# Bi-SQUID Versus dc SQUID in Flux-Driven Traveling-Wave Parametric Amplifier
## 2 Effect of Finite Size of Artificial Cells
### Two-Line Amplifier Design  
The flux-driven JTWPA with two-line amplifier design suggested in [16] consists of two coupled separate artificial waveguide lines. The first artificial LC line, where both the signal and idler waves propagate, is based on using a serial array of dc SQUIDs coupled inductively to the second superconducting LC transmission line carrying the pumping wave. The latter is linear (non-Josephson) and therefore is free from any limitation on the pump wave amplitude and from energy leakage to higher harmonics as well. The pump signal applies rf magnetic flux \(\Phi_{rf}\) to the SQUID cells and, due to the cells' nonlinear properties, provides modulation of the dipole cell inductances \(L_{cell}\) (with a depth \(m\)) in the wave manner through influence of the maximum superconducting current \(\left|I_{s}\right|_{max}\) of the dc SQUID cell. The inductance modulation can be provided either on the pump frequency when a dc flux bias \(\left|\Phi_{dc}\right|<\Phi_{0}/2\) is additionally applied:  
\[L_{cell}=L_{cell}(\Phi_{dc}\
eq 0)\cdot\left[1+\text{m}_{1}\cdot\cos\left( \Omega_{p}t-k_{p}x\right)\right] \tag{6}\]  
or on the double frequency when no dc flux is applied (\(\Phi_{dc}=0\)):  
\[L_{cell}=L_{cell}(\Phi_{dc}=0)\cdot\left[1+\text{m}_{2}\cdot\cos\left(2\Omega_ {p}t-2k_{p}x\right)\right] \tag{7}\]  
However, in the latter case, the modulation depth \(\text{m}_{2}\) is considerably less than \(\text{m}_{1}\) due to the approximately \(\left|\cos\right|\)-like dependence [20] of the maximum superconducting current \(\left|I_{s}\right|_{max}\) of dc SQUID on the applied magnetic flux \(\Phi_{c}\). At a low loop inductance (as assumed in [16]), \(\left|I_{s}\right|_{max}\approx 2L_{c}|\cos(\varphi_{c}/2)|\), where \(\varphi_{c}=2\pi\Phi_{c}/\Phi_{0}\), and hence at \(\left|\Phi_{dc}\right|<\Phi_{0}/2\)  
\[\left|I_{s}\right|_{max}\approx 2I_{c}\left[\cos(\varphi_{dc}/2)-\sin(\varphi_{ dc}/2)\cdot(\varphi_{rf}/2)-0.5\cdot\cos(\varphi_{dc}/2)\cdot(\varphi_{rf}/2)^{2} \right],\]  
where \(\varphi_{dc}=2\pi\Phi_{dc}/\Phi_{0}\) and \(\varphi_{rf}=2\pi\Phi_{rf}/\Phi_{0}\) are the normalized values of the applied dc and rf magnetic fluxes. This expression shows that dc flux changes the maximum superconducting current (from \(2I_{c}\) to \(2I_{c}\cos(\varphi_{dc}/2)\)) and hence the dipole inductance of dc SQUID and modulation of the inductance from (7) to (6) with a modulation depth about four times higher. Figure 1: If \(Z_{0}\) is the wave impedance of a long artificial line of \(\Gamma\)-cells, the value \(Z_{0}\) does not change with cell addition. The inductance modulation waves (6) and (7) yield in the signal frequency mixing either to \(\omega_{p}\pm\omega_{s}\) or to \(2\omega_{p}\pm\omega_{s}\) corresponding to the three-wave and four-wave operation modes, when \(\omega_{s}\) is the signal frequency and the idler frequency is \(\omega_{i}=\omega_{p}-\omega_{s}\) or \(\omega_{i}=2\omega_{p}-\omega_{s}\), respectively. In both cases, the signal and idler wave frequencies can be optimally located in the vicinity of \(\sim\)\(\omega_{cut}/3\), where the reflection factor is moderate, \(|\Gamma|\simeq 0.15\), and the phase velocities of the waves are quite close to each other: \(v_{s}\approx v_{i}\approx 1/\sqrt{LC}\) (with using dimensionless coordinate \(x\) normalized on the cell length \"\(a\)\"), and therefore they can be tuned closely to the phase velocity of the pump wave propagating in the second line. For example, when one sets the pump frequency as \(\omega_{p}\simeq(3/4)\cdot\omega_{cut}\) in three-wave mode or \(2\omega_{p}\simeq(3/4)\cdot\omega_{cut}\) in four-wave mode, the signal and idler frequency band can range from \(\omega_{s}\simeq\omega_{cut}/4\) to \(\omega_{s}\simeq\omega_{cut}/2\), that enables eliminating wave components at frequencies \(\omega_{p}+\omega_{s,i}>\omega_{cut}\) and \(2\omega_{p}+\omega_{s,i}>\omega_{cut}\), respectively, and the reflection factor at signal frequency \(\omega_{s}\) and idler frequency \(\omega_{i}\) decreases to \(|\Gamma|\lesssim 0.2\).