# Bi-SQUID Versus dc SQUID in Flux-Driven Traveling-Wave Parametric Amplifier
## 3 Increase in Dynamic Range Using bi-SQUID Cells  
The dynamic range (DR) of an amplifier can be defined as the maximum-to-minimum ratio of the output signal power within the range of the linear relation between the output signal and input signal, starting from the minimum value of the output signal amplitude restricted by the root-mean-square (_rms_) value of the output noise. Although the gain of the flux-driven JTWPA is not restricted by the pump wave depletion, the output signal power is inevitably limited by nonlinear distortions as the _rf_ current amplitude increases and nears the maximum superconducting current of the cells (as inductances) in use, namely dc SQUIDs suggested in [16]. In fact, a linear wave-type equation with a periodically modulated reactive parameter, which was derived in [16] for the magnetic flux \(\Phi(x,t)\) in the frame of distributed line approximation (using dimensionless coordinate \(x\) normalized on the cell spacing \"\(a\)\"), gives an exponential increase in amplitudes with \(x\) of both the signal and idle traveling flux waves \(\Phi_{s}(x,t)\) and \(\Phi_{i}(x,t)\). However, to obtain an output power, one needs to know the amplitude of the current traveling waves \(I_{s}(x,t)\) and \(I_{i}(x,t)\). The relation between the flux and current amplitudes, \(\Phi_{a}\) and \(I_{a}\), is linear at low values: \(k\Phi_{a}=Ll_{a}\), where \(k\) is the wave number (reduced to the reciprocal of the cell spacing), \(L\) is the running inductance (dipole inductance of SQUID-cell) and \(k\Phi_{a}\) is amplitude of the flux per unit length. But this ratio becomes nonlinear with an increase in the wave amplitudes and then the current wave amplitude  
Figure 2: Reflection factor \(|\Gamma(\omega)|\) for the artificial waveguide of lumped L and C elements which is connected to resistor \(R=\rho\equiv\sqrt{L/C}\); optimal values of both the pump wave frequency \(\omega_{p}\) in 3-wave mode (or double frequency \(2\omega_{p}\) in 4-wave mode) and the frequency range for signal and idler waves. can be written as \(I_{a}=k\Phi_{a}/L-\delta I_{a}(\Phi_{a})\). This leads to the power gain reduction by factor \(\eta_{p}=[\delta I_{a}/(k\Phi_{a}/L)]^{2}\) rising the current amplitude nearing the maximum superconducting current value of the SQUID cells in use. The linear domain limit determines the dynamic range of the amplifier. The end point of the linear range can be characterized by the signal amplitude corresponding to the power gain reduction by 1 dB; this is called the 1 dB compression point. Increasing the critical current \(l_{c}\) of the used Josephson junctions to increase the maximum superconducting current of the SQUID cells cannot be an acceptable solution for extending the linear domain, because the critical current magnitude determines some other normalized parameters answering for the amplifier's performance. For example, one has to increase geometric loop inductance in order to increase the flux applied by the pumping wave, but one has to decrease the normalized value of the loop inductance  
\[l=2\pi I_{c}L/\Phi_{0} \tag{8}\]  
by decreasing the junction critical current to increase both the modulation depth of the maximum superconducting current of the cell and the characteristic Josephson junction inductance  
\[L_{J0}=\Phi_{0}/(2\pi I_{c}) \tag{9}\]  
and hence the modulation depth of the dipole cell inductance. To increase the linear domain of the flux-driven JTWPA at the tradeoff-assigned magnitude of the junction critical current, one needs to modify the cells in use in order to increase the linear domain of the cell characteristics describing the relation between transport current via the cell \(I_{b}\) produced by the signal and idler waves and the corresponding phase drop \(\Delta\varphi\) on the cell. The phase drop answers the flux \((\Phi_{0}/2\pi)\Delta\varphi\) per length of the waveguide line composed of the cells.