The spatial-temporal distribution of voltage in a JJ is described by the equation (see chapter 9 in [31]):

(1)

where c0 is the (Swihart) velocity of EMWs in the TL formed by the JJ and L□ is the inductance of JJ per square. Jz is the current density through the JJ, which has Cooper pair and quasiparticle (QP) components,

(2)

Here, Jc0 is the Josephson critical current density, η is the Josephson phase difference, and rQP = RQPab is the QP resistance per unit area.

Active patch antenna model of a junction

Equation 1 is the equation for an active TL [37] with a distributed feed-in current density Jz. Therefore, a JJ has many similarities with the microstrip patch antenna. However, there are three main differences:

(i) The feed-in geometry. A patch antenna has a point-like feed-in port, through which the oscillating current is applied [34-36]. The FFO is biased by a dc current distributed over the whole JJ area.

(ii) The excitation scheme. A patch antenna is a linear oscillator pumped by a harmonic signal. In contrast, a JJ is biased by a dc-current and the oscillatory component is generated inside the JJ via the ac-Josephson effect and the flux-flow phenomenon.

(iii) The slow propagation speed of EMWs inside the JJ, c0 ≪ c. This is caused by a large kinetic inductance of superconducting electrodes. For Nb-based JJs, c/c0 ≈ 40 (see the estimation in section Discussion). For atomic-scale intrinsic JJs in layered cuprates, c0 can be almost 1000 times slower than c [32]. Because of that, the wavelength inside the JJ is much smaller than in free space, λ ≪ λ0. Therefore, a JJ corresponds to a patch antenna with an extraordinary large effective permittivity, = (c/c0)2.

The dynamics of a JJ is described by a nonlinear perturbed sine-Gordon equation,

(3)

It follows from Equation 1 and Equation 2, taking into account the ac-Josephson relation, V = (Φ0/2π)∂η/∂t. Equation 3 is written in a dimensionless form with space, = x/λJ, normalized by λJ,and time, = ωpt, by the Josephson plasma frequency, ωp. Here α is the QP damping factor, and = Jb/Jc0 is the normalized bias current density, which originates from the ∂2V/∂y2 term in Equation 1 [38]. In what follows, “tilde” will indicate dimensionless variables, = ω/ωp and = λJk. The definition of and the interconnection between different variables are clarified in the Appendix section.

Radiative resistance of a patch antenna

A rectangular patch antenna has two radiating slots, which correspond to the left and right edges of the JJ in Figure 1a. The slots can be considered as magnetic current lines (magnetic dipoles) [39]. Radiation from the antenna is determined by the radiative impedance, Zrad. For a patch with a very thin insulator (as is the case for a tunnel JJ), the radiative admittance of one slot, 1/Zrad1 = G1 + iB1, contains a large imaginary part B1, caused by the large capacitance. However, at the cavity mode resonance the imaginary contributions from the two slots cancel out [34,36,39] and the radiative impedance becomes real. Therefore, at the resonance the radiation power from one slot is

(4)

where |v(0,a)| is the amplitude of voltage oscillations at the slot (x = 0,a) and G1 is the radiative conductance of the single slot. Low-Tc JJs are operating at sub-terahertz frequencies, for which the wavelength in free space is large, λ0 ≫ b ≫ d. In this limit [36,39],

(5)

where Z0 = ≃ 376.73 (Ω) is the impedance of free space.

To calculate the total radiation power from both slots one has to take into account the mutual radiative conductance, G12, and the array factor AF [36]. G12 is originating from a cross product of electric and magnetic fields generated by different slots. For λ0 ≫ b ≫ d it is equal to [36,40]

(6)

Here, J0 is the zeroth-order Bessel function, k0 = 2π/λ0 is the wave number in free space, and the angle Θ is defined in Figure 1b. For the n-th cavity mode,

(7)

the argument of J0 becomes (c0/c)πnsinΘ. Since c0 ≪ c, k0a is small. Expanding in Equation 6, J0(x) ≃ 1 − x2/4 (for x ≪ 1), we obtain:

(8)

It is seen that the mutual conductance for a JJ with thin electrodes (slow c0) is not negligible and can be as big as the single-slot conductance G1, Equation 5.

The array factor takes into account the interference of electromagnetic fields from the two slots in the far field. It depends on the separation between the slots, a, the relative phase shift, β, and the direction (φ,Θ). Since radiation from a patch antenna is induced by magnetic current lines, it is more intuitive to consider the interference of magnetic fields, H1 + H2 = AFH1. For the geometry of Figure 1a and Figure 1b, it can be written as [36,40]

(9)

Odd-number cavity modes have antisymmetric voltage oscillations but symmetric magnetic currents, β = 0. This leads to a constructive interference with the maximum AF = 2 perpendicular to the patch along the z-axis. For even modes its vice versa, β = π, and a destructive interference leads to a node, AF = 0, along the z-axis.

