, 〈|v_{pe}|^{2}〉 = v_{q}^{2} M(t) and v_{q}^{2 }≅ [eε_{p0}/m(ω_{0} −Ω_{e})]^{2}, where ε_{p0}^{2 }= η_{0}P_{r}Gsin^{2}θ_{d}/4πh^{2}; η_{0} = 377 Ω is the free space intrinsic impedance, P_{r} is the radiated power of the HF transmitter; G = G_{0} − L_{D}, G_{0} is the antenna gain of the transmitter and L_{D} is the D region absorption loss; h and θ_{d} are the altitude and the magnetic dip angle of the heated region.

#### 2. Which is set up via a beat wave approach

Consider that the HF heater array is split into two sub-arrays, transmitting CW X-mode HF waves at two slightly different frequencies f_{01} and f_{02}, where f_{01} = f_{0}, f_{02} = f_{0} + f_{1}, and f_{1} is the beat-wave frequency in the VLF band; the wave field **E**_{p} of the two sub-array heaters is **E**_{p} = **E**_{p1} + **E**_{p2} = ($x\u0302$ − i $y\u0302$)(E_{p0}/2)[1 + exp(−iω_{1}t)]exp[i(k_{0}z–ω_{0}t)] + c.c., where E_{p0} = ε_{p0}/√2 is the field amplitude of each heater beam. The electron quiver velocity **v**_{pe} = −i($x\u0302$ − i $y\u0302$) [eE_{p0}/2m(ω_{0} − Ω_{e})][1 + exp(−iω_{1}t)]exp[i(k_{0}z–ω_{0}t)] + c.c., which leads to the power modulation (PM) function M_{B}(t) = 1 + cosω_{1}t, i.e., 〈|v_{pe}|^{2}〉 = v_{q}^{2} M_{B}(t).

The fractional energy loss term on the LHS of Eq. (12) involves both elastic and inelastic collisions. The main processes involved in the inelastic collisions in the energy range of interest (<6 eV) are the rotational and vibration excitation of N_{2} and O_{2}. Loss through optical excitation is neglected. The fractional electron heat loss rate δ(T_{e})ν_{e}(T_{e}) = (δ_{el} + δ_{r} + δ_{v}) ν_{e}, a sum of three main causes from the elastic collisions with the neutral particles and from the rotational and vibration excitations of the neutral particles, is determined to be^{21,29}

where a normalized temperature χ = T_{e}/T_{e0} is introduced, ν_{en0} = ν_{en}(T_{e} = T_{e0}) and a reference equilibrium electron temperature T_{e0} = 1500 K is assumed; and

### B. Numerical model of VLF wave radiated by heater modulated electrojet

In the numerical analysis, dimensionless variables and parameters are introduced: ν_{en}/ν_{en0} = χ^{5/6}, τ = ν_{en0}t, η= (ν_{en0}/v_{t0})z, R = (ν_{en0}/v_{A})r, ξ = τ − R, |c**B**(**r**, t)/E_{0}| = *B*(ξ), ω_{10} = ω_{1}/ν_{en0}, T_{e0} = 1500 K is set, v_{t0} = (T_{e0}/m)^{1/2 }= 1.5× 10^{5 }m/s, η = (eE_{0}/mΩ_{e}v_{t0})^{2}, b = (ν_{en0}/Ω_{e})^{2}, and α = (v_{q}/v_{t0})^{2}; and set θ = 0 in Eq. (11) to simplify the analyses. Thus, Eqs. (12) and (11) are normalized to the dimensionless forms

and

where d_{τ} = d/dτ and d_{η} = d/dη, ζ = T_{n}/T_{e0}, Δ = χ_{b}^{5/6 }+ 8.38χ_{b}^{−1/2} + μ_{I}(χ_{b})] (χ_{b} – ζ), and χ_{b} = T_{eb}/T_{e0}; *Λ*_{0} = (5 *V* ω_{pe}^{2}Ω_{e}b^{3/2}/24πcv_{A}^{2}R).

