Influence of optical fiber location behind an apodized phase mask on Bragg grating reflection efficiencies at Bragg wavelength and its harmonics

An apodized fiber Bragg grating formation using a phase mask with variable duty cycle is numerically analyzed. In particular, an impact of position of an optical fiber behind the phase mask with Gaussian apodization profile on Bragg grating reflection efficiencies at Bragg wavelength and its harmonics is extensively studied. It is shown that reflection efficiency of each harmonic strongly depends on the optical fiber location with respect to the adjacent Talbot planes during the grating inscription. An analytical formula for calculation such periodical changes of reflection strength is proposed. It is also proved, that the smaller optical fiber diameter the higher fluctuations of reflectivity for particular harmonic occur. Results presented for such general case (i.e. phase mask with variable duty cycle with all non-zero diffraction orders) directly correspond to less complex structures, such as uniform phase masks and those with variable groove depth. They are also useful in optimization of Bragg wavelength and harmonic reflection efficiencies as well as in deep understanding of apodized FBG formation using aforementioned phase masks.