LA POLÍTICA Y LA EDUCACIÓN

As an example, the procedure followed to design an ABR defect resonator with first- order Bragg gratings is outlined below, in the specific case of TE polarized modes. For a given choice of the azimuthal number m and the wavelength λ0, the width of

each layer in the cylindrical structure and the TE field components can be found simultaneously using the Bessel function transfer matrix formalism described in Sec- tion 2.2. Beginning in layerj = 1, the boundary condition requiring finiteness of the

fields at ρ = 0 (Eq. 2.42) dictates that the vector of constants describing the Hz(ρ)

field be given by _ C D   1 =  1 0   1 , (3.16)

where we have taken C1 = 1 for simplicity. Given a numerical value for this vector,

and knowing γ₁ = 2πn1/λ0, the radial dependence of the field Hz(ρ) within layer

j = 1 is fixed. The field, along with the radial positions of its zeros and extrema, can be found using a numerical routine for evaluating Bessel functions, such as those included in Matlab. Since we desire to design an optical mode having a concentration of intensity at the position of the annular defect within the cylindrical Bragg reflector, the placement of the first dielectric interface, located at a radius ρ=R1, is dictated

by the design rules for increasing TE field amplitude, given by Eqs. 3.8-3.9. After the radial position of the first dielectric interface is known, the refractive index of the dielectric is changed. The transfer matrix T1 enforcing the continuity of the

tangential field components at the interface is then used in order to calculate the vector of constants in the j = 2 layer, as per Eqs. 2.37 and 2.39. After numerical evaluation of the field’s zeros and/or extrema within the j = 2 layer, the second dielectric interface is appropriately placed, and so on. The above steps are repeated for the desired number of periods of the inner Bragg reflector. The annular defect is then introduced. The inner defect radius is again chosen according to the design rules for increasing field, however the outer radius is chosen according to the design rules for decreasingfield, given by Eqs. 3.10-3.11. Finally, within the outer Bragg reflector, the radial positions of the interfaces are chosen according to decreasing field. The number of periods in the outer Bragg reflector is chosen to achieve sufficient damping of thefield at large radii, in order to produce a low-loss radially confined mode.

In order to illustrate the results of this design approach, we consider the particular example of a TE polarized defect mode designed to have an azimuthal numberm = 7 at a wavelength λ0 = 1.55 µm, within an ABR structure having high index layers

withn= 2.8, and low index layers withn= 1.56. These values of the refractive index and polarization have been chosen for their relevance toward the semiconductor ABR devices which will be discussed in greater detail in Chapters 4 and 5. The normalized

Hz(ρ) mode profile is plotted as a function of the radial coordinate in Fig. 3.3(a), superimposed with the designed refractive index profile. For the case shown, the

Figure 3.3: (a) TE profileHz(ρ)of an ABR defect mode havingm = 7,λ0 = 1.55µm,

first-order high index layers withn = 2.8, and first-order low index layers with n= 1.56. The radial refractive index profile is superimposed to illustrate the placement of the dielectric interfaces with respect to the zeros and extrema of thefield. (b) Width of the low and high index layers versus layer index. The sixth high index layer is the defect layer, and has a width approximately twice that of the high index layers within the Bragg reflectors.

index. The defect is the sixth high index layer going radially outward from ρ = 0. The inner and outer Bragg reflectors consist of 5 and 10 periods, respectively. The

field has peak amplitude within the defect layer, and decays exponentially within a relatively few number of periods of the Bragg reflectors on either side, due to the large refractive index contrast. Note that within each reflector layer, thefield completes an "effective" quarter wavelength oscillation, i.e. the width of each layer is equivalent to the distance between successive zeros and extrema of the field. Furthermore, the defect layer spans the distance between two successive zeros of thefield, an "effective" half wavelength. The width of each low and high index layer is plotted in Fig. 3.3(b). The layer width is seen to decrease as a function of radius, in particular for the low index layers, and approaches a constant value at large radii, equivalent to the conventional "quarter-wavelength" thickness of a 1D Bragg reflector.

It is worthwhile to note that in addition to supporting modes with peak inten- sity within a high index defect, the Bragg confinement mechanism also admits the possibility of guiding light within a low index defect, a scenario impossible for total- internal-reflection based devices [50].

The procedure followed for the design of ABR structures for TM polarized modes is identical to that outlined above, but makes use of the appropriate transfer matrix in Eq. 2.38 and the design rules in Eqs. 3.12-3.15.