High linearity superconducting radio frequency magnetic field detector

Abstract

A superconducting quantum interference devices (SQUID) comprises a superconducting inductive loop with at least two Josephson junction, whereby a magnetic flux coupled into the inductive loop produces a modulated response up through radio frequencies. Series and parallel arrays of SQUIDs can increase the dynamic range, output, and linearity, while maintaining bandwidth. Several approaches to achieving a linear triangle-wave transfer function are presented, including harmonic superposition of SQUID cells, differential serial arrays with magnetic frustration, and a novel bi-SQUID cell comprised of a nonlinear Josephson inductance shunting the linear coupling inductance. Total harmonic distortion of less than −120 dB can be achieved in optimum cases.

Claims

1. A transducer for transducing a flux Φ into an electrical signal, comprising: a first Josephson junction circuit, having a first output responsive to a nonlinear impedance which depends on the flux Φ in a periodic manner, with periodicity Φ.sub.1=h/2e; a second Josephson junction circuit, having a second output responsive to a nonlinear impedance which depends on the flux Φ in a periodic manner, with periodicity Φ.sub.2=h/2e, where h is Planck's constant and e is the charge on an electron, a combiner, configured to harmonically superpose at least the first output and the second output to increase a linearity with respect to the periodic flux Φ of the superposed first output and second output around a predetermined value of the flux Φ.

2. The transducer according to claim 1, further comprising: a third Josephson junction circuit, having a third output responsive to a nonlinear impedance which depends on the flux Φ in a periodic manner, with periodicity Φ.sub.3=h/2e, the combiner being configured to harmonically superpose at least the first output, the second output, and third output to increase a linearity with respect to the periodic flux Φ, of the superposed first output, second output, and third output around a predetermined value of the flux Φ.

3. The transducer according to claim 2, wherein the first Josephson junction circuit, the second Josephson junction circuit, and the third Josephson junction circuit are provided in a non-periodic spatial array structure.

4. The transducer according to claim 1, wherein the first Josephson junction circuit has a transduction area a.sub.1, and the second Josephson junction circuit has a transduction area a.sub.2, wherein transduction area a.sub.1 is different from transduction area a.sub.2.

5. The transducer according to claim 1, wherein the superposed first output and second output around the predetermined value of the flux Φ has a power at least 6 dB higher than a larger of a power associated with the flux Φ coupled to the first output and the second output around the predetermined value of the flux Φ.

6. The transducer according to claim 1, wherein the first Josephson junction circuit and the second Josephson junction circuit are magnetically frustrated, wherein the periodicity Φ.sub.1 is out of phase with respect to the periodicity Φ.sub.2.

7. The transducer according to claim 1, wherein the superposed first output and second output around a predetermined value of the flux Φ have suppressed even harmonics of the periodicity of the flux Φ.

8. The transducer according to claim 1, wherein the first Josephson junction circuit and the second Josephson junction circuit each have a bias current, wherein the respective bias current of the first Josephson junction circuit is different from the bias current of the second Josephson junction circuit.

9. The transducer according to claim 1, wherein the first Josephson junction circuit comprises a bi-SQUID, having a modified coupling inductance modified by shunting with a non-linear inductive element.

10. The transducer according to claim 1, wherein the first Josephson junction circuit comprises a first Josephson junction having a critical current I.sub.c1, a second Josephson junction having a critical current I.sub.c1, a loop having a loop inductance, and a shunt Josephson junction having a critical current I.sub.c3, and a shunt inductance L=Φ/2π√{square root over (I.sub.c.sup.2+I.sub.sh.sup.2)}, where I.sub.sh is a shunt current passing through the shunt Josephson junction.

11. The transducer according to claim 1, having a gain of at least 3 dB with respect to the flux Φ and the superposed first output and second output around a predetermined value of the flux Φ has a linearity of at least 101 dB.

12. A method for transducing a flux Φ into an electrical signal, comprising: providing a plurality of Josephson junction circuits, each having an output responsive to a nonlinear impedance which depends on a respective flux Φ.sub.0 in a periodic manner, with periodicity Φ.sub.0=h/2e, where h is Planck's constant and e is the charge on an electron; and harmonically superposing the respective outputs of the plurality of Josephson junction circuits to increase a linearity with respect to the periodic flux Φ.sub.0.

13. The method according to claim 12, wherein the plurality of Josephson junction circuits are provided in a non-periodic spatial array structure.

14. The method according to claim 12, wherein the plurality of Josephson junction circuits each has a transduction area a, at least two of the respective transduction area a being different.

