Effects of hydrostatic pressure on the photonic band structure and quality factor of an L3 cavity in a photonic crystal slab

Results in Physics 16 (2020) 102947

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Effects of hydrostatic pressure on the photonic band structure and quality factor of an L3 cavity in a photonic crystal slab

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 I C LE I N FO

A B S T R A C T

Keywords: Photonic crystal slabs Photonic band structure Pressure Quality factor

In this work, we investigate the effects of hydrostatic pressure on the photonic band structure and quality factor of the fundamental mode in an L3 cavity using the guided-mode expansion method in two-dimensional photonic crystal slabs. The two-dimensional photonic crystal is composed of air holes with circular cross-sections arranged in a hexagonal lattice. Here, the dielectric constant of the slab is a function of hydrostatic pressure and temperature. Upon increasing pressure at a given temperature, the photonic band structure exhibits a shift towards higher frequencies for the TE-like and TM-like modes of the photonic crystal slab. In the L3 cavity, the lateral displacement of the holes must also be considered as the quality factor of the fundamental mode increases with the displacement of the holes. However, there is evidence that the confinement of the L3 cavity decreases when both the hydrostatic pressure and the symmetric lateral shift of the holes increase.

1. Introduction Photonic crystals (PCs) are artificial structures that exhibit spatial periodicity of the dielectric constant and magnetic permeability [1,2]. The electromagnetic properties of PCs are determined by the classical Maxwell equations because PCs are analogous to a solid crystal with an electronic band structure [3]. One of the main features of PCs is the photonic band gap (PBG), wherein light propagation is forbidden [4]. The structural color is caused by interference, diffraction or dispersion in nanoarchitecture, thus exhibiting design flexibility and compatibility with camouflage materials [5]. One-dimensional photonic crystals (1DPC) manufactured with a specific color were theoretically proposed and investigated experimentally by D. Qi et al. [6,7]. These authors reported that when changing the thickness of the surface layer in the 1DPC, they obtained four different colors: khaki, brown, navy and cyan. The concept of a surface graphical photonic crystal (SGPC) is proposed in [8] to produce angle-insensitive visible-infrared compatible camouflage. The SGPC comprises two specially designed parts: a quasi-periodic 1D-PC (Ge/ZnS) with an arithmetic sequence in the physical thickness for each period and a graphical ZnS surface. Theoretical results reveal that the incident angles exert negligible influence on the designed colorful digital camouflage. The PBG width is sensitive to the refractive index, layer thickness, reference wavelength and periodicity. When the optical thickness is equal to one-quarter of the reference

wavelength, and there is a great difference in refractive index of the materials of the 1D-PC, the theoretical and experimental results report an increase in the PBG width [9]. High-efficiency narrow-band filters are possible through a quasi-one-dimensional PC with a mirror-symmetric heterostructure [10]. The defect mode appears within the PBG at a transmittance of 95.9% and with a quality factor of 705 at 1550 nm. The tuning of the defect mode in the transmission spectrum and of the narrow band filter quality may be achieved when both the thickness of the materials and structure periodicity are increased. The photonic band structure (PBS) in two-dimensional PCs (2D-PC) depends on the materials that constitute the PC, as well as on the crystalline structure [11]. We can also highlight important 2D-PC applications, such as broadband selective absorbers/emitters for solar thermophotovoltaic applications [12]. The results obtained by alternating hafnium oxide and titanium oxide materials in tetragonal lattices embedded in an antireflection-coated tungsten film show that high solar collection efficiency can be obtained with this type of heterostructure. The larger PBGs allow the design of devices such as all-dielectric coaxial waveguides [13], mirror fibers [14], optical nanoswitches [15,16], optical transistors [17,18], photonic crystal (or hollow-core) fibers [19,20], optical cloaking [21,22], superprisms and superlenses [23,24], among others. To enhance the PBGs, the symmetric topology structure of the PC is reduced by the introduction of fractal structures or sequences. Applying the plane wave expansion

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

https://doi.org/10.1016/j.rinp.2020.102947 Received 7 December 2019; Received in revised form 11 January 2020; Accepted 13 January 2020 Available online 30 January 2020 2211-3797/ © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).

