# Formation of Degenerate Band Gaps in Layered Systems

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## Abstract

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## 1. Introduction

## 2. Formation of Degenerate Band Gaps

#### 2.1. Formation of Degenerate Band Gaps Inside a Passing Band

**Figure 1.**One-dimensional stratified PC with layers made of uniaxial materials. Thick arrows indicate orientations of the optical axes of the layers (in the plane of the layers). The left arrow k is the wave vector of an incident wave.

**Figure 2.**Dispersion curves of a PC with two uniaxial layers of the same thickness $d$ per unit cell. (${\Lambda}=2d$) Parameters of the first layer: ordinary dielectric permittivity ${\mathrm{\epsilon}}_{1}^{ord}=2.0$, extraordinary one ${\mathrm{\epsilon}}_{1}^{ext}=5.0$. Parameters of the second layer: ordinary dielectric permittivity ${\mathrm{\epsilon}}_{2}^{ord}=3.0$, extraordinary one ${\mathrm{\epsilon}}_{1}^{ext}=8.0$. Solid lines correspond to the frequencies of the pass bands, dashed lines correspond to the frequencies of band gaps. (

**a**) The optical axes of all the layers are collinear; (

**b**) The optical axes of adjacent layers are misaligned by an angle $\alpha =0.5$ rad.

**Figure 3.**Isofrequency curves for magnetophotonic crystal with two layers of the same thickness $d$ in a period, ${\Lambda}=2d$. ${k}_{0}d=0.9$ is a wave number in vacuum. The parameters for the first, anisotropic, layer of the period are: ordinary dielectric permittivity ${\mathrm{\epsilon}}_{1}^{ord}=8.0$, extraordinary one ${\mathrm{\epsilon}}_{1}^{ext}=2.0$. The parameters for the second, gyrotropic, layer of the period are: diagonal dielectric permittivity ${\mathrm{\epsilon}}_{2}=3.0$, non-diagonal components of dielectric tensor for the case of non-zero external magnetic field ±ig = ±0.3i. Black curves correspond to the case of zero external magnetic field, red curves correspond to non-zero external magnetic field. The two black curves correspond to s- and p-polarizations. Solid curves correspond to propagating waves, dotted curves correspond to band gaps. The optical axes of anisotropic layers are parallel to the plane of incidence ($Oyz$).

#### 2.2. Formation of Degenerate Band Gaps Inside a Brillouin Band Gap

**Figure 4.**Dispersion curves are calculated for a PC with two layers of the same thickness $d$ in a period, ${\Lambda}=2d$. ${k}_{0}$ is a wave number in vacuum. The parameters for the first layer of the period are: ordinary dielectric permittivity ${\mathrm{\epsilon}}_{1}^{ord}=5.0$, extraordinary one ${\mathrm{\epsilon}}_{1}^{ext}=7.8$. The parameters for the second layer of the period are: ordinary dielectric permittivity ${\mathrm{\epsilon}}_{2}^{ord}=7.5$, extraordinary one ${\mathrm{\epsilon}}_{2}^{ext}=5.4$. In the left figure for $\mathrm{Re}\left({k}_{Bl}\right){\Lambda}$ solid lines correspond to pass bands, dashed lines correspond to band gaps.

**Figure 5.**Bloch dispersion curves (red lines) for a PC with the parameters as in Figure 4, but with misaligned optical axes of adjacent layers. A DBG appears at the intersection of the Brillouin BGs. At the frequencies of the DBG both real and imaginary parts of the Bloch wave numbers for different dispersion curves are equal. Black lines being the same as in Figure 4 are shown for reference. The angle between optical axes of the adjacent layers is 0.08 rad. In the left figure for $\mathrm{Re}\left({k}_{Bl}\right){\Lambda}$ solid lines correspond to pass bands, dashed lines correspond to band gaps.

**Figure 6.**Bloch dispersion curves (red lines) for a PC with two layers of the same thickness $d$ in a period. The optical axes of adjacent layers are misaligned by an angle of 0.3 rad. The parameters for the first layer of the period are: ordinary dielectric permittivity ${\mathrm{\epsilon}}_{1}^{ord}=5.0$, extraordinary one ${\mathrm{\epsilon}}_{1}^{ext}=5.4$. The parameters for the second layer of the period are: ordinary dielectric permittivity ${\mathrm{\epsilon}}_{2}^{ord}=7.5$, extraordinary one ${\mathrm{\epsilon}}_{2}^{ext}=7.8$. A DBG does not appear at the intersection of Brillouin BGs. The figure shows that the intersection of dispersion curves for different dispersion branches is not the only condition for a DBG to appear. Black lines corresponding to the case of collinear optical axes (the same as in Figure 4) are shown for reference. In the left figure for $\mathrm{Re}\left({k}_{Bl}\right){\Lambda}$ solid lines correspond to pass bands, dashed lines correspond to band gaps.

## 3. Properties of Degenerate Band Gaps

#### 3.1. Linkage Between Brillouin and Degenerate Band Gaps. Formation of the So-Called “Degenerate Band Edge” and Frozen Mode

**Figure 7.**Frequency dependence ${k}_{0}d$ on $\mathrm{Re}\left({k}_{Bl}\right){\Lambda}$ for a multilayer PC ($d$—thicknesses of all layers, ${\Lambda}=2d$ is a PC period). The solid lines correspond to propagating waves, the dashed lines correspond to BGs.

**1**and

**3**are Brillouin BGs,

**2**is a DBG.

**А**is a DBG boundary,

**B**is a Brillouin BG boundary.

**Figure 8.**Frequency dependence ${k}_{0}d$

**on**$\mathrm{Re}\left({k}_{Bl}\right){\Lambda}$ for a multilayer PC ( $d$—thicknesses of all layers, ${\Lambda}=2d$ is a PC period). The solid lines correspond to propagating waves, the dashed lines correspond to BGs.