The total emission power is

(10)

where the plus/minus signs are for odd/even modes, respectively. For equal amplitudes, |v(0)| = |v(a)|,

(11)

with the effective radiative resistance

(12) Determination of voltage amplitudes

To calculate Prad, we need voltage amplitudes at the JJ edges. Within the TL model of patch antennas, v(x) is obtained by decomposition into a sum of cavity eigenmodes [34]. For JJs, a similar approach is used for the analysis of Fiske steps [16,29-31]. To separate dc and ac components, we write

(13)

Here, k = 2π(Φ/Φ0)/a is the phase gradient induced by the external field, where Φ is the flux in the JJ. ω = 2πΦ0Vdc is the angular Josephson frequency proportional to the dc voltage Vdc. The last term, ϕ, represents the oscillatory component induced by cavity modes and fluxons. This term generates the ac voltage, which we aim to determine:

(14) Small-amplitude, multimode analysis

In the small-amplitude limit, ϕ ≪ 1, a perturbation approach can be used. A linear expansion of Equation 3 yields [16,29,31],

(15)

Here, is the excess dc current with respect to the ohmic QP line. It is caused by the second term on the right-hand side, which enables nonlinear rectification of the Josephson current. The excess dc current is defined as

(16)

The oscillatory part is described by the equation

(17)

A comparison with Equation 1 shows that this is the active TL equation in which the supercurrent wave, sin(kx + ωt), acts as a distributed (x,t)-dependent drive.

To obtain ϕ, a decomposition into cavity eigenmodes is made [15,16,29,31], similar to the TL analysis of patch antennas [34-36]:

(18)

Note that Equation 18 does not include the dc term, n = 0, which is accounted for in instead, so that ϕ generates solely ac voltage, as described by Equation 14. Substituting Equation 18 in Equation 17 and taking into account the orthogonality of eigenfunctions, one obtains

(19) (20) (21)

From Equation 14, voltage amplitudes at radiating slots are:

(22) (23) Excess current

Without geometrical resonances, the dc current, well above the field-dependent critical current, I ≫ Ic(H), is determined by the QP resistance, I = V/RQP. In dimensionless units, I/Ic0 = αV/Vp, where Vp = Φ0ωp/2π is the voltage at plasma frequency. At resonances, a partial rectification of the oscillating supercurrent occurs, leading to the appearance of Fiske steps in the I–V curves. The excess dc current, obtained from Equation 16, is [16,29,31]

(24)

Figure 2a shows calculated I–V characteristics of a JJ with a = 5λJ, α = 0.1 and at a magnetic field corresponding to Φ = 5Φ0 in the JJ. Blue symbols represent the direct numerical simulation of the sine-Gordon Equation 3 for up and down current sweep. The red line shows the analytic solution with the excess current given by Equation 24. The agreement between exact (without linearization) numeric and (approximate) analytic solutions is quite good. It is seen that a series of Fiske steps appear in the I–V. Vertical grid lines mark positions of cavity mode resonances, ω/c0 = kn. Fiske steps appear at this condition because of the vanishing of term in the denominator of gn, Equation 19. The main step occurs at the double resonance condition, ω/c0 = kn = k. It happens at n = 2Φ/Φ0 and leads to the vanishing of (k − kn) in the denominators of Equation 20 and Equation 21. The condition, ω/c0 = k, is referred to as the velocity matching because at this point the velocity of the fluxon chain (or phase velocity of the current wave in Equation 17) reaches c0[16].

Figure 2: (a) Simulated current–voltage characteristics of a junction with L = 5λJ, Φ/Φ0 = 5 and α = 0.1. Blue symbols represent the full numeric solution of the sine-Gordon equation (up and down current sweep). The red line represents the approximate (perturbative) analytic solution, I = V/RQP + ΔI. (b) Excess dc current, ΔI(V), at Fiske steps. The thick red line represents the multimode analytic solution, Equation 24. Thin blue, black, and olive lines show single-mode solutions for n = 9, 10, and 11, respectively. Vertical grid lines in (a) and (b) mark Fiske step voltages. Voltages are normalized by (a) the plasma frequency voltage, Vp, and (b) the lowest Fiske step voltage, V1.

Single-mode analysis

Figure 2b shows the excess current, ΔI/Ic0 versus V, normalized by the n = 1 Fiske step voltage, V1 = Φ0c0/2a. Such normalization clearly shows that the main resonance occurs at n = 2Φ/Φ0 = 10. The thick red line represents the full multimode solution, Equation 24. Thin blue, black, and olive lines represent a single eigenmode contribution for n = 9, 10, and 11, respectively. A perfect coincidence with the red line indicates that for underdamped JJs, α ≪ 1, it is sufficient to consider just a single mode. This greatly simplifies the analysis.