Results in Physics 16 (2020) 102947

F. Segovia-Chaves and H. Vinck-Posada

determined by the electromagnetic ⎯→ ⎯ H (→ r ) magnetic field given by

method, H.F. Zhang et al. [25,26] theoretically studied PBG properties and defect modes in two-dimensional fractal plasma photonic crystals. In these works, the authors considered dielectric cylinders arranged in a square lattice with Thue-Morse and Fibonacci sequences, finding that these structures not only favor obtaining a cutoff frequency region but also that the quasi-localised states can be tuned at different sequence orders. Previous studies have focused on the tuning of the PBG in 2DPCs based on the dependence of the dielectric function on external agents, such as temperature [27], hydrostatic pressure [28], and magnetic fields [29]. For example, F. Segovia et al. [30,31] report the effects of temperature and pressure on the PBS of a 2D-PC composed rods with circular and triangular cross-sections arranged in a hexagonal lattice. The results reveal that a noticeable shift of the PBS (to higher frequencies) may be observed as the hydrostatic pressure increases. In addition, the emergence of new PBGs due to an increasing magnetic field in square and hexagonal lattices comprising GaAs rods are studied in [32]. Photonic crystals embedded in planar waveguides are known as PC slabs. These structures allow light to be guided and confined within the three spatial directions [33], wherein light propagation in the perpendicular direction is controlled by total internal reflection, while within the 2D-PC, propagation is achieved through distributed Bragg reflection [34]. The main purpose of this paper is to study the effects of hydrostatic pressure on the PBS and the quality factor of the fundamental mode in an L3 cavity. In addition, this study focuses on the dependence of the dielectric function of the slab on both pressure and temperature [35]. First, we estimate the effects of pressure on the PBS for a symmetrical PC slab comprising holes in a hexagonal lattice. Then, after removing three lattice holes (an L3 cavity), the quality factor is calculated at different pressure values. To perform these calculations, we used the guided-mode expansion (GME) method proposed by D. Gerace [36,37]. This work is structured as follows. Section 2 presents the theoretical model with the main GME method equations used. Finally, Sections 3 and 4 denote the numerical results and the corresponding conclusions, respectively.

→ ∇ ×

wave

equation

for

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

the

(1)

where ω and c represent the angular frequency and velocity of light, respectively [4]. In the xy plane, the translational symmetry of 2D-PC is ⎯→ ⎯ determined by the spatial periodicity of the ∊ (→ r ) = ∊ (→ r + R ) di⎯→ ⎯ electric constant with R lattice vectors. By using the GME method, we can calculate the PBS for the PC slab, which is grounded on expanding ⎯→ ⎯ the H (→ r ) field over the basis of the guided modes of the homogeneous dielectric slab (see Fig. 1(b)):

⎯→ ⎯ H (→ r)=

∑∑

⎯ guided ⎛→ →⎞ ⎯→ → → cα ⎜ k + G ⎟ H → k +G ( r ) ⎝ ⎠

(2) G α → where G is the two-dimensional reciprocal lattice vector, and α labels the guided modes [36]. The guided modes of the homogeneous di→ electric slab depend on the g wave vector, and photonic modes in the → PC slab depend on the k Bloch vector, which is confined within the first Brillouin zone. When substituting Eq. (2) in Eq. (1), we get a linear eigenvalue problem, which can be expressed as

∑

ω2 cμ c2

Hμν c ν =

μ

(3)

where Hμν are the matrix elements given by

Hμν =

→

⎯→ ⎯ ∗

→

⎯→ ⎯

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

(4) → In Eq. (4), the inverse of the ∊ ( r ) dielectric function expands within a set of planes waves; then

