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Abstract: A comprehensive, complete theory suitable for a long period grating (LPG) on/in a planar or channel waveguide is described on the basis of coupled mode theory. The theory aims at key characteristic parameters with respect to the effect of the LPG, including resonance wavelength, transmission power of the guided mode in the waveguide, full-width at half-maximum (FWHM) of the power attenuation band in transmission spectrum of the guided mode, and coupling coefficient between a guided and a coupled mode. The type of the grating including relief and index modulation, different polarization state combinations of the guided mode and the resonant mode, and slanted grating structure are considered in this theory. The influences of the coupling between two guided modes and between a guided mode and a substrate mode are discussed. The role of cladding layer in a long period waveguide grating is demonstrated in particular. Finally, the expressions for the sensitivity of resonance wavelength as a function of external temperature, pressure and surrounding refractive index are derived. [Copyright &y& Elsevier]