## 1 Introduction  
The use of Ge for the design and fabrication of photo-receivers for high-speed optical fiber communication has become prevalent in recent years. Due to its broadband absorption range from roughly 300 nm to 1600 nm wavelengths, widespread adoption of Ge for short-range multi-mode fiber (MMF) communication at 850 nm and mid-to-long range single-mode fiber (SMF) communication at 1310 nm and 1550 nm have been achieved. Moreover, PDs made from Ge are intrinsically CMOS fabrication process compatible, enabling low-cost solution for mass production. In the literature, remarkable progress on Ge PDs with low dark current, high responsivity, and high bandwidth, have been realized in both waveguide and normal-incidence geometries [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. However, when applying Ge PDs to photo-receiver modules, efficient optical coupling remains a challenging topic. For waveguide geometry, optical couplers such as grating coupler or inverted taper coupler are usually needed, which introduce additional fiber-to-coupler insertion loss and a stringent alignment requirement [14, 15, 16, 17]. For normal-incidence geometry, the trade-off between bandwidth, determined by RC time constant and transit time constant, and external responsivity, determined by PD size and PD thickness, impose a complication in optimizing PD design parameters [8, 14]. With the demand for increasingly higher data rates over time, the size of a PD needs to be shrunk to reduce the capacitance and therefore attain the desired receiver bandwidth, consequently reducing the optical aperture size of the PD. Therefore, it is becoming harder and harder to achieve low coupling loss and high alignment tolerance when the photo-receiver is packaged in an optical module. One approach to increase the optical aperture size is to incorporate a microlens (ML), in which parameters such as radius of curvature, focal length, contact angle, and refractive index should be carefully designed to derive a high optical performance [18]. At the same time, when fabricating the ML, one needs to consider (1) materials that are reliable and possess proper contraction coefficient and viscosity, and (2) fabrication processes that are efficient and low cost. Depending on the applications, choosing suitable materials and fabrication processes to develop a high-performing ML can be a challenging topic. There have been numerous works demonstrating various approaches including reflow [19], ink-jet [20], imprint [21], and grayscale lithography [22], to name a few. In the reflow method, a layer of material is coated on the substrate and patterned via optical lithography to form a flat-top pillar. The pillar is then heated to a specific temperature for well-controlled time duration to form the reflow lens. Note that this method has been applied in mass production of MLs for sensors and imagers. In the ink-jet method, the material is placed in a pipe and then dispensed from a dropper onto the substrate. The size and shape of the lens are dependent upon the volume of droplets and the contact angle between the material and the substrate. Note that for the ink-jet method, highly accurate and stable-yet-fast dispensing tools are needed to overcome the issues of poor alignment and low throughput. In the imprint method, the lenses are formed using a mold as the master replica. Note that for this method, how to fabricate a precise mold and how to de-mold after the imprint are crucial. In the grayscale lithography method, direct patterning the photoresist with a grayscale mask to match the lens shape is first applied. Then the photoresist pattern is transferred to the layer below the photoresist through dry etching. Note that while this method can be handled by a mass production facility in a straightforward manner, the cost of gray-scale mask is high, and the dry etch process would cause roughness over the ML surface, reducing its optical quality due to light scattering.