Simple plane wave implementation for photonic crystal calculations

Shangping Guo; Sacharia Albin

doi:10.1364/OE.11.000167

Optics Express
Vol. 11,
Issue 2,
pp. 167-175
(2003)
•https://doi.org/10.1364/OE.11.000167

Simple plane wave implementation for photonic crystal calculations

Shangping Guo and Sacharia Albin

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Shangping Guo and Sacharia Albin, "Simple plane wave implementation for photonic crystal calculations," Opt. Express 11, 167-175 (2003)

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Abstract

A simple implementation of plane wave method is presented for modeling photonic crystals with arbitrary shaped ‘atoms’. The Fourier transform for a single ‘atom’ is first calculated either by analytical Fourier transform or numerical FFT, then the shift property is used to obtain the Fourier transform for any arbitrary supercell consisting of a finite number of ‘atoms’. To ensure accurate results, generally, two iterating processes including the plane wave iteration and grid resolution iteration must converge. Analysis shows that using analytical Fourier transform when available can improve accuracy and avoid the grid resolution iteration. It converges to the accurate results quickly using a small number of plane waves. Coordinate conversion is used to treat non-orthogonal unit cell with non-regular ‘atom’ and then is treated by standard numerical FFT. MATLAB source code for the implementation requires about less than 150 statements, and is freely available at http://www.lions.odu.edu/~sguox002.

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Fig. 1. (a) Conversion of a triangular lattice with a circular cylinder ‘atom’; (b) Conversion of a triangular lattice with an elliptical cylinder ‘atom’; (c) TE band structure of a 2D triangular lattice with elliptical air holes in GaAs. Data used: R_x = 0.28a, R_y = 0.14a, ε_a=13, ε_b=1.0.

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Fig. 2. Band structure of a 3D diamond lattice using 343 plane waves for this calculation, the inset shows the unit cell of the diamond lattice.

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Fig. 3. Convergence of TM mode. (a) Convergence of the first band. (b) The iteration errors for the first 10 bands. A uniform mesh with different resolution is used to represent the unit cell, and each grid is averaged by a 10×10 submesh.

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Fig. 4. Convergence of TE mode. (a) Convergence of the first band. (b) The iteration errors for the first 10 bands. A uniform mesh with different resolution is used to represent the unit cell, and each grid is averaged by a 10×10 submesh.

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Fig. 5. Eigen frequency convergence as a function of grid resolution for TM mode in a 2D triangular lattice. 225 plane waves are used for this calculation. Line with ‘o’: grid is averaged by a 10×10 submesh; line with ‘+’:not averaged.

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Fig. 6. Eigen frequency convergence as a function of grid resolution for TE mode in a 2D triangular lattice. 225 plane waves are used for this calculation. Line with ‘o’: grid is averaged by a 10×10 submesh; line with ‘+’: not averaged.

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Fig. 7. Convergence of defect frequency for TM mode using different supercell size in a square lattice with the center rod being removed

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Tables (2)

Table 1. Comparison of several methods

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Table 2. Defect frequency of TM mode in a 2D square lattice using a 7×7 supercell

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Equations (15)

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\nabla \times \frac{1}{ε (r)} \nabla \times \vec{H} (r) = \frac{ω^{2}}{c^{2}} \vec{H} (r)

\vec{H} (r) = \sum_{\vec{G}, λ} h_{G, λ} {\hat{e}}_{λ} e^{i (\vec{k} + \vec{G}) \cdot \vec{r}}

ε (r) = \sum_{\vec{G}} ε_{G} e^{i \vec{G} \cdot \vec{r}} ε_{G} = \frac{1}{V} \underset{Ω}{∰} ε (r) e^{- i \vec{G} \cdot \vec{r}} d Ω

\sum_{G'} ∣ \vec{k} + \vec{G} ∣ ∣ \vec{k} + \vec{G'} ∣ ε^{- 1} (\vec{G} - \vec{G'}) [\begin{matrix} {\hat{e}}_{2} \cdot {\hat{e}}_{2}^{'} & - {\hat{e}}_{2} \cdot {\hat{e}}_{1}^{'} \\ - {\hat{e}}_{1} \cdot {\hat{e}}_{2}^{'} & {\hat{e}}_{1} \cdot {\hat{e}}_{1}^{'} \end{matrix}] [\begin{matrix} h_{1, G'} \\ h_{2, G'} \end{matrix}] = \frac{ω^{2}}{c^{2}} [\begin{matrix} h_{1}, G \\ h_{2}, G \end{matrix}] .

ε (G) = ε_{b} δ (G) + (ε_{a} - ε_{b}) \frac{2 π R^{2}}{A} \frac{J_{1} (G R)}{G R} = ε_{b} δ (G) + 2 (ε_{a} - ε_{b}) f \frac{J_{1} (G R)}{G R}

f = \frac{V o l_{a t o m}}{V o l_{c e l l}} .

