Generation of tunable Terahertz out-of-plane radiation using Josephson vortices in modulated layered superconductors

We show that a moving Josephson vortex in spatially modulated layered superconductors generates out-of-plane THz radiation. Remarkably, the magnetic and in-plane electric fields radiated are of the same order, which is very unusual for any good-conducting medium. Therefore, the out-of-plane radiation can be emitted to the vacuum without the standard impedance mismatch problem. Thus, the proposed design can be more efficient for tunable THz emitters than previous proposals, for radiation only propagating along the ab-plane.


I. INTRODUCTION
The recent growing interest in terahertz (THz) science and technology is due to its many important applications in physics, astronomy, chemistry, biology, and medicine, including THz imaging, spectroscopy, tomography, medical diagnosis, health monitoring, environmental control, as well as chemical and biological identification 1 . This range of the electromagnetic spectrum sits between 0.3 and 30 THz, which corresponds to 1000-10 µm (wavelength), 1.25-125 meV (energy) or 14-1400 K (temperature). The THz gap, that is still hardly reachable for both electronic and optical devices, covers temperatures of biological processes and a substantial fraction of the luminosity remanent from the Big Bang 1 .
High-temperature Bi 2 Sr 2 CaCu 2 O 8+δ superconductors have a layered structure that allows the propagation of electromagnetic waves (called Josephson plasma oscillations 2,3,4,5,6 ) with Josephson plasma frequency ω J . This is drastically different from the strong damping of electromagnetic waves in low temperature superconductors. The Josephson plasma frequency lies in the THz range (see, e.g., 7,8,9 ). Indeed, tunable filters of THz radiation have been proposed using the Josephson vortex lattice as a tunable photonic crystal 10 . Moreover, detectors of THz radiation have been very recently proposed using surface Josephson plasma waves 11 . A possible way to generate THz radiation in Bi 2 Sr 2 CaCu 2 O 8+δ and related compounds is to apply an in-plane magnetic field H ab and an external current J c perpendicular to the superconducting layers (i.e., along the c-axis). Josephson vortices (JVs) induced by H ab and driven fast by the c-axis current emit THz radiation (e.g., 7,9 ). However, it was shown 12,13,14 that the radiation propagates only along the plane of motion of the JVs and decays in the c-direction. This THz radiation is characterized by a huge impedance mismatch resulting in a very small fraction of THz wave intensity emitted from the sample 9 .
This impedance mismatch is a very important problem restricting possible applications 15 .
To avoid this problem, we propose a new class of THz emitters based on JVs moving through in-plane modulated layered superconductors, including both the strongly anisotropic high-T c Bi 2 Sr 2 CaCu 2 O 8+δ single crystals and artificial stacks of Josephson junctions (SJJ), e.g., Nb-Al-AlO x -Nb. In-plane spatial variations of the Josephson maximum c-axis current J c can be obtained by using either irradiation of a standard Bi 2 Sr 2 CaCu 2 O 8+δ sample covered by a modulated mask (see, e.g., 16 ) or pancake vortices controlled by an out-of-plane magnetic field 17 .
In order to pass through the superconductor-vacuum interface without a significant decrease of the amplitude, the electric and magnetic components of the propagating wave have to be of the same order of magnitude. This feature is inherent for the out-of-plane Josephson plasma waves (JPW), propagating both along and perpendicular to the layers with short wavelength along the c-axis. For such waves, the transmission coefficient is about unity 18 . The out-of-plane JPW can be emitted, for instance, by a fast moving Josephson vortex if its velocity V exceeds a certain threshold value V min . However, this out-of-plane Cherenkov-type radiation always completely reflects from the sample boundary and thus cannot be emitted into the vacuum. Indeed, the longitudinal wave vector q for the Cherenkov radiation is related to the wave frequency ω by q = ω/V and is much larger than the maximum possible value ω/c for waves in vacuum. This problem can be solved if the out-of-plane Cherenkov radiation propagates through a modulated layered superconductor. The out-of-plane Cherenkov wave interacting with periodic inhomogeneities generates new modes with wave vectors q m = q − 2πm/a, where a is the spatial period of the modulations and m is an integer. Thus, the wave vector q 1 = q − 2π/a can meet the condition q 1 < ω/c for vacuum waves and is emitted from a sample without an impedance mismatch. It is important to stress that the Cherenkov radiation generated by any relativistic particle in any medium undergoes a complete internal reflection (since q > ω/c) and, thus, we propose this general way of emitting any Cherenkov-type radiation to vacuum. Here, we predict this out-of-plane Cherenkov radiation, and derive the modes propagating in a modulated superconductor and emitted into the vacuum.
Our proposal mainly concerns bulk layered superconductors, where it is important to have radiation propagating not only along the ab-planes, but also along the c-axis. For the case of thin films having a thickness much smaller than the in-plane London penetration depth (about 200 nm), the radiation damped along the c-axis can also give some contribution to the waves emitted from the wide sample side, which is parallel to the ab-plane. This can even increase the fraction of the out-of-plane radiation discussed in this paper. In other words, we show that there are two contributions to the out-of-plane radiation along the c-axis: damped waves and propagating waves. The latter one is the real outof-plane radiation. The former one (damped waves) can contribute to the out-of-plane radiation only if the sample is sufficiently thin. However, we will not consider this case in detail because the power of the emitted radiation decreases for decreasing thickness.

