Photonic band structure computations

Daniel Hermann; Meikel Frank; Kurt Busch; Peter Wölfle

doi:10.1364/OE.8.000167

Optics Express
Vol. 8,
Issue 3,
pp. 167-172
(2001)
•https://doi.org/10.1364/OE.8.000167

Photonic band structure computations

Daniel Hermann, Meikel Frank, Kurt Busch, and Peter Wölfle

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Daniel Hermann, Meikel Frank, Kurt Busch, and Peter Wölfle, "Photonic band structure computations," Opt. Express 8, 167-172 (2001)

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Abstract

We introduce a novel algorithm for band structure computations based on multigrid methods. In addition, we demonstrate how the results of these band structure calculations may be used to compute group velocities and effective photon masses. The results are of direct relevance to studies of pulse propagation in such materials.

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Media 1: MOV (813 KB)

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Fig. 1. Graphical illustration of the V-cycle used in the multigrid iteration of Eq. (5) depicting four levels of grids. A movie (881.mov(0.8M)) shows the multigrid iteration using two levels for the third band at the x-point in the TM-polarized case (see Fig. 2).

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Fig. 2. Photonic band structure for a square lattice of dielectric rods (∊_a =13, r/a=0.45) in air (∊_b =1) for TM-polarization (blue) and TE-polarization (green).

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Fig. 3. Group velocity of the first three bands (first band: black; second band: red; third band: green) for a square lattice of dielectric rods (∊_a =13, r/a=0.45) in air (∊_b =1) for TM-polarized light. The corresponding band structure is displayed in Fig. 2.

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Tables (1)

Table 1. Comparison of numerical data obtained from PWM (upper part) and the multigrid method (lower part) using different numbers N of plane waves and different mesh-sizes (128×128 and 256×256), respectively. The values for the defect give an estimate of the absolute error of the Bloch-functions [7]. The computations have been performed at the X-point for TM-polarization using a 466 MHz Celeron processor.

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Equations (13)

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\frac{1}{∊ (\vec{r})} (\partial_{x}^{2} + \partial_{y}^{2}) E (\vec{r}) + \frac{ω^{2}}{c^{2}} E (\vec{r}) = 0,

\partial_{x} + (\frac{1}{∊ (\vec{r})} \partial_{x} H (\vec{r})) + \partial_{y} (\frac{1}{∊ (\vec{r})} \partial_{y} H (\vec{r})) + \frac{ω^{2}}{c^{2}} H (\vec{r}) = 0 .

E_{\vec{k}} (\vec{r} + {\vec{a}}_{i}) = e^{i \vec{k} {\vec{a}}_{i}} E_{\vec{k}} (\vec{r})

H_{\vec{k}} (\vec{r} + {\vec{a}}_{i}) = e^{i \vec{k} {\vec{a}}_{i}} H_{\vec{k}} (\vec{r}),

ℒ_{\vec{k}} U_{i} = Λ_{i} U_{i},

\frac{1}{V} \int_{w s c} d^{2} r E_{n \vec{k}}^{*} (\vec{r}) ∊ (\vec{r}) E_{m \vec{k}} (\vec{r}) = δ_{n m}

\frac{1}{V} \int_{w s c} d^{2} r H_{n \vec{k}}^{*} (\vec{r}) H_{m \vec{k}} (\vec{r}) = δ_{n m},

Λ_{i} = \frac{〈 ℒ_{\vec{k}}^{1} U_{i}^{1}, U_{i}^{1} 〉}{〈 U_{i}^{1}, U_{i}^{1} 〉},

E_{\vec{k}} (\vec{r}) = e^{i \vec{k} \vec{r}} u_{\vec{k}} (\vec{r}),

(Δ + 2 i \nabla \cdot \vec{k} - {\vec{k}}^{2}) u_{\vec{k}} (\vec{r}) + \frac{ω_{\vec{k}}^{2}}{c^{2}} ∊ (\vec{r}) u_{\vec{k}} (\vec{r}))) = 0 .

\hat{H} (\vec{k}) u_{\vec{k} + \vec{q}} (\vec{r}) + (2 \vec{q} \cdot \vec{Ω} - {∣ \vec{q} ∣}^{2}) u_{\vec{k} + \vec{q}} (\vec{r}) + \frac{ω_{\vec{k} + \vec{q}}^{2}}{c^{2}} ∊ (\vec{r}) u_{\vec{k} + \vec{q}} (\vec{r}) = 0 .

ω_{\vec{k} + \vec{q}} = ω_{\vec{k}} + (1 st order kp - PT) + (2 nd order kp - PT) + \dots

= ω_{\vec{k}} + \vec{q} \cdot {\vec{υ}}_{\vec{k}} + \vec{q} \cdot M_{\vec{k}} \cdot \vec{q} + \dots .

N	band 1	band 2	band 3	band 4	CPU-time
400	0.15251	0.19350	0.32085	0.37111	3.6s
800	0.15199	0.19179	0.32006	0.36865	22.1s
1200	0.15175	0.19103	0.31977	0.36742	70.6s
1600	0.15163	0.19066	0.31962	0.36678	99.8s
128×128	0.15068	0.18770	0.31847	0.36203	16.2s
(Defect)	(0.00016)	(0.00153)	(0.00039)	(0.00867)
256×256	0.15071	0.18778	0.31851	0.36219	47.6s
(Defect)	(0.00006)	(0.00065)	(0.00037)	(0.00541)

Abstract

Supplementary Material (1)

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Figures (3)

Tables (1)

Equations (13)

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