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TM band diagram of a waveguide in a honeycomb photonic lattice composed by triangular-shaped rods
Optik - International Journal for Light and Electron Optics 202 (2020) 163595
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
TM band diagram of a waveguide in a honeycomb photonic lattice composed by triangular-shaped rods
T
Francis Segovia-Chavesa,b,*, Herbert Vinck-Posadaa, Erik Navarro-Baróna 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 TM band diagram Honeycomb lattice Waveguide Plane wave expansion method
Using the plane wave expansion and the supercell technique, we calculate the TM band diagram of a waveguide in a honeycomb lattice that composed rods with triangular cross-sections embedded in air. For this study, we use GaAs rods with a dielectric function dependent on pressure and temperature. We report the existence of a defect band within the photonic band gap that presents large confinement for small wave vector values along the line defect. However, for large wave vector values, the defect band approaches the regular crystal bands, thus scattering a greater amount of energy. When the triangles are rotated, greater confinement and smaller scattering of energy is observed than that for unrotated triangles. Additionally, we report that the TM band diagram and the defect band may be tuned to higher frequencies by increasing the applied pressure.
1. Introduction Photonic crystals (PC) are artificial structures with spatial periodic variation between high and low dielectric constant regions [1]. A fundamental property of PCs is the existence of frequency ranges known as photonic band gaps (PBGs), wherein light cannot propagate throughout the crystal's structure [2,3]. Among the most important applications of PBGs are cavities, which are formed when the translational symmetry of the PC breaks. In fact, important confined modes originate within PBGs in defective PCs (viz. point and line defects) in lasers [4,5], waveguides [6,7], and multiplexers [8,9]. A series of works have reported on PBG tuning as well as on photonic band structures (PBSs) through the application of external agents such as temperature and hydrostatic pressure. PBGs can be tuned based on the