15. The method according to claim 12, wherein the harmonically superposed respective outputs have a power at least 3 dB higher than a power associated with the flux 1 coupled to the plurality of Josephson junction circuits.

16. The method according to claim 12, wherein the plurality of Josephson junction circuits comprise at least two magnetically frustrated Josephson junction circuits, having respectively out of phase periodicity Φ.sub.0.

17. The method according to claim 12, wherein the plurality of Josephson junction circuits each have a bias current, wherein at least two of the respective bias currents differ.

18. The method according to claim 12, wherein at least one Josephson junction circuit comprises a first Josephson junction having a critical current I.sub.c1, a second Josephson junction having a critical current I.sub.c1, a loop having a loop inductance, and a shunt Josephson junction having a critical current I.sub.c3, and a shunt inductance L=Φ/2π√{square root over (I.sub.c.sup.2+I.sub.sh.sup.2)}, where I.sub.sh is a shunt current passing through the shunt Josephson junction.

19. A transducer, comprising: an irregular array of Josephson junction circuits, each Josephson junction circuit comprising an input configured to receive a bias current, a first Josephson junction having a critical current I.sub.c1, a second Josephson junction having a critical current I.sub.c1, a loop having a loop inductance, and a shunt Josephson junction in parallel with the loop, having a critical current I.sub.c3, and a shunt inductance L=Φ/2π√{square root over (I.sub.c.sup.2+I.sub.sh.sup.2)}, where I.sub.sh is a shunt current passing through the shunt Josephson junction; and a harmonic combiner, configured to superpose respective outputs of the irregular array of Josephson junction circuits, to at amplify a flux coupled to the respective loops of the irregular array of Josephson junction circuits, and to produce a linearized superposed output.

20. The transducer according to claim 19, having a harmonic distortion with respect to flux of less than about −101 dB.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The foregoing summary, as well as the following detailed description of the embodiments of the present invention, will be better understood when read in conjunction with the appended drawings. For the purpose of illustrating the invention, there are shown in the drawings embodiments which include those presently preferred. As should be understood, however, the invention is not limited to the precise arrangements and instrumentalities shown.

(2) FIGS. 1A, 1B and 1C show basic SQUID structures of the prior art.

(3) FIG. 2 shows an iterative algorithm for successively linearizing the voltage across a serial array of SQUID cells by adjusting the distribution of areas of the cells.

(4) FIG. 3 shows a differential array structure consisting of two series arrays of SQUIDs with current biasing I.sub.b=I.sub.c; in one array, each cell is biased by magnetic flux Φ.sub.0/2, and an additional flux bias of Φ.sub.0/4 is applied to all cells to set the operating point.

(5) FIGS. 4A and 4B show a schematic of a bi-SQUID and its periodic voltage response, as contrasted with that of a conventional SQUID without the shunt junction.

(6) FIG. 5 shows the model dependence of the linearity (in dB) vs. the critical current Ica of the shunt junction for a bi-SQUID, for several values of inductance L.

(7) FIG. 6 shows the model dependence of the linearity on the output signal amplitude.

(8) FIGS. 7A and 7B show the circuit schematic of a bi-SQUID cell for amplifier and antenna applications.

(9) FIG. 8 shows the experimental dependence of voltage on signal current (proportional to flux) for a fabricated bi-SQUID.

(10) FIG. 9A shows the experimental current dependence of an array of 12 bi-SQUIDs.

(11) FIG. 9B shows the differential voltage output of two arrays of 12 bi-SQUIDs, with magnetic frustration of Φ.sub.0/2 between the two arrays.

(12) FIG. 10A shows the found optimal distribution of the cell areas along a parallel SQIF.

(13) FIG. 10B shows the Voltage response (solid line) of the differential circuit of two parallel SQIFs with the found cell area distribution at optimal magnetic frustration, in which the response linearity within the shaded central area is as high as 101 db, and the frustrated SQIF responses are shown by dashed lines.

(14) FIGS. 11A and 11B show the linearity of the differential voltage response versus both the number N of SQIF cells (FIG. 11A) and the spread in the cell areas at N=36 (FIG. 11B).

(15) FIG. 12 shows a two-dimensional differential serial-parallel SQIF structure.

(16) FIG. 13A shows the voltage response of a parallel array of N=6 junctions coupled by inductances with normalized value l=1 at different shunting resistors R.sub.sh connected in parallel to the coupling inductances, in which the dashed line shows the voltage response of the array in the limit of small coupling inductance.

