Digitized grating period

A laser, comprising:

• a first region that includes an active region;
• and
• a second region coupled with the first region, wherein the second region includes an optical waveguide to carry light generated in the active region, wherein the optical waveguide includes a grating, and wherein the grating includes:
• a first unit cell having a first period and a first length, wherein the first period includes a first grating element width and a first space between adjacent grating elements, and wherein the first length includes a number of first periods;
• and
• a second unit cell having a second period and a second length, wherein the second period is different than the first period and includes a second grating element width and a second space between adjacent grating elements, and wherein the second length includes a number of second periods,
• wherein the grating has a Bragg wavelength and the Bragg wavelength is given by a digitized grating period which is a period sum divided by a sum of the first length and the second length where the period sum is a sum of the first period multiplied by the first length and the second period multiplied by the second length.