Question

How to calculate the lattice constant of photonic crystal?

Answer 1

In general there is not an analytical relationship between the features of a crystal's photonic band structure (e.g. the band gap range or the dispersion of a defect waveguide mode) and it's parameters (e.g. hole radius, permittivity ). The band structure needs to be calculated numerically. One option is the open source software, "mpb" but similar results could also be achieved with COMSOL or other commercial software packages.

In terms of a starting point for your design, Molding the Flow of Light by Joannopoulos might be a helpful reference (it's available in PDF for free). PhysRevB.62.8212 might be a particularly helpful reference for designing defect waveguides in photonic crystal slabs.

Finally, the conversion from normalized frequency units to absolute wavelength is given by 

lambda_0 = a / f

where f is the frequency in units of (c/a) where c is the speed of light and a is the lattice constant. You can use that to determine the necessary lattice constant for operation at 532 nm.