Effects of pressure and rotation on photonic crystal slabs composed by triangular holes arranged in a hexagonal lattice

Optik - International Journal for Light and Electron Optics 207 (2020) 164382

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Original research article

Effects of pressure and rotation on photonic crystal slabs composed by triangular holes arranged in a hexagonal lattice

T

Francis Segovia-Chavesa,b,*, Herbert Vinck-Posadaa a b

Grupo de Superconductividad y Nanotecnología, Departamento de Física, Universidad Nacional de Colombia, AA 055051 Bogotá, Colombia Grupo de Física Teórica, Programa de Física, Universidad Surcolombiana, AA 385 Neiva, Colombia

A R T IC LE I N F O

ABS TRA CT

Keywords: Photonic crystal slabs Hexagonal lattice Guided-Mode Expansion method Photonic band structure Pressure

In this work, we use the Guided-Mode Expansion method to calculate the photonic band structure in a photonic crystal slab composed by triangular air holes arranged in a hexagonal lattice. Furthermore, this study focuses on the dependence of the dielectric function of the slab on hydrostatic pressure. The results reveal that when pressure increases, the photonic band structure shifts to higher energies at constant temperatures and constant triangle side lengths. In addition, the width of the TE-like photonic band gap also increases as pressure increases. An increase in band gap width when comparing the photonic band gap for the lattice with unrotated holes against that with rotated holes is observed.

1. Introduction Metamaterials have resulted in great impact in recent years due to their promising applications in the fields of optics [1,2] and photonics [3,4], such as high efficiency filters [5,6], optical signal processing [7,8] and invisibility [9,10]. Photonic crystals (PCs) are a particular type of metamaterials wherein the dielectric function is periodic in space [11,12]. The key feature of PCs is the appearance of non-accessible energy zones in photonic band structures (PBS), known as photonic band gaps (PBG), which are a consequence of dielectric function periodicity and can be used to confine stationary modes through the insertion of defects that disturb the crystalline lattice [13]. When the optical contrast is disturbed by an external agent, such as the hydrostatic pressure, the PBS of the PC may be tuned. Using the Plane Wave Expansion method, Segovia et al. [14] report that the PBS shifts to high energies due to the variation in GaAs dielectric constant that is caused by the increasing hydrostatic pressure, in a two-dimensional PC (2D-PC) composed of cylindrical holes. Recently, [15] considered a 2D-PC composed by air holes with cross-sections in the shape of isosceles triangles immersed in a GaAs background. They reported an increase in the PBG width caused by increasing the internal angle of the triangles, coupled with the PBS shifting due to increasing pressure. In this paper, we are interested in calculating the PBS of a PC slab constituted by a 2D-PC composed by air holes with crosssections in the shape of equilateral triangles arranged in a hexagonal lattice embedded in a GaAs background. For the purposes of this paper, we consider that the dielectric constant of the semiconductor depends on pressure [16]. Furthermore, the PBS is calculated through the Guided-Mode Expansion (GME) method [17]. This paper is organised as follows: in Section 2, we provide the general expressions of the theoretical model used during our analysis. In Section 3, we present the numerical results and the discussion related to PBS calculation. Finally, conclusions are presented in Section 4.

⁎ Corresponding author at: Grupo de Física Teórica Universidad Surcolombiana, and Grupo de Superconductividad y Nanotecnología Universidad Nacional. E-mail address: [email protected] (F. Segovia-Chaves).

https://doi.org/10.1016/j.ijleo.2020.164382 Received 31 December 2019; Accepted 9 February 2020 0030-4026/ © 2020 Elsevier GmbH. All rights reserved.

Optik - International Journal for Light and Electron Optics 207 (2020) 164382

F. Segovia-Chaves and H. Vinck-Posada

Fig. 1. PC slab with a thickness of d composed by a hexagonal lattice of triangular holes with a side length of L and a lattice constant of a .

