Optical Resonant Cavities in Volume Bragg Gratings

We propose narrow-band filter in volume Bragg grating (VBG) with bandwidth less than ten picometers. Two recorded Bragg gratings with the same modulation amplitudes and slightly different resonant wavelengths form moir\'{e} pattern with average carrier spatial frequency and slowly varying envelope of modulation amplitude. Each semi-period of modulation is just apodized reflective VBG; however two of them together form narrow-band transmission Fabry-Perot cavity due to phase $\pi $-shift as result of sign change of slowly varying envelope. We fabricated first moir\'{e} VBG filter in photo-thermo-refractive glass with resonance wavelength near 1550 nm, aperture size 5 mm, bandwidth 50 pm and 95{\%} maximum transmittance. We considered also case when carrier Bragg grating wave vector does not coincide with moir\'{e} pattern wave vector which allows creating filters with tunable one-period envelope profile from sinusoidal function to cosinusoidal one. Doubled resonant cavity with cosinusoidal profile demonstrates flattop transmission peak. Analytical expression for tunable bandwidth was found. Robust solid-state moir\'{e} VBG filters tolerant to high-power laser irradiation with tunable filtering characteristics are suggested as optical elements for laser design and spectroscopy applications.