(17) FIG. 13B shows the voltage response of the differential circuit of two frustrated parallel arrays of N=6 Josephson junctions at l=1 and R.sub.sh=R.sub.N, where R.sub.N is the Josephson junction normal resistance.

(18) FIG. 14 shows an active electrically small antenna based on a two-dimensional differential serial-parallel SQIF-structure.

(19) FIG. 15 shows the dependence of the normalized low-frequency spectral density of the resistor voltage noise on number N of resistors R.sub.N connected in parallel by coupling inductances at different normalized values l of the inductances.

(20) FIG. 16 shows the dependence of the normalized spectral density of the resistor voltage noise in a parallel array of 30 resistors R.sub.N on normalized frequency at different normalized values l of the coupling inductances.

(21) FIG. 17 shows a normalized transfer factor B=dV/dΦ for a parallel array of Josephson junctions versus number of junctions N at different normalized values l of coupling inductances.

(22) FIG. 18 shows the normalized voltage response amplitude V.sub.max for two parallel arrays of Josephson junctions coupled correspondingly by unshunted inductances (lower curve) and by the optimally shunted inductances (upper curve) versus normalized inductance value.

(23) FIG. 19A shows a serial array with stray capacitances and typical I-V curves of a serial array of 10 DC SQUIDs calculated using the RSJ model in the presence of stray capacitances and without capacitances.

(24) FIG. 19B shows the experimentally measured I-V curve of a serial array of 20 SQUIDs fabricated using standard niobium integrated circuit technology.

(25) FIG. 20A shows a schematic of a system subject to stray capacitances in the SQUID array structures fabricated using standard niobium technology with two screens (upper and lower screens) in the case of a continuous double screen.

(26) FIG. 20B shows a schematic of a first system which reduces stray capacitances in the SQUID array structures fabricated using standard niobium technology with two screens (upper and lower screens) in the case of an individual double screen.

(27) FIG. 20C shows a schematic of a second system which reduces stray capacitances in the SQUID array structures fabricated using standard niobium technology with two screens (upper and lower screens) in the case of an intermittent (dashed) double screen.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

I. Bi-SQUID

(28) As described above, a major part of the invention comprises a new SQUID cell, the bi-SQUID. The DC SQUID, modified by adding a Josephson junction shunting the loop inductance, provides extremely high linearity with the proper selection of parameters. This is somewhat surprising, since a Josephson junction presents a nonlinear inductance. However, the junction nonlinearity is able to compensate the nonlinearity of the device in order to achieve an improved linearity close to 120 dB for significant loop inductances (which are necessary to achieve large coupling to external signals). It is to be understood by those skilled in the art that any other nonlinear reactance that functions in a similar way would have a similar effect on reducing the nonlinearity of the system transfer function.

(29) The linearity dependence of the shunt junction I.sub.c3 on critical current at different inductances of the SQUID loop is shown in FIG. 5. The linearity is calculated using a single-tone sinusoidal flux input (of amplitude A/A.sub.max=0.2, where A.sub.max corresponds to the flux amplitude Φ.sub.0/4), and measuring the total harmonic distortion in dB. This result shows that the linearity is sharply peaked for each value of l=LI.sub.c/Φ.sub.0, but with different optimized values of I.sub.c3. Very large values of linearity as high as ˜120 dB are achievable. FIG. 6 shows how the linearity parameter varies as a function of the signal amplitude for other parameters fixed. The linearity decreases as the signal approaches the maximum value.

(30) A serial array of bi-SQUIDs can be implemented to increase the dynamic range up to a value comparable with the response linearity. Moreover, a serial SQIF providing a single (non-periodic) voltage response with a single triangular dip at zero magnetic flux can be implemented.

(31) Single bi-SQUIDs, serial arrays of bi-SQUIDs, and a prototype of an active electrically small antenna based on a bi-SQUID-array were designed, fabricated and tested, using a 4.5 kA/cm.sup.2 Nb HYPRES process (Hypres Inc., Elmsford N.Y.). The layout design of the chips with these elements was made before the completion of the numerical simulations aimed at the optimization of the circuit parameters, in particular before obtaining the results presented in FIG. 5 were obtained. Therefore the critical currents of all Josephson junctions in bi-SQUIDs were chosen equal (I.sub.c1=I.sub.c2=I.sub.c3=I.sub.c) while the optimal shunt shunting junction critical current should be somewhat less for the implemented inductance parameter l=1.4.

