Temperature effects on the confinement of light in a photonic crystal slab

Journal Pre-proof Temperature effects on the confinement of light in a photonic crystal slab Francis Segovia-Chaves, Herbert Vinck-Posada

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S0030-4026(20)30127-3

DOI:

https://doi.org/10.1016/j.ijleo.2020.164293

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IJLEO 164293

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Optik

Received Date:

12 December 2019

Accepted Date:

23 January 2020

Please cite this article as: Francis Segovia-Chaves, Herbert Vinck-Posada, Temperature effects on the confinement of light in a photonic crystal slab, (2020), doi: https://doi.org/10.1016/j.ijleo.2020.164293

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Temperature effects on the confinement of light in a photonic crystal slab Francis Segovia-Chavesa,b , Herbert Vinck-Posadaa Departamento de F´ısica, Universidad Nacional de Colombia, AA 055051 Bogot´ a, Colombia b Programa de F´ısica, Universidad Surcolombiana, AA 385 Neiva, Colombia

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Abstract

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In this paper, we numerically investigated the temperature effects on photonic band structure and quality factor using the guided-mode expansion method in two-dimensional photonic crystal slabs. We considered that the slab is composed of silicon (Si) and that the two-dimensional photonic crystal is composed of air holes arranged in a hexagonal lattice. We report that by increasing the temperature, thermo-optical effects cause a short frequency shift of the photonic band structure for transverse-electric-like and transverse-magnetic-like modes. When removing an air hole in the hexagonal lattice, we report a confined degenerate mode within the TE-like photonic band gap, which exhibits a shift at lower frequencies with increase in temperature. Moreover, we identified a greater confinement mode in the cavity because of the increase in the quality factor with increase in temperature. We hope that this work will be taken into account for the development of new perspectives in the design of optical devices. Keywords: Photonic crystal slabs, hexagonal lattice, temperature, quality factor.

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1. Introduction

Photonic crystals (PCs) are artificial structures by which light propagation is manipulated. PCs are characterized by a periodicity of the refractive index with a period comparable to the wavelength of light in the material [1,2]. PCs are similar to semiconductor materials with a photonic band structure in which light propagation is prohibited in a certain frequency range [3]. These prohibited frequencies regions are known as photonic band gaps Preprint submitted to Elsevier

December 12, 2019

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(PBGs) [4]. PCs are examined within the framework of Maxwell’s classical electromagnetic theory. To solve these equations, we used numerical methods such as plane wave expansion [5,6], finite differences in the time and frequency domains [7,8] and transfer matrix [9]. The photonic band structure can be possibly tuned to a desired frequency range by controlling the optical properties of the constituent materials of the PC using an external agent that alters the optical contrast. Pressure and temperature are external agents that can be applied on the PC. F. Segovia et al. reported on two-dimensional photonic crystals (2D-PCs), which show the effects of pressure in square and hexagonal lattices for dispersers with circular and triangular cross-sections [10,11]. They reported a shift at higher frequencies of the photonic band structure because of an increase in optical contrast with increase in pressure. The effects of temperature because of thermal and thermo-optical expansion were investigated by Elsayed et al. [12] who reported an enhancement in the PBG for temperatures higher than room temperature because of an increase in the contrast between the refractive indexes of the Si dispersers and the air background. Moreover, we achieved light confinement by breaking the spatial periodicity of the lattice on the PC and identified technological applicability in optical switching [13, 14], waveguides [15, 16], resonant cavities [17, 18] and Fabry-Perot resonators [19, 20]. The primary purpose of this study is to investigate the effects of temperature on symmetric PC slabs, which comprise a 2D-PC embedded in a waveguide. In the 2D-PC, we consider a hexagonal lattice of air holes within the slab core of Si. The guided-mode expansion (GME) method proposed by Gerace [21] was used to calculate the photonic band structure and the confined modes when we removed a hole from the hexagonal lattice. The rest of the paper is arranged as follows. In Section 2, we present the main equations of the GME method for the problem of interest. In Section 3, the photonic band structure and the quality factor in the PC slab for different temperature variations are presented. Finally, in Section 4, the conclusions are provided. 2. Theoretical Model

Fig. 1 (a) shows the PC slab in which the structure of interest is a hexagonal lattice of air holes. The thickness of the slab is d, the radius of the holes is R and the lattice constant is a. Fig. 1 (b) shows a homogeneous planar waveguide, in which the dielectric constants are 1 , 2 and 3 , which 2

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represent the lower cladding, core and upper cladding, respectively.

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Figure 1: a) d is the thickness of PC slab composed of hexagonal lattice of air holes of radius R and lattice constant a. b) Homogeneous dielectric slab of lower cladding dielectric constants (1 ), core (2 ) and upper cladding (3 ) dielectric constants.

