It is well known that a dielectric microdisk cavity can support high Q whispering gallery modes, which can be classified according to their polarization (TE or TM), their azimuthal mode number m and their radial mode number l [50, 51]. A TE whispering gallery mode TE(m, l) has 2m nodes in the azimuthal direction and l -1 nodes in the radial direction. It is also doubly degenerate and can be classified into an even or odd mode according to its mirror reflection symmetry. By coupling the microdisks together as in Fig. (3.lb), we can form the even and odd CROW bands from such whispering gallery modes. In this section, we study the CROW bands formed by the TE(7, 1) whispering gallery modes.
In these calculations, we choose the radius r of the micro disk to be 30 FDTD cells and use three parameters for the inter-microdisk spacing R, which is normalized as R/2r and takes the value of R/2r
=
1.1, 1.17 and 1.23. r / ,,\ is used as the unit of frequency.The TE(7, 1) even mode and odd mode of a single microdisk are shown in Fig.
3.8. The frequency and Q factor of the even TE(7, 1) mode are found to be 0.645 and 1500. For the odd TE(7, 1) mode, the frequency and the Q factor are 0.639 and 1200 respectively. The degeneracy of the two TE(7, 1) modes is broken, due to the deviation of the dielectric microdisk from an ideal circular shape in our 2D simulation.
As in the case of photonic crystal defect cavities, such degeneracy splitting will not cause significant change on the dispersion and mode characteristics of the CROW band, since the even and odd CROW modes remain orthonormal to each other.
The even CROW bands are calculated for three microdisk spacing parameter R/2r. Shown in Fig. 3.9 is the mode profile of a waveguide mode with R/2r
=
1.1 and K=
0.57r / R. It is qualitatively the same as that of the even T E(7, 1) mode in Fig. (3.8a). This suggests that the tight binding approximation is still valid, even for the close distance between the microdisks.The CROW loss can be characterized by its effective Q, as defined in Sec. 2.3.2.
We find that the effective Q factors of the even CROW bands depend strongly on
(a)
(b)
Figure 3.8: The TE(7, 1) whispering gallery modes in a single microdisk cavity. The even mode is shown in a) and the odd mode is shown in (b).
Figure 3.9: The even microdisk CROW mode formed by coupling the even TE(7, 1) modes together. We use R/2r ::::; 1.1 and K
=
0.51r / R.6000 5000 4000
a
Q)
:g3000
--
Q)LU
2000 1000
00 0.2 0.4 0.6 0.8 1
K R/n:
Figure 3.10: The effective Q factors of the even microdisk CROW modes with R/2r
=
1.17.
the Bloch vector K. The Qe11(K) of the CROW band with R/2r
=
1.17 is shown in Fig. 3.10 as a function of K. The fact that effective Q depends on K is not surprising and can be explained intuitively. The TE(m, l) mode has an azimuthal dependence of eimcp [50], which means that its radiation loss has similar angular dependence. As K, the Bloch vector of the CROW modes varies, the radiation field from different individual resonators interferes constructively or destructively with each other, which consequently causes the radiation loss of the CROW modes to increase or decrease and deviate from that of the single microdisk resonator.The dispersion relations for the "even" CROW modes with three different values of R/2r are shown in Fig. 3.11. The error bars in Fig. 3.11 refer to the frequency uncertainty due to the radiation decay of the CROW modes, which is estimated to be wK/2Qeff as in section 2.1.1. Within the limit of these frequency errors, the numerical data agree well with the tight-binding results.
These frequency errors are also taken into account when we fit the numerical results into Eq. (3.19) using the least-square method. We no longer treat the numeri-
0.642
0.64
I
T .,l --- f / i 1 l
* / / 1
/ l
t,✓
\S 0.638 T
J,,Y
o.636 ___..---·r 1
f..--- i ..L
!
1
I_, l~*--R-/2_r_=_1-.1-, E_v_e_n -M-ic-ro-di-sk_C_R_O_W.L-.1 0.634
0
0.636
0.634 0
0.642
0.636
0.634 0
0.2 0.4 0.6 0.8
KR/ rr
I * R/2r = 1.17, Even Microdisk CROW I
0.2 0.4 0.6 0.8
KR/ rr
I * R/2r = 1.23, Even Microdisk CROW I
0.2 0.4 0.6 0.8
KR/ rr
(a)
(b)
(c)
Figure 3.11: The dispersion of the even microdisk CROW band. The FDTD results are represented by asterisks. The error bars refer to the frequency error caused by the finite decay rate of the CROW modes and are estimated to be w/2Qeff· The solid lines are the least square fits of the numerical results using Eq. 3.19. The dispersion diagrams for R/2r
=
1.1, 1,17 and 1.23 are shown respectively in a), b) and c).cally calculated mode frequencies equally and weigh them by the frequency deviation of WK/2Qeff· The coupling coefficient 11;1 obtained from this fitting are respectively -4.5 X 10-3, -2.5 X 10-3 and -1.3 X 10-3 for different values of
R/2r =Ll,
1.17 and 1.23. As expected, 11;1 decreases as the inter-microdisk spacing increases.Figure 3.12: The odd microdisk CROW mode formed by coupling the odd TE(7, 1) modes together. We use R/2r
=
1.1 and K=
0.51r / R.The odd CROW bands are also calculated using the same set of values for R/2r.
The CROW mode shown in Fig. 3.12 is calculated with R/2r = 1.1 and K = 0.51r / R, and as before, is similar to the odd TE(7, 1) mode shown in Fig. (3.8b). The odd CROW bands are shown in Fig. 3.13 for R/2r
=
1.1, 1.17 and 1.23. Again after considering the frequency deviation due to the decay of CROW modes, the numerical results agree well with the theoretical fits. The results for 11;_1 obtained from the theoretical fitting are respectively 4.8 x 10-3, 2. 9 x 10-3 and 1.4 x 10-3 forR/
2r=
1.1, 1.17 and 1.23.
In the case of coupled microdisks, it is difficult to calculate the coupling coefficient using the overlap integral in Eq. (3.20). One reason is that the electromagnetic field outside the microdisk depends strongly on the boundary conditions of the computa- tional domain, especially in the regions far from the microdisk. This is quite different from the case of the defect cavity in 2D photonic crystal, where the photonic crystal can effectively block much of the influence of the absorbing boundaries. Another
0.648~---;_--~ _-_-_-_-_--~ _-_-_-_-_--~ _-_-_-_---:__--~ _-_-_--.~
I * R/2r = 1.1, Odd Microdisk CROW I
0.6471-~, ..
"1 ..
o.646
~t
T~:: ~~~[ (a)
0.642 ~
'"-1:
0.641
t~)'
0.647 0.64 . 0.645
~0.644 0.643 0.642
0.2 0.4 0.6 0.8
KR In
0.641 I * R/2r = 1.17, Odd Microdisk CROW l
~0.644 0.643 0.642 0.641
I
*0.2 0.4 0.6 0.8
K R/1t
R/2r = 1.23, Odd Microdisk CROW I
0.2 0.4 0.6 0.8
KR/ re
(b)
(c)
Figure 3.13: The dispersion of the odd microdisk CROW band. The FDTD results are represented by asterisks. The error bars refer to the frequency error caused by the finite decay rate of the CROW modes and are estimated to be w/2Qeff· The solid lines are the least square fits using Eq. (3.19). The dispersion diagrams for R/2r
=
1.1, 1,17 and 1.23 are shown respectively in (a), (b) and (c).
reason is that the electromagnetic field does not decay exponentially away from the microdisk, and therefore creates a normalization problem.