(32) FIG. 7A shows the schematic equivalent circuit of the bi-SQUID for both the fabricated single bi-SQUID and the serial array of bi-SQUIDs, for amplifier applications. To apply magnetic flux, a control strip line coupled magnetically with an additional transformer loop was used. The coupling loop with high inductance L.sub.ex is connected in parallel to inductance L.sub.in and therefore practically does not change the interferometer inductance. FIG. 7B shows the corresponding equivalent circuit of the Bi-SQUID for an electrically small antenna.

(33) The voltage response of the bi-SQUID to applied flux (as measured in current units) is shown in FIG. 8. The applied bias current was slightly more than 2I.sub.c for the bi-SQUID. The shunt junction critical current is not optimal at the implemented inductance parameter l=1.4. As a result, the observed voltage response is not perfectly linear, although it shows a clear triangular shape. The measured transfer function closely coincides with simulations, however. As for the small hysteresis at the flux value close to ±Φ.sub.0/2, this indicates that effective inductance parameter of a single-junction SQUID l*≡l.Math.i.sub.c3≡2πLI.sub.c3/Φ.sub.0 is more than 1 and hence the static phase diagram becomes hysteretic.

(34) The voltage response of the 12-element bi-SQUID array is presented in FIG. 9A, and looks virtually identical to that for a single bi-SQUID. The applied bias current was slightly more than the critical current of the array. The voltage response linearity of both bi-SQUID and bi-SQUID serial array can be further improved by means of differential connection of two identical bi-SQUIDs or serial arrays oppositely frustrated by half a flux quantum. This improvement results from cancellation of all even harmonics of the individual responses. FIG. 9B shows the voltage response of the differential scheme of two serial arrays of 12 bi-SQUIDs frustrated by half a flux quantum as well as the source responses of the arrays. The arrays are biased about 10% above their critical currents.

II. SQIF-Based Differential Structures

(35) The differential scheme of two parallel SQIFs oppositely frustrated by an applied magnetic field δB (see FIG. 3) is able to provide extremely linear voltage response in case of a proper choice of the SQIF structure. In the limit of vanishing inductances l of the interferometer cells, one can use an analytical relation for the parallel SQIF response [3]-[5]:
V(B)=V.sub.c√{square root over ((I.sub.b/I.sub.c).sup.2−|S.sub.K(B)|.sup.2)}  (5)

(36) where S.sub.K(B) is the structure factor:

(37) $\begin{array}{cc}{S}_K\left(B\right)=\frac{1}{K}\underset{k=1}{\overset{K}{.Math.}}\exp \left(i\frac{2\pi }{{\Phi }_0}B\underset{m=1}{\overset{k-1}{.Math.}}{a}_m\right),& \left(6\right)\end{array}$

(38) where I.sub.b is the bias current, I.sub.c is the total critical current, K is the number of Josephson junctions, and a.sub.m is the area of the m-th interferometer cell. For sufficiently large K, one can use integration instead of summation, and relation (5) can be transformed as follows:

(39) $\begin{array}{cc}S\left(B\right)=\frac{1}{L}{\int }_0^L\mathrm{dz}.Math.\exp \left(i\frac{2\pi }{{\Phi }_0}B{\int }_0^{z}a\left(x\right)\mathrm{dx}\right).& \left(7\right)\end{array}$

(40) A solution for the specific distribution of the interferometer cell areas a(x) along the SQIF-structure (0<x<L) to make the differential circuit voltage response
ΔV(B)=V(B+δB)−V(B−δB)  (8)
close to the linear relation
ΔV(B)=k.Math.B  (9)

(41) in a signal region −α.Math.δB<B<α.Math.δB, where α≤1 is sought.

(42) Relations (5)-(9) allow derivation of master equations and minimizing the resulting functional to obtain an optimal distribution a(x). One can use an iterative algorithm to find the problem solution, starting from some initial approximation (see FIG. 2). In the case of finite inductances l of the interferometer cells, the SQIF response V(B) has to be calculated by means of numerical simulation, using in particular the well known software PSCAN [18].

(43) The problem can have more than one solution. Various analytical approximations for the problem solution at l=0 are found; the best one is as follows:
a(x)/a.sub.Σ=1.2−0.48 sin.sup.3(πx),  (10)

(44) where a.sub.Σ—total area of the parallel SQIF.

