The excitation of Terahertz (THz) radiation by the interaction of an ultrashort laser pulse with the modes of a miniature corrugated plasma waveguide is considered. The axially corrugated waveguide supports the electromagnetic modes with appropriate polarization and subluminal phase velocities that can be phase matched to the ponderomotive potential associated with the laser pulse, making significant THz generation possible. This process is studied via full format Particle-in-Cell simulations that, for the first time, model the nonlinear dynamics of the plasma and the self-consistent evolution of the laser pulse in the case where the laser pulse energy is entirely depleted. It is found that the generated THz is characterized by lateral emission from the channel, with a spectrum that may be narrow or broad depending on the laser intensity. A range of realistic laser pulse and plasma parameters is considered with the goal of maximizing the conversion efficiency of optical energy to THz radiation. As an example, a fixed drive pulse (0.55 J) with a spot size of 15 μm and a duration of 15 fs produces a THz radiation of 37.8 mJ of in a 1.5 cm corrugated plasma waveguide with an on axis average density of 1.4 × 1018 cm−3.

Terahertz radiation (THz) lies between microwave and infrared wavelengths in the electromagnetic (EM) spectrum and spans frequencies from 300 GHz to 20 THz. A wide variety of applications for THz radiation1 can be found including time domain spectroscopy (TDS),2 medical and biological imaging,3,4 and remote detection5–7 among others. Most airports, for instance, use millimeter wave/THz scanners for security checking. Existing small scale THz sources based on laser-solid interactions are limited to μJ/pulse levels due to material damage8 although a recent discovery using optical rectification (OR)9–11 in organic crystals can exceed this limit. The need for higher power sources has led to the consideration of THz generation via laser-plasma interactions,12–14 in which THz peak energies of tens of μJ can be achieved. Higher energy THz pulses can be generated at accelerator facilities via synchrotron15 or transition radiation.16 Such facilities are relatively large and expensive to operate. This motivates research interest in small-scale, high efficiency terahertz sources.

Research has been actively conducted to investigate THz radiation generation by laser pulses propagating in plasmas since it was first demonstrated by Hamster et al.17,18 In this case, the source of the radiation is the current driven by the ponderomotive force of a laser pulse. However, generation of radiation by a laser pulse propagating through a uniform plasma is generally minimal. In order for electromagnetic modes to efficiently couple to the driving source, which travels at the group velocity of the laser pulse, the plasma must be inhomogeneous or immersed in a strong background magnetic field. A THz generation scheme involving laser pulses (or possibly electron beams) propagating through axially corrugated plasma channels has been proposed by Antonsen et al.19,20 In this scheme, the corrugated plasma channel acts as a slow wave structure that supports electromagnetic modes that can be phase matched with the driver. The scheme offers the possibility of much higher efficiency of THz generation under conditions of full laser pulse depletion. This prospect is investigated here.

In this paper, we report theoretical, numerical, and simulation results of ponderomotively driven THz generation by laser pulses propagating through corrugated plasma waveguides. Such waveguides have already been realized in the laboratory.21,22 The experimental set up is shown in Fig. 1(a). A Nd:YAG laser pulse is line-focused onto a cluster jet creating a plasma that hydrodynamically expands, leading to the formation of a channel with a transversely parabolic density profile. The periodic structure is created by spatially modulating the laser intensity on the back side of the axicon or by periodically obstructing the cluster flow. A second ultra-short Ti:Sapphire laser pulse is then injected into the channel following the channel formation pulse. The second pulse drives the terahertz generation. Shown in Fig. 1(b) is a snap shot of the experimentally generated axially modulated plasma density profile.

FIG. 1.

(a) Diagram of an experimental setup for generating an axially corrugated plasma channel.21,22 (b) Snapshot of an experimentally generated modulated plasma channel (only 3 periods of many are shown).

FIG. 1.

(a) Diagram of an experimental setup for generating an axially corrugated plasma channel.21,22 (b) Snapshot of an experimentally generated modulated plasma channel (only 3 periods of many are shown).

Close modal

The organization of this paper is as follows: In Sec. II, we introduce the mechanisms underlying THz generation in corrugated plasma waveguides. A mathematical model of the channel is considered, and the dispersion relation is analyzed. In Section III, we conduct full format Particle-in-Cell (PIC) simulations to investigate the self-consistent evolution of the laser pulse in the strongly nonlinear regime and optimize the conversion efficiency from the laser pulse to THz energy. Tunability of the THz spectrum is also discussed. A numerical method to approximately find the frequencies of modes and coupling to the vacuum is also provided. The dependence of THz radiation on plasma density, driver pulse intensity, laser pulse duration, channel length, and other channel parameters is investigated and discussed in detail. Two different types of plasma channels are considered and compared. In Section IV, we present our conclusions and discuss future directions.

We consider the interaction of a drive pulse with a tenuous plasma. The central frequency of the drive pulse is significantly higher than the generated electromagnetic waves. The pulse produces a cycle-averaged low frequency ponderomotive force on the electrons, which induces an electron current that can produce radiation. The force on electrons is the gradient of the ponderomotive potential of the pulse, F p = V p , where Vp = mc2a2/4, the normalized laser vector potential is a = eEa/mcω0, m is the electron mass, e is the electron charge, c is the speed of light in vacuum, ω0 is the laser carrier frequency, and Ea is the electric field amplitude with both spatial and temporal dependences. The plasma ions are relatively heavy and can be treated as a stationary background during the interaction. In order for the excited current to generate electromagnetic radiation, the plasma must be inhomogeneous so that in no reference frame, the driven currents are static.

Further to have significant THz emission, the phase velocity of the excited channel modes must match the group velocity of the laser pulse in the plasma. In a uniform plasma, the modes typically have superluminal phase velocities according to the dispersion relation ω 2 = k 2 c 2 + ω p 2 , where ω p = 4 π e 2 N / m is the plasma frequency. However, the phase velocity in a corrugated plasma channel can be subluminal, allowing phase matching. In particular, the periodic spatial modulations of the channel parameters cause the electromagnetic modes of the channel to have a Floquet type representation, which implies subluminal partial waves.23 Because of the superposition of spatial harmonics, these EM modes can have components with subluminal phase velocities. Thus, significant THz emission can be achieved.