When only temporal modulation is considered, the spatial derivative term on the RHS of Eq. (16a) is set to zero, which reduces Eq. (16a) to

Three cases of modulation frequency f_{1} = 1, 3, and 10 kHz are presented as examples. The numerical analysis is carried out for electrojet appearing near 90 km, where T_{eb} = 200 K, T_{n} ≅ T_{i} ≅ 190 K, and E_{0} = 50 mV/m. A heater frequency of ω_{0}/2π = 3.2 MHz is used; ν_{en0} = 9 × 10^{5 }s^{−1}, M_{n}/m = 5.52 × 10^{4}, Ω_{e}/2π = 1.4 MHz ≫ ν_{en0}, and v_{q} = 1.56× 10^{4}ε_{p0} m/s, where ε_{p0} is in V/m, and c is the speed of light in free space. Assume ε_{p0} = 1.5 V/m, which leads to v_{q} = 2.33 × 10^{4 }m/s, α = 0.024, and consider rectangular wave modulation having 50% duty cycle, i.e., M_{R}(τ) = [1/2+ ∑_{k=1} sinc(kπ/2) cos k(ω_{10}τ + π/2)].

Equation (16c) is solved numerically subject to the initial condition χ(0) = 1. Time functions of modulated electron temperature produced by the X-mode heater at the three modulation frequencies are obtained, as shown in Fig. 1(a). The results are substituted into Eq. (16b) to obtain the radiation fields *B*(ξ) in the three cases, which are presented in Fig. 1(b), in which the vertical axis has an arbitrary unit. As shown, in the cases of 1 and 3 kHz modulations, the on periods are long enough for the heating to reach a steady state level; but the off period in the 3 kHz modulation is too short for cooling, thus the temperature modulation amplitude drops. Both heating and cooling times are too short in the case of 10 kHz modulation; it takes the modulation three periods to reach a steady state which settles at a much lower modulation amplitude. Consequently, the radiation intensity decreases with the increase of the modulation frequency, as demonstrated in Fig. 1(b), in which each signal also contains harmonic components. Taking a Fourier transform, the time harmonic component at the modulation frequency (i.e., the first harmonic component) is extracted; the dependency of the amplitude on the modulation frequency is presented in Fig. 1(c). As shown, the amplitude of the VLF signal, generated via electrojet modulation, decreases monotonically with the frequency.

The numerical results indicate that in the VLF band, the second term on the LHS of Eq. (16c) is much less than ω_{10}χ, i.e., δ(T_{e})ν_{e}(T_{e}) ≪ ω_{1}, thus, Eq. (16c) may be solved approximately by setting χ = χ_{E} + δχ, where χ_{E} = T_{eE}/T_{e0} is determined by the relation

and δχ is governed by the equation

d_{τ} δχ ≅ (2/3)χ_{E}^{5/6}α∑_{k=1} M_{k} cos k(ω_{10}τ + φ_{k}).

Thus, the induced time harmonic electron temperature perturbation δχ is obtained to be

## III. VLF WAVE GENERATION BY BEATING OF TWO HF HEATER WAVES

The momentum equations of electrons and ions are given as

and

where Ω_{i} = eB_{o}/M is the ion cyclotron radian frequency and ν represents the collision frequency. We now combine Eqs. (18) and (19) into one; in the resultant equation, those terms contributing directly to the nonlinear beating are equated to be

where **J** = −ne(**v**_{e} − **v**_{i}); the electron and ion inertial terms and the ion convective and pressure terms, which do not directly contribute to the nonlinear beating, are neglected; the relations mn_{e}ν_{ei} = Mn_{i}ν_{ie}, n_{e} ≅ n_{i} = n, ν_{en} ≪ Ω_{e}, and ν_{in} ≪ Ω_{i} are applied to simplify the equation. The two terms on the RHS of Eq. (20) correspond to the ponderomotive and thermal pressure forces.

Equation (20) is then solved to obtain the nonlinear beating current density **J**_{B} = **J**_{BP} + **J**_{BT} at the difference frequency of the two HF heater waves, where **J**_{BP} and **J**_{BT} are driven by the ponderomotive and thermal pressure forces, respectively. The result is

where **V**_{pe} is the electron quivering velocity induced by the HF fields and the electron density is n_{e} = n_{0} + Δn_{e}; here, n_{0} is the background electron density and Δn_{e} is the density of irregularities; 〈 〉 represents a VLF bandpass filter; P_{e} = nT_{e} and T_{e} is governed by the electron thermal energy equation. Only temporally modulated heating is considered, the convective and diffusion terms are set to zero in the electron thermal energy equation, which becomes

where δ(T_{e})ν_{e}(T_{e}) is the average fraction of energy loss rate in (elastic and inelastic) collisions with the neutrals. Let T_{e} = T_{e0} + 〈δT_{e}〉 + ⋯⋯ and 〈|**V**_{pe}|^{2}〉 = Wexp(−iω_{1}t) + c.c., the induced temperature perturbation 〈δT_{e}〉, which oscillates at the beat frequency ω_{1}, is obtained to be

where ν_{el} = ν_{ei} + ν_{en} is the elastic collision frequency and δν = δ(T_{e0})ν_{e}(T_{e0}) + 2ν_{ei}(m/M).