(45) FIGS. 10A and 10B show both the cell area distribution (10) (FIG. 10A) and the differential circuit voltage response (FIG. 10B). Linearity of the voltage response within the shaded central area is as high as 101 dB. To estimate the linearity a sin-like input signal was applied, and the spectrum of the output signal then studied. A ratio of the basic harmonic to the maximal higher one was used to characterize the response linearity. It was found that a very high linearity can be obtained using a relatively small number N of SQIF cells with areas fitted to (10). FIG. 11A shows that the linearity increases rapidly with the number N and at N>35 reaches a plateau where the linearity is as high as 101 dB. As for the impact of technological spread in the cell areas, FIG. 11B shows that the tolerable spread is about 4% at N=36; and then the linearity decreases with the spread value. Approximately the same result was obtained for the spread in critical currents of Josephson junctions. A further increase of N can be used to decrease the impact of the technological spread in the SQIF circuit parameters as well as to increase the dynamic range proportional to √{square root over (N)} up to the linearity level obtained.

(46) Both the dynamic range and the output signal amplitude can be additionally increased by connection of the differential SQIF structures in series, i.e., by providing a two-dimensional differential serial-parallel SQIF structure (see FIG. 12). The number K of the elements connected in series is responsible for the output signal amplitude, while the total number of Josephson junctions N*=N.Math.K is responsible for the dynamic range of the structure. By varying the number of elements connected in parallel (N) and in series (K), one can change the impedance of the structure over a wide range.

III. Effects of Real Junctions and Coupling Inductances

(47) At the same time, there are several problems which should be solved to realize the potentially high performance of the amplifier or antenna. First of all, one should note that the optimal specific structure of the parallel SQIF reported in [5] was determined based on the ideal RSJ model of Josephson junctions and for the case of vanishing coupling inductances (L=0). Deviations of junctions and inductors from ideal theoretical behavior will hinder the linearity of the real structure fabricated. There are two general approaches to the problem solution: (i) to provide the closest approach of the experimental Josephson-junction characteristics to the ones given by the RSJ model and (ii) to synthesize an optimal SQIF structure founded in experimental Josephson-junction characteristics by means of numerical simulation technique (for example by software PSCAN [18]) and an iterative algorithm (FIG. 2). Indeed, an optimal strategy may be based on a combination of schemes.

(48) In particular, as for the coupling inductance L, the negative influence of the finite value of L on the voltage response linearity can be reduced by shunting resistors R.sub.SH connected in parallel to the inductances. Due to the fact that the impedance of the RL circuit becomes low enough at the Josephson oscillation frequency, the parallel array voltage response approaches that for smaller and smaller inductance with the decrease of R.sub.SH down to some optimal resistance value depending on the normalized inductance l; further increase in R.sub.SH leads to some other linearity distortions. Therefore, the most effective method is synthesis of an optimal SQIF structure with the cell area distribution a(x) optimized for the finite value of l. In this case one should use a high performance numerical simulation technique (e.g., PSCAN software [18]) for calculation of the SQIF voltage response V(Φ)) in every cycle of the iterative algorithm (FIG. 2), which has to be used to solve the master equation.

(49) The shunting technique efficiency is confirmed by results of numerical simulations presented in FIGS. 13A and 13B. One can see that at R.sub.SH≈0.1R.sub.N (where R.sub.N is Josephson junction normal resistance), the voltage response of the parallel array of 6 junctions with l≡2πI.sub.CL/Φ.sub.0=1 approaches that for vanishing coupling inductances. As a consequence, the required linear voltage response of the differential scheme of two parallel SQIFs with N=20 and coupling inductances l=1 each shunted by resistor R.sub.SH=0.1R.sub.N are observed.

IV. Advantages of SQIF-Like Structures

(50) In the case of a serial SQIF including N DC SQUIDs, the thermal noise voltage V.sub.F across the serial structure is proportional to square root of N, while the voltage response amplitude V.sub.max(Φ) and the transfer factor B=∂V/∂Φ both are about proportional to N. This means that the dynamic range D=V.sub.max(Φ))/V.sub.F increases as N.sup.1/2. As for the parallel SQIF, in the case of vanishing coupling inductances (l=0), the dynamic range is also proportional to square root of number of junctions N. In fact, the thermal noise voltage V.sub.F across the parallel structure decreases with the square root of N, while the voltage response amplitude V.sub.max(Φ) remains constant and the transfer factor B=∂V/∂Φ increases as about N.

(51) A SQIF-like structure is characterized by a superior broadband frequency response from DC up to approximately 0.1.Math.ω.sub.c, where ω.sub.c is characteristic Josephson frequency [13]. Therefore, a further increase in characteristic voltage V.sub.c of Josephson junctions by implementation in niobium technology with higher critical current density, or by use of high-T.sub.c superconductors, promises an extension of the frequency band up to several tens of gigahertz. Moreover, the SQIF eliminates high interference, and it sufficiently decreases the well known saturation problem of SQUID-based systems. Therefore, SQIF-based systems can easily operate in a normal lab environment.