A. Brief summary of the results
For electromagnetic waves in any conducting media, the electric field E is very weak with respect to the magnetic field H: E ≪ H. Also, for in-plane radiation: E ≪ H. Thus, only a small fraction (∼ E/H) of the radiation can leave the sample. This is the so-called "impedance mismatch" problem that has severely limited progress in this field for years. Now, we are also considering c-axis short-wavelength out-of-plane radiation. This radiation has a strong enough in-plane electric field E to overcome the superconducting-vacuum interface. Indeed, E and the magnetic field both are of the same order of magnitude, similar to the one for waves propagating in the vacuum. This solves the impedance mismatch problem. Thus, we propose to use c-axis shortwavelength out-of-plane radiation, propagating in a periodically modulated Bi 2 Sr 2 CaCu 2 O 8+δ sample, to overcome the severe impedance mismatch problem which limits the application of layered superconductors for THz emitters.
The spatial modulations of the maximum Josephson current allows the emitted waves to shift their wavenumbers (within a wide range ω J /V ) towards the narrow spectral window ∼ ω J /c for waves propagating in the vacuum. Thus, the emitted waves can pass the superconductor-vacuum interface within a narrow frequency window ∼ ω J V /c. This offers the possibility to select a narrow frequency window from the initiallybroad THz radiation produced by the JVs. Also, this should allow to achieve superradiance via the stabiliza-tion of the square Josephson lattice moving in a periodically modulated Bi 2 Sr 2 CaCu 2 O 8+δ samples 19 . Other recent novel ways to control vortex motion are also attracting considerable attention 20,21 .

II. MODEL
We consider an infinite layered superconductor as in Fig. 1b. Following 22 , we assume that the superconducting layers are extremely thin, so that the spatial variations of the phase of the superconducting order parameter and the electromagnetic field inside the layers in the direction perpendicular to the layers can be neglected. We choose the xy-plane to be parallel to the crystallographic ab-plane and the c-axis along the z-axis. Superconducting layers are numbered by the subscript l.
The electric E and magnetic H fields have components, The gauge-invariant phase difference ϕ l between (l + 1)th and lth superconducting layers is described by coupled sine-Gordon equations 18,22,23 , Here D is the spatial period of the layered structure, λ ab is the London penetration depth along the c-axis, the operator ∆ l is defined as is the Josephson frequency, ε is the interlayer dielectric constant, and the modulation factor µ(x) = µ(x + a) is a periodic function with spatial period a. For simplicity, we neglect the relaxation term related to the quasiparticle current, which is very small at low temperatures. The coupled sine-Gordon equations (1) describe both the Josephson vortices in layered superconductors and the emitted Josephson plasma waves. In the last case Eq. (1) should be linearized, i.e., sin(ϕ l ) replaced by ϕ l .