The HF transmitter array is split into two sub-arrays, transmitting CW heaters at slightly different frequencies f_{01} and f_{02} along the geomagnetic zenith, where f_{01} = f_{0}, f_{02} = f_{0} + f_{1}, and f_{1} is the beat-wave frequency in the VLF band, the wave field **E**_{p} of overlapped heaters is

where “±” correspond to O/X-mode heaters; E_{p0} is the field amplitude of each heater beam, ψ is the phase difference of two HF waves transmitted by the two sub-arrays; and ω_{0} and k_{0} are the heater radian frequency and wavenumber, respectively. The induced electron velocity is derived to be

Thus, W = [eE_{p0}/m(ω_{0} ± Ω_{e})]^{2}e^{iψ}. With the aid of Eqs. (23) and (25), Eq. (21) becomes

where **J**_{BP}(**r**) = (e/2Ω_{e})($y\u0302$∂_{x} + $x\u0302$∂_{y}) (n_{e}W) and **J**_{BT}(**r**)= −i(2e/3Ω_{e}) [1 − i(δν/ω_{1})][ν_{el} ω_{1}/(ω_{1}^{2 }+ δν^{2})] ($y\u0302$∂_{x} − $x\u0302$∂_{y}) (n_{e}W); the (ω_{0} ± Ω_{e})^{−2} dependence of W indicates that X-mode heaters are more effective than O-mode heaters to generate the beat-wave current.

The radiation of the time harmonic current source (26) is evaluated via the vector potential ** A**(

**r**, t), whose phasor function

**A**(

**r**) is given by

^{18}

where k is the wavenumber of the radiation and dV′ = dx′dy′dz′ is the differential volume of the induced current distribution at **r** = **r**′. Choosing the coordinates such that the x and z axes are on the meridian plane and setting η and ξ as the horizontal and vertical (upward) axes in the meridian plane, then, (η, ξ) are related to (x, z) by η = x sin θ_{d} + z cos θ_{d} and ξ = x cos θ_{d} − z sin θ_{d}, where θ_{d} is the magnetic dip angle; and ∂_{x} = sinθ_{d} ∂_{η} + cosθ_{d} ∂_{ξ}, dV′ = dη′dy′dξ′, and (ξ_{1}, ξ_{2}) is the altitude region of the VLF source current.

Set n_{e} = n_{0}(ξ) + Δn_{ex}(η, ξ) + Δn_{ey}(y, ξ), where n_{0} is the background plasma density distribution and Δn_{ex}(η, ξ) and Δn_{ey}(y, ξ) are the density irregularities of different polarizations, hence, ($y\u0302$∂_{x} ± $x\u0302$∂_{y}) (n_{e}E_{p0}^{2}) = $y\u0302$[cos θ_{d} ∂_{ξ} (n_{e}E_{p0}^{2}) + sinθ_{d} E_{p0}^{2}∂_{η}Δn_{ex}] ± $x\u0302$ E_{p0}^{2}∂_{y}(Δn_{ey}) and ∫($y\u0302$∂_{x′} + $x\u0302$∂_{y′}) (n_{e}E_{p0}^{2}) dV′ = $y\u0302$ cos θ_{d} A_{c}[(n_{e}E_{p0}^{2})|ξ_{2} − (n_{e}E_{p0}^{2})|ξ_{1}] ± $x\u0302$∫(E_{p0}^{2}∂_{y′}Δn_{ey}) dV′, where A_{c} is the effective cross section area of the HF heaters.

In the overdense situation, i.e., HF heaters are reflected below the f_{o}F2 layer (f_{0} < f_{o}F2/f_{x}F2 for the O/X-mode heater, respectively), we may assume a linear density profile with a scale length L below the reflection layer and n_{0} is modeled to be n_{0}(ξ) = n_{c}[1 + (ξ − ξ_{c})/L], where n_{c} = n_{0}(ξ_{c}) is the electron density at the altitude ξ_{c} of the reflection layer. In the underdense situation, n_{0}(ξ) = n_{c}[1 ± (ξ − ξ_{c})/L_{1,2}] for ξ ≶ ξ_{c}, where L_{1}/L_{2} are the scale lengths below/above the f_{o}F2 layer and n_{c} is the density at the f_{o}F2 layer. The amplitude E_{p0} of the HF heater is also a function of ξ; E_{p0}(ξ) can be approximated by an airy function in the overdense case and by a WKB solution in the underdense case; Δn_{e} = Δn_{ex}(η, ξ) + Δn_{ey}(y, ξ) = ∫ n_{x}(ξ, κ) exp[iκ(η sin θ_{d}+ ξ cos θ_{d})]dκ/2π + ∫ n_{y}(ξ, κ) e^{iκy}dκ/2π + c.c.