For a resonance at mode n,

(25)

and

(26) (27)

where

(28) Large-amplitude case

The described above perturbative approach is valid only for small amplitudes. Simulations in Figure 2a are made for an underdamped JJ, α = 0.1. In this case the quality factor of high-order cavity modes is large,

and |gn| is not small. Since ϕ appears within the sin η term in Equation 3, the maximum possible amplitude of |gn| is π. This reflects one of the key differences between FFO and patch antenna. The patch antenna is a linear element in which the voltage amplitude is directly proportional to the feed current. A FFO is essentially nonlinear. The amplitude of Josephson phase oscillations will not grow beyond |gn| = π. Instead, higher harmonic generation will occur.

Full numerical simulations of the sine-Gordon equation (Equation 3), shown by blue symbols in Figure 2a, reveal that the amplitude of oscillations reach π at the end of the velocity-matching step. This causes a premature switching out of the resonance before reaching the resonant frequency. It is somewhat miraculous that the agreement with the perturbative solution (red line in Figure 2a) is so good. Apparently, it works remarkably well far beyond the range of its formal applicability, |gn| ≪ 1.

A general single-mode solution for an arbitrary amplitude was obtained by Kulik [30]. The amplitude at the resonance, , is given by the first solution of the implicit equation [31],

(29)

where J0 is the zeroth-order Bessel function. This equation can be easily solved numerically. It is also possible to obtain an approximate analytic solution by expanding J0(x) ≃ 1 − x2/4 for small x. With such expansion, Equation 29 is reduced to a quadratic equation with the solution

(30)

For overdamped JJs, α ≫ 1, it reduces to the small-amplitude result of Equation 25, |gn| = . For underdamped JJs, it qualitatively correctly predicts saturation of the amplitude for α→0, although at a value of 4 instead of π. Thus, Equation 30 provides a simple and sufficiently good approximation for a significantly broader range of damping parameters than Equation 25.

Input resistance

For the practically most important velocity matching mode, kn = k, from Equations 19–21 it follows, Bn = 1, Cn = 0, Fn = 1, leading to a remarkably simple result,

(31)

This equation has a straightforward meaning illustrated by the equivalent circuit in Figure 1c. A JJ is a source of a spatially distributed oscillating current, Jz = Jc0sin(ωt + kx), with a fixed amplitude, Jc0, but spatially dependent phase, kx. It couples to the cavity mode via some effective input impedance Zin. Zin depends on ω, kn and k and is, in general, complex. However, since the phase of the current wave is strongly varying along the junction, it is hard to define the phase shift between current and voltage. Therefore, in what follows, I will be talking about the input resistance, Rin = |Zin|, defined via the relation

(32)

From Equation 26 it follows,

(33)

Figure 3a–c shows, respectively, Bn, Cn, and Rin/RQP = Fn versus n for the case from Figure 2. Lines are obtained for continuous variation of n in Equation 20 and Equation 21, and circles represent the actual cavity modes with integer n. From Figure 3c, it is seen that Rin has a distinct maximum at the velocity matching condition n = 2Φ/Φ0 = 10. At this point, , the wave numbers of the cavity mode and the current wave coincide, leading to a perfect coupling along the whole length of the JJ. Therefore, Rin = RQP and v = Ic0RQP. For other modes, kn ≠ k, the coupling with Josephson current oscillations is much weaker. As seen from Figure 3c, it is oscillating with n. For the particular case with integer Φ/Φ0, Rin vanishes for all even modes. This leads to the absence of corresponding Fiske steps in Figure 2a.

Figure 3: Panels (a) and (b) show mode-number dependence of coefficients Bn and Cn, given by Equation 20 and Equation 21, for the case from Figure 2 with Φ/Φ0 = 5. Panel (c) shows the corresponding oscillatory dependence of the input resistance, Equation 28 and Equation 33. (d) Input resistance for Φ/Φ0 = 5 (olive), 5.25 (blue) and 5.5 (red). The large Rin enables good coupling of the cavity mode to the Josephson current.