III. NONLOCAL SINE-GORGON EQUATION
In zero approximation with respect to the modulation µ, the travelling wave solution of Eq. (1), ϕ l = φ l (ζ = x − V t), represents the JV moving with a constant velocity. The main phase difference φ = φ 0 (ζ) occurs at the central junction, where nonlinearity plays a crucial role, while equations for junctions with l = 0 can be linearized. However, the magnetic flux of a JV spreads over a large number of these junctions, l ∼ λ ab /D ≫ 1. Thus, the equation for the central junction cannot be decoupled from others. This determines a complicated non-local structure of the JV in layered superconductors.
Following the approach by Gurevich 24 , the closed-form non-local sine-Gordon equation for φ 0 (ζ) can be derived (see Appendix) , where K 0 (x) is the modified Bessel function, ε is the penetration depth of the magnetic field along layers, and Φ 0 is the flux quantum. Strictly speaking, equation (3) is derived for a JV moving in a weaker junction with a critical current J w c < J c but describes qualitatively even the stack of identical junctions (see Appendix). Equation (3) is reduced to the usual local sine-Gordon equation only for λ c ≪ λ J , i.e., if the kernel K 0 in the integral in Eq. (3) is a sharper function of ζ ′ than ∂ 2 φ/∂ζ ′2 . For Bi 2 Sr 2 CaCu 2 O 8+δ as well as for artificial layered superconductors, the opposite strongly non-local limit, λ c ≫ λ J , is realized. In this case, a soliton-like solution, of Eq. (3) was obtained in Ref. 24 for a fixed JV, V = 0, with the soliton size It is important to stress that the soliton size L (rather than λ J ) coincides with the well-known estimate of the JV core (L = γD, γ = λ c /λ ab ) in a layered superconductor. Analytical and numerical analysis proves that the size L(V ) of the moving JV remains of the same order at any allowed vortex velocities Note that, due to non-locality, the maximum vortex velocity V c in a layered superconductor is much smaller than the maximum vortex velocity c sw = λ J ω J (so-called Swihart velocity) in a single junction 25 .

IV. JOSEPHSON PLASMA WAVES
In homogeneous (µ = 0) layered superconductors, the linearized coupled sine-Gordon equations admit wave solutions of the form, with the dispersion relation, for the transverse wave vector k(q, ω). This relation coincides with the spectrum obtained in 18 in the particular limit where the breaking of the charge neutrality effect (that we can easily take into account) is neglected. The electric and magnetic fields exhibit the same spatio-temporal dependence as in Eq. (7) for the phase difference ϕ l ; while their amplitudes, H 0 , E x0 , and E z0 , are related to ϕ 0 via According to Eq. (8), the JPW can propagate if ω > ω J . For homogeneous layered superconductor, JVs moving with constant velocity V can excite waves with q = ω/V (Cherenkov radiation 7,12,13,14 ). Substituting ω = qV in Eq. (8) and noticing that the right-hand side of this equation is less than unity, we obtain the limiting vortex velocity If the vortex velocity is larger than this threshold value, V > V min , then Im(k) = 0 and the out-of-plane waves can be excited. The characteristic angle θ of the propagating radiation (Cherenkov cone) is determined by three standard conditions: (i) Eq. (8), (ii) ω = qV , and (iii) the minimum wavelength k −1 < ∼ D/π. For V = V min the Cherenkov cone is obviously closed, θ = 0. At lower vortex velocities, V < V min , the Cherenkov radiation with Im(k) = 0 (decaying from the central junction) 12 can propagate only along the abplane 26 (see also 27 where the Cherenkov radiation in two coupled junctions is studied).
As was shown above, the maximum vortex velocity V c ∼ ω J L, at which the moving soliton is stable, is of the same order as V min . Below, the out-of-plane Cherenkov radiation generated by a vortex moving in a junction weaker than others is derived (while the question if such radiation exists in a system of identical junctions remains open 7 ). A subset of weaker intrinsic Josephson junctions in Bi 2 Sr 2 CaCu 2 O 8+δ -based samples can be made using either (i) the controllable intercalation technique 28 , (ii) Chemical Vapor Deposition (CVD) (see, e.g., 29 ), or (iii) via the admixture of Bi 2 Sr 2 Cu 2 O 6+δ and Bi 2 Sr 2 Ca 2 Cu 3 O 10+δ 30 . Also, such a system can be easily created using artificial stacks of layers of low temperature superconductors.