The magnetic flux density of the radiation is given by

The density irregularities in the background plasma may be enhanced by the HF heaters via filamentation instability,^{30} which, in turn, enhances the beat-wave generation.

In the VLF band, δν ≪ ω_{1}; the radiation amplitude in Eq. (28) is proportional to

where r_{0} is the distance to the ground, J_{B0} is the amplitude of the heater induced current density at the beat frequency, *V* = *AH* is the volume of the induced current density distribution, *A* is the cross section area in the horizontal plane, and *H* is the depth of the distribution in the vertical direction.

The ratio of the thermal nonlinearity to ponderomotive nonlinearity, i.e., |J_{BT}/J_{BP}|, is approximately to 4ν_{el}/3ω_{1}; this is in agreement with the previous result of Stenflo^{31} on analyzing the nonlinear coupling for stimulated scattering in the ionosphere. Apparently, this ratio is very small in the E and F regions. In the D region from 80 to 90 km, this ratio can be near 1. Therefore, we have to compare this ratio with the beat current induced via thermal nonlinearity in the D region to that induced via ponderomotive nonlinearity in the F region, i.e., the ratio r_{F}J_{D}*V*_{D}/r_{D}J_{F}*V*_{F} which is given by

where the subscripts D and F represent D and F regions and P is the power density. Because (4ν_{el}/3ω_{1}) ∼ 1, (P_{D}A_{D}/P_{F}A_{F}) ∼ 1, (*H*_{D}/*H*_{F})(r_{F}/r_{D}) < 1, in general, and (N_{eD}/N_{eF}) ≪ 1, lead to r_{F}J_{D}*V*_{D}/r_{D}J_{F}*V*_{F} ≪ 1, with the conclusion that the beat wave is generated mainly via ponderomotive nonlinearity in the F region.

## IV. EXPERIMENTS AND RESULTS

### A. Generation of VLF radiation by electrojet modulation

Results of three experiments,^{12} carried out at Gakona, Alaska in 2010 and 2011 by transmitting X-mode HF heaters along the geomagnetic zenith at full power (3.6 MW) with different electrojet modulation schemes, are presented in the following: (1) On April 4, 2010 from UTC 07:11 to 07:31 (23:11 to 23:31, Alaska daylight time), the 12 × 15 array of the HAARP HF transmitter was split into two 6 × 15 sub-arrays, transmitting at 3.2 MHz. One was run at CW full power (1.8 MW) and the other run with full power at 100% PM by rectangular waves of 5 kHz, 8 kHz, and 13 kHz, each modulation case was run for 2 min; (2) On April 9, 2010 from UTC 20:43 to 21:03, full power HF heater of 3.2 MHz was run at 50% PM by rectangular waves of 5 kHz, 8 kHz, and 13 kHz; and (3) On July 25, 2011 from UTC 09:42 to 09:49, a beat wave approach was applied for VLF generation at a single beat frequency for the 60 s period. The 12 × 15 array of the transmitter was split into two 6 × 15 sub-arrays, one sub-array transmitted full power at 4.2 MHz and the other full power at 4.2 MHz + f_{1}, where f_{1} was changed sequentially in the order of 2 kHz, 3.5 kHz, and 5.5 kHz. The background magnetic induction variation Δ**B** was monitored by the fluxgate magnetometer located at Gakona.

