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In the current work long Josephson junctions are being studied. Magnetic flux distribution is the physical measure for phase difference of the wave functions in the superconducting layers of the junction. The current phase relation, in most cases, can be considered as an odd strict 2π-periodic function, and hence, it can be presented in Fourier series of sinuses. It is well-known from the physical experiment that with a sufficient degree of precision, a number of physical systems are reliably described with the contribution of only first two harmonics. The adequate mathematical model for the distribution of the magnetic flux is then the double sine-Gordon equation with Neumann boundary conditions at the ends of the junction. Even in the stationary case, the boundary problem is highly nonlinear and the only tool for its comprehensive study is numerical methods. The aim of the present work is to show that in the case of zero external current, the stationary equation turns out to be a fully integrable model, derived from a variation principle with a cosine potential. In our work analytical solutions for the magnetic flux distributions described in the terms of Jacobi elliptic sinuses are derived. Analytical studies in this case serve to further numerically study of this multiparametric nonlinear boundary problem, which is so important in the applied nanophysics.In the current work long Josephson junctions are being studied. Magnetic flux distribution is the physical measure for phase difference of the wave functions in the superconducting layers of the junction. The current phase relation, in most cases, can be considered as an odd strict 2π-periodic function, and hence, it can be presented in Fourier series of sinuses. It is well-known from the physical experiment that with a sufficient degree of precision, a number of physical systems are reliably described with the contribution of only first two harmonics. The adequate mathematical model for the distribution of the magnetic flux is then the double sine-Gordon equation with Neumann boundary conditions at the ends of the junction. Even in the stationary case, the boundary problem is highly nonlinear and the only tool for its comprehensive study is numerical methods. The aim of the present work is to show that in the case of zero external current, the stationary equation turns out to be a fully integrable model,...

The flux-flow oscillator (FFO) based on a long Josephson junction has been implemented as a broadband tunable terahertz (THz) source to open space. For this purpose, the transmitting slot antenna has been coupled to the oscillator. Additionally, an elliptical lens with a diameter of 10 mm has been matched to the antenna, forming a narrow output beam of the THz emission. Two designs for the antenna, integrated with the oscillator and developed for operation at different frequency ranges of 0.32-0.55 THz and 0.4-0.7 THz, have been investigated. The FFO has been phase locked to an external reference oscillator by utilizing a harmonic mixer. Its linewidth in the phase-locking regime is determined by the phase noise of the reference oscillator and the number of harmonics used and has been measured to be less than 0.1 MHz. A free-running FFO linewidth from about 2 MHz to several MHz, depending on the operating point, has been obtained. Output emission to open space has been measured by a superconducting integrated spectrometer located in a separate cryostat. The FFO operation as an external source with the achieved emission power and spectral characteristics has demonstrated its applicability for different tasks and purposes where tunable THz sources are required.

The transition from the model of a long Josephson junction of variable width to the model of a junction with a coordinate-dependent Josephson current amplitude is effected through a coordinate transformation. This establishes the correspondence between the classes of Josephson junctions of variable width and quasi-one-dimensional junctions with a variable thickness of the barrier layer. It is shown that for a junction of exponentially varying width the barrier layer of the equivalent quasi-one-dimensional junction has a distributed resistive inhomogeneity that acts as an attractor for magnetic flux vortices. The curve of the critical current versus magnetic field for a Josephson junction with a resistive microinhomogeneity is constructed with the aid of a numerical simulation, and a comparison is made with the critical curve of a junction of exponentially varying width. The possibility of replacing a distributed inhomogeneity in a Josephson junction by a local inhomogeneity at the end of the junction is thereby demonstrated; this can have certain advantages from a technological point of view., Comment: 9 pages, 6 figures

We have investigated numerically a system consisting of a long Josephson junction embedded into a microstrip resonator. For such a configuration the Josephson vortex dynamics in the junction is driven by the oscillating currents in the resonator. We have calculated the complex impedance of the junction at the resonant frequency. The Q factor of the resonator and the change of the resonant frequency of the whole system can then be easily calculated knowing the parameters of the resonator without Josephson junction. The fluxon dynamics and the complex impedance of the junction are simulated for different values of the junction length, damping, dc bias current, and dc magnetic field. This allows us to explain the behavior of the impedance of the junction and different loss mechanisms. In particular, we predict a nonmonotonic dependence of the both real and imaginary parts of the impedance on the amplitude of the oscillating current in the resonator for some range of parameters. Our results show that such a combination of active and passive superconducting elements forms a rather interesting nonlinear physical system which might be useful for development of tunable/switchable superconducting resonators and suggests an experimental technique for detecting trapped vortices in long junctions. (C) 2002 American Institute of Physics.