We consider the corrugated plasma waveguides to be cylindrically symmetric, with electron densities N(r, z) described by the following:

N ( r , z ) N 0 = { n 0 + ( n 1 n 0 ) r 2 r c 2 r r c n 1 r 0 r r 0 r c r c < r < r 0 0 r r 0 ,
(1)

where N0 is a normalization density. The quantities n0, n1, r0, and rc are all potentially periodic functions of z with period λm and the modulation wavenumber km = 2π/λm. The quantity n0(z) is the normalized on-axis density, and n 1 ( z ) is the normalized density at r = rc. For our studies, we take n 0 = 1 + δ sin ( k m z ) and n 1 = n ¯ 1 + δ 1 sin ( k m z ) , respectively, δ is the density modulation amplitude of the on-axis density n0, n ¯ 1 is the average density at r = rc, and δ1 is the density modulation amplitude of n1. The density has a parabolic transverse profile to guide the laser pulse during propagation. The quantity rc is the radius at which the density reaches n1, and then the density decreases linearly to zero from rc to r0. The quantities rc and r0 may also be axially modulated, r c , 0 = r ¯ c , 0 + Δ c , 0 cos ( k m z + θ c , 0 ) . Figures 2(a) and 2(b) are false color images of two different density profiles, based on parameters given in the caption. Both profiles are similar to those realized experimentally.21,22 The channel in Fig. 2(a) has its peak density off axis, while the channel in Fig. 2(b) has its peak density on axis. We refer to these two types as the “on-axis peak” (Fig. 2(b)) and the “off-axis peak” (Fig. 2(a)) channels, respectively.

FIG. 2.

False color image of two example electron density profiles generated by Eq. (1): (a) an “off-axis peak” channel with maximum density at the lateral edges. (b) An “on-axis peak” channel with maximum density at the center. Here, red indicates high density and blue indicates low density. All parameters of the two types of channels are displayed in Table I. In (b), and both the cut-off radius and channel radius are also modulated as r c [ μ m ] = 22.5 7.5 sin ( k m z ) and r0[μm] = rc + 15. Radially average density profiles of each type of channels are shown on the left and right, respectively.

FIG. 2.

False color image of two example electron density profiles generated by Eq. (1): (a) an “off-axis peak” channel with maximum density at the lateral edges. (b) An “on-axis peak” channel with maximum density at the center. Here, red indicates high density and blue indicates low density. All parameters of the two types of channels are displayed in Table I. In (b), and both the cut-off radius and channel radius are also modulated as r c [ μ m ] = 22.5 7.5 sin ( k m z ) and r0[μm] = rc + 15. Radially average density profiles of each type of channels are shown on the left and right, respectively.

Close modal

We review the linear theory of channel modes presented in Ref. 23. The plasma is taken to be a cold fluid with a linear response. Radially polarized, azimuthally symmetric TM modes (Er, Ez, and Bθ) of the channel are considered. In this case, an approximate wave equation assuming | · E | | E | / r c (Refs. 19 and 23) for the radial electric field Er can be derived

( 1 c 2 2 t 2 + 2 z 2 + 1 r r r r 1 r 2 ) E r = ω p 0 2 c 2 N ( r , z ) N 0 E r ,
(2)

where ωp0 is the plasma frequency evaluated for the normalization density N0.

In the case in which the electron density profile is parabolic as r, i.e., it is characterized by the first line of Eq. (1), and we take δ1 = δ, we can find analytic expressions for the eigenmodes. Specifically, the γth order radial eigenmode of the channel is described by

E r ( r , z , t ) = E 0 f ( z ) H γ ( r / w c h ) exp ( r 2 w c h 2 ) exp ( i ω t ) ,
(3)

where wch is the mode width given by 4 / w c h 4 = ω p 0 2 ( n 1 n 0 ) / r c 2 c 2 . The function Hγ is the γth polynomial defined as n = 1 n = 2 γ 1 α n ( r / w c h ) n with coefficients determined by α n / α n 2 = 4 ( n 1 2 γ ) / ( n 2 1 ) and α1 = 1. Given the dependence of the fields on the radial coordinates, the function f(z) satisfies the following Mathieu equation:

d 2 f d z 2 + k 0 2 f = ω p 0 2 c 2 δ sin ( k m z ) f ,
(4)

where the value of k0 is related to the frequency by

ω 2 ( ω p 0 2 + 8 γ c 2 w c h 2 ) = k 0 2 c 2 .
(5)

The dispersion relation is found by solving Eq. (4) with Floquet boundary conditions, f ( z + λ m ) = exp ( i k z λ m ) f ( z ) . This then determines the dependence k0(kz), which is inserted in Eq. (5) and gives ω(kz) the dispersion relation.

Shown in Fig. 3 is the dispersion relation for the lowest (γ = 1) radial mode of the model channel with parameters given in the caption. The dependence of ω on phase advance demonstrates the characteristic periodicity of frequencies in kz for periodic structures. The laser pulse, represented by a straight line in the plot, moves at its group velocity (vgc) in the plasma channel. At places where the dashed pulse line and the dispersion relation curve intersect, phase matching occurs, and THz excitation can be expected at these frequencies.

FIG. 3.

Dispersion relation curves of the lowest (fundamental) radial mode of a corrugated channel evaluated by Eq. (5) with N0 = 1.4 × 1018 cm−3, δ1 = δ = 0.9, n ¯ 1 = 3 , and rc = 30 μm. The straight line (red) corresponds to the laser pulse moving at the speed of light, ω = kzc.

FIG. 3.

Dispersion relation curves of the lowest (fundamental) radial mode of a corrugated channel evaluated by Eq. (5) with N0 = 1.4 × 1018 cm−3, δ1 = δ = 0.9, n ¯ 1 = 3 , and rc = 30 μm. The straight line (red) corresponds to the laser pulse moving at the speed of light, ω = kzc.

Close modal

Equations (4) and (5) apply as long as wch < rc, that is, the THz or laser spot size is smaller than the channel width. Equation (5) is a good approximation to the dispersion relation regarding the cycle-averaged density profile and assuming that the transverse parabolic shape extends to infinity. However, to determine the exact frequencies of the excited modes, one can numerically evaluate the wave equation, Eq. (2), using an exact electron density profile. A more accurate calculation of the dispersion relation, including the comparison with Eq. (5), is provided in the  Appendix.