1 = ∊ (→ r)

∑

→→

ηG eiG · r

(5)

G

with Fourier coefficients ηG given by 2. Theoretical model

ηG =

1 A

→→

∫ ∊−1 (→r ) e−iG · r d2r

(6) → where A is the unit cell area and r = (x , y ) [34]. The dispersion relation of the homogeneous dielectric slab is calculated by solving the implicit equations, which are obtained by applying continuity conditions from tangential components to the transverse-electric (TE polarization) and transverse-magnetic (TM polarization) fields,

Fig. 1(a) displays the PC slab in the xy plane composed of air holes arranged in a hexagonal lattice on a free-standing high-index slab of thickness d. The radius of the holes in the hexagonal lattice is R with a lattice constant a. Fig. 1(b) denotes the homogeneous planar waveguide along the z-axis, with dielectric constants for the lower cladding (∊1), core (∊2 ) and upper cladding (∊3). Based on electromagnetic theory and the linearity of Maxwell’s equations, light propagation within a linear, isotropic, non-magnetic medium and in the absence of sources is

q (χ1 + χ3 ) cos (qd ) + (χ1 χ3 − q2) sin (qd ) = 0

(7)

χ χχ q2 q ⎛ χ1 + 3 ⎞ cos (qd ) + ⎜⎛ 1 3 − 2 ⎟⎞ sin (qd ) = 0 ∊2 ⎝ ∊1 ∊3 ⎠ ∊2 ⎠ ⎝ ∊1 ∊3 ⎜

⎟

ω2

(8)

ω2

ω2

where χ1 = g 2 − ∊1 2 , q = ∊2 2 − g 2 and χ3 = g 2 − ∊3 2 . c c c This paper focuses on symmetrical PC slab surrounded by air (∊1 = ∊3 = 1.0 ). The slab is a semiconductor (GaAs) whose dielectric constant depends on hydrostatic pressure (P) and temperature (T), is given by

∊2 (P , T ) = (∊0 + AeT / T0 P ) e−αP

(9)

with ∊0 = 12.446, A = 0.21125, T0 = 240.7 K, and α = 0.00173 kbar−1 [35]. The hexagonal lattice of air holes only exists inside the slab core, with Fourier coefficients given by 2

(

)

1 2πR 1 ⎧ ⎪ ∊2 + a2 3 1 − ∊2 ηG = ⎨ 4πR 1 − 1 J (RG ) ⎪ a2 3 G ∊2 1 ⎩

Fig. 1. a) PC Slab with thickness d composed of air holes of radius R arranged in a hexagonal lattice with a lattice constant a. b) Homogeneous dielectric slab with lower cladding (∊1), core (∊2 ) and upper cladding (∊3 ) dielectric constants.

where 2

(

)

J1 (RG )

is

the

→ G =0 → G ≠ 0 Bessel

(10) function

of

order

1

and

Results in Physics 16 (2020) 102947

F. Segovia-Chaves and H. Vinck-Posada

Fig. 2. a) Photonic band structure for a PC slab with a hexagonal lattice composed of air holes for TE-like (black line) and TM-like (green line) modes. TE-like photonic band structure for b) P = 30 kbar (blue line) and c) P = 70 kbar (red line). The orange lines represent the light cone, and the PBG is shown in grey. The values used in the simulations are T = 4 K, d = 0.5a , and R = 0.3a . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

