  We developed a broadband two-layer anti-reflection (AR) coating for use on a
sapphire half-wave plate (HWP) and an alumina infrared (IR) filter for the
cosmic microwave background (CMB) polarimetry. Measuring the faint CMB B-mode
signals requires maximizing the number of photons reaching the detectors and
minimizing spurious polarization due to reflection with an off-axis incident
angle. Sapphire and alumina have high refractive indices of 3.1 and are highly
reflective without an AR coating. This paper presents the design, fabrication,
quality control, and measured performance of an AR coating using
thermally-sprayed mullite and Duroid 5880LZ. This technology enables large
optical elements with diameters of 600 mm. We also present a newly developed
thermography-based nondestructive quality control technique, which is key to
assuring good adhesion and preventing delamination when thermal cycling. We
demonstrate the average reflectance of about 2.6% (0.9%) for two observing
bands centered at 90/150 (220/280) GHz. At room temperature, the average
transmittance of a 105 mm square test sample at 220/280 GHz is 83%, and it will
increase to 90% at 100 K, attributed to reduced absorption losses. Therefore,
our developed layering technique has proved effective for 220/280 GHz
applications, particularly in addressing dielectric loss concerns. This AR
coating technology has been deployed in the cryogenic HWP and IR filters of the
Simons Array and the Simons observatory experiments and applies to future
experiments such as CMB-S4.