THz generation in corrugated plasma waveguides is simulated using the full format PIC code TurboWAVE.24 The simulations, performed in 2D planar geometry, feature a finite sized plasma channel illuminated by an ultrashort, intense laser pulse incident from the simulation boundary. Figure 4(a) shows an example of an off-axis density peak plasma channel of 10 periods with a modulation wavelength of λm = 50 μm. To quantify the radiation emitted from the plasma channel, we calculate the Poynting flux and its spectral density through each of the surfaces outside the plasma region indicated by the dashed lines. This captures emission in the forward, backward, and lateral directions. The initial 2D simulations are performed in the lab frame in a domain of dimensions 205.9 × 753.8 μm with 1024 × 20 480 cells in the x and z directions, respectively. A laser pulse, with parameters detailed in Table I, traverses the plasma channel from left to right. The initial normalized vector potential is a0 = 0.4 (pulse energy UL = 66 mJ) with a0 defined as a0 = eE0/mcω0, where E0 is the peak field amplitude. Figure 4(b) shows a false color image of the transverse component of Poynting flux PxEzBy after the laser pulse traverses the channel. The excited plasma wave can be observed as the rapid oscillations of Px inside the channel; the alternate positive and negative values indicate a small average flux. However, at both lateral boundaries (top and bottom of the image), one can observe the THz emission in the form of the red and blue streaks in the lateral Poynting flux.

FIG. 4.

(a) Diagram of the simulation setup. The diagnostic box is set outside the plasma channel, and the Poynting flux through each surface is calculated. The plasma channel consists of 10 modulation periods with N0 = 1.4 × 1018 cm−3 and all other parameters shown in Table I. (b) A snap shot of the transverse component of Poynting flux EzBy in PIC simulations after the laser propagates through the channel from left to right, with red and blue streaks indicating that lateral THz radiation is generated.

FIG. 4.

(a) Diagram of the simulation setup. The diagnostic box is set outside the plasma channel, and the Poynting flux through each surface is calculated. The plasma channel consists of 10 modulation periods with N0 = 1.4 × 1018 cm−3 and all other parameters shown in Table I. (b) A snap shot of the transverse component of Poynting flux EzBy in PIC simulations after the laser propagates through the channel from left to right, with red and blue streaks indicating that lateral THz radiation is generated.

Close modal
TABLE I.

Typical parameters.a

Laser pulse Central wavenumber λ 800 nm Spot size rL 15 μm Pulse duration τ, FWHM 50 fs Normalized vector potential a0 0.4
Channel type  λm[μm]  r ¯ c [ μ m ]   r ¯ 0 [ μ m ]   δ  n ¯ 1   δ1  Δc  θc  Δ0  θ0 
Off-axis peak (Fig. 2(a) 15  30  40  0.9  0.9 
On-axis peak (Fig. 2(b) 15  22.5  37.5  0.7  1.3  0.1  7.5  π/2  7.5  π/2 
Laser pulse Central wavenumber λ 800 nm Spot size rL 15 μm Pulse duration τ, FWHM 50 fs Normalized vector potential a0 0.4
Channel type  λm[μm]  r ¯ c [ μ m ]   r ¯ 0 [ μ m ]   δ  n ¯ 1   δ1  Δc  θc  Δ0  θ0 
Off-axis peak (Fig. 2(a) 15  30  40  0.9  0.9 
On-axis peak (Fig. 2(b) 15  22.5  37.5  0.7  1.3  0.1  7.5  π/2  7.5  π/2 
a

Note: Unless otherwise stated, all parameters are identical to those in Table I.

To investigate the spectrum of the lateral THz emission, the fields are Fourier transformed in time. The radiated energy per unit length U through each diagnostic surface in Fig. 4(a) can be obtained in the form of a spectral density in ( ω , r ) , where r denotes the 2D spatial coordinates (x, z)

d U d ω = d A n ̂ · S ( ω , r ) ,
(6a)
U = 0 d ω d U d ω ,
(6b)

where n ̂ is the unit vector normal to the surface A. The spectral density is given by

S ( ω , r ) = c 8 π 2 ( E ( ω , r ) × B * ( ω , r ) + c . c . ) .
(7)

To quantify the THz radiation emitted across each diagnostic surface, we calculate the z component of the spectral density, Sz, for the left and right diagnostic boundaries and the x component, Sx, for the lateral boundary. Figure 5(b) is the laterally radiated spectral density, Sx, from a PIC simulation of the off-axis type density profile. The low frequency, broad band THz radiation observed at the entrance of the channel, shown in Fig. 5(a), is due to resonant transition radiation as discussed in Refs. 25–27. Lateral THz radiation28 is also observed in the corrugated plasma channel and characterized by a coherent, narrow band spectrum as shown in Fig. 5(c). The frequencies of the first three excited THz modes based on the phase matching condition predicted by the simplified model of Sec. II C are 13.18 THz, 15.3 THz, and 16.5 THz, respectively. In this case, the simulation results in Fig. 5(b) show that the THz radiation leaves the channel, creating an intensity pattern with maxima at 9 separate locations along the longitudinal distance where the fundamental mode emission dominates.

FIG. 5.

(a) Radiated THz energy across the lateral diagnostic boundary shows two different mechanisms of generating THz. (b) Simulation results of radiated THz spectral density across the lateral diagnostic boundary using the same channel parameters in Fig. 4(a) and a0 = 0.4. Besides the low frequency, broad band THz radiation when the laser pulse crosses the plasma interface,25 lateral THz radiation is also observed. (c) Radiated THz spectrum (only channel modes are considered in this plot) shows that different channel modes are excited. The frequency of each excited mode matches well with the phase matching condition in Fig. 3.

FIG. 5.

(a) Radiated THz energy across the lateral diagnostic boundary shows two different mechanisms of generating THz. (b) Simulation results of radiated THz spectral density across the lateral diagnostic boundary using the same channel parameters in Fig. 4(a) and a0 = 0.4. Besides the low frequency, broad band THz radiation when the laser pulse crosses the plasma interface,25 lateral THz radiation is also observed. (c) Radiated THz spectrum (only channel modes are considered in this plot) shows that different channel modes are excited. The frequency of each excited mode matches well with the phase matching condition in Fig. 3.