The N-S and E-W components of the radiation magnetic field from the first two experiments and the radiation spectral power density from the third experiment are presented in the first row of Fig. 2. The time averaged field amplitude of each component in experiment 3 is obtained by the square root of the integration of the corresponding spectral power density in column C of Fig. 2 over the corresponding frequency band. Averaged magnetic amplitudes δ**B** of the VLF radiations generated in the experiments, obtained by the vector sum of the averaged N-S and E-W components of the wave magnetic flux densities, are presented in the second row of Fig. 2; the background magnetic variations Δ**B** in the corresponding experimental periods are presented in the third row of Fig. 2 as a reference of the background condition. As shown, the intensity of the VLF radiation has a decreasing dependency on the radiation frequency. The experimental results confirm the numerical simulation presented in Fig. 1(c), which has a similar frequency dependency. Such a decreasing frequency dependency is explained to be due to the decrease of the heating and cooling times of the electron plasma with the increase of the modulation frequency. While the radiation amplitudes at the same frequency in the three experiments are quite different, it is also shown that there are significant differences in the geomagnetic variations, which are approximately 120 nT, 20 nT, and 25 nT deviation from the background geomagnetic induction during the three experiments. The geomagnetic variation is mainly caused by the appearance of the electrojet and the intensity of the VLF radiation generated by the electrojet modulation scheme is proportional to the electrojet current. The comparison of Figs. 2(a) and 2(b) indicates that dependence of the VLF radiation intensity on the electrojet is stronger than linear dependence and the comparison of Figs. 2(b) and 2(c) indicates that the electrojet modulation by power modulated HF heater is still about 5 dB more effective than that by the beat wave approach after being calibrated by the (ω_{0} – Ω_{e})^{−2} dependence of Q_{H} in Eq. (13).

### B. Beat wave generation of VLF radiation in the absence of electrojet

#### 1. Experiment background conditions

Nighttime and daytime experiments on beat wave generation of VLF waves were conducted on March 22 and 29, 2011 from UTC 07:05 to 07:45 (23:05 to 23:45, Alaska daylight time) and from UTC 20:00 to 20:40 (12:00 to 12:40, Alaska daylight time), respectively.^{24} Although meridional midnight at Gakona, AK nominally occurs at UTC 09:40, the solar zenith angle was 110.1°–113.0° during the March 22 experimental period, indicating the ionosphere was already in the earth's shadow below approximately 400 km. Diagnostic instruments at Gakona, Alaska were used. The background conditions monitored by the 30 MHz riometer are presented in Fig. 3(a), in which a rectangular box in each plot specifies the experimental period. There is no/significant 30 MHz riometer absorption (0/0.3 dB) in the nighttime/daytime experimental period. It suggests that D-region was absent/present in the nighttime/daytime experiment. Representative nighttime/daytime ionograms are presented in Fig. 3(b). The ionospheric current produced magnetic flux density appears as the deviation of the background magnetic flux density, which was recorded by the Gakona magnetometer to refer the intensity and variation of the ionospheric current in the experimental periods. Presented in Fig. 3(c) are the magnified segments of the Gakona's H, D, and Z plots for the experimental periods. As shown, the magnetic deviations of the three components in both experimental periods are small, less than 20 nT in vector sum, indicating that the ionospheric current was weak during the experiments.

#### 2. Experiments and results

Similar to that on April 4, 2010, the HAARP HF transmitter was operated by splitting it into two 6 × 15 sub-arrays, which were run at CW full power (1.8 MW each); one sub-array transmitted at 3.2/3.37 MHz and the other at 3.2/3.37 MHz + f in the nighttime/daytime experimental period, where f = 3.5 kHz, 5.5 kHz, 7.6 kHz, 9.5 kHz, 11.5 kHz, 13.6 kHz, 15.5 kHz, 17.5 kHz, 19.6 kHz, and 21.5/21.7 kHz. The experiment was conducted with the X-mode HF heater directed along the geomagnetic zenith and run at 2 min on and 2 min off for each f. The nighttime riometer and ionogram presented in Fig. 3 show that there were no significant D and E layers present and the HAARP operating frequency of 3.2 MHz was greater than foF2 (≈2.2 MHz) as well as fxF2 (≈3 MHz) in the experiment.

Radiation was generated at all ten beat frequencies f. The receiver recorded both North-South (N-S) and East-West (E-W) components of the power spectral density. The two components are combined to obtain the total power spectral density of each radiation, which is presented in Fig. 4(a). In the plots, the intensities of the radiation and the background noise are marked by a circle and a cross. As shown, the recorded radiation has good signal to noise ratio (S/N). The time averaged field amplitude of each component at a frequency is given by the square root of the integration of the corresponding spectral power density over the corresponding frequency band. The magnitudes of the vector sums of the averaged N-S and E-W components of the wave magnetic flux densities are presented in Fig. 4(b). As shown, the radiation amplitude of the nighttime experiment increases continuously with the frequency from 3.5 to 11.5 kHz. After the radiation amplitude drops from the maximum at 11.5 kHz with an increase of the frequency, it becomes relatively flat in the frequency range from 13.6 to 21.5 kHz. A similar frequency dependency is also observed in the daytime experiment, except that the maximum of the radiation amplitude appears at 7.6 MHz. The radiation amplitude measured on the ground is expected to be inversely proportional to the cubic of the distance between the generation region and the ground. The ionograms presented in Fig. 3(b) indicate that the generation region of the VLF radiation in the nighttime experiment is much higher than that in the daytime experiment; on the other hand, the radiation amplitude of the daytime experiment is in fact smaller than the corresponding one of the nighttime experiment. This is because the D region absorption in the daytime experimental period, as shown in Fig. 3(a), is large and the beat wave generation in the nighttime experiment was performed with the X-mode heater wave frequency f_{0} > fxF2, which allows the generation region to be large.