G=

2πj 2 a

( ) +(

2π (2m − j ) 2 a 3

)

light within the PBG, the following calculations only focus on TE-like modes, wherein the tuning of the PBS may be achieved for a range of frequencies of interest when increasing the hydrostatic pressure without modifying the photonic structure. Figs. 2(b) and (c) denote the effects on PBS upon increasing hydrostatic pressure to 30 kbar and 70 kbar, respectively. As the pressure increases, we can observe that the PBS shifts towards the higher-frequency region as opposed to the PBS reported at 0 kbar. When the pressure increases to 30 and 70 kbar, the dielectric constants of the semiconductor material are 12.01 and 11.21, respectively. The shift experienced by the PBS with pressure corresponds to the electromagnetic variational theorem because the electric field of the high-frequency modes concentrates a higher fraction of energy in regions where the dielectric constant is low [4]. The periodicity of the 2D-PC is then disturbed by removing holes, thus causing light to confine itself around the cavity. Fig. 3(a) shows a top view of the PC slab after removing three horizontal holes (L3 cavity) at 0 kbar. The L3 cavity studied in this work exhibits a lateral displacement s of the holes at the edges of the defect. Then, we use the supercell technique to calculate the fundamental mode confined within the structure. Fig. 3(b) presents the fundamental mode (ω0 ) within the PBG for a TElike photonic band structure for a lateral shift of 0.15a and at three

with integers j and m [30]. Finally, the

quality factor (Q) for a single cavity is the energy lost per cycle versus the energy stored. Using the GME method, the Q-factor is calculated for a photonic mode with a frequency of ωk by estimating the imaginary part of the frequencies (Im (ωk2) ); thus,

Q=

ωk 2Im (ωk )

where Im (ωk ) =

(11)

Im (ωk2)/2Re (ωk ) .

3. Numerical results and discussion For the numerical calculations included in this work, constant values were considered for temperature (4 K), slab thickness (0.5a ), and hole radius (0.3a ). Fig. 2(a) denotes the PBS in dimensionless frequency units (ωa/2πc ) at 0 kbar (∊2 = 12.65), and we employed up to 109 plane waves and four guided modes. In the PC slab, photonic modes can be classified mainly into even (TE-like) and odd (TM-like), which are confined vertically and located below the light cone (white region). The results reveal the existence of a PBG for the TE-like modes in the region within 0.26 ⩽ ωa/2πc ⩽ 0.34 . Because of the importance of confining 3

Results in Physics 16 (2020) 102947

F. Segovia-Chaves and H. Vinck-Posada

Fig. 3. a) Top view PC slab with L3 cavity at P = 0 kbar. b) PBS at P = 0 kbar (black line), P = 30 kbar (blue line) and P = 70 kbar (red line). The horizontal lines represent the ω0 fundamental modes at each given pressure value. The values used in the simulations are T = 4 K, d = 0.5a, R = 0.3a and s = 0.15a . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 4. a) Intensity E 2 of the L3 cavity for the fundamental mode with s = 0.15a . b) Quality factor of the fundamental mode as a function of the symmetric lateral shift. The values used in the simulations are T = 4 K, d = 0.5a, R = 0.3a and P = 0 kbar.

Fig. 5. Quality factor of the fundamental mode in function of the symmetric lateral shift at P = 0 kbar (black line), P = 30 kbar (blue line) and P = 70 kbar (red line). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

given pressure values (0, 30 and 70 kbar). In this Figure, we may observe that when pressure increases, the position of the fundamental mode shifts towards higher frequencies. That is, at pressures of 0, 30 and 70 kbar, the modes are located at ω0 = ωa/2πc ≃ 0.267, 0.273 and ⎯→ ⎯ 0.282, respectively. Fig. 4(a) show the ∣ E ∣2 intensity at the point of symmetry M (k x a = 0, k y a = 2π / 3 ) of the fundamental mode at P = 0 kbar and s = 0.15a. In this figure, it can be observed that the results match those reported in the scientific literature [38,39], where the fundamental mode exhibits strong field intensity in the center of the cavity, which is very important for practical applications. The dependence of the Q-factor on the lateral shift s of the fundamental mode is calculated by Eq. (11) at a given pressure of 0 kbar. According to the results, when s increases, the Q-factor reaches a maximum value of Q≃