Close modal

We now explore the dependence of THz generation on plasma density in the case of the off-axis peak channel of Fig. 2(a). Figure 6 displays the radiated spectral density d U / d ω , for several average, on-axis plasma densities. The central frequency of the emission peak increases with increasing density, in accord with the dependence of the channel mode frequencies on density. For all the plasma densities considered in Fig. 6, the fundamental mode provides the dominant contribution to the radiation energy. As shown in Fig. 6, the radiation generated for this channel peaks around a plasma density of 1.75 × 1017 cm−3. As discussed in Ref. 19, the energy extracted from the drive pulse is converted into both electromagnetic radiation (EM) and plasma waves (PW). For example, the simulation predicts that the THz energy radiated in a 1.5 cm channel for the 1.4 × 1017 cm−3 density is 0.048 mJ, while the energy extracted from the laser pulse over the same distance is 4.1 mJ (∼1.17% of depletion). To efficiently deplete the laser pulse energy within a shorter channel, a higher plasma density is preferred. Therefore, to efficiently generate THz radiation, there must be a trade-off between the increasing density to increase the plasma current and the decreasing density to increase the lateral output coupling as shown in Fig. 6.

FIG. 6.

Radiated THz spectral density d U / d ω for different plasma densities showing the dependence of central THz frequency on plasma density.

FIG. 6.

Radiated THz spectral density d U / d ω for different plasma densities showing the dependence of central THz frequency on plasma density.

Close modal

The spatial and spectral dependences of the laterally radiated THz are illustrated for different laser intensities in Fig. 7. Ponderomotively driven radiation is expected to scale with the laser amplitude as U L 2 a 0 4 (quadratic in ponderomotive potential) for a0 ≪ 1. One can see that the THz energy follows this scaling for small a0 but is enhanced above this scaling for a0 > 1. Because the relativistic ponderomotive potential scales as a 0 2 / γ , the observed enhancement is even stronger than might be expected. The enhancement phenomenon is accompanied by a broadening of the THz spectrum to the point that individual modes can no longer be identified. For large a0, we find that the higher frequency channel modes are excited by nonlinear currents. This, along with the broadening of the spectrum, provides the enhancement over the linear scaling. For this short channel (10 periods), the modification of the temporal profile of the laser pulse is small. Thus, the broad spectrum can be attributed to nonlinearity in the excited plasma response.

FIG. 7.

(a) Laterally radiated THz spectrum for different laser intensities a0 = 0.2 (pulse energy, 16.7 mJ; black), a0 = 0.4 (pulse energy, 66 mJ; blue), a0 = 0.8 (pulse energy, 0.267 J; green), and a0 = 2.0 (pulse energy, 1.66 J; red). All other parameters are given in Table I. Note that the scale factor changes as laser intensity increases. (b) Radiated THz energy ETHz and efficiency η for different laser intensities. For a0 < 1, ETHz scales quadratically with laser intensity, and η is independent of a0 as expected. However, for a0 > 1, scaling is enhanced since higher order channel modes are excited due to nonlinear currents.

FIG. 7.

(a) Laterally radiated THz spectrum for different laser intensities a0 = 0.2 (pulse energy, 16.7 mJ; black), a0 = 0.4 (pulse energy, 66 mJ; blue), a0 = 0.8 (pulse energy, 0.267 J; green), and a0 = 2.0 (pulse energy, 1.66 J; red). All other parameters are given in Table I. Note that the scale factor changes as laser intensity increases. (b) Radiated THz energy ETHz and efficiency η for different laser intensities. For a0 < 1, ETHz scales quadratically with laser intensity, and η is independent of a0 as expected. However, for a0 > 1, scaling is enhanced since higher order channel modes are excited due to nonlinear currents.

Close modal

Our goal is to optimize the conversion efficiency of optical laser pulse energy to THz. Channel lengths in our simulations are limited by computation time, and so, we are not able to simulate every channel for a distance long enough to substantially deplete the laser pulse. We thus first consider short channels (10 periods) and define an efficiency that is the fraction of the depleted laser energy ( | Δ E L a s e r | ) transferred to THz, η = E T H z / | Δ E L a s e r | . By maximizing this efficiency, less power is expended driving the plasma oscillations, thus freeing it to drive THz over longer distances. We note that this efficiency does not depend on the laser intensity for a0 < 1, based on the linear theory for which both ETHz and ΔELaser scale as a 0 4 . However, for large a0, higher frequency THz modes are excited by nonlinear currents, enhancing the efficiency scaling. We have achieved an energy conversion efficiency of approximately 3% for the THz generation as displayed in Fig. 7(b), and this can be further optimized by varying the corrugated plasma density profiles. For example, the conversion efficiency for the weakly relativistic case can be efficiently enhanced by finding an optimum plasma density as discussed in regard to Fig. 6. Further, as the laser pulse propagates through the channel, its energy depletes. The accompanying spectral red-shifting results in compression and increases the pulse amplitude,29,30 which contributes to the enhancement of conversion efficiency. This phenomenon will be discussed in detail in Sec. III E.

The frequencies of the linear channel modes are determined by the plasma profile. Thus, the pulse duration does not affect the THz mode frequency, but it does determine the amplitude of the driving current at each frequency. THz emission can be expected for frequencies given by the intersections of the channel dispersion curves and the laser pulse “light line.” An additional requirement for the generation of THz is that the temporal spectrum of the laser pulse envelope includes the mode frequency. If the laser pulse has a Gaussian temporal profile exp [ ( t z / c ) 2 / τ p 2 ] with 2 l n ( 2 ) τ p as the pulse duration (FWHM), the amplitude of the ponderomotive driver (for fixed a0) at a mode frequency ω is given by τ p exp ( ω 2 τ p 2 / 4 ) , and thus, minimal radiation is expected for frequencies ωτp ≫ 1. Therefore, the value of τp can be adjusted to excite a specific range of channel mode frequencies. For excitation of the fundamental mode of 13.18 THz, the desired value is τp = 16.9 fs corresponding to a FWHM pulse duration of 28.4 fs. The simulation results further verify our estimation. Shown in Fig. 8 is a comparison of the radiated spectral density d U / d ω for pulse durations of 100 fs, 30 fs, and 15 fs, respectively. The initial normalized vector potential a0 = 0.4 is kept fixed for all 4 cases. For the case of a 100 fs laser pulse, the amplitude of the ponderomotive driver for any channel mode is small, such that minimal THz generation is observed. For the fundamental mode (13.18 THz) of the same plasma channel, the desired pulse duration is 30 fs, and the simulation result shows that the THz radiation maximizes at this frequency. In addition, higher order radiation is observed as the pulse duration is shortened to 15 fs as shown in Fig. 8, where there is an enhancement of the channel mode near 20 THz.