V. OUT-OF-PLANE CHERENKOV RADIATION
In order to find the relation between the amplitudes of the emitted waves inside a superconductor and the phase difference, φ, in the central junction, we use the standard equation 24,25 : where J ± x are the currents at the top and bottom edges of the central contact. Equation (14) is valid if D ≪ 2λ ab , that is, if the magnetic flux through the central junction is small compared to Φ 0 . In other words, the gradient of the gauge invariant phase difference φ along the central junction occurs due to the gradient of the phase χ ± of the superconducting order parameter in the layers forming the central junction rather than the trapped magnetic flux. Using Eq. (14) and the Maxwell equation where φ(q) is the Fourier transform of φ(ζ).
When φ(q) is known, Eq. (15) determines the magnetic field distribution of the emitted out-of-plane Cherenkov radiation. For simplicity, we use the Fourier transform φ(q) = φ 0 (q) of the solution for a fixed vortex, Using this last equation and integrating H 0 (q, qV ) exp (iqx − iqV t + ik(q, qV )z) over q min < q for the travelling out-of-plane waves we derive the expression for the magnetic field H Cher of the radiation: where The magnetic field distribution (17) in the emitted Cherenkov waves is shown in Fig. 1a.
Due to the rather unusual dispersion relation (8), i.e., the decrease of k(q, ω = qV ) with increasing q, as well as due to the spatial extension of a vortex, the generated electromagnetic waves are located outside the Cherenkov cone (Fig. 1a), which is drastically different from the Cherenkov radiation of a fast (point-like) relativistic particle. The new type of radiation predicted here could be called outside-the-cone Cherenkov radiation.  (17). The "running" coordinate, x−V t, is measured in units of γD/(π v 2 β 2 − 1), while the out-ofplane coordinate y is normalized by D/(π v 2 β 2 − 1), where β = JcDw/J w c D. The moving vortex emits radiation propagating forward. This radiation forms a cone determined by the vortex velocity V . (b) Geometry of the problem: in a weaker junction (located between the two blue superconducting planes in (b)) a c-axis current J c drives a Josephson vortex with velocity V , which is higher than the minimum velocity Vmin of the propagating electromagnetic waves. Red strips in (b) schematically show outside-the-cone Cherenkov radiation.

VI. IMPEDANCE MISMATCH
It is important to emphasize that a fast moving vortex emits mostly the JPWs with short out-of-plane lengths, i.e., with k(q) about k(q min ) = π/D (see Fig. 1a). According to Eq. (11), the ratio E x /H is about for these short waves. For this estimate we use commonly accepted parameters for Bi2212 samples (ω J /2π ∼ 1 THz, depending on doping and temperature, λ ab = 2000Å , and D = 15Å). This result, unusual for conducting media, suggests that there is no impedance mismatch for the out-of-plane radiation. However, as was mentioned above, the Cherenkov radiation can never pass through the sample boundary because it has a large longitudinal wave vector q = ω/V ≫ ω/c. In order to decrease the longitudinal wave vector q, the spatial modulations of the critical current can be used. To analyze this analytically we use the perturbation scheme H = H (0) + µH (1) + .. with µ(x) = µ cos(2πx/a), µ ≪ 1, in Eq. (1).
In zeroth-order approximation, at the sample boundary, the mode generated by a vortex completely reflects back to the superconductor sample as with a damping coefficient The amplitudes H + 0,0 , H − 0,0 , and H damp vac of the modes are determined by the continuity of the electric and magnetic fields at the sample boundary.
To the first approximation, using the linear relations between φ l and H l , as well as equation (1), we obtain the following equation: The solution of this equation consists of the sum of the following modes: propagating in vacuum with wave vector The continuity of E x and H for the longitudinal wave vector q − 2π/a define the amplitudes of the waves H ± 1,0 , H − 1,1 , and H prop vac , as well as the transition coefficient T for the emitted waves from the sample, for |p − 2π/aq min | < V /c and T = 0 otherwise. Here, p = q/q min and v = V /V min are the dimensionless longitudinal wave vector and vortex velocity. Because of β ∼ 1, the transmission coefficient T ∼ 1 for out-of-plane radiation in a narrow frequency region The quite narrow window of the transmitted waves occurs due to the broad spectrum, of the Cherenkov radiation emitted by a Josephson vortex with respect to the spectrum of waves propagating in the vacuum: A periodic spatial modulation can shift all the wave vectors towards the spectral window of the vacuum waves, while it cannot affect the width of the spectrum. Thus, the surface cuts a narrow strip from the broad Cherenkov radiation, allowing these waves to pass the interface. Note that changing the period a (e.g., changing the distance between pancake vortices via the c-axis magnetic field) allows to tune the frequency window for the emitted radiation. In order to make the frequency window ∆ω wider, one can employ, e.g., appropriate aperiodic modulations µ(x) = κ2 κ1 cos κx dκ with κ 1 and κ 2 incommensurate. Also, µ(x) could be modulated as a 1D quasicrystal 31 .

VII. CONCLUSIONS
We propose how to generate out-of-plane THz radiation in a controllable frequency range. We show that the standard severe mismatch problem can be overcome here for out-of-plane radiation using spatially modulated samples. Moreover, recent studies 32,33 of Josephson vortex arrangements in small samples (there, the interaction of vortices with sample boundaries acts similar to an additional potential) suggest a way to obtain a square vortex lattice, which is important for superradiance 34 . Thus, spatial modulations of J c can result in a more ordered vortex flow or even a flowing square vortex lattice generating superradiance. Of course this problem requires more detailed studies, which will be presented in the future.