The coupling of a cavity mode to the current wave in the JJ depends on magnetic field and flux in the JJ (via the parameter k). This is illustrated in Figure 3d for Φ/Φ0 = 5 (olive line, the same as in Figure 3c), 5.25 (blue), and 5.5 (red). Although the oscillatory behavior of Fiske step amplitudes is well known [16,29,31], the interpretation of such behavior in terms of the input resistance makes a clear connection to the analysis of patch antennas, for which Rin is one of the most important parameters. From this point of view, geometrical resonances with large voltage amplitudes appear only for modes coupled to the current source (Josephson oscillations) via a large input resistance, Equation 32. As seen from Figure 3d, the best coupling with maximum, Rin = RQP, occurs for the velocity-matching step, n = 2Φ/Φ0. Modes with Rin = 0 are not coupled to Josephson oscillations and, therefore, are not excited at all. In particular, there is no coupling to any mode in the absence of an applied field, Rin(H = 0) = 0. This is why Fiske steps do not appear at zero field.

Inclusion of radiative losses in a cavity mode analysis

Finally, in order to calculate radiative characteristics, we need to take into consideration radiative losses. In the previous section, only QP losses in a pure cavity eigenmode were considered. Yet, pure eigenmodes, En ∝ cos(knx), Hn ∝ sin(knx), do not emit any radiation because they do not produce ac magnetic fields at the edges Hn(0,L) = 0 [36]. Consequently, the Pointing vector is zero. In other words, eigenmodes have infinite radiative impedance, Zrad(0,L) = E(0,L)/H(0,L) = ∞. Therefore, despite large electric fields, the radiated power Prad ∝ E2/Zrad is zero [10].

Radiative losses can be included using the equivalent circuit sketched in Figure 1c. Voltage oscillations at the JJ edges are produced by the oscillating supercurrent via the input resistance, Equation 32. The generated electromagnetic power is distributed between internal losses, characterized by the dissipative resistance, Rdis, and radiative losses to free space, characterized by the radiative resistance Rrad. They are connected by the transmission line impedance,

(34)

Here Zsurf is the surface impedance of the electrodes, GQP = 1/RQP is the quasiparticle conductance, L is the inductance, and C is the capacitance of the JJ. The bars indicate that the quantities are taken per unit length. For not very high frequencies and temperatures, the surface resistance of Nb electrodes is small (as will be discussed below). For tunnel JJs, GQP is also small. In this case,

(35)

It is very small because b ≫ Λ ≫ d and can be neglected for all practical cases. Therefore, in Figure 1c we may consider that the dissipative and radiative resistances are connected in parallel. Analysis of patch antennas [36] and numerical calculations for JJs with radiative boundary conditions [10] show that radiative losses can be simply included in the cavity mode analysis by introducing the total quality factor, Qtot, of the cavity mode with parallel dissipative and radiative channels,

(36)

Here, Qdis is associated with all possible dissipative losses, such as QP resistance in the JJ as well as surface resistance in electrodes and dielectric losses while Qrad represents radiative losses,

(37)

Using definitions of α and Q, we can introduce a total damping factor

(38)

where the total resistance is

(39)

Thus, to include radiative losses, α and RQP in the equations above should be replaced by αtot and Rtot. For the n-th cavity mode resonance we obtain,

(40)

For the most important velocity matching resonance from Equation 31, we obtain

(41)

with Rrad and Rtot defined in Equation 12 and Equation 39.

Power efficiency

The total power dissipated in a JJ is given by the product of dc voltage and dc current,

(42)

Here, the left factor is the dc voltage, and the right one is the total dc current. It contains the QP current (first term) and the rectified excess current, ΔI, (second term). The latter is written using Equation 27 at the resonance condition . It is important to note that the nonlinear rectification occurs only inside the JJ. Therefore, the damping parameter αdis within the JJ is used for both terms. The first term in Equation 42 describes dissipative dc losses, which generate only heat, Pheat = V2/2Rdis. The second term in Equation 42 describes the total power consumed by the cavity mode, Pcav = VΔI. Only this term is participating in radiation. From Equation 39 and Equation 40, we obtain a well-known connection between the radiated power and the power consumed solely by the cavity mode,

(43)

As usual, the maximum emission power is achieved at the matching condition Rrad = Rdis. In this case, exactly one half of the cavity mode power is emitted and another half is dissipated. This is typical for antennas [36] and is consistent with direct simulations for JJs with radiative boundary conditions [10]. Yet, the overall power efficiency is reduced by the “leakage” QP current in Equation 42, which just produces heat. For the I–V curves in Figure 2a, the ohmic QP current is more than twice ΔI at the velocity matching step. Therefore, the total power efficiency, Prad/Ptot, for such moderately underdamped JJ will not exceed 50/3 ≃ 17%. Since the leakage current decreases with increasing RQP, strongly underdamped JJs are necessary for reaching a power efficiency of approx. 50%. This is the case for Nb tunnel JJs [9] and for high-quality intrinsic JJs in Bi-2212 high-Tc cuprates, for which the quality factor may exceed several hundreds [32] and ΔI can be several times larger than the leakage QP current [9,32].