FIG. 8.

Radiated THz spectral density d U / d ω for different laser pulse durations, 100 fs, 50 fs, 30 fs, and 15 fs, respectively.

FIG. 8.

Radiated THz spectral density d U / d ω for different laser pulse durations, 100 fs, 50 fs, 30 fs, and 15 fs, respectively.

Close modal

We now investigate for the first time THz generation in corrugated plasma waveguides of sufficient length to deplete the laser pulse. As the laser pulse propagates through the plasma channel, the pulse envelope is modified and its energy is depleted through conversion into both electromagnetic radiation and plasma waves.19,31–34 For example, the energy of a laser pulse with a0 = 2.0 and a pulse duration of 15 fs propagating through the 10 period channel shown in Fig. 4(a) is depleted by 1.12%. At the same time, the central frequency red-shifts from 375 THz to 371 THz. These changes in energy and central frequency are consistent with action conservation:29–32 as the pulse depletes, the spectrum red-shifts and the normalized vector potential a0 associated with the laser pulse increases. As shown in Sec. III C, the pulse depletion rate increases dramatically with intensity.

We thus simulate an 800 nm laser pulse of a duration of 15 fs, a transverse spot size of 15 μm, and a0 = 2.0 (pulse energy 0.55 J). The channel parameters are the same as in Fig. 4(a), except the channel length extended to 1.5 cm. The simulation is conducted with a moving window of length 750 μm to collect all the THz emission following the pulse. The 2D moving frame has a size of 205.9 × 753.8 μm with 1024 × 20 480 cells in the x and z directions, respectively. The energy stored in the laser pulse is displayed in Fig. 9 as a function of propagation distance. Within the propagation distance of 1.5 cm, 80% of the pulse energy is depleted. Fig. 9 also displays the radiated THz energy versus propagation distance. The rate ( d U / d z ) of THz energy generation increases with distance as the normalized vector potential a0 increases during propagation due to action conservation. As a result, after propagation of 1.5 cm, more than 8% of the total pulse energy is converted into THz radiation.

FIG. 9.

Pulse energy depletion (black solid line) of a laser with initial a0 = 2.0 (0.55 J) during propagation in a corrugated plasma channel. The simulation result shows that around 80% of the energy stored in the laser pulse is depleted in a distance of 1.5 cm. Scaling of Radiated THz energy (blue dashed line) versus propagation distance shows that the generated THz energy is higher than the linear scaling with distance.

FIG. 9.

Pulse energy depletion (black solid line) of a laser with initial a0 = 2.0 (0.55 J) during propagation in a corrugated plasma channel. The simulation result shows that around 80% of the energy stored in the laser pulse is depleted in a distance of 1.5 cm. Scaling of Radiated THz energy (blue dashed line) versus propagation distance shows that the generated THz energy is higher than the linear scaling with distance.

Close modal

For comparison, THz generation using a relatively low intensity laser pulse with a pulse duration of 30 fs, a transverse spot size of 15 μm, and a0 = 0.4 (pulse energy, 44.5 mJ) is also simulated. In this case, we found the optimum plasma density for short propagation distance (10 periods) to be 1.4 × 1017 cm−3 as shown in Sec. III B. The lower plasma density and laser intensity lead to a much longer pulse depletion length than for that of the previous case displayed in Fig. 9. Consequently, due to computational restrictions, we were not able to simulate a channel long enough to substantially deplete the pulse energy. Instead, we simulate pulse propagation through a corrugated plasma channel for a distance of 1.5 cm. The simulation results displayed in Fig. 10 indicate that only 10% of the energy stored in the laser pulse is depleted within that distance. The variation with z of the depletion rate is due to the mismatch between the transverse pulse width and the matched width for the guiding channel. This leads to variations in spot size and intensity, with the depletion rate depending strongly on intensity. Shown in Fig. 10 is the radiated THz energy versus plasma channel length. About 50 μJ of THz energy is generated within a 1.5 cm interaction distance. As a result, it can be concluded that the conversion efficiency for a0 = 0.4 is much lower than that of a higher intensity pulse. However, a much longer plasma channel is needed to deplete the pulse energy for a0 = 0.4.

FIG. 10.

Pulse energy depletion (black solid line) of a laser with initial a0 = 0.4 (44.5 mJ) during propagation in a corrugated plasma channel. The simulation result shows that only 10% of the energy stored in the laser pulse is depleted in a distance of 1.5 cm. Scaling of Radiated THz energy (blue dashed line) versus plasma channel length indicates that only 1.1% of the depleted laser energy is converted into THz radiation in this case.

FIG. 10.

Pulse energy depletion (black solid line) of a laser with initial a0 = 0.4 (44.5 mJ) during propagation in a corrugated plasma channel. The simulation result shows that only 10% of the energy stored in the laser pulse is depleted in a distance of 1.5 cm. Scaling of Radiated THz energy (blue dashed line) versus plasma channel length indicates that only 1.1% of the depleted laser energy is converted into THz radiation in this case.

Close modal

In this section, we consider the corrugated plasma channel of the type shown in Fig. 2(b). The channel length is 10 periods with a modulation wavelength of 50 μm. The other channel parameters are δ = 0.7 , n ¯ 1 = 1.3 , δ 1 = 0.1 (see Eq. (1)). In order to match the density profile shown in Fig. 1(b), both cut-off radius and channel radius are also modulated according to r c [ μ m ] = 22.5 7.5 cos ( k m z ) and r0[μm] = rc + 15. Simulation results for laser pulses, with normalized vector potential a0 = 0.4 and a0 = 2.0, are shown in Fig. 11. Both laser pulses have the same transverse spot size of 15 μm and a pulse duration (FWHM) of 50 fs with a central wavenumber of 800 nm. The case with a0 = 0.4 shows that narrow band THz radiation is excited around the fundamental frequency of 13.6 THz. As displayed in Table II, the amount of generated THz energy and conversion efficiency for this channel are significantly higher than the case shown in Fig. 5 for the same laser pulse. This could be explained by the excitation of a higher electron current due to the relatively greater density inhomogeneity experienced by the driver. In addition, the axially averaged density profile has a lower radial barrier that allows the generated THz waves to escape the channel. For a higher intensity laser pulse with a0 = 2.0, the generated THz shown in Fig. 11 is characterized by a different spectrum compared with Fig. 7(a). Although the amount of THz energy is still enhanced relative to a0 = 0.4, the spectrum is confined in a relatively narrow band near the fundamental frequency, while in the case of Fig. 7(a), higher order THz modes are significantly generated and consequently modify the spectrum.