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Using the GME method, we calculate the photonic band structure in PC → − − slabs. This method is based on the fact that the magnetic field ( H (→ r )) is expanded because of the guided modes obtained from homogeneous dielectric slabs (Fig. 1 (b)); thus, XX → − → − → − → − → − − →( r ) = c ( k + G ) H guided H− (→ r) (1) − → G

− → k +G

γ

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− → − → where G , k and γ represent the reciprocal lattice vector 2D, the Bloch vector and the guided mode, respectively. Therefore, photonic modes on the → − PC slab are restricted in the first Brillouin zone for k , and the guided modes − in the homogeneous dielectric slab depend on the wave vector → g. Light propagation is determined using Maxwell’s electromagnetic theory. In the case of isotropic, periodic and linear media, the master equation for → − → H (− r ) is given by the following equation: → − ∇×

− → − → − → 1 → ω2 → − H (− r) ∇ × H ( r ) = → − 2 c ( r )

(2)

→ − − − where (→ r ) = (→ r + R ) represents the periodicity of the dielectric function of → − PC with R being the lattice vectors in the xy plane [22]. When replacing Eq. 3

P ω2 (1) in (2), we obtain a linear eigenvalue problem, µ Hµν cν = 2 cµ , where c ω and c are the angular frequency and speed of light, respectively. Moreover, the matrix element Hµν is calculated using the following equation: Z i h→ i − → −∗ → − → − → 1 h→ − − − ∇ × H ( r ) · ∇ × H ( r ) d→ r (3) Hµν = ν µ → − ( r )

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X − →→ 1 i G ·− rk = η e G − (→ r) G

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Eq. (3) is then expanded into a set of plane waves, which are the inverse of → − − (→ r ) with Fourier coefficients η( G ); thus,

(4)

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− →− 1 R −1 → →  (− r )e−i G · r k d2 r in which A is the unit cell area. For where ηG = A homogeneous dielectric slabs, when applying the conditions of continuity of the tangential components of the transverse-electric (TE) and transversemagnetic (TM) fields, the dispersion relation is obtained. This relation is obtained by solving the following implicit equations:

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q(χ1 + χ3 )cos(qd) + (χ1 χ3 − q 2 )sin(qd) = 0

(5)

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χ1 χ3 q 2 q χ1 χ3 (6) ( + )cos(qd) + ( − 2 )sin(qd) = 0 2 1 3 1 3 2 r r r 2 2 ω ω ω2 where χ1 = g 2 − 1 2 , q = 2 2 − g 2 and χ3 = g 2 − 3 2 . c c c In this study, we will focus on a symmetric PC slab surrounded by air (1 = 3 = 1.0). Moreover, we consider that the medium of the slab is Si, where the refractive index of the Si is temperature-dependent, according to the thermo-optical effect; hence, n = n0 (1 + β∆T )

(7)

where n0 is the refractive index at room temperature, β is the thermo-optical coefficient and ∆T is the temperature variation [23]. We assume that the thickness of the slab changes with ∆T : d = d0 (1 + α∆T ) 4

(8)

3. Numerical results and discussion

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where d0 and α represent the thickness at room temperature and the medium thermal expansion coefficient, respectively. Note that Fourier coefficients for the hexagonal lattice of air holes are calculated using the following equations:    2 → − 1 1 2πR   + √ 1− G =0  Si a2 3 Si  (9) ηG = → − 1 4πR   √ 1− J1 (RG) G 6= 0  2 Si a 3G s 2  2 2πj 2π(2m − j) √ where Si is the refractive index of Si, G = + a a 3 (with j and m being integers) and J1 (RG) is the Bessel function of order 1 [24].

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In the following numerical calculations, we consider that the thickness of the slab (d0 = 0.5a) and the holes radius (0.3a) is constant. For air holes, the refractive index is 1 and β = 1.0 × 10−6 K−1 . For the Si slab, n0 =3.5, β = 1.86 × 10−4 K−1 and = 0.5 × 10−6 K−1 . In the PC slab, the photonic modes are classified into even (TE-like) and odd (TM-like); moreover, these are vertically confined and located below the light cone. Fig. 2 (a) shows the photonic band structure in dimensionless frequency units (ωa/2πc) for a temperature variation ∆T = 0◦ C . We observe that only in the TE-like mode, there is a PBG in the region 0.264 ≤ ωa/2πc ≤ 0.346. When ∆T increases, the refractive index of Si increases, thus causing a shift at lower frequencies of photonic band structure, as shown in Fig 2 (b). Moreover, the width of PBG remains constant because it increases ∆T , as shown in Table 1.