13.57 × 10 4 at s = 0.21a. Fig. 5 displays the effects of hydrostatic pressure on the quality factor of the fundamental mode as a function of the symmetric lateral shift. We found that as pressure increases (30 and 70 kbar), the Q-factor reaches its maximum value at s = 0.21a , which is lower than the results obtained at P = 0 kbar. That is, at P = 30 kbar and 70 kbar, the Q-factor at s = 0.21a is Q≃ 12.89 × 10 4 and Q≃ 12.13 × 10 4 , respectively. Therefore, the cavity is less confined as pressure increases for a symmetric lateral shift of 0.21a. However, for the lateral shift values between 0 and 0.1a , in Fig. 5, we can distinguish a region where the cavity confinement is greater owing to the Q-factor increasing with pressure, as shown in the inset (a). For values exceeding s = 0.18a , the confinement decreases. That is, the Q-factor decreases as 4

Results in Physics 16 (2020) 102947

F. Segovia-Chaves and H. Vinck-Posada

Table 1 Q-factor values for different pressure and lateral shift values.

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Q-factor Shift lateral (s/a)

Pressure (kbar )

0.0

0.1

0.19

0

7.41 × 103

17.66 × 103

89.29 × 10 4

30

7.65 × 103

18.25 × 103

88.26 × 10 4

70

7.84 × 103

18.67 × 103

86.27 × 10 4

pressure increases (see inset (b)). Table 1 reports the Q-factor values for the fundamental mode at three given values of s (0.0a, 0.1a and 0.19a ) and P (0, 30 and 70 kbar). 4. Conclusions This study uses the GME method to calculate the dependence of the photonic band structure on hydrostatic pressure in a regular PC slab composed by air holes arranged in a hexagonal lattice. We found that an increase in pressure causes the photonic band structure to shift towards higher frequencies for TE-like modes. When considering an L3 cavity, we report that when the hydrostatic pressure and symmetric lateral shift increase, the quality factor of the fundamental mode decreases, which also implies a decrease in cavity confinement. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgement 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”. We thank J. P. Vasco for his assistance in the GME method. F.S.-Ch. also acknowledges to Vicerrectoría de Investigación, Universidad Surcolombiana Neiva-Huila. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, athttps://doi.org/10.1016/j.rinp.2020.102947. References [1] Yablanovitch E. Inhibited spontaneous emission in solid state physics and electronics. Phys Rev Lett 1987;58:2059. https://doi.org/10.1103/PhysRevLett. 58.2059. [2] John S. Strong localization of photons in certain disordered dielectric superlattices. Phys Rev Lett 1987;58:2486. https://doi.org/10.1103/PhysRevLett. 58.2486. [3] Aschcroft N, Mermin D, Wei D. Solid State Physics. Cengage Learning Asia; 2016. Revised Edition. [4] Joannopoulos J, Johnson S, Meade R. Photonic crystals: molding the flow of light. Princenton University Press; 2007. [5] Chen Y, Liu W. Design and analysis of multilayered structures with metal-dielectric gratings for reflection resonance and color generation. Opt Lett 2012;37:4–6. https://doi.org/10.1364/OL.37.000004. [6] Qi D, Wang X, Cheng Y, Gong R, Li B. Design and characterization of one-dimensional photonic crystals based on ZnS/Ge for infrared-visible compatible stealth applications. Opt Mater 2016;62:52–6. https://doi.org/10.1016/j.optmat.2016.09. 024. [7] Qi D, Wang X, Cheng Y, Chen F, Liu L, Gong R. Quasi-periodic photonic crystal fabry-perot optical filter based on Si/SiO2 for visible-laser spectral selectivity. J Phys D: Appl Phys 2018;51:225103 https://doi.org/10.1088/1361-6463/aabf83. [8] Qi D, Chen F, Wang X, Luo H, Cheng Y, Niu X, Gong R. Effective strategy for visibleinfrared compatible camouflage: surface graphical one-dimensional photonic

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