Modeling of phase volume diffractive gratings, part 1: transmitting sinusoidal uniform gratings

TLDR: In this paper, a detailed model of the diffraction of plane and Gaussian beams on plane uniform phase Bragg gratings based on Kogelnik's theory of coupled waves is presented.

Abstract: A detailed model of the diffraction of plane and Gaussian beams on plane uniform phase Bragg gratings based on Kogelnik's theory of coupled waves is presented. The model describes transmitting gratings with arbitrary orientation in a plane-parallel plate taking into ac- count spectral width and angular divergence of laser beams along with material dispersion of a photosensitive medium. The model results are compared with experimental data for high-efficiency Bragg gratings in a photothermorefractive PTR glass. © 2006 Society of Photo-Optical Instrumenta- why our modeling is illustrated for those variations of grat- ing parameters that are typical for PTR VBGs: the refrac- tive index is in the range of 1.5 at wavelengths ranging from 0.4 to 2.7 m, refractive index modulation is up to 1000 ppm 10 3 , and grating thickness is from 0.2 to 20 mm. The goal of this work is to reduce Kogelnik's theory to practical formulas that enable practical modeling and de- sign of diffractive optical elements based on VBGs. This part of modeling considers diffraction of plane monochro- matic, divergent, and polychromatic laser beams on uni- form sinusoidal lossless transmitting volume gratings and compares the model with experimental results in PTR Bragg gratings. Further parts will describe modeling of re- flecting volume gratings holographic mirrors as well as the application of both transmitting and reflecting grating for spectral beam combining for different types of lasers.

Figures

Fig. 5 Selectivity of transmitting Bragg gratings for 0=1085 nm, nav=1.4867: a dependence of diffraction efficiency on deviation from Bragg angle and wavelength with grating thickness in millimeters: 2.0 for curves 1 and 2 and 1.0 for curve 3; refractive index modulation in parts per million: 250 for curves 1 and 3 and 125 for curve 2; spatial frequency 1086 mm−1; b dependence of angular selectivity half width at first zero HWFZ on spatial frequency for optimal refractive index modulation with grating thickness in millimeters: curve 1, 0.5; curve 2, 2.0; curve 3, 5.0; and curve 4, 10; and c dependence of spectral selectivity HWFZ on spatial frequency for optimal refractive index modulation with grating thickness in millimeters: curve 1, 0.5; curve 2, 2.0; curve 3, 5.0; and curve 4,10.

Fig. 6 Ratio of angular selectivity HWFZ of normal transmitting VBG in air to that inside a medium for 0=1085 nm and nav =1.4867.

Fig. 7 Effect of material dispersion on the Bragg angle inside a grating medium. The angular dispersion factor is a partial derivative of the Bragg angle over the wavelength resulting from material dispersion of refractive index normalized to absolute value of Bragg angle. The refractive index and material dispersion are determined conform to Ref. 30.

Fig. 1 Propagation of optical rays through a volume Bragg grating: Nf and Nf,ex, normals to the front surface for incident Ii and reflected Ir beams; Nb, normal to the back surface for the transmitted It and diffracted Id beams; Ki, Kim, and Kdm, wave vectors of incident beam in air, and incident and diffracted beams in the medium; KG grating vector; , grating inclination; i, r, im, t, and d, angles of incidence, reflection, incidence in medium, transmission, and diffraction; m, Bragg angle; and m * , incident Bragg angle.

Fig. 2 Possible orders of Bragg diffraction inside medium: Ii and Id, incident and diffracted beams; Ki, wave vector of incident beam; KG, grating vector; m, Bragg angle; m * , incident Bragg angle.

Fig. 8 Selectivity of transmitting VBG for divergent polychromatic beams with central wavelength at 1085 nm: a dependence of DE on detuning from Bragg condition, for a monochromatic wave, DE is 100%; incident beam divergence HWe−2M in milliradians: curve 1, 0.04; curve 2, 0.2; curve 3, 0.4; and curve 4, 0.8; grating angular selectivity is 0.4 mrad; beam HWe−2M spectral width in nanometers: curve 1, 0.1; curve 2, 0.5; curve 3, 1.0; and curve 4, 2.0; grating spectral selectivity is 1.0 nm; C dependence of grating DE on the beam divergence for angular selectivity HWFZ of grating in milliradians: curve 1, 0.12; curve 2, 0.4; curve 3, 1.2; and curve 4, 4.0; shown by dotted arrows; and c dependence of grating DE on the beam spectral width with spectral selectivity HWFZ of grating in nanometers: curve 1, 0.1; curve 2, 1.0; and curve 3, 10; shown by dotted arrows, where the dotted line corresponds to DE for a beam with divergence or spectral width equal to the grating selectivity.