Sinusoidal Response of dc SQUIDs For rf Power Measurements

Current, power, and attenuation measurements with rf SQUIDs are based on the fact that the voltage from the microwave readout circuit can be made a nearly sinusoidal function of the magnetic flux threading the SQUID. We point out here that an asymmetric dc SQUID with sufficiently low inductance can achieve a very sinusoidal output voltage with good modulation depth. The spectral purity of the sinusoid can be substantially better than that obtained with rf SQUID systems. The purity improves with increasing asymmetry of the junction critical currents, and decreasing values of the LIc product, where L is the SQUID inductance and Ic is the smaller of the critical currents. Results of several calculations are presented. Substantial improvement in SQUID methods of rf current, power, or attenuation measurement may thus be possible with use of such dc SQUIDs.


Introduction
In the early 1970s, Kamper et al. [1][2][3][4][5]1 recognized that superconducting quantum interference devices (SQUIDs), with their periodic response to magnetic flux <1>, had the potential for being used in a completely new kind of current, power, or attenuation measurement. In the systems developed, the rf impedance of a single-junction SQUID (rf SQUID) was measured. The impedance variations were reflected into a resonant circuit and measured as variations in phase or amplitude of the output rf voltage. If the voltage response is purely sinusoidal in flux with period equal to the flux quantum 253 (1) the time-averaged voltage response to a sinusoidally varying flux of amplitude <l>m takes the form (2) Equation (1) is achieved approximately with use of an overdamped SQUID and a broadband microwave readout circuit [4]. By determining the zeroes of the response experimentally as the flux amplitude is varied, and equating them to the zeroes of the Bessel function J o , one can determine <l>m or its equivalent in terms of current.
The periodic response to an impressed flux is, however, not ideally sinusoidal, and this necessitates an elaborate series of corrective procedures [2][3][4][5][6][7]. The purpose of this note is to point out that a dc SQUID (a superconducting loop containing two Josephson junctions-see fig. 1) with asymmetric critical currents and sufficiently small inductance and capacitance can have a considerably purer sinusoidal voltage output with adequate modulation. Thus one of the major sources of difficulty in realizing a Josephson junction device for rf measurements can potentially be removed. Although the SQUID inductance is small, the current in the input coil necessary to couple flux into the SQUID can be kept at a reasonable level. The noise in such SQUIDs is also expected to be small.
The threshold characteristic (Le., maximum zerovoltage current vs. flux) of a symmetric dc SQUID is not at all sinusoidal. Figure 2 shows an example for a 4-pH SQUID with equal critical currents of 55 J..LA. It has long been known, however, that the -- voltage developed across a symmetric SQUID has a sinusoidal appearance when the bias current is sufficiently greater than the maximum critical current of the device. This is observed experimentally and from calculations (see, e.g., [8] and references therein).
That an asymmetric dc SQUID can exhibit a sinusoidal threshold characteristic has been noted earlier by us [9] as well as by Fulton et al. [10]. No specific application was seen at the time. Intuition suggests, however,· that this sinusoidal character should be reflected in the voltage developed across the SQUID, and calculations bear this out, as we shall presently show.
The equations describing the threshold behavior of a dc SQUID are [9] cos c1>2= -cos c1>1/(a+f3 cos c1>1), a solution only for cJ>1 very close to (2n + l)1T 12 where n is any integer or zero. Choosing cJ>1 ;::::;:1T12 and noting that the requirements of a < 1 and f3 < 1 requires /32 < 1 (although /31 may still be of the order of unity), we find from eqs (4) and (5) that (6) This establishes the approximate sinusoidal behavior of the threshold characteristic of a sufficiently asymmetric dc SQUID with low inductance.
In figure 3 we show the computed threshold behavior of an asymmetric dc SQUID, having equal inductances L1 =L2=2.0 pH, and unequal critical currents 101 = 100 }.LA, 1 02 = 10 }.LA. We also plot the perfect sinusoid of eq (6) for comparison. Calculations show that the spectral purity of the threshold characteristic can be improved by making the inductances asymmetric as well. This remarkable purity of the threshold characteristic is carried over into the voltage across the SQUID.
The dynamic equations describing a dc SQUID are (8) Here the e's are the junction capacitances and the R's are resistances shunting the junctions. Ib is the bias current injected as shown in figure 1. To calculate the voltage across the SQUID, we solve eqs (7) and (8)    The solid circles in figure 5 show the spectrum of the symmetric SQUID of figure 4. The spectral purity of the voltage of this SQUID is better than that reported for an rf SQUID [4]. The second harmonic is about 26 dB below (5 percent 00 the fundamental, and the third harmonic is 50 dB down (0.3 percent). Figure 6 shows the time-averaged voltage vs. of the voltage is not as large as with the symmetric SQUID, however. We find this to be generally true -asymmetric SQUIDs have superior spectral purity but less depth of modulation than symmetric SQUIDs with the same bias current and the same total inductance, critical current, and shunt resistance.
The computed I-V curves of figures 7(a,b) show the differences in modulation for the two SQUIDs  however, the larger capacitance of this example greatly reduces the modulation depth. It is thus important that the junction capacitance be kept as low as possible so that the capacitive impedance does not shunt out the resistance. Perhaps the principal virtue of the low-inductance SQUIDs, and especially the asymmetric lowinductance SQUIDs, lies in the fact that the harmonics above the second are very greatly reduced and are probably negligible. This is important because the earlier work with rf SQUIDs [5] found that the second harmonic could effectively be nulled, and that the higher harmonics constituted the principal problem. That the second harmonic is also substantially reduced is of course an improvement. Since the basic source of systematic error in the measurement of rf attenuation is caused by harmonic distortion of the sinusoidal response of the system [4], the SQUIDs discussed here should have a distinct advantage over the rf SQUIDs.
