3-1 Shifting dielectric rod defects in the photonic crystal slab
An air-hole formed defect waveguide has two mechanisms of the light confinement:
effects of the PBG and the index-guiding. Instead, there is only one bandgap effect of a dielectric rod waveguide needs to be considered; therefore it is simpler to analyze. Thus, simulation results of the dielectric rod type are discussed first in this chapter.
In the dielectric rod type structure, a TM polarization is easier to generate the PBG, as shown in Fig. 3.1. In the case of a PC slab, due to the lack of the translational symmetry in the vertical direction, there is no purely TE or TM mode, but rather the TE-like (even) or the TM-like (odd) mode. The band diagram of a PC slab possesses a light line. When the fields are spatially bounded, such as being localized around the defects, then the frequencies form a discrete set. The propagation modes, which under the light line are discrete because their energy is localized within the PCW, called as the guided modes. The region above the light line corresponds to the continuum of the extended modes. Modes in this region will radiate the energy outward, called the radiation modes. In the application of optical communications, we mainly consider the guided modes only.
In this section, the PWEM and the FDTD method are used to simulate and to design the PC slab with the radius of rods and the height being 0.2a and 2a, respectively, where a is the lattice constant of the crystal. The radius of the defect rods is 0.13a in both square and triangular lattice lattices. The dielectric constant of the dielectric rods is 12, which corresponds to dielectric constant of silicon at 1.55μm [see Fig. 3.2].
Fig. 3.1 Dispersion relations of a perfect dielectric rod PC slab with square lattices of the (a) TM-like (the magnetic field polarized along the waveguide) and the (b) TE-like EM waves.
The gray line indicates the light line, and the shade region is the PBG. (c) The 1st Brillouin zone for a square lattice with the irreducible zone.
Fig. 3.2 The refractive index of silicon at different wavelength.
3-1-1 Shifting a point defect
First, making the PC different from symmetry by tuning the location of a point defect away from the original center and investigates the influence on the eigenfrequency. Moving the point defect along the ±y (or ±x) direction in the square lattice will shift the eigenfrequency (ω0) of a point defect toward the higher frequency as shown in Fig. 3.3.
Fig. 3.3 The eigenfrequency of the point defect with a defect rod located at different positions along the y-axis.
The electric field mainly localized in the dielectric rod when there is no shift of the defect. As shifting the defect rod away from the center, the portion of fields localized at the dielectric region is reduced, thus the electric field extends to the air region. Figure 3.4(a) shows how the field distribution varies with the shift degree of the defect along y-axis. The less concentration of the electric field in the dielectric substrate makes a buleshift [23]. And in Fig. 3.4(b), there is almost unchanged of the field distribution along the x-axis.
Fig. 3.4 The electric fields of the point defect made of a dielectric rod along the (a) y-axis and the (b) x-axis. The defect rod is gradually shifted toward the y direction.
3-1-2 Shifting a line defect
Next, we will investigate a PC slab with all line defects moving along the y-axis in the square lattice. The result shown in Fig. 3.5(a) indicates that the dispersion curves shift completely to the higher frequency, which suggests that the coupling coefficients Pm between defect rods are unaffected as shifting the line defects. The frequency shift is primarily dominated by the variation of eigenfrequency (ω0) of the point defect as moving the rod.
Otherwise, as the defect rods are longitudinally shifted along the x-axis, the results are
presented in Fig. 3.5(b). The dispersion curve bends down in large wavevectors. This is due to the reason that both ω0 and P0 would increase and cancel out the effect of changing the eigenfrequency ω1 in the section of small wavevectors. Shifts of dispersion curves merely change apparently when the higher-order terms Pmcos(mka) become important.
Fig. 3.5 Dispersion curves of a single waveguide slab with all the defect rods are moved toward the (a) y and (b) x directions.
3-1-3 Shifting line defects of directional couplers in triangular and square lattices
In this section we will discuss the effects as changing the separation between two PCWs of the slab. Transversely shifting two defect rows gets Fig. 3.6, which shows how the dispersion relation varies as changing the structure of the DC. Here, we discuss the square lattice structure first. As simultaneously moving two line defects off the center and keeping the separation between PCWs fixed, we find that the dispersion curves shift toward the higher frequency, as shown in Fig. 3.6(a). In addition, the separation of the two dispersion curves does not change, i.e., the coupling length keeps fixed. On the other hand, reducing the separation of two line defects to decrease the coupling between PCWs pushes the dispersion curves apart (see Fig. 3.6(b)), and thus increases the coupling length. However, symmetrically enlarging the separation of the waveguides not only shifts the dispersion curves toward the higher frequencies but also makes two dispersion curves closer (see Fig. 3.6(c)).
Fig. 3.6 Dispersion relations of the shifted DCs, whose defect rods are moved transversely (a) with fixed separation, (b) to approach and (c) to apart from each other.
