Resonant electromagnetic modes are analyzed inside a dielectric cavity of equilateral triangular cross section and refractive index n, surrounded by a uniform medium of refractive index ${\mathit{n}}^{\prime}$. The field confinement is determined only under the requirements needed to maintain total internal reflection of the internal electromagnetic fields, matched to external evanescent waves. Two-dimensional electromagnetics is considered, with no dependence on the coordinate perpendicular to the cross section, giving independent TE and TM polarizations. Generally, the mode spectrum becomes sparse and the minimum mode frequency increases rapidly as the index ratio $\mathit{N}=\mathit{n}\u2215{\mathit{n}}^{\prime}$ approaches 2. For specified quantum numbers and N, the TM modes are lower in frequency than the TE modes. Quality factors are estimated by supposing that evanescent boundary waves leak cavity energy at the triangle vertices; diffractive effects are not included. At an index ratio that is large compared with a mode’s cutoff ratio, this method predicts greater field confinement for TE polarization and higher quality factors than for TM polarization.

Junho Yoon, Sung-Jae An, Kwanghae Kim, Ja Kang Ku, and O'Dae Kwon Appl. Opt. 46(15) 2969-2974 (2007)

References

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Definitions of the Parameters of the Plane Waves Labeled by Wave Vectors ${\mathbf{k}}_{l}$, within the Triangular Cavity^{
a
}

$\text{wave}$

1

2

3

4

5

6

${\alpha}_{l}$

α

$-\alpha +240\xb0$

$\alpha +240\xb0$

$-\alpha +120\xb0$

$\alpha +120\xb0$

$-\alpha $

${\mathbf{k}}_{l}$

${\mathbf{k}}_{1}$

${R}^{2}\bullet {\mathbf{k}}_{6}$

${R}^{2}\bullet {\mathbf{k}}_{1}$

$R\bullet {\mathbf{k}}_{6}$

$R\bullet {\mathbf{k}}_{1}$

${\mathbf{k}}_{1}(-\alpha )$

${\alpha}_{l}$ is the angle that each ${\mathbf{k}}_{l}$ makes to the x axis. R is the operator for rotation through $+120\xb0$ around the $z\u0302$ axis.

Table 2

Relations between the Incident and Reflected Wave Amplitudes on the Lower Boundary $\left({b}_{0}\right)$, the Upper Right Boundary $\left({b}_{1}\right)$, and the Upper Left Boundary $\left({b}_{2}\right)$^{
a
}

Boundary

Incident

${\theta}_{i}$

Reflected

${b}_{0}$

3, ${A}_{3}$

${\theta}_{i,3}=\alpha -30\xb0$

4, ${A}_{4}={A}_{3}{e}^{i{\Delta}_{3}}$

${b}_{0}$

5, ${A}_{5}$

${\theta}_{i,5}=\alpha -150\xb0$

2, ${A}_{2}={A}_{5}{e}^{i{\Delta}_{5}}$

${b}_{0}$

6, ${A}_{6}$

${\theta}_{i,6}=90\xb0-\alpha $

1, ${A}_{1}={A}_{6}{e}^{i{\Delta}_{6}}$

${b}_{1}$

1, ${A}_{1}$

${\theta}_{i,3}$

2, ${A}_{2}={A}_{1}{e}^{i{\Delta}_{3}}$

${b}_{1}$

3, ${A}_{3}$

${\theta}_{i,5}$

6, ${A}_{6}={A}_{3}{e}^{i{\Delta}_{5}}$

${b}_{1}$

4, ${A}_{4}$

${\theta}_{i,6}$

5, ${A}_{5}={A}_{4}{e}^{i{\Delta}_{6}}$

${b}_{2}$

5, ${A}_{5}$

${\theta}_{i,3}$

6, ${A}_{6}={A}_{5}{e}^{i{\Delta}_{3}}$

${b}_{2}$

1, ${A}_{1}$

${\theta}_{i,5}$

4, ${A}_{4}={A}_{1}{e}^{i{\Delta}_{5}}$

${b}_{2}$

2, ${A}_{2}$

${\theta}_{i,6}$

3, ${A}_{3}={A}_{2}{e}^{i{\Delta}_{6}}$

The net reflection phase shifts are ${\Delta}_{l}={\delta}_{l}-{k}_{ly}\frac{a}{\sqrt{3}}$, where ${\delta}_{l}=\delta \left({\theta}_{i,l}\right)$ is the Fresnel reflection phase shift. See Fig. 1 for the geometrical reasoning behind this table.