The dependence of the first critical field of long linear and annular Josephson junctions on the width of the surrounding stripline, the so called idle region, is investigated experimentally. The stripline modifies the effective Josephson length λeff in the junction. The experimental data are compared with the theory by Caputo et al. [J. Appl. Phys. 85, 7291 (1999)] and good agreement is found. The dependence of the first critical field on the width of the surrounding stripline can be well explained using the same λeff for both annular and linear junctions.

Series-biased arrays of long Josephson junction fluxon oscillators can be phase locked by mutual coupling to a high-Q, linear distributed resonator. A simplified model of such a device, consisting of junctions described by the particle-map perturbation theory approach which are capacitively coupled to a lumped, linear tank circuit, reproduce the essential experimental observations at a very low computational cost. A more sophisticated model, consisting of partial differential equation descriptions of the junctions, again mutually coupled to a linear tank, substantially confirm the predictions of the simplified model. In the particle-map model, the locking range in junction bias current increases linearly with the coupling capacitance; in the partial differential equation (p.d.e.) model, this holds up to a certain maximum value of the capacitance, after which a saturation of the locking range is observed. In both models, for a given spread of junction lengths, the existence of a minimum value of the capacitance for locking to a tank with a given resonant frequency is evidenced. Journal of Applied Physics is copyrighted by The American Institute of Physics.

The height of the velocity‐matching step (VM step) of a flux‐flow type Josephson oscillator is studied systematically by numerical simulations. It is defined by the phase velocity of the propagating fluxons equaling that of the electromagnetic wave in the junction structure. The height of VM step (γm) is an important parameter for the practical application of such a device. The results presented on γm are important data in order to save computer time in the simulation of electromagnetic properties, such as the output voltage, at the velocity‐matching condition. The detailed simulations were performed using the perturbed sine‐Gordon equation (PSGE) with a uniformly distributed bias current γ. The PSGE was solved numerically by using an implicit finite‐difference scheme. The distributions of electromagnetic fields in an overlap, long Josephson junction in the flux‐flow state are mapped. The parameters influencing γm are discussed. We show that γm increases almost linearly with an external magnetic field βe ...

A pinning force acting upon a periodic fluxon chain in a long Josephson junction with a commensurable periodic lattice of local inhomogeneities is calculated by means of a perturbation theory. The pinning force for a single fluxon is also calculated. Dynamics of a chain and of a single fluxon are also studied. The experimental part of the paper reports results of measurements of a critical bias current Ic versus external magnetic field H. The measurements have been performed for long overlap Nb‐NbOx‐Pb junctions Ic(H) curve were observed at multiple values of H. Agreement between the theory and experiment for different spatial periods of the fluxon chain is satisfactory.

We report a circuit that integrates an underdamped long Josephson junction with overdamped single-flux-quantum (SFQ) circuits. We confirm that the resonant soliton modes in the long junction are not affected by SFQ cells coupled to the junction, and demonstrate that the radiation frequency and linewidth of the soliton resonances can be measured with SFQ T-flip-flops. Our experimental results also show that a 4π quantum mechanical phase leap at the end of the long junction, which is due to the reflection of a soliton, creates two single flux quanta propagating in the overdamped Josephson transmission line.

Measurements were made on a two-stack long Josephson junction with very similar parameters and electrical access to the thin middle electrode. Mutually phase-locked fluxon-antifluxon states were observed. The observed propagation velocity is in agreement with the theoretical prediction. The I-V curves recorded using the bias current in one of the junctions as fixed parameter show that these states continue to also exist when the other junction is oppositely biased. This is found in numerical simulations based on the experimental junction parameters as well. We propose a model in which waves in the junctions coexist with fluxons. (C) 1996 American Institute of Physics.