(52) An approach to synthesis of multi-SQUID serial structures has been reported, capable of providing periodic high linearity voltage response [11, 12]. The approach is based on the formation of serial structures which are capable of providing periodic triangular voltage response to a magnetic field B. Using interferometer cells with a harmonic voltage response, one can synthesize a serial array consisting of many groups of identical interferometers, each group providing a specific spectral component of the resulting voltage response of the array. According to estimations, the response linearity reaches 120 dB, if the number of the groups is as high as about 165. The second way to synthesize a highly linearity array structure is through implementation of a differential scheme of two serial arrays of DC interferometers biased by current I.sub.b=I.sub.C (critical current biasing), where I.sub.C is the interferometer critical current.

(53) According to an embodiment, a more advanced system is provided comprising one- and two-dimensional multi-element structures characterized by SQIF-like high linearity voltage response. The structures are based on use of a differential scheme of two magnetically frustrated parallel SQIFs, with both a specific cell area distribution a(x) along array and a critical current biasing (see FIG. 3). Optimization of the cell area distribution allows an increase of the voltage response linearity up to the levels required. This optimization can be performed numerically by solution of a master equation with the aid of an iterative algorithm.

(54) A multi-element structure synthesized according to the present embodiments can be used, for example, to provide high performance amplifiers. The proposed two-dimensional structure can also used as an active antenna device. The efficiency of the antenna can be significantly increased by combining it with a reflecting parabolic antenna. By varying the number of elements connected in parallel (N) and in series (K), one can set the impedance to a value needed to optimally match the antenna load used.

(55) The high expectation for the multi-element SQIF-like structures is based on estimations based on idealized structures, as well as on the voltage response characteristics calculated with use of RSJ model. However, the true characteristics of the actually realized array structures may be different. Limitations imposed by finite coupling inductances and stray capacitances are discussed below.

(56) The finite value of coupling inductances 1 between Josephson junctions in a parallel array is of importance for all principal characteristics of the array, because of limitations on the coupling radius.

(57) The finite coupling radius limits an increase of both the dynamic range and the transfer factor dV/dΦ with increase of number of junctions N. To study the noise characteristics in a clearer and more powerful manner, one can perform numerical simulation of a parallel array of the inductively coupled resistors R.sub.N, each connected to an individual source of white-noise current.

(58) FIG. 14 shows an active electrically small antenna based on two-dimensional differential serial-parallel SQIF-structure (the filled structure in central part of chip). SQIF sections are connected by strips of normal metal. The chip contains a regular matrix of identical blocks of parallel SQIFs, to provide a homogeneous magnetic field distribution in the center part of the chip. The inset shows such a block with a parallel SQIF. The shown SQIF is topology-oriented for a high-Tc superconductor technology.

(59) FIG. 15 shows the dependence of the low-frequency spectral density S.sub.v(0) of the resistor voltage noise on the number of resistors N at different values of normalized coupling inductance l. The data are presented for normalized frequency ω/ω.sub.c=10.sup.−3 corresponding closely to the signal frequency range in a SQUID/SQIF amplifier (here ω.sub.c is characteristic Josephson frequency). Within the coupling radius, the spectral density S.sub.v(0) decreases as 1/N and then it comes to a constant value when number N becomes more than coupling radius depending on l.

(60) FIG. 16 shows the spectral density S.sub.v(ω) versus normalized frequency ranged from 0.01 to 1 for parallel array of 30 resistors R.sub.N. At both coupling inductances l=3 and l=1, the spectral density S.sub.v(ω) monotonically increases with frequency and remains constant at l=0.001. It reflects a decrease in coupling radius with frequency for both inductances l=3 and l=1, as well as the fact that the coupling radius at l=0.01 exceeds the size of the array of 30 elements in the entire frequency range.

(61) One can see that implementation of noiseless resistors R.sub.SH=0.1R.sub.N shunting the inductances l=1 stops both the coupling radius decrease and the noise spectral density increase at ω/ω.sub.c≥0.1 (see dashed line in FIG. 16). A proper account of the respective noises of the shunting resistors will lift the curve about two times as high, as if the spectral density for the noise current sources connected to basic resistors R.sub.N becomes more by factor k≈4-5.