Transversely shifting two PCWs of a DC in the triangular lattice and keeping the separation fixed, one gets the dispersion relation as in Fig. 3.7. At the wavevector smaller than that of the decoupling point, two curves of the odd mode and the even mode show blue shifted with the coupling length unchanged, similar as the result of Fig. 3.6(a). Generally, tuning PCWs of a DC in the square and the triangular lattices, we will obtain alike conclusions, therefore, discussions of other conditions are abbreviating in this section.
Fig. 3.7 The dispersion relation of a triangular lattice DC with two defect waveguides are shifted transversely with fixed separation.
Besides the two dispersion curves of the symmetric DC cross at one point, there are other phenomena. The frequency of the even mode is higher than that of the odd mode for lower wavevector. As the wavevector becomes higher than the decoupling point, the odd mode owns higher frequency, namely, mode switch occurs at the decoupling point, as shown in Fig.
3.8.
Fig. 3.8 Change of parity in a symmetric dielectric rod DC.
Next, we inspect the results of shifting all defect rods longitudinally in Fig. 3.9. In the square lattice structure, there is no decoupling point [see Fig. 3.9(a)]. After longitudinally moving line defects, two dispersion curves come approaching to each other. When gradually shifting the defect rods, two dispersion curves begin to cross at one point at high frequency.
As rods keep on moving further, the decoupling point will move toward the lower frequency or the smaller wavevector k. Conversely, in the triangular lattice DC, the decoupling point moves toward the higher frequency or the larger wavevector k and eventually two dispersion curves do not cross [see Fig. 3.9(b)]. Whether the dispersion curves cross or not depends on the coupling strength, as mentioned in Section 2-3, only when the ratio 2β α >1 [see Fig.
3.10] that the decoupling point will appear at the wavevector k=[cos ( 2−1 − β α)] a.
Fig. 3.9 Dispersion relations of shifting double defect rod channels longitudinally in the (a) square lattice and the (b) triangular lattice PC slabs.
Fig. 3.10 The values of (a) α, β and (b) 2β α as defect rods shifted in x direction in a square lattice DC.
3-2 Shifting air-hole defects in the photonic crystal slab
Because an air-hole slab is relatively simple to fabricate, this type of DCs has become the most widely used in the practical devices. In PC slab waveguides, not only the PBG provides a mechanism to restrict light propagating in the channel, but also the air-claddings provides better confinement to decrease the radiation loss during the light propagation [25].
Hence, in this section we will simulate the DC slab composed of a thin planar dielectric substrate with air-holes and surrounded by air.
In the air-hole DC, the electric fields of TE polarized light are localized in the dielectric substrate to create a bandgap; the comparative result is in Fig. 3.11.
Fig. 3.11 Dispersion relations of a perfect triangular lattice air-hole PC slab of the (a) TE-like and the (b) TM-like polarization incident EM waves, and (c) the 1st Brillouin zone for a triangular lattice with the irreducible zone.
With the intention to obtain a larger PBG, an air-hole slab is usually arranged with triangular lattice. From the simulation data of Fig. 3.12, we can clearly find that there is almost no PBG in the square lattice air-hole slab. Therefore, in the following simulations,
we only did on the air-hole PC slab with triangular lattice encircled by air. T he radius of the air holes, the thickness and dielectric constant of the slab are 0.3a, 0.55a and 12, respectively.
The radius of the defect hole in the waveguide is 0.44a.
Fig. 3.12 Dispersion relations of a perfect air-hole PC slab with square lattices. The incident EM wave is TE polarization.
3-2-1 Shifting a point defect
We will consider the shift in the eigenfrequency and the redistribution of the electric field in this section upon shifting the point defect along the ±y (or ±x) axis in the triangular lattice PC slab. The eigenfrequency (ω0) blue shifts as the defect hole progressively moved away from the center shown in Fig. 3.13. Due to the localization nature of Ey polarization, the field distribution has less influence or is inert subjected to displacement of the defect hole than that of the extended Ex component as shown in Fig. 3.14(a) with maximum Ex(0.6a) ~ 0.15 (ΔEx ~ 0.1 on moving defect) and in Fig. 3.14(d) with maximum Ey(0.8a) ~ 0.3 (ΔEy ~ 0.3 on moving defect). On the other hand, the smaller change of the extended or smoother Ex field as moving the defect hole leads to less variation of eigefrequency of the shifted point defect in Ex than in Ey as in Fig. 3.13.
Fig. 3.13 The eigenfrequency of the air-hole point defect located at different positions shifted along the y and x axes, respectively.
Fig. 3.14 (a) Ex and (b) Ey of an air-hole point defect shifted toward the y-axis at z=0 plane.
(c) Ex and (d) Ey of an air-hole point defect shifted toward the x-axis at z=0 plane.
