Inverse-Designed 90-Degree Silicon Nitride Bends for the C Band

We demonstrate the use of inverse design to achieve high-efficiency, compact <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$90^{\circ }$</tex-math></inline-formula> bends with radii as small as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$6 \,{\mu }\mathrm{m}$</tex-math></inline-formula> with silicon nitride waveguides. These bends are designed for the TE polarization of single-mode waveguides with a cross-section of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$850 \,{\mathrm{nm}}$</tex-math></inline-formula> × <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$400 \,{\mathrm{nm}}$</tex-math></inline-formula>, optimized for the C-band. Both simulations and experimental results confirm that the freeform bends obtained through inverse design outperform conventional circular and Euler bends across four footprint sizes ranging from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$6$</tex-math></inline-formula> × <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$6 \,{\mu }\mathrm{m}^{2}$</tex-math></inline-formula> to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$21$</tex-math></inline-formula> × <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$21 \,{{\mu }\mathrm{m}}^{2}$</tex-math></inline-formula>. All inverse-designed bends were fabricated using an e-beam lithography process and designed with a minimum feature size of 160 nm, ensuring maximum compatibility with commercial optical lithography processes and viability for large-scale production. Simulations for the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$6 \,{\mu }\mathrm{m}$</tex-math></inline-formula> radius bends show that the best freeform bend exhibits an average loss of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$1.62 \,{\mathrm{dB}}$</tex-math></inline-formula> across the entire C-band, compared to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$2.88 \,{\mathrm{dB}}$</tex-math></inline-formula> for circular bends and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$3.46 \,{\mathrm{dB}}$</tex-math></inline-formula> for Euler bends. Experimental results align well with simulations, with measured losses of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$1.89 \,{\mathrm{dB}}$</tex-math></inline-formula>, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$2.86 \,{\mathrm{dB}}$</tex-math></inline-formula>, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$3.44 \,{\mathrm{dB}}$</tex-math></inline-formula> for the freeform, circular, and Euler bends, respectively. For the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$11 \,{\mu }\mathrm{m}$</tex-math></inline-formula> bending radius, one of the freeform structures demonstrated low losses of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$0.18 \,{\mathrm{dB}}$</tex-math></inline-formula>, representing reductions by factors of up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$6.6$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$1.8$</tex-math></inline-formula> times compared to Euler and circular bends, respectively. For the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$16 \,{\mu }\mathrm{m}$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$21 \,{\mu }\mathrm{m}$</tex-math></inline-formula> radii, the best freeform bends achieved average losses of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$0.08 \,{\mathrm{dB}}$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$0.11 \,{\mathrm{dB}}$</tex-math></inline-formula> per <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$90^{\circ }$</tex-math></inline-formula> bend across the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$1530 \,{\mathrm{nm}}$</tex-math></inline-formula> to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$1565 \,{\mathrm{nm}}$</tex-math></inline-formula> wavelength range. These results highlight the potential of inverse design to enable significantly more compact routing in silicon nitride photonic chips, facilitating the development of high-density photonic circuits.