2. Theoretical model The object of interest herein is a PC slab with a thickness of d surrounded by air (ε¯1 = 1.0 ), with a 2D-PC in the xy plane comprising air holes with cross-sections in the shape of equilateral triangles with side lengths of L , arranged in a hexagonal lattice embedded in a GaAs background, as shown in Fig. 1. ⎯→ ⎯ → The PBS is calculated using the GME method wherein the magnetic field H ( r ) expands on a basis of the guided modes (γ ) of the → homogeneous dielectric slab, with the Bloch vector constrained to the first Brillouin zone k [17]. Then:

⎯→ ⎯ → H→ k (r ) =

→

∑ ∑ cγ ( k

→ ⎯→ ⎯ → + G ) H →guided → (r ) k +G

(1) ⎯→ ⎯ → ⎯→ ⎯ → → where the reciprocal lattice vector is G . In the 2D-PC, periodicity is determined by the ε ( r ) = ε ( r + R ) dielectric function (with R 2 → → ⎯→ ⎯ → ⎯→ ⎯ ⎯→ ⎯ 1 ω r ) = 2 H (→ r ) [13]. lattice vectors) [18], where the H ( r ) field is provided by the solution of the master equation, ∇ × → ∇ × H (→ c ε(r ) The linear eigenvalue problem used to calculate the photonic band structure, is given by G

∑ Hμν c ν = μ

γ

ω2 cμ c2

(2)

where ω and c represent the angular frequency and the speed of light, respectively. The Hμν matrix elements are obtained through

Hμν =

→

→

⎯→ ⎯

⎯→ ⎯

∫ ε (1→r ) [ ∇ × H μ*(→r )]·[ ∇ × Hν (→r )] d→r

(3)

→ The guided modes with a wave vector of g are obtained by solving the dispersion relation for the transverse-electric and transversemagnetic fields, which are given by 2qχ1 cos(qd) + (χ12 − q2)sin(qd) = 0

(4)

χ2 2q q2 cos(qd) + ( 12 − 2 )sin(qd) = 0 ε¯1 ε¯2 ε¯1 ε¯2

(5)

where ε¯j (j = air, GaAs) represents the spatial average, χ1 = function expands to a set of plane waves [19],

ηG =

1 A

1 ε (→ r)

= ∑G ηG

g2 − →→ eiG · r∥

ω2 ε¯1 2 c

and q =

ω2 ε¯2 2 c

− g 2 . In the xy plane, the reverse dielectric

, where the ηG Fourier coefficients are given by

→→

∫ ε−1 (→r ) e−iG · r d2r ∥

(6)

where A is the area of the unit cell [20]. For the two-dimensional lattice, ηG is given by

1 L2 ⎛ 1 ⎧ ⎞ ⎪ εGaAs (P , T ) + 2a2 1 − εGaAs (P , T ) ⎝ ⎠ εG = ⎨ L2 1 ⎞ [I (Gx , Gy ) + I (−Gx , Gy )] ⎪ 2 ⎛1 − εGaAs (P , T ) ⎠ ⎩ 2a ⎝ ⎜

⎜

⎟

⎟

→ G =0 → G ≠0 (7)

where Gy L

I (Gx , Gy ) =

⎛ 2i e⎝ 2 3 Gy L ⎜

G L − x ⎞ 4 ⎠

⎟

3

⎡ i ⎢e ⎣

3 LG y Gx L sinc ⎜⎛ 4

⎝

3 Gy L ⎞

− 4

G L ⎤ − sinc ⎛ x ⎞ ⎥ 4 ⎠ ⎝ ⎠ ⎦

⎟

(8)

In addition, the dielectric function of the GaAs depends on hydrostatic pressure (P) and temperature (T), then

εGaAs (P , T ) = (ε0 + AeT / T0 P ) e−αP

(9) 2

Optik - International Journal for Light and Electron Optics 207 (2020) 164382

F. Segovia-Chaves and H. Vinck-Posada

Fig. 2. TE-like (black line) and TM-like (red line) photonic band structures. Slab thicknesses are (a) d = 0.3a and (b) d = 0.5a . The green lines represent the light cone and the grey region represents the PBG for the TE-like modes. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

with ε0 = 12.446, A = 0.21125, T0 = 240.7 K, and α = 0.00173 kbar −1 [16]. When considering rotating the holes at an θ angle, Eq. (6) may be rewritten as follows:

ε¯G =

1 A

∫ ε−1 (→r ′) e−i (G x′+G y′) d→r x

y

(10)

→ where x ′ and y′ are the coordinates of the rotated reference system and the G components are given by Gx = Gx cos θ + Gy sin θ