Two overlapped HF heaters of slightly different frequencies can modulate the electrojet at beat frequency to generate VLF waves similar to that employing an amplitude modulation (AM) heater. However, the intensity of the VLF radiation generated by the ionospheric current modulation scheme has a decreasing dependency on the modulation (radiation) frequency due to the decrease of the heating and cooling times. Thus, the increasing dependency appearing in the low frequency part and the relatively insensitive dependency in the high frequency part, shown in Fig. 4, indicate that the ionospheric current (electrojet) modulation mechanism is not likely to contribute significantly on the VLF wave generation in this beat wave approach. The change from increasing to decreasing frequency dependency after the radiation frequency exceeds a critical value seems to depend on the background plasma conditions.

## V. CONCLUSIONS

A conventional approach to set up an ionospheric VLF transmitter is to use the ionospheric electrojet as a virtual antenna and to apply an intensity modulated HF heater to induce an AC in the electrojet for VLF radiation. Theory has been formulated to establish a numerical model. Experiments have been conducted at HAARP to explore this VLF transmitter. The radiation of the transmitter has been detected by the ground VLF receivers to demonstrate its feasibility. However, the response of the electrojet to the modulation goes down as the modulation frequency increases, causing the radiation intensity to fall off with an increase of the modulation frequency, as indicated by the numerical simulation and verified by the experimental results. Moreover, this approach to generate VLF radiation relies on the appearance of the electrojet; the signal quality and intensity of the radiation depend on the stability and strength of the electrojet, which often varies in time and does not appear all the time in the polar ionosphere.

On the other hand, theory shows that two overlapped HF heaters of slightly different frequencies can modulate the electrojet at beat frequency to generate VLF waves similar to that employing an AM heater. In addition, this beat wave approach, via the nonlinearity and inhomogeneity of the ionospheric plasma, can also generate VLF radiation in the absence of the electrojet. Experiments to explore this beat wave approach have been conducted.

Two beat wave generation experiments were conducted, one in the nighttime in the absence of D/E layers and one in the daytime with significant D-layer absorption. X-mode HF heaters of slightly different frequencies were transmitted at CW full power. In the nighttime experiment, the ionosphere was underdense to the HF heaters. VLF waves at ten beat frequencies ranging from 3.5 to 21.5/21.7 kHz were generated. The frequency dependencies of the daytime/nighttime radiation amplitudes are quite similar, but the nighttime radiation is much stronger than the corresponding one generated in the daytime experiment, consistent with the facts that the D region absorption reduced the heater wave intensity significantly in the daytime experiment. Moreover, the nighttime radiation was generated in the underdense situation attained with relatively low heater frequency, which were identified in the previous experiment^{23} to be the favorable beat wave generation conditions.

In conclusion, a more reliable ionospheric VLF transmitter is to apply two overlapped HF heater waves of slightly different frequencies for electrojet modulation. The experimental results show that this approach can modulate the electrojet at beat frequency to generate VLF radiation similar to that employing an AM heater wave. The experimental results also show that VLF radiation with good S/N can still be generated in a weak electrojet situation. The results depend on the HF heater intensity rather than the electrojet strength. Moreover, the radiation was generated in much larger frequency band than generated by the modulated electrojet. These combined characteristics suggest that there is merit in further investigation of the beat wave approach as a reliable, broad band ionospheric VLF transmitter.

## ACKNOWLEDGMENTS

I am grateful to Dr. Lee Snyder for helpful discussion and to John Labenski, BAE Systems-Technology Solutions, for recording VLF radiation. This work was supported by the High Frequency Active Auroral Research Program (HAARP), AFRL at Hanscom AFB, MA, and by the Office of Naval Research, Grant No. ONR-N00014-10-1-0856.