Theory for calculating approximate values for the propagation constants of an optical rib waveguide by weighting the refractive indices

Microwave simulation of optoelectronic bending loss in presence of dielectric discontinuity

“…11 keeping only the radiating terms, gives e~j k3r 4nr exp jk^y' cos 6 + R' sin 0 cos 0") (12) where r = r/r and $" = </ >' - (13) 329 After some minor manipulation eqn. 1 then gives in the far field where we observe that R' = a on S, and we put is _ I gjkj(y' cos 0 + R' sin 0 cos 4>")-jm<f>" AAJ' = 2ne imnl2+jk3y ' cos °J m (fc 3 R' sin 0) (15) where J m denotes the Bessel function of order m. Here dKJdR' denotes the value of dK/dR' on R! = a.…”

“…The special procedure for finding the radiation is as follows. The functions E^y) and H^y) are assumed to be the same as for a semivectorial straight waveguide solver [23] which is known to give good agreement with other computer programs [14,15,16,17]. This also produces values of the waveguide propagation constants ft.…”

Ultra-Low Threshold Vertical-Cavity Surface-Emitting Lasers for USAF Applications

“…Coupling occurs only between the radial and axial equations for a given vector potential A z or F z : the two potentials remain uncoupled in (14) and (15). But A z and F z will be coupled later by boundary conditions when we solve for hybrid modes.…”

“…We take the latter approach, and approximate each equation in (11) with two coupled ordinary differential equations. We accomplish this by extending the WIM-which was first developed to calculate waveguide modes in horizontal-cavity ridge-waveguide lasers [15,16]-to address the eigenmodes of cylindrical cavities. This technique has the advantage of giving the best separable solution to (11) in the variational sense, and allows us to estimate the field profile, optical confinement factor, resonant wavelength, and threshold material gain of each cavity mode.…”

“…We now present boundary conditions and solutions for the axial and radial equations (14) and (15). Recall that the unknowns in this formalism are, for each mode, the functions…”