FIG. 11.

Radiated THz spectrum d U / d z for two different laser intensities a0 = 0.4 (black solid line) and a0 = 2.0 (blue dashed line). The plasma density profile used in the simulations is shown in Fig. 2(b) and has a local on-axis peak of the channel. The simulation result shows that a narrow band THz spectrum is excited and higher THz energy is preferentially generated in this type of channel.

FIG. 11.

Radiated THz spectrum d U / d z for two different laser intensities a0 = 0.4 (black solid line) and a0 = 2.0 (blue dashed line). The plasma density profile used in the simulations is shown in Fig. 2(b) and has a local on-axis peak of the channel. The simulation result shows that a narrow band THz spectrum is excited and higher THz energy is preferentially generated in this type of channel.

Close modal
TABLE II.

Comparison of generated THz energy for different laser intensities and density profiles.

Channel type a0 Pulse duration [fs] On-axis density [×1018 cm−3] Channel length [mm] Energy depletion into the channel (%) THz energy [mJ] efficiency η
Fig. 2(a)   0.4  30  0.14  15  10.11  0.048  1.11% 
0.4  50  1.4  0.5  0.0175  3.36 e-5  0.28% 
2.0  15  1.4  15  81.8  38  8.44% 
2.0  50  1.4  0.5  0.39  0.2  3.25% 
Fig. 2(b)   0.4  30  1.4  0.5  0.033  0.0016  12.03% 
0.4  50  1.4  0.5  0.024  0.0013  8.17% 
0.4  50  0.14  0.5  0.006  4.68 e-4  11.7% 
2.0  30  1.4  15  48.2  16  3% 
2.0  50  1.4  0.5  0.4  0.6493  9.84% 
Channel type a0 Pulse duration [fs] On-axis density [×1018 cm−3] Channel length [mm] Energy depletion into the channel (%) THz energy [mJ] efficiency η
Fig. 2(a)   0.4  30  0.14  15  10.11  0.048  1.11% 
0.4  50  1.4  0.5  0.0175  3.36 e-5  0.28% 
2.0  15  1.4  15  81.8  38  8.44% 
2.0  50  1.4  0.5  0.39  0.2  3.25% 
Fig. 2(b)   0.4  30  1.4  0.5  0.033  0.0016  12.03% 
0.4  50  1.4  0.5  0.024  0.0013  8.17% 
0.4  50  0.14  0.5  0.006  4.68 e-4  11.7% 
2.0  30  1.4  15  48.2  16  3% 
2.0  50  1.4  0.5  0.4  0.6493  9.84% 

We have investigated THz generation in corrugated plasma channels accounting for nonlinear excitation of plasma waves and laser pulse depletion. Theoretical analysis and Full format PIC simulations were conducted. A range of laser pulse parameters and plasma channel structures were considered with the goal of maximizing the conversion efficiency of optical pulse energy to THz energy.

Table II displays the simulation results for different pulse and plasma parameters for the two types of corrugated channel displayed in Fig. 2. Most of the simulations were conducted using a 10-period channel to investigate the conversion efficiency of the depleted optical pulse energy to generated THz. For these simulations, the pulse energy only depleted a small percentage. Three examples of longer channels were conducted to examine the consequences of significant pulse depletion, and the results are included in Table II. Our general conclusions are as follows: Generally, THz generation increases with laser amplitude a0. For fixed a0, THz generation at frequency ω maximizes, where ωτp ∼ 1, where τp is the pulse duration. This applies for both channel types. For a short channel with only 10 periods (channel length 0.5 mm), simulation results indicate that more THz energy is generated for the on-axis peak density channel type shown in Fig. 2(b). Efficient THz generation involves a trade-off between the increasing density to increase the plasma current and the decreasing density to increase the lateral output coupling. However, since the excited THz mode depends on the channel structure according to the dispersion relation discussed in Sec. II C, lower plasma density also modifies the generated THz spectrum. In addition, to efficiently deplete the laser pulse energy within a shorter channel, a higher plasma density is preferred.

As an example, we choose a laser pulse with a0 = 2.0 and a pulse duration of 30 fs. The channel type shown in Fig. 2(b) is used with an averaged on axis electron density of 1.4 × 1018 cm−3 and the channel length is extended to 1.5 cm. The simulation results as displayed in Table II show after the laser pulse propagates through the channel for 1.5 cm, around 48% of the laser pulse energy is depleted and 16 mJ of this energy is converted to THz energy with a narrow spectrum (Fig. 11). The conversion efficiency is around 3% and less than the case shown in Sec. III E. This is probably due to the fact that the channel displayed in Fig. 2(b) no longer remains a transverse parabolic structure capable of guiding. In fact, laser energy leaks laterally, and the wave action is not conserved as the laser pulse propagates through the channel. Therefore, the normalized vector potential a0 decreases with the propagation distance and less THz energy is generated.

In order for the present mechanism to be a useful high power source of THz radiation, the spectrum should be tunable. Since the excited THz consists of a superposition of the channel modes of the corrugated plasma structure, all the parameters used in Eq. (1) can be tuned to modify the THz spectrum. More specifically, varying the averaged on axis density N0 is the most straightforward way to tune the frequency in the experiment. For example, Figure 6 shows that varying the average density N0 from 1.75 × 1017 cm−3 to 1.4 × 1018 cm−3 results in a shift in the central frequency from 5 THz to 14 THz. For the channel type of Fig. 2(a), the THz spectrum changes dramatically with laser intensity going from a narrow spectrum at a0 = 0.4 to a broad spectrum at a0 = 2.0 with enhanced energy as well. For the channel type of Fig. 2(b), the spectrum remains strongly peaked over the same range of laser amplitudes. Since the generated THz waves are emitted laterally, one needs an optical system to collect the radiation. This might be realized with a conical mirror to focus the THz radiation to one direction for practical uses. Overall, the mechanism, using realistic corrugated plasma structures, presented in this paper provides a potential high power source of THz with a tunable spectrum and a conversion efficiency of over 8%.

The authors would like to acknowledge Dr. Daniel Gordon for the use of TurboWAVE and thank Luke Johnson and Thomas Rensink for fruitful discussions. Part of the simulations was performed on NERSC Edison and Deepthought2 clusters at UMD. This work was supported by the Office of Naval Research and the U.S. Department of Energy.