**Figure 9.**Frequency dependence ${k}_{0}d$ on $\mathrm{Re}\left({k}_{Bl}\right){\Lambda}$ for a multilayer PC ($d$—thicknesses of all layers, ${\Lambda}=2d$ is a PC period). The solid lines correspond to propagating waves, the dashed lines correspond to BGs.

**Figure 10.**The dependence of the coefficient $b$ in (3) on the distance Δ (along the frequency axis) between the boundaries A and B of degenerate and Brillouin BGs. The parameters for the PCs are cited in the text. The squares are the result of numerical simulations, the solid line is a trend line.

#### 3.2. Linkage Between Anisotropic and Gyrotropic Degenerate Band Gaps

**Figure 11.**The dependence of ${\Delta}{k}_{0}^{\mathrm{G}\mathrm{A}}$ on ${\Delta}{k}_{0}^{\mathrm{A}}$ at a fixed ${\Delta}{k}_{0}^{\mathrm{G}}$ for a bilayer PC. The solid curve is calculated using the “Pythagorean theorem”, and points are calculated numerically by using the transfer matrix method. The parameters for the first layer of the PC are: ${\mathrm{\epsilon}}_{x}^{first}=5.8$, ${\mathrm{\epsilon}}_{y}^{first}=5.6$, ${g}^{first}=2\times {10}^{-3}$; the parameters for the second layer are: ${\mathrm{\epsilon}}_{x}^{second}=8.43$, ${\mathrm{\epsilon}}_{y}^{second}=6.84$, ${g}^{second}=0$. The thicknesses of both the layers are the same. ${k}_{0}^{\ast}$ is the central frequency of the BGs.

**Figure 12.**Dispersion curves of a PC with four layers in a unit cell. A DBG exists for $\mathrm{Re}{k}_{\mathrm{B}\mathrm{l}}<0$ whereas at $\mathrm{Re}{k}_{\mathrm{B}\mathrm{l}}>0$ a DBG is closed.

#### 3.3. Absence of the Optical Borrmann Effect at the Boundary of a Degenerate Band Gap

**Figure 13.**The results of computer simulation: the amplitude of the electric field distribution for a Bloch wave in a PC at frequencies near band edges:

**1**—below a BG and

**2**—above a BG. The curve at the top is the function $\mathrm{\epsilon}\left(x\right)$.

**Figure 14.**(Top figure) Dispersion curves of the PC with two anisotropic layers in a period. (Bottom figure) The relative part of the energy of Bloch waves concentrated in the second anisotropic layer of the PC unit cells. Solid and dotted curves correspond to two different Bloch modes. The degenerate BG forms around the frequency ${k}_{0}\left({d}_{1}+{d}_{2}\right)\approx 1.76$. Only the parts of curves that lie in pass bands are shown.

**Figure 15.**(Top figure) Dispersion curves of the PC with one anisotropic and one gyrotropic layer in a period. (Bottom figure) The relative part of the energy of the Bloch waves concentrated in the gyrotropic PC layers. Solid and dotted curves correspond to two different Bloch modes. The degenerate BG forms around the frequency ${k}_{0}\left({d}_{1}+{d}_{2}\right)\approx 1.89$. Only the parts of curves that lie in pass bands are shown.

## 4. Formation of the Tamm States Based on Degenerate Band Gap

**Figure 16.**System under consideration. The unit cell of the first PC consists of a uniaxial crystal (${\mathrm{\epsilon}}_{xx}=2.7$, ${\mathrm{\epsilon}}_{yy}=5.0$) and a magnetooptical layer (${\mathrm{\epsilon}}_{diag}=3.0$, ${\mathrm{\epsilon}}_{off\_diag}=i\alpha =0.02i$ and ${\mathrm{\epsilon}}_{off\_diag}=0$ at zero magnetization). The unit cell of the second PC consists of two isotropic layers with permittivities ${\mathrm{\epsilon}}_{1}=3.1$ and ${\mathrm{\epsilon}}_{2}=3.4$. The thickness of each layer equals $d$.

**Figure 17.**Propagation coefficient of the system under consideration. The dotted line corresponds to zero magnetization; the solid line corresponds to the magnetized case.

## 5. Anisotropy of Admittance

## Acknowledgments

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## Appendix

_{1}is real and ${\alpha}_{1}$ is pure imaginary, and the numerator in Equation (A9) can be zero. It means closing of the DBG. It is worth mentioning that the formula for ${\Delta}{k}_{0}^{\text{GA}}$ depends on the choice of Bragg conditions. If we take the second pair of Bragg c Equation (A6) instead of Equation (A5) then Equation (A9) is replaced by

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**MDPI and ACS Style**

Ignatov, A.I.; Merzlikin, A.M.; Levy, M.; Vinogradov, A.P.
Formation of Degenerate Band Gaps in Layered Systems. *Materials* **2012**, *5*, 1055-1083.
https://doi.org/10.3390/ma5061055

**AMA Style**

Ignatov AI, Merzlikin AM, Levy M, Vinogradov AP.
Formation of Degenerate Band Gaps in Layered Systems. *Materials*. 2012; 5(6):1055-1083.
https://doi.org/10.3390/ma5061055

**Chicago/Turabian Style**

Ignatov, Anton I., Alexander M. Merzlikin, Miguel Levy, and Alexey P. Vinogradov.
2012. "Formation of Degenerate Band Gaps in Layered Systems" *Materials* 5, no. 6: 1055-1083.
https://doi.org/10.3390/ma5061055