Formulas

Junction impedance. Above the superconducting gap the junction acts as a real tunnel resistance R_n shunted by the self-capacitance C_j. With τ = Rn Cj and ω = 2πf,

• Zj(ω) = Rn(1 − jωτ) / (1 + ω²τ²)
• Re Zj = Rn / (1 + ω²τ²), Im Zj = −ωτ Rn / (1 + ω²τ²)
• Conjugate Zj* = Re Zj + j |Im Zj| (the target match impedance).

Reflection coefficient and coupling efficiency. Conjugate matching gives Γ = 0:

• Γ(f) = (Zrad − Zj*) / (Zrad + Zj)
• Power coupling efficiency ec(f) = 1 − |Γ|². Rafferty et al. write ec = 1 − |Γ| (an amplitude-style definition); this calculator uses the standard power form so that ∫ ec df directly gives the antenna noise bandwidth.
• Antenna noise bandwidth: ΔfN = ∫ ec(f) df, integrated over the plotted frequency range.

Radiation-impedance models.

• Constant: Zrad(f) = R0 (purely real, frequency-independent). Use for a quick "what if" estimate.
• Lorentzian (parallel-RLC equivalent): Zrad(f) = Rpeak / [1 + j Q (f/f0 − f0/f)]. Re Z_rad peaks at f₀; Im Z_rad crosses zero at f₀ with negative slope (inductive below, capacitive above). FWHM ≈ f₀/Q.
• 3D half-wave dipole (3D transmon). Dipole of total length L in free space. f0 = c0 / (2L √εeff) with ε_eff ≈ 1 inside a 3D cavity. Classical thin-dipole peak resistance Rpeak ≈ 73 Ω; Q ≈ 4 captures the fundamental's bandwidth crudely.
• 2D single-ended aperture loop (Babinet). Island embedded in a groundplane behaves like the aperture dual of a one-wavelength wire loop antenna at f0 = c0 / (p √εeff), where p is the island perimeter and ε_eff ≈ ½(1 + ε_r) ≈ 6 on Si/sapphire. From Babinet's principle Zw Za = η²/4 ≈ 35 531 Ω²; with a one-wavelength wire loop having Rw ≈ 130–240 Ω the aperture peak is Rpeak ≈ 100–200 Ω. Default 150 Ω, Q ≈ 4.
• 2D differential folded-dipole aperture. Two-pad differential qubit is the aperture dual of a folded half-wave dipole. f0 = c0 / (2L √εeff). Folded-dipole wire has Rw ≈ 292 Ω, so aperture Rpeak = 35 531 / 292 ≈ 122 Ω (Rafferty et al. simulate ≈80–150 Ω). Default 100 Ω, Q ≈ 4.

All three shape presets are textbook lumped approximations and ignore details that matter at the 5–20% level: finite trace thickness, substrate finite size, dielectric resonances, and proximity to ground tabs. For quantitative work use full-wave simulation (e.g. HFSS, CST, openEMS); see Rafferty et al. (Phys. Rev. Appl. 16, 054012, 2021 / arXiv:2103.06803) for the numerical Z_rad(f) of several specific Xmon, differential, and 3D-transmon geometries.