A thermally assisted magnetic recording head is disclosed having a spot size converter with at least one secondary waveguide adjoining a top or bottom surface of a primary waveguide. Each waveguide has tapered sides but the secondary waveguide is tapered at a greater angle over a shorter taper distance in order to couple propagated light into the primary waveguide before the front end of the taper. The secondary waveguide terminates in a ridge with a fixed width w3 of about 50-170 nm that is between the front end of the taper and the air bearing surface (ABS). The ridge enables transverse magnetic (TM) transmission mode efficiency above 90% even with a typical process misalignment in the cross-track and height directions. The primary waveguide has a front section with width w2 between an end of its tapered sides and the ABS where w2 is substantially larger than w3.