{\vec{a}}_{1} = a_{1 x} \hat{x} + a_{1 y} \hat{y} + a_{1 z} \hat{z}

{\vec{a}}_{2} = a_{2 x} \hat{x} + a_{2 y} \hat{y} + a_{2 z} \hat{z},

{\vec{a}}_{3} = a_{3 x} \hat{x} + a_{3 y} \hat{y} + a_{3 z} \hat{z}

\vec{r} = ({m a}_{1 x} + {n a}_{2 x} + {l a}_{3 x}) \hat{x} + ({m a}_{1 y} + {n a}_{2 y} + {l a}_{3 y}) \hat{y} + ({m a}_{1 z} + {n a}_{2 z} + {l a}_{3 z}) \hat{z}

∣ r^{2} ∣ = {\vec{r}}^{T} [g] \vec{r}

[g] = [\begin{matrix} a_{1 x} & a_{1 y} & a_{1 z} \\ a_{2 x} & a_{2 y} & a_{2 z} \\ a_{3 x} & a_{3 y} & a_{3 z} \end{matrix}] [\begin{matrix} a_{1 x} & a_{2 x} & a_{3 x} \\ a_{1 y} & a_{2 y} & a_{3 y} \\ a_{1 z} & a_{2 z} & a_{3 z} \end{matrix}]

ε (\vec{r} + {\vec{r}}_{0}) \leftrightarrow e^{i \vec{G} \cdot {\vec{r}}_{0}} ε_{G} .

\sum_{r_{i}} ε (\vec{r} + {\vec{r}}_{i}) \Leftrightarrow \sum_{r_{i}} e^{i \vec{G} \cdot {\vec{r}}_{i}} ε_{G},

ε (\vec{G}) = 3 f (ε_{a} - ε_{b}) (\frac{\sin GR - GR \cos GR}{{(GR)}^{3}}) \cos (\vec{G} \cdot \vec{r_{0}})

Method	Grid	# of PWs	Band 1 at X	Band 2 at X	Band 3 at X	Band 4 at X	CPUtime(s)*
Analytical FT	-	169	0.15071	0.18778	0.31852	0.36231	2.60
Analytical FT	-	225	0.15071	0.18778	0.31851	0.36229	5.06
Our FFT	243×243	169	0.15073	0.18781	0.31855	0.36240	33.10
PWM [9]	40×40	-	0.15163	0.19066	0.31962	0.36678	99.80
Multigrid [9]	256×256	-	0.15071	0.18778	0.31851	0.36219	47.60

N	# of PWs	Defect freq.	Iteration Error	CPU Time(s)*
4	81=9×9	0.421996	3.6769e-02	0.49
5	121=11×11	0.415577	6.8329e-02	1.10
6	169=13×13	0.410824	3.1510e-02	2.18
7	225=15×15	0.407251	2.7674e-02	4.39
8	289=17×17	0.405432	1.6684e-02	7.45
9	361=19×19	0.404108	2.1813e-02	12.98
10	441=21×21	0.399563	2.8045e-02	20.38
11	529=23×23	0.398201	1.7524e-02	32.63
12	625=25×25	0.397168	5.4067e-03	51.49
13	729=27×27	0.396306	2.4535e-03	78.79
14	841=29×29	0.395868	2.7855e-03	112.89
15	961=31×31	0.395736	1.3450e-03	165.65
16	1089=33×33	0.395492	1.6496e-03	238.34
17	1225=35×35	0.395005	2.2181e-03	331.06

Method	Grid	# of PWs	Band 1 at X	Band 2 at X	Band 3 at X	Band 4 at X	CPUtime(s)*
Analytical FT	-	169	0.15071	0.18778	0.31852	0.36231	2.60
Analytical FT	-	225	0.15071	0.18778	0.31851	0.36229	5.06
Our FFT	243×243	169	0.15073	0.18781	0.31855	0.36240	33.10
PWM [9]	40×40	-	0.15163	0.19066	0.31962	0.36678	99.80
Multigrid [9]	256×256	-	0.15071	0.18778	0.31851	0.36219	47.60

N	# of PWs	Defect freq.	Iteration Error	CPU Time(s)*
4	81=9×9	0.421996	3.6769e-02	0.49
5	121=11×11	0.415577	6.8329e-02	1.10
6	169=13×13	0.410824	3.1510e-02	2.18
7	225=15×15	0.407251	2.7674e-02	4.39
8	289=17×17	0.405432	1.6684e-02	7.45
9	361=19×19	0.404108	2.1813e-02	12.98
10	441=21×21	0.399563	2.8045e-02	20.38
11	529=23×23	0.398201	1.7524e-02	32.63
12	625=25×25	0.397168	5.4067e-03	51.49
13	729=27×27	0.396306	2.4535e-03	78.79
14	841=29×29	0.395868	2.7855e-03	112.89
15	961=31×31	0.395736	1.3450e-03	165.65
16	1089=33×33	0.395492	1.6496e-03	238.34
17	1225=35×35	0.395005	2.2181e-03	331.06

Abstract

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Equations (15)

Optics Express