(62) FIG. 17 shows the dependence of the normalized transfer factor B=dV/dΦ for a parallel array of Josephson junctions versus number of junctions N at different normalized values l of coupling inductances. The dashed curve shows the transfer factor dependence for l=1 when all the coupling inductances are shunted by resistors R.sub.SH=R.sub.N. The observed saturation in the transfer factor, depending on N, is reached when number of junctions exceeds the coupling radius at frequency ω/ω.sub.c˜1.

(63) In such a way, increases in dynamic range D=V.sub.max(Φ)/V.sub.F with the number N of Josephson junctions in a parallel array are limited by the coupling radius at finite coupling inductances. Shunting of the inductances for improving linearity of the differential SQIF voltage response does not really change the dynamic range. In fact, the observed increase in the voltage response amplitude V.sub.max(Φ) (see FIG. 18) is compensated by an increase in V.sub.F owing to the noise of the shunts.

(64) In the case of an unloaded serial array of DC SQUIDs, the dynamic range does actually increase with the number N of interferometer cells. Nevertheless, in reality, stray capacitances and load impedance are both able to substantially change the I-V curve of the array, and hence the amplitude V.sub.max and form of the array voltage response. The decrease in V.sub.max leads to a proportional decrease in dynamic range. The change in the voltage response curve reduces linearity of the whole array structure.

(65) FIGS. 19A and 19B shows the typical impact of the stray capacitances on I-V curve of the serial array of DC SQUIDs. The contribution of the stray capacitance of each SQUID increases with the SQUID position from ground to signal terminal. Stray capacitances cause the I-V curve to appear similar to a hysteresis curve, as well as to form one or even more undesired features on the I-V curve. The features shown result from a phase-locking phenomenon. In the solid line labeled a of FIG. 19B, the features of the I-V curves of the array cells do not coincide because of different “capacitive loads.” In the dashed line labeled b, the features of all the individual I-V curves coincide because of mutual phase-locking of the Josephson-junction oscillations.

(66) The fabrication of serial arrays based on standard niobium technology using two superconducting screens is accompanied by undesirably high stray capacitances (see FIG. 20A). To essentially decrease the impact of the capacitance, individual double screening may be used for each SQUID as shown in FIG. 20B and FIG. 20C. Both schemes are characterized by the I-V curve b in FIG. 13B, but the latter one provides lower inductances of the strips which connect the SQUID cells.

V. Conclusion

(67) Advantages of one- and two-dimensional SQIF-like structures for microwave applications as high-performance amplifying devices are readily apparent from their ability to provide an increase in dynamic range with a number of elements as well as high linearity when employing a properly specified array structure. Linearity can be especially enhanced using cells comprising the bi-SQUID structure. At the same time, there are some fundamental limitations imposed by finite coupling inductances, stray capacitances and parasitic couplings. Therefore, implementation of high-performance devices preferably employs careful and detailed analysis of the multi-element array structure, taking into consideration all the true parameters including all parasitic parameters and couplings. A differential scheme comprising two magnetically frustrated parallel SQIFs is developed to obtain a highly linear single-peak voltage response. The response linearity can be increased up to 120 dB by means of a set of properly specified cell area distribution of the SQIFs. The high linearity is attainable with a relatively small number of junctions. Such a circuit provides a high-performance two-dimensional serial-parallel SQIF-like array. Varying the number of elements connected in parallel, and in series, permits setting the impedance value needed to solve the problem related to negative impact of the load used. The synthesized structures can be used to design high-efficiency amplifiers and electrically small active antennae for use in the gigahertz frequency range. The efficiency of the antenna can be significantly increased by combination with a reflecting parabolic antenna.

(68) It should be appreciated that changes could be made to the embodiments described above without departing from the inventive concepts thereof. It should be understood, therefore, that this invention is not limited to the particular embodiments disclosed, but it is intended to cover modifications within the spirit and scope of the present invention as defined by the appended claims.