3-2-2 Shifting a line defect
From the results done by the PWEM, we found the dispersion curves of an air-hole PC slab shift toward the higher frequency as moving all line defects along the y-axis. The result shown in Fig. 3.15(a) indicates that the dispersion curves shift toward the higher frequency that are primarily dominated by the variation of eigenfrequency (ω0) of the point defects as moving the hole. The dispersion curves of different y-shifted PCWs show basically parallel to one another except for that of the PCW having y > 0.1a. As keeping on shifting further,
the defect hole is close to the nearby hole and there is no sufficient space to allow the electric field wholly localized in the dielectric region. Therefore, the electric field will overflow to the air region that makes the high-order coupling coefficient P2 become larger than P1. Since the sign of P2 and P1 are both positive the dispersion curve should bend down under this circumstance. The results of longitudinally moving defect holes along the x-axis are presented in Fig. 3.15(b). The slope of a dispersion curve, which is the group velocity and defined as dϖ dk, slightly increases with the extent of shift. This trend is mainly due to the increase of P1 that leads this term P1cos(ka) in Eq. (2.3) affecting the eigenfrequency ω1.
Fig. 3.15 Dispersion curves of a single PCW air-hole slab with all the defect holes shifted toward the (a) y and (b) x directions.
3-2-3 Shifting line defects of directional couplers
In this section, we will simulate different conditions of transversely moving two line defects. Figure 3.16 shows how the dispersion curves vary with altering the separation of two line defects. Simultaneously shifting two rows of air-hole defects away from the center and keeping the distance between two PCWs fixed, we obtain that the dispersion curves are shifted toward the higher frequency, as shown in Fig. 3.16(a). The dispersion curves of the DC only blue shift with similar coupling lengths. And the decoupling point also keeps at the same wavevector due to the value of 2β α is not changed [see Fig. 3.17] but its eigenfrequency blue shifts accordingly. Next, by reducing the separation of two line defects symmetrically to increase the coupling coefficients between PCWs, dispersion curves will shift apart, i.e., the coupling length decreases, and the decoupling point shifts toward the higher wavevector [see Fig. 3.16(b)] since the value of 2β α becomes smaller [see Fig.
3.18], whereas, enlarging the separation of two line defects will give the opposite trend, as shown in Fig. 3.16(c) and Fig. 3.19.
Fig. 3.16 Dispersion curves of the shifted DCs, whose defect holes are moved transversely (a) with separation fixed, (b) to approach, and (c) to apart from each other.
Fig. 3.17 The values of (a) α, β and (b) 2β α as defect holes shifted simultaneously in the y direction.
Fig. 3.18 The values of (a) α, β and (b) 2β α as defect holes shifted to approach in the y direction.
Fig. 3.19 The values of (a) α, β and (b) 2β α as defect holes shifted to apart from each other in the y direction.
Second, we investigate the influence of moving all defect holes along the x direction.
After longitudinally shifting double line defects, two dispersion curves move toward the higher frequency [see Fig. 3.20] and the decoupling point also moves toward the higher wavecector. As gradually shifting the air-hole defects, two dispersion curves finally no longer cross due to 2β α ≤1 [see Fig. 3.21]. Contribution from Ey field dominates the influence on coupling coefficients, as shown in Fig. 3.14.
Fig. 3.20 Dispersion curves of moving all the defect holes along the x direction in the triangular lattice DC.
Fig. 3.21 The values of (a) α, β and (b) 2β α as defect holes shifted in x direction in the triangular lattice DC.
3-3 Coupling lengths calculated by the FDTD method
In this section, we anticipate obtaining the EM waves propagating conditions in the DC by the FDTD method by considering an EM wave with frequency 0.37264c/a is incident into the dielectric rod DC with square lattice, as shown in Fig. 3.22(a). Without shifting the PCWs, the coupling length is approximately equal to 18.8a [see Fig. 3.22(b)]. As
transversely shifting two defect waveguides 0.2a closer, the coupling length reduces to 8.2a [see Fig. 3.22(c)]. The coupling lengths derived by the FDTD method are consistent with the results calculated by the PWEM [see Fig. 3.23].
Fig. 3.22 (a) A dielectric rod DC with square lattice. Propagating EM waves in two waveguides (b) without any shift and (c) transversely shifted 0.2a closer.
Fig. 3.23 The coupling lengths of different shift of waveguides in the dielectric rod DC.
Next, we survey the conditions of an EM wave transfers in the air-hole DC, the device structure is shown as Fig. 3.24(a). The frequency of the incident wave is 0.285c/a having the coupling length of about 3.5a [see Fig. 3.24(b)]. As shifting the defect rows 0.1a closer to each other, the coupling length reduces to 4.6a [see Fig. 3.24(c)]. Comparing the results of the PWEM, as shown in Fig. 3.25, the FDTD method obtains similar coupling lengths.
Fig. 3.24 (a) The air-hole DC with triangular lattice. Propagating EM waves in two waveguides (b) without any shift, (c) transversely shifted 0.1a to apart from each other.
Fig. 3.25 The coupling lengths of different shift in the air-hole DC.