A spatially modulated waveguide Bragg grating mirror is suspended over a substrate by plurality of fingers extending laterally away from the waveguide centerline. The positions of the fingers are coordinated with the positions of crests and valleys of amplitude or phase modulation of the Bragg grating, to avoid disturbing the Bragg grating when it is tuned by heating. When the Bragg grating is heated, the heat flows through the fingers creating a quasi-periodic refractive index variation along the Bragg grating due to quasi-periodic temperature variation created by the heat flow from the grating through the supporting fingers. Due to coordination of the positions of supporting fingers with positions of the crests and valleys of modulation, the optical phase coherence is maintained along the Bragg grating, so that the spectral lineshape or filtering property of the Bragg grating is substantially preserved.