REFERENCES

(69) Each of the following is expressly incorporated herein by reference: [1] V. K. Kornev, I. I. Soloviev, N. V. Klenov, and O. A. Mukhanov, “Synthesis of High Linearity Array Structures,” Superconducting Science and Technology (SUST), vol. 20, 2007, p. 5362-S366. [2] V. K. Kornev, I. I. Soloviev, N. V. Klenov, and O. A. Mukhanov, “High linearity Josephson-junction array structures,” Physica C, vol. 468, 2008, p. 813-816. [3] J. Oppenlaender, Ch. Haeussler, and N. Schopohl, “Non-Φ.sub.0-periodic macroscopic quantum interference in one-dimensional parallel Josephson junction arrays with unconventional grating structure”, Phys. Rev. B, Vol. 63, 2001, p. 024511-1-9. [4] V. Schultze, R. I. Isselsteijn, H.-G. Meyer, J. Oppenlander, Ch. Häussler, and N. Schopohl, “High-T.sub.c superconducting quantum interference filters for sensitive magnetometers,” IEEE Trans. Appl. Supercond., v. 13, No. 2, p. 775-778 (2003). [5] V. K. Kornev, I. I. Soloviev, J. Oppenlaender, Ch. Haeussler, N. Schopohl, “Oscillation Linewidth and Noise Characteristics of Parallel SQIF,” Superconductor Science and Technology (SUST), Vol. 17, Issue 5, 2004, p. 5406-S409. [6] V. K. Kornev, A. V. Arzumanov, “Numerical Simulation of Josephson-Junction System Dynamics in the Presence of Thermal Noise,” Inst. Physics Conf. Ser., No 158, IOP Publishing Ltd, 1997, p. 627-630. [7] G. V. Prokopenko, S. V. Shiov, I. L. Lapitskaya, S. Kohjiro, M. Maezawa, and A. Shoji, “Study of multi-Channel RF Amplifier Based on DC SQUID for 3-5 GHz Band,” IEEE Trans. on Applied Superconductivity, vol. 15, No. 2, (2005), p. 741-744. [8] K. D. Irwin, M. E. Huber, IEEE Trans. Appl. Supercond. AS-11 (2001), 1265. [9] M. Mueck, “Increasing the dynamic range of a SQUID amplifier by negative feedback,” Physica C 368 (2002) 141-145. [10] J. Oppenländer, Ch. Haussler and N. Schopohl, Phys. Rev. B 63, 024511 (2001). [11] O. A. Mukhanov, V. K. Semenov, D. K. Brock, A. F. Kirichenko, W. Li, S. V. Rylov, J. M. Vogt, T. V. Filippov, Y. A. Polyakov, “Progress in the development of a superconductive high-resolution ADC,” Extended Abstracts of ISEC'99, Berkeley Calif., p. 13-16, June 1999. [12]O. A. Mukhanov, V. K. Semenov, W. Li, T. V. Filippov, D. Gupta, A. M. Kadin, D. K. Brock, A. F. Kirichenko, Yu. A. Polyakov, I. V. Vernik, “A superconductor high-resolution ADC,” IEEE Trans. Appl. Supercond., vol. 11, p. 601-606, March 2001. [13] V. K. Kornev, A. V. Arzumanov, “Numerical Simulation of Josephson-Junction System Dynamics in the Presence of Thermal Noise,” Inst. Physics Conf. Ser., No 158, IOP Publishing Ltd, 1997, p. 627-630. [14] T. V. Filippov, S. Pflyuk, V. K. Semenov, and E. Wikborg, “Encoders and decimation filters for superconductor oversampling ADCs,” IEEE Trans. Appl. Supercond., vol. 11, p. 545-549, March 2001. [15] M. Mueck, M.-O. Andre, J. Clarke, J. Gail, C. Heiden, Appl. Phys. Letters, 72 (1998) 2885. [16] G. V. Prokopenko, S. V. Shitov, I. L. Lapitskaya, V. P. Koshelets, and J. Mygind, “Dynamic characteristics of S-band dc SQUID amplifier,” Applied Superconductivity Conf. (ASC'04), Jacksonville Fla., October 2004, report 4EF10. [17] G. V. Prokopenko, S. V. Shitov, I. V. Borisenko, and J. Mygind, “A HTS X-band dc SQUID amplifier: modelling and experiment,” Applied Superconductivity Conf. (ASC'04), Jacksonville Fla., October 2004, report 4EF12. [18] Polonsky S.; Shevchenko P.; Kirichenko A.; Zinoviev D.; Rylyakov A., “PSCAN'96: New software for simulation and optimization of complex RSFQ circuits”, IEEE transactions on applied superconductivity, 1996 Applied Superconductivity Conference. Part III, Pittsburgh, Pa., 1997, vol. 7 (3), no 2 (1517 p.), pp. 2685-2689. [19] Kornev, V. K., I. I. Soloviev, N. V. Klenov, and O. A. Mukhanov. “Bi-SQUID: a novel linearization method for dc SQUID voltage response”, Superconductor Science and Technology 22, no. 11 (2009): 114011.