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Table 1: PBG position for different temperature variations.

Temperature (◦ C) 0 125 175 400

PBG 0.264 ≤ ωa/2πc ≤ 0.346 0.258 ≤ ωa/2πc ≤ 0.34 0.256 ≤ ωa/2πc ≤ 0.338 0.246 ≤ ωa/2πc ≤ 0.328

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Figure 2: a) Photonic band structure for one PC slab with a hexagonal lattice of air holes: TE-like (black line) and TM-like (red line). The green lines represent the light cone and the grey area is PBG for TE-like. b) Photonic band structure TE-like for ∆T = 0◦ C (black line), ∆T = 125◦ C (blue line), ∆T = 175◦ C (green line) and ∆T = 400◦ C (orange line).

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The light confinement in a cavity is possible by breaking the periodicity of 2D-PC. In Fig. 3 (a), a top view of the hexagonal lattice is shown when the air hole is removed for ∆T = 0◦ C. Using the supercell technique with period L = 7a, we found inside the PBG a localized mode as shown in Fig. 3 (b). However, in the inset of Fig. 3 (b), we can see two modes inside the PBG with very close frequency values ωa/2πc ' 0.292 (mode 1) and 0.295 → − 2 (mode 2). Fig. 3 (c) shows the intensity | E √ | for the degenerate mode at the point of symmetry M (kx a = 0, ky a = 2π/ 3). By increasing ∆T , the position of the modes is tuned, which exhibit a change of position at lower frequencies. For ∆T = 125◦ C, the positions of the modes are ωa/2πc ' 0.285 and 0.288. Similarly, for ∆T = 175◦ C, the modes are located at ωa/2πc ' 0.283 and 0.286, while at ∆T = 400◦ C, the shift of the modes is greater than ωa/2πc ' 0.272 and 0.276.

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Figure 3: a) PC slab top view with an air hole removal in the hexagonal lattice centre. b) Photonic band structure (TE-like). The horizontal lines inside the PBG represent the localized modes. c) Intensity E 2 of the localized modes. The values used in the simulations are ∆T = 0◦ C with a supercell length L = 7a.

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Note that the energy loss per cycle versus the energy stored in a cavity is determined by the quality factor (Q). This factor can possibly be calculated using the GME method for the photonic frequency mode ωk ; thus, Q=

ωk 2Im(ωk )

(10)

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where Im(ωk ) = Im(ωk2 )/2Re(ωk ) is the imaginary part of the frequencies (Im(ωk2 )). As mentioned before, we determine that, in the cavity of interest, the confinement of energy is greater for mode 1 where the Q-factor increases with increase in ∆T , as shown in Table 2. From the reported results, we observe that the Q-factor decreases because it increases the temperature variation. When we increase the size of the supercell by L = 9a, we find

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Table 2: Q-factor values for the confined modes.

Temperature (◦ C)

Mode 2 326.187 325.299 324.669 320.743

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Mode 1 437.93 441.549 442.755 447.923

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0 125 175 400

Q-factor

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three modes with frequencies inside the PBGωa/2πc ' 0.287 (mode 1), 0.293 (mode 2) and 0.298 (mode 3), as shown in Fig. 4 (a). The intensity profile → − | E |2 for these three modes is shown in Fig. 4 (b), where the cavity Q-factor exhibits a non-monotonous behaviour with increase in ∆T , as shown in Table 3. For ∆T = 175◦ C, the modes are located at ωa/2πc ' 0.278 (mode 1), 0.284 (mode 2) and 0.29 (mode 3).

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Figure 4: a) Photonic band structure (TE-like). The horizontal lines inside the PBG represent the localized modes. b) Intensity E 2 of the localized modes. The values used in the simulations are ∆T = 0◦ C with a supercell length L = 9a.

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Table 3: Q-factor values for the confined modes with a supercell L = 9a.

Temperature (◦ C)

Q-factor Mode 1 Mode 2 Mode 3 449.735 621.562 983.483 588.305 525.587 1621.43

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4. Conclusions

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Using the GME method, we calculate the photonic band structure in a PC slab, which is constituted by a hexagonal lattice of air holes. The slab is composed of Si and, considering the thermo-optical effect, we observe an increase in its dielectric constant with increase in temperature variation. The increase in temperature causes a position change of the photonic modes at lower frequencies, thus ensuring that the PBG width remains constant. By removing a gap from the hexagonal lattice, a degenerate mode is located in the cavity with a Q-factor that increases with the rise in temperature variation. The results confirm that the Q-factor exhibits a non-monotonous behavior to temperature by increasing the supercell size in the defective hexagonal. lattice.

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