In this appendix, we will discuss the numerical method to calculate the exact frequencies of the axially averaged excited modes. Equation (2) can be further written as

( 1 r r r r 1 r 2 ) E r + ( k c 2 k p 2 ( r ) ) E r = 0 ,
(A1)

where the cut off wavenumber kc of EM modes and plasma wavenumber kp are defined as k c 2 = ω 2 / c 2 k 0 2 and k p 2 ( r ) = ω p 2 ( r ) / c 2 , respectively. Since we consider radially polarized TM modes, Er must vanish on axis, i.e., Er(0) = 0. We know that outside the channel, N(r > r0) = 0, and thus Eq. (A1) yields Bessel's differential equation. The far field Er outside the channel must match the properties of an outgoing wave, which has the form of the first kind Hankel function H 1 ( 1 ) ( k c r ) and asymptotically behaves as E r 1 r exp ( i k c r ) . The boundary condition allows us to know the ratio of Er to its derivative outside the channel. As a result, we can numerically integrate Eq. (A1) using the shooting method to determine the kc values that satisfy the on axis boundary condition Er(0) = 0.

To calculate the radial eigenmodes numerically for an arbitrary, but given the transverse density profile, one can numerically evaluate Er by Eq. (A1) using the shooting method for a set of k and determine what k satisfies the on axis boundary condition Er(0) = 0. For mathematical simplicity, we set Φ = r E r , β ( r ) = k p 2 ( r ) and Eq. (A1) yields

2 Φ r 2 1 r Φ r + ( k c 2 β ( r ) ) Φ = 0 .
(A2)

One can also find that as r → 0, Φ ∼ ar2 + b; the on axis boundary condition is satisfied only if b → 0. The finite difference (FD) shooting method can be implemented by

Φ j 1 = Φ j + 1 ( 2 j 1 ) + 2 j [ h 2 ( k c 2 β j ) 2 ] Φ j 1 + 2 j ,
(A3)

where h is the step size.

To find out the number of different radial modes, i.e., kc, which can be supported by a channel with finite transverse size, one can scan the parameter kc in Eq. (A3) and apply the Nyquist Theory illustrated in Fig. 12. F(s) is an analytic function defined in a closed region of the complex s-plane shown on the left. As s travels a clockwise path in the s-plane, F(s) encircles the origin on the complex F(s)-plane N times

N = Z P ,
(A4)

where Z and P denote the number of zeros and poles of the function F(s) in the closed region, respectively. For our shooting method, as shown in Eq. (A3), Φ(0) has no poles and the result yields N = Z.

FIG. 12.

Nyquist theory: Cauchy's principle.

FIG. 12.

Nyquist theory: Cauchy's principle.

Close modal

This method predicts the frequency of radiation for any given transverse density profile. For example, Fig. 13 shows that for a density profile shown in Fig. 4(a), Φ(0) encircles the origin twice as kc scans from 0.301 μm−1 to 0.402 μm−1, which implies that the channel can support two radial eigenmodes according to Cauchy's principle. Further, one can apply linear interpolation to narrow the range of kc for each mode to find the exact value of kc for the field to satisfy the boundary condition. Figures 14(a) and 14(b) are two figures indicating the range of kc during the interpolation to find the exact value kc for first and second radial eigenmodes, respectively.

FIG. 13.

Φ(0) encircles origin (red dot) twice as kc scans from 0.301 μm to 0.402 μm. The density profile in this case is as follows: N0 = 1.4 × 1018 cm−3, δ1 = δ = 0.9, n ¯ 1 = 3 , rc = 30 μm, and r0 = 40 μm.

FIG. 13.

Φ(0) encircles origin (red dot) twice as kc scans from 0.301 μm to 0.402 μm. The density profile in this case is as follows: N0 = 1.4 × 1018 cm−3, δ1 = δ = 0.9, n ¯ 1 = 3 , rc = 30 μm, and r0 = 40 μm.

Close modal
FIG. 14.

(a) kc of the fundamental radial mode is found to be between 0.301 μm−1 and 0.303 μm−1. (b) kc of the second order radial mode is found to be between 0.36 μm−1 and 0.37 μm−1. Applying the linear interpolation can help narrow the range of kc, and we find kc = 0.30195 μm−1 for the first radial mode and kc = 0.3638 μm−1 for the second radial mode.

FIG. 14.

(a) kc of the fundamental radial mode is found to be between 0.301 μm−1 and 0.303 μm−1. (b) kc of the second order radial mode is found to be between 0.36 μm−1 and 0.37 μm−1. Applying the linear interpolation can help narrow the range of kc, and we find kc = 0.30195 μm−1 for the first radial mode and kc = 0.3638 μm−1 for the second radial mode.

Close modal

The calculation of kc for the radial eigenmodes is displayed in Table III and matches closely with the estimate k c = ω p 0 2 / c 2 + 8 γ / w c h 2 from Eq. (5).

TABLE III.

Exact values of numerically calculated kc and estimation from Eq. (5) for different radial modes.