In 1982 M. Cromar of this laboratory made a preliminary study on the suitability of a thin-film dc SQUID as the detector element in an rf attenuator-calibrator system (unpublished). Using a resistively shunted symmetric SQUID whose LIe product was considerably larger than <1>0, he showed that at low signal frequencies, the zeroes of the response approximated the zeroes of the 10 Bessel function closely enough that accuracy at rf frequencies comparable to that of the earlier measurements with rf SQUIDs might be realized. This research was not pushed further.
The microfabricated dc SQUIDs are expected to have better thermal and temporal stability than the single-junction rf SQUIDs used earlier. The overall circuitry necessary to attain the output voltage should also be simpler with the dc SQUID. Imperfections in microwave components were found to be a major contributor to the harn .onic distortion in the rf SQUID system [5]. Such components would not be used in the readout scheme for a dc SQUID.
The 4-pH inductances used here are small, but SQUIDs with lower inductance have been fabri-258 cated. The low inductance of the proposed SQUIDs presents a potential problem of sufficient coupling, however. For example, if the mutual inductance between a 4-pH SQUID and the input line were only 4-pH, about 500 /-LA would be required in the line to produce one flux quantum in the SQUID. If one desires 200 nulls in the voltage response (100 flux quanta), about 50 rnA must flow in the line in this case. Although higher than typical, this value is still below what a superconducting stripline could support. Higher values of mutual inductance will decrease the maximum current needed. The mutual inductance to an input coil carrying the signal can in fact be made much larger than the self-inductance of the SQUID. Planar coupling to low-noise, low-inductance SQUIDs is the subject of much current research [11][12][13]. Of course, one may also choose to work with higherinductance SQUIDs, accepting somewhat less spectral purity if the critical currents are kept at the same values, in order to decrease the maximum current needed in the stripline or to decrease the degree of coupling to the SQUID.
The effect of noise is another consideration. It is beyond the scope of this note to undertake a detailed study of noise effects, which depend upon the readout method used. Ideally, of course, it is desirable that the noise of the SQUID plus its readout system be dominated by the intrinsic noise of the SQUID. Readout schemes with this in view are under active investigation; Ketchen [11] discusses several of them. For a readout system operating at a frequency of about 100 kHz, 11/ noise is not a consideration. However, even at lower frequencies where 11/ noise dominates, a new readout scheme [14] for significantly reducing the noise has been developed.
Properly fabricated dc SQUIDs have the lowest noise figures of any devices [11]. White noise decreases with decreasing inductance, which is favorable for the SQUIDs under consideration. The parameter 2'7TkT 110<1>0 is sometimes used to characterize the noise. If we take 10=0.11 rnA, which is the average of the critical currents used in the preceding examples, we find the value 0.002 at 4 K for this noise parameter. The ultra-low-noise SQUIDs operate near this value. The spectral density S v of the voltage fluctuations in a resistor R is given by 4kTR. For shunt resistances of about 1 n, the voltage noise power spectrum is 2X 10-22 V 2 /Hz at 4 K. The flux noise density S<t> is approximately Svll aV/a<l>1 2 • The ultra-low-noise SQUIDs are built to be biased at that value of flux that gives the greatest energy sensitivity, which occurs at the largest value of 1 aV /a<l> I. However, in the present case the flux will be swept over many periods of <1>0, so that each value of flux contributes almost equally. Since the voltage is nearly sinusoidal, 1 aV /a<1> 12 varies almost as sine-squared, which we may replace by 1/2 times an amplitude, to a good approximation. Thus Sq,z2kTR<l>51(7TV o )2, where Vo is the voltage amplitude. For the symmetric SQUID of figure 4, VozS.S J-tV so that Sq,z lO-I3<1>51Hz. These values are not far from values quoted for low-noise SQUIDs [13-1S]. Other expressions for the voltage noise power spectrum are available [14,IS] which take into account noise contributions from the circulating current in the SQUID. The noise values from these expressions are not significantly different from those calculated above because of the very small inductance. Although these arguments do not establish that noise will not be a significant problem affecting the accuracy of determining the voltage nulls in rf measurements, they show that the SQUIDs proposed here share characteristics of low-noise SQUIDs, and thus are encouraging.
The purpose of this note has been to suggest a new scheme for rf measurements using SQUIDs, and to demonstrate its potential advantages. A more extensive analysis of feasibility could include the following: simulations of the effect of noise upon the accuracy with which the zeroes of the response can be determined; studies of the magnitude and effects of the parasitic inductance associated with the shunt resistors; inclusion of the circuit carrying the signal-source, coupling coil, and load-to determine whether the nonlinear impedance reflected into the circuit by the SQUID is important; determination of the optimum choice of inductance, critical currents, asymmetry, and bias current to obtain the best combination of spectral purity and depth of modulation, or in short the greatest accuracy possible.
The considerations presented here suggest that thin-film, low inductance, low critical current, dc SQUIDs, especially those with asymmetric critical currents, would be superior to rf SQUIDs in rf current, power, or attenuation measurements.

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This work has benefitted from discussions with several of my colleagues, to whom I express my thanks.