Table 3

Properties of Some of the Lower TM Modes for $\mathit{N}=3.2$, Labeled by $(m,n)$ or by Underlined Indexes $(\underset{\u0331}{m},\underset{\u0331}{l})$, Where Compared with Results of Ref. [12], in Parentheses

Properties of Some of the Lower TE Modes for $\mathit{N}=3.2$, Labeled by $(m,n)$ or by Underlined Indexes $(\underset{\u0331}{m},\underset{\u0331}{l})$, Where Compared with Results of Ref. [12], in Parentheses

Mode Frequencies and Free-Space Wavelengths in the Range Around $1.3\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}1.6\text{\hspace{0.17em}}\mu \mathrm{m}$, for Cavities with $\mathit{N}=3.2$ and Edge Lengths a^{
a
}

a$\left(\mu \mathrm{m}\right)$

Mode

$ka$

λ$\left(\mu \mathrm{m}\right)$

f (THz)

2

${\mathrm{TM}}_{3,5}$

14.341

1.5675

191.25

2

${\mathrm{TE}}_{1,3}$

12.563

1.7893

167.54

5

${\mathrm{TM}}_{6,10}$

33.517

1.6767

178.79

5

${\mathrm{TM}}_{8,10}$

35.320

${1.5911}^{*}$

188.41

5

${\mathrm{TM}}_{7,11}$

37.711

${1.4902}^{*}$

201.17

5

${\mathrm{TM}}_{9,11}$

39.502

${1.4227}^{*}$

210.72

5

${\mathrm{TM}}_{8,12}$

41.889

${1.3416}^{*}$

223.46

5

${\mathrm{TM}}_{10,12}$

43.685

${1.2865}^{*}$

233.03

5

${\mathrm{TE}}_{5,9}$

32.040

1.7540

170.92

5

${\mathrm{TE}}_{7,9}$

34.815

1.6142

185.72

5

${\mathrm{TE}}_{6,10}$

36.755

1.5290

196.07

5

${\mathrm{TE}}_{8,10}$

39.011

1.4406

208.10

5

${\mathrm{TE}}_{7,11}$

41.135

1.3662

219.43

5

${\mathrm{TE}}_{9,11}$

43.202

1.3008

230.46

The modes are labeled by $m,n$. Wavelengths marked with an asterisk fall about 2% below peaks in PL data, Ref. [3]

Tables (5)

Table 1

Definitions of the Parameters of the Plane Waves Labeled by Wave Vectors ${\mathbf{k}}_{l}$, within the Triangular Cavity^{
a
}

$\text{wave}$

1

2

3

4

5

6

${\alpha}_{l}$

α

$-\alpha +240\xb0$

$\alpha +240\xb0$

$-\alpha +120\xb0$

$\alpha +120\xb0$

$-\alpha $

${\mathbf{k}}_{l}$

${\mathbf{k}}_{1}$

${R}^{2}\bullet {\mathbf{k}}_{6}$

${R}^{2}\bullet {\mathbf{k}}_{1}$

$R\bullet {\mathbf{k}}_{6}$

$R\bullet {\mathbf{k}}_{1}$

${\mathbf{k}}_{1}(-\alpha )$

${\alpha}_{l}$ is the angle that each ${\mathbf{k}}_{l}$ makes to the x axis. R is the operator for rotation through $+120\xb0$ around the $z\u0302$ axis.