Gy = −Gx sin θ + Gy cos θ

(11)

3. Numerical results and discussion In the following numerical results, both temperature (T = 4 K) and the side lengths of the triangular holes (L = 0.8a) are held constant. Fig. 2 denotes the PBS in dimensionless frequency units (ωa/2πc ) at a pressure of 0 kbar. In the PC slab, photonic modes are classified as even (TE-like) and odd (TM-like), which are vertically confined and located under the light cone (white area). In Fig. 2(a) and (b), slab thicknesses are 0.3a and 0.5a respectively. The results obtained reveal the presence of a PBG for the TE-like PBS, wherein, for d = 0.3a , the PBG is located at 0.309 ≤ ωa/2πc ≤ 0.366. However, when the slab thickness increases to d = 0.5a , a redshift may be observed, with the PBG now located at 0.268 ≤ ωa/2πc ≤ 0.323. As the hydrostatic pressure increases without modifying the photonic structure, the dielectric constant of the GaAs decreases, thereby causing the TE-like PBS to shift towards higher frequency regions, as shown in Fig. 3(a) and (b), wherein slab thicknesses are 0.3a and 0.5a respectively. Furthermore, the PBG width also increases as the pressure increases. Table 1 reports the PBG positions at different pressure values for two given slab thickness values (d = 0.3a and 0.5a ). Next, we present the effects exerted on the PBS when changing the angle of orientation of the triangles in the two-dimensional photonic lattice, as shown in Fig. 4(a) wherein d = 0.5a, P = 0 kbar and θ = 20∘. Fig. 4(b) displays how PBG width increases by rotating the triangles: at θ = 20∘ and 30 ∘, the PBG is located at 0.267 ≤ ωa/2πc ≤ 0.335 and 0.267 ≤ ωa/2πc ≤ 0.341, respectively. An increase in the width of the PBG located within 0.211 ≤ ωa/2πc ≤ 0.27 is observed, as shown in Fig. 4(b). When increasing both the

Fig. 3. TE-like photonic band structure at pressures of 0 kbar (black line), 30 kbar (orange line) and 70 kbar (blue line). Slab thicknesses are (a) d = 0.3a and (b) d = 0.5a . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) 3

Optik - International Journal for Light and Electron Optics 207 (2020) 164382

F. Segovia-Chaves and H. Vinck-Posada

Table 1 Position of TE-like PBGs at different pressure and slab thickness values. Pressure (kbar)

0 30 70

PBG

d = 0.3a

d = 0.5a

[0.309, 0.366] [0.316, 0.375] [0.326, 0.386]

[0.268, 0.323] [0.275, 0.33] [0.284, 0.341]

Fig. 4. (a) Top view of the 2D-PC with triangles rotated at θ = 20∘ . (b) TE-like photonic band structure at θ = 0∘ (black line), θ = 20∘ (red line) and θ = 30∘ (green line). (c) TE-like photonic band structure at pressure values of 0 (black line) and 70 kbar (blue line) with rotation angles of 0 ∘ and 30 ∘ . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

pressure (0–70 kbar) and the angle of rotation (0 ∘–30 ∘), not only PBS shifts but also a more noticeable PBG width increase is observed, as shown in Fig. 4(c).

4. Conclusions Through the GME method, this study calculated the PBS for the TE-like and TM-like modes in a PC slab composed by triangular air holes arranged in a hexagonal lattice embedded in a GaAs background. The results of this study reveal that the dielectric function of the GaAs decreases with pressure at constant temperatures, causing the PBS to shift to higher frequencies, wherein TE-like PBG widths increase when both the applied pressure along with the angle of rotation of the triangles increase.

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Optik - International Journal for Light and Electron Optics 207 (2020) 164382

F. Segovia-Chaves and H. Vinck-Posada

Acknowledgments F.S.-Ch. and H. V.-P gratefully acknowledge funding by COLCIENCIAS projects: “Emisión en sistemas de Qubits Superconductores acoplados a la radiación. Código 110171249692, CT 293-2016, HERMES 31361” and “Control dinámico de la emisión en sistemas de Qubits acoplados con cavidades no-estacionarias, HERMES 41611”. F.S.-Ch. also acknowledges to Vicerrectoría de Investigación, Universidad Surcolombiana Neiva-Huila. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

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