Mode number kc[μm−1], exact kc[μm−1], Eq. (5)
1st  0.30195  0.3024 
2nd  0.36384  0.3652 
Mode number kc[μm−1], exact kc[μm−1], Eq. (5)
1st  0.30195  0.3024 
2nd  0.36384  0.3652 
1.
M.
Sherwin
,
C.
Schmuttenmaer
, and
P.
Bucksbaum
, in
Report of a DOE-NSF-NIH Workshop, Arlington
, VA, 2004 (
2004
), Vol. 12, p.
14
.
2.
M. C.
Nuss
and
J.
Orenstein
, in
Millimeter-Wave Spectroscopy of Solids
, edited by
G.
Gruener
(
Springer Verlag
,
Heidelberg
,
1997
), and references therein.
3.
D. M.
Mittleman
,
M.
Gupta
,
R.
Neelamani
,
R. G.
Baraniuk
,
J. V.
Rudd
, and
M.
Koch
,
Appl. Phys. B: Laser Opt.
68
,
1085
(
1999
).
4.
Y.
Liu
,
F.-F.
Yin
,
N.-k.
Chen
,
M.-L.
Chu
, and
J.
Cai
,
Med. Phys.
42
,
534
(
2015
).
5.
D. M.
Mittleman
,
R. H.
Jacobsen
, and
M. C.
Nuss
,
IEEE J. Sel. Top. Quantum Electron.
2
,
679
(
1996
).
6.
G.
Nusinovich
,
P.
Sprangle
,
C.
Romero-Talamas
, and
V.
Granatstein
,
J. Appl. Phys.
109
,
083303
(
2011
).
7.
J.
Isaacs
,
C.
Miao
, and
P.
Sprangle
,
Phys. Plasmas
23
,
033507
(
2016
).
8.
E.
Budiarto
,
J.
Margolies
,
S.
Jeong
,
J.
Son
, and
J.
Bokor
,
IEEE J. Sel. Top. Quantum Electron.
32
,
1839
(
1996
).
9.
C.
Vicario
,
A. V.
Ovchinnikov
,
S. I.
Ashitkov
,
M. B.
Agranat
,
V. E.
Fortov
, and
C. P.
Hauri
,
Opt. Lett.
39
,
6632
(
2014
).
10.
X.
Wu
,
C.
Zhou
,
W. R.
Huang
,
F.
Ahr
, and
F. X.
Kärtner
,
Opt. Express
23
,
29729
(
2015
).
11.
J. A.
Fülöp
,
Z.
Ollmann
,
C.
Lombosi
,
C.
Skrobol
,
S.
Klingebiel
,
L.
Pálfalvi
,
F.
Krausz
,
S.
Karsch
, and
J.
Hebling
,
Opt. Express
22
,
20155
(
2014
).
12.
W. P.
Leemans
,
C. G. R.
Geddes
,
J.
Faure
,
C.
Tóth
,
J.
van Tilborg
,
C. B.
Schroeder
,
E.
Esarey
,
G.
Fubiani
,
D.
Auerbach
,
B.
Marcelis
,
M. A.
Carnahan
,
R. A.
Kaindl
,
J.
Byrd
, and
M. C.
Martin
,
Phys. Rev. Lett.
91
,
074802
(
2003
).
13.
K.-Y.
Kim
,
A.
Taylor
,
J.
Glownia
, and
G.
Rodriguez
,
Nat. Photonics
2
,
605
(
2008
).
14.
A.
Gopal
,
S.
Herzer
,
A.
Schmidt
,
P.
Singh
,
A.
Reinhard
,
W.
Ziegler
,
D.
Brömmel
,
A.
Karmakar
,
P.
Gibbon
,
U.
Dillner
,
T.
May
,
H.-G.
Meyer
, and
G. G.
Paulus
,
Phys. Rev. Lett.
111
,
074802
(
2013
).
15.
T.
Nakazato
,
M.
Oyamada
,
N.
Niimura
,
S.
Urasawa
,
O.
Konno
,
A.
Kagaya
,
R.
Kato
,
T.
Kamiyama
,
Y.
Torizuka
,
T.
Nanba
,
Y.
Kondo
,
Y.
Shibata
,
K.
Ishi
,
T.
Ohsaka
, and
M.
Ikezawa
,
Phys. Rev. Lett.
63
,
1245
(
1989
).
16.
U.
Happek
,
A. J.
Sievers
, and
E. B.
Blum
,
Phys. Rev. Lett.
67
,
2962
(
1991
).
17.
H.
Hamster
,
A.
Sullivan
,
S.
Gordon
,
W.
White
, and
R. W.
Falcone
,
Phys. Rev. Lett.
71
,
2725
(
1993
).
18.
H.
Hamster
,
A.
Sullivan
,
S.
Gordon
, and
R.
Falcone
,
Phys. Rev. E
49
,
671
(
1994
).
19.
T. M.
Antonsen
,
J. P.
Palastro
, and
H. M.
Milchberg
,
Phys. Plasmas
14
,
033107
(
2007
).
20.
T. M.
Antonsen
,
Phys. Plasmas
17
,
073112
(
2010
).
21.
B. D.
Layer
,
A.
York
,
T. M.
Antonsen
,
S.
Varma
,
Y.-H.
Chen
,
Y.
Leng
, and
H. M.
Milchberg
,
Phys. Rev. Lett.
99
,
035001
(
2007
).
22.
G. A.
Hine
,
A. J.
Goers
,
L.
Feder
,
J. A.
Elle
,
S. J.
Yoon
, and
H. M.
Milchberg
,
Opt. Lett.
41
,
3427
(
2016
).
23.
A. J.
Pearson
,
J.
Palastro
, and
T. M.
Antonsen
,
Phys. Rev. E
83
,
056403
(
2011
).
24.
D. F.
Gordon
,
IEEE Trans. Plasma Sci.
35
,
1486
(
2007
).
25.
C.
Miao
,
J. P.
Palastro
, and
T. M.
Antonsen
,
Phys. Plasmas
23
,
063103
(
2016
).
26.
C.
Miao
,
J. P.
Palastro
, and
T. M.
Antonsen
, in
6th International Particle Accelerator Conference (IPAC 15)
(
JACoW
,
2015
), p.
2661
, Paper No. WEPWA068.
27.
C.
Miao
,
J. P.
Palastro
,
A. J.
Pearson
, and
T. M.
Antonsen
,
AIP Conf. Proc.
1777
,
080009
(
2016
).
28.
C.
Miao
,
J. P.
Palastro
,
A. J.
Pearson
, and
T. M.
Antonsen
, in
2015 40th International Conference on Infrared, Millimeter, and Terahertz waves (IRMMW-THz)
(
2015
), pp.
1
2
.
29.
W.
Zhu
,
J. P.
Palastro
, and
T. M.
Antonsen
,
Phys. Plasmas
20
,
073103
(
2013
).
30.
W.
Zhu
,
J. P.
Palastro
, and
T. M.
Antonsen
,
Phys. Plasmas
19
,
033105
(
2012
).
31.
T. M.
Antonsen
, Jr.
and
P.
Mora
,
Phys. Rev. Lett.
69
,
2204
(
1992
).
32.
P.
Mora
and
T. M.
Antonsen
, Jr.
,
Phys. Plasmas
4
,
217
(
1997
).
33.
B.
Shadwick
,
C.
Schroeder
, and
E.
Esarey
,
Phys. Plasmas
16
,
056704
(
2009
).
34.
C.
Benedetti
,
F.
Rossi
,
C. B.
Schroeder
,
E.
Esarey
, and
W. P.
Leemans
,
Phys. Rev. E
92
,
023109
(
2015
).