Table 2

Relations between the Incident and Reflected Wave Amplitudes on the Lower Boundary $\left({b}_{0}\right)$, the Upper Right Boundary $\left({b}_{1}\right)$, and the Upper Left Boundary $\left({b}_{2}\right)$^{
a
}

Boundary

Incident

${\theta}_{i}$

Reflected

${b}_{0}$

3, ${A}_{3}$

${\theta}_{i,3}=\alpha -30\xb0$

4, ${A}_{4}={A}_{3}{e}^{i{\Delta}_{3}}$

${b}_{0}$

5, ${A}_{5}$

${\theta}_{i,5}=\alpha -150\xb0$

2, ${A}_{2}={A}_{5}{e}^{i{\Delta}_{5}}$

${b}_{0}$

6, ${A}_{6}$

${\theta}_{i,6}=90\xb0-\alpha $

1, ${A}_{1}={A}_{6}{e}^{i{\Delta}_{6}}$

${b}_{1}$

1, ${A}_{1}$

${\theta}_{i,3}$

2, ${A}_{2}={A}_{1}{e}^{i{\Delta}_{3}}$

${b}_{1}$

3, ${A}_{3}$

${\theta}_{i,5}$

6, ${A}_{6}={A}_{3}{e}^{i{\Delta}_{5}}$

${b}_{1}$

4, ${A}_{4}$

${\theta}_{i,6}$

5, ${A}_{5}={A}_{4}{e}^{i{\Delta}_{6}}$

${b}_{2}$

5, ${A}_{5}$

${\theta}_{i,3}$

6, ${A}_{6}={A}_{5}{e}^{i{\Delta}_{3}}$

${b}_{2}$

1, ${A}_{1}$

${\theta}_{i,5}$

4, ${A}_{4}={A}_{1}{e}^{i{\Delta}_{5}}$

${b}_{2}$

2, ${A}_{2}$

${\theta}_{i,6}$

3, ${A}_{3}={A}_{2}{e}^{i{\Delta}_{6}}$

The net reflection phase shifts are ${\Delta}_{l}={\delta}_{l}-{k}_{ly}\frac{a}{\sqrt{3}}$, where ${\delta}_{l}=\delta \left({\theta}_{i,l}\right)$ is the Fresnel reflection phase shift. See Fig. 1 for the geometrical reasoning behind this table.

Table 3

Properties of Some of the Lower TM Modes for $\mathit{N}=3.2$, Labeled by $(m,n)$ or by Underlined Indexes $(\underset{\u0331}{m},\underset{\u0331}{l})$, Where Compared with Results of Ref. [12], in Parentheses

Properties of Some of the Lower TE Modes for $\mathit{N}=3.2$, Labeled by $(m,n)$ or by Underlined Indexes $(\underset{\u0331}{m},\underset{\u0331}{l})$, Where Compared with Results of Ref. [12], in Parentheses

Mode Frequencies and Free-Space Wavelengths in the Range Around $1.3\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}1.6\text{\hspace{0.17em}}\mu \mathrm{m}$, for Cavities with $\mathit{N}=3.2$ and Edge Lengths a^{
a
}

a$\left(\mu \mathrm{m}\right)$

Mode

$ka$

λ$\left(\mu \mathrm{m}\right)$

f (THz)

2

${\mathrm{TM}}_{3,5}$

14.341

1.5675

191.25

2

${\mathrm{TE}}_{1,3}$

12.563

1.7893

167.54

5

${\mathrm{TM}}_{6,10}$

33.517

1.6767

178.79

5

${\mathrm{TM}}_{8,10}$

35.320

${1.5911}^{*}$

188.41

5

${\mathrm{TM}}_{7,11}$

37.711

${1.4902}^{*}$

201.17

5

${\mathrm{TM}}_{9,11}$

39.502

${1.4227}^{*}$

210.72

5

${\mathrm{TM}}_{8,12}$

41.889

${1.3416}^{*}$

223.46

5

${\mathrm{TM}}_{10,12}$

43.685

${1.2865}^{*}$

233.03

5

${\mathrm{TE}}_{5,9}$

32.040

1.7540

170.92

5

${\mathrm{TE}}_{7,9}$

34.815

1.6142

185.72

5

${\mathrm{TE}}_{6,10}$

36.755

1.5290

196.07

5

${\mathrm{TE}}_{8,10}$

39.011

1.4406

208.10

5

${\mathrm{TE}}_{7,11}$

41.135

1.3662

219.43

5

${\mathrm{TE}}_{9,11}$

43.202

1.3008

230.46

The modes are labeled by $m,n$. Wavelengths marked with an asterisk fall about 2% below peaks in PL data, Ref. [3]