A nonuniform magnetic field has been applied to a long Josephson junction to create two separated vortex and antivortex regions. Such a configuration is the Josephson version of the so‐called fluxonic diode, an electrical dual to the semiconductor p‐n junction diode. Volt‐ampere relations of a junction subjected to this magnetic field are compared with those obtained using a unidirectional magnetic field. The study, supported by simulation, shows a number of unique characteristics including rectification and fluxon injection across the junction. Both are necessary phenomena for realization of a bipolar transistor dual.

For creation of a noisy non-stationary spectrometer operating in the sub-THz frequency range, the emitting power and spectral characteristics of a flux-flow oscillator based on a long Josephson junction are considered. The effect of bias current profiles on the spectral linewidth of a long overlap Josephson junction is investigated and compared with the inline junction geometry. It has been demonstrated that the spectral linewidth can be increased by a factor of three with a moderate reduction of the emitted power.

The current-voltage and spectral characteristics of a flux flow oscillator (FFO) based on a long Josephson junction are studied. The investigations are performed in the range of small bias currents and magnetic fields where the FFO radiates a quasi-chaotic signal with extremely large radiation linewidth, and the displaced linear slope (DLS) is observed at the current-voltage characteristic. Using direct numerical simulation of the sine-Gordon equation, it is shown that for large lengths of the Josephson junction or in the case of partial matching of the FFO with external waveguide system, the DLS with extremely large linewidth is transformed into Fiske steps with very narrow linewidth. As for Fiske steps, the appearance of regime of chaotic oscillations can be explained by multiple reflections of the traveling waves from junction ends rather than simply by excitation of the internal oscillation modes in the “soft” fluxon chain at weak magnetic fields.

The current-driven vortex motion in a resistively shunted long Josephson junction is calculated. It is shown that the force on the vortex exerted by an ac Josephson current or a bias current routinely assumed for studying Josephson transmission lines is a Lorentz force, whereas the driving force on the vortex acted by a transport dc Josephson current is a London force, which could have an expression apparently similar to that of a Lorentz force with an opposite sign.

The dc and ac magnetic properties of a melt-textured YBa2Cu3Ox ring with a perpendicular crack is studied. It is found that the crack behaves as a resistively shunted very long Josephson junction. However, besides the expected edge currents, certain inner currents have to occur in the junction. This anomaly may be a general feature of a very long junction with similar values of Josephson and London penetration depths, which is a case not yet studied theoretically.

Numerical as well as experimental studies have been made of the oscillation stability for the travelling‐wave‐type oscillator, which utilizes a vortex motion in a long Josephson junction, i.e., a flux flow. For the Josephson oscillator, a steep current step in a dc I‐V characteristic improves the oscillation stability. Two kinds of effects which degrade the steepness of the current step have been investigated for the long junction of the overlap geometry. One is the self‐field effect which makes the current step inclined, and the other is the resonant motion of the vortices which induces a staircaselike structure on the current step. We have discussed the method for the reduction of these effects and proposed a novel junction geometry without these effects, i.e., an overlap junction with a projection on one junction side. With the new geometry, we have obtained current steps whose slope is about one hundred times steeper than those obtained with the conventional overlap geometry and whose height attains a...

A theoretical study is made of a travelling‐wave‐type oscillator, which utilizes a flux flow in a long Josephson junction for use as a local oscillator in the integrated superconducting receiver system. An internal electromagnetic field of the oscillator junction in the flux‐flow state is investigated both numerically and analytically. It is shown that the voltage amplitude of the internal oscillation increases gradually in the direction of the flux flow and reaches a maximum value at the junction end. An equivalent circuit of the oscillator is also obtained, which gives dependences of the emitted radiation on frequency, magnetic field, and load. It is shown that the output power attains the value of the order of 10−6 W in the frequency range between 100 and 500 GHz, and that the output power and the radiation frequency can be controlled by both the bias voltage and the applied magnetic field. These theoretical results explain quantitatively the experimental ones with a Pb‐alloy long junction of length 24 λJ.

Results of extensive digital simulations of a long Josephson junction subject to the presence of dc and rf driving currents are reported. The dynamic behavior of this system is mapped as a function of rf amplitude and frequency of the parameter set which leads to fluxon oscillations in the absence of an rf drive. In response of the rf excitation the junction is shown to exhibit a variety of behaviors including periodic motion in which the fluxon oscillation is phase locked to the rf frequency, chaotic motion in which spatial structure persists, and spatially uniform periodic and chaotic motion.

A long Josephson junction dc biased on a zero‐field singularity and emitting radiation at microwave frequencies is irradiated with external microwave power. This power can be supplied either by a room‐temperature oscillator or by another long junction. We find that the oscillations of the junction can coherently lock to the external signal for frequency intervals ranging from 500 kHz to 50 MHz. The dependence of the width of these intervals of coherence on the external microwave power is measured for the case in which the power is generated by a room‐temperature oscillator.

The transmission, reflection, and absorption characteristics for fluxons propagating in long Josephson junction transmission lines which are connected by resistive networks have been studied by means of computer simulation. Although fluxon scattering behavior is qualitatively similar to that expected for electromagnetic waves, the existence of threshold resistances which depend upon applied bias current and junction loss is observed. For the case of two long junctions coupled by a serious resistor, it is found that variations in bias current, junction loss, and resistor value determine whether a fluxon incident on the resistor is either absorbed, or produces a reflected fluxon, a transmitted fluxon, or both a reflected and a transmitted fluxon. It is found that three junction segments can be connected in a configuration which functions as a fluxon multiplexer. Finally, the time delay (that is, the time required for a fluxon to attain steady state energy) in a typical low‐loss tunnel junction line is found...

An oscillator which utilizes the effect of the vortex motion in long Josephson tunnel junctions, i.e., flux flow, has been presented in millimeter and submillimeter wave region. An electromagnetic wave generated by the oscillator is detected with a small tunnel junction as a detector with a refined coupling configuration. Quantitative evaluation of the detected power showed that the detected power attained the value of 10−6 W in the frequency range between 100 and 400 GHz, which is far superior to previous results. Frequency and magnetic field dependences of the present system were also measured, which showed that the output power was able to be controlled by the dc magnetic field. The present oscillator will be promising as the local oscillator in the integrated Josephson receiver systems.

The propagation of magnetic flux pulses along the insulating barrier of a long Josephson junction is investigated both theoretically and experimentally. The theoretical study includes applications of both (i) the recently developed ’’inverse‐scattering‐transform method’’ (ISTM) to the corresponding sine‐Gordon equation in characteristic (light cone) coordinates and (ii) Whitham’s method (WM) of averaged Lagrangian analysis to the sine‐Gordon equation in laboratory coordinates. As the number of solitons (flux quanta) in the pulse becomes large, the ISTM becomes numerically unwieldy while WM becomes more accurate; thus these two analytical tools are complementary. WM has the advantage of being readily modified to account for small dissipative effects. Our experimental observations of magnetic‐flux propagation were entirely restricted to the ’’large‐amplitude limit’’ in which the average of the ac Josephson current is effectively zero. In this limit, WM indicates pulse propagation with linear dissipation. Th...

We have explored numerically the effects on soliton propagation of a third order damping term in the modified sine-Gordon equation. In Josephson tunnel junctions such a term corresponds physically to quasiparticle losses within the metal electrodes of the junction. We find that this loss term plays the dominant role in determining the shape and stability of the soliton at high velocity. Applied Physics Letters is copyrighted by The American Institute of Physics.

Within the framework of the sine‐Gordon model of a long Josephson junction, we have calculated a threshold value of bias current density which admits capture of a fluxon by an effective potential well generated by a step of inductance in the junction. Along with a kinematic approximation, a nontrivial dynamical correction is found. We have also calculated analytically energy emitted by a fast fluxon moving past the step.

The dispersion relation of small electromagnetic waves applied in the uniform flux-flow state of a long Josephson junction has been obtained. The results contain exponentially growing waves due to the parametric interaction between the applied fields and the moving vortices, which indicates the possibility of a Josephson-junction frequency converter of the traveling-wave type.