Modified crossed Dragone optical design of the LiteBIRD low-frequency telescope

LiteBIRD is a JAXA-led international project aimed at measuring the cosmic microwave background (CMB) polarization with high sensitivity to detect polarization $B$ modes. This detection would provide evidence of inflation. LiteBIRD will observe the full sky for three years at the L2 Lagrange point of the Earth-Sun system across 34-448 GHz, and is expected to launch in the Japanese fiscal year of 2032. The Low-Frequency Telescope (LFT) will observe in the 34-161 GHz range implementing a modified crossed Dragone (MCD) reflective optical design optimized for high optical performance across a wide $18^{\circ} \times 9^{\circ}$ field-of-view (FOV). In this paper, we report the LFT optical design details including its optimization and optical performance assessed using optical simulations. The MCD design consists of a paraboloidal primary and a hyperboloidal secondary reflector with polynomial correction terms up to 7th order, achieving Strehl ratios $\ge 0.97$ at 161 GHz across the FOV. The Mueller QU (UQ) cross-polarization response is $\le -26.9$ dB at 34 GHz. The simulated beam sizes are $< 78^{\prime}$ at 34 GHz. The simulated sidelobe response for the direct and diffuse triple reflection sidelobes are estimated to be $< -57$ dB, and for the focused triple reflection sidelobe $< -37$ dB at 34 GHz. The LFT optical design satisfies all the optical requirements and specifications for the project, and is compatible with the LiteBIRD science goals.