Properties of the quarter-wave Bragg reflection waveguide: theory

Abstract

The Bragg reflection waveguide (BRW), or one-dimensional photonic crystal waveguide, has recently been proposed for a wide spectrum of applications ranging from particle acceleration to nonlinear frequency conversion. Here, we conduct a thorough analytical investigation of the quarter-wave BRW, in which the layers of the resonant cladding have a thickness corresponding to one quarter of the transverse wavelength of a desired guided mode. An analytical solution to the mode dispersion equation is derived, and it is shown that the quarter-wave BRW is polarization degenerate, although the TE and TM mode profiles differ significantly as the external Brewster's angle condition in the cladding is approached. Analytical expressions for waveguide properties such as the modal normalization constants, propagation loss, and overlap factors between the mode and each waveguide layer are derived, as are dispersion and tuning curves.

Dispersion tailoring of the quarter-wave Bragg reflection waveguide

“…For this analysis it is assumed that the cladding consists of an infinite number of periods, although the results are valid for structures with a reasonably large number of periods [20]. In addition, it is assumed that the waveguide is symmetric about the core.…”

“…(2) are separated by the external Brewster angle condition in the cladding, at which energy localization is lost. We define here a TM symmetry factor (6) and for the remainder of this paper we will refer to the "even" (ρ < 1) and "odd" (ρ > 1) TM modes in reference to their symmetry, as discussed in [20]. The quarter-wave BRW (QtW-BRW) is one in which each cladding layer has an optical thickness of λ /4 with respect to the transverse propagation vector:…”

“…As derived in [20], the effective index in this case is independent of polarization for slab waveguides and is equal to (8) for the fundamental mode, where the subscript on ω 0 signifies the nominal angular frequency at which the quarter-wave condition is satisfied, as opposed to the variable ω that will be used later in this paper. In addition, the matrix elements and eigenvalue reduce to …”

“…(3), (12) resulting in a matrix element perturbation (13) A similar analysis gives (14) . [20]. Finally, using Eq.…”

“…This ensures that the mode is not evanescent in any layer, a condition that would otherwise prevent the possibility of a π /2 phase thickness. As n 1 > (n co , n 2 ) by definition, this requirement dictates that 0 < n eff < min(n co , n 2 ) [20], resulting in the constraints (25) We choose a core composition of Al 90 Ga 10 As and a core thickness of 350 nm. Material dispersion data is shown in Table 1 (27) with (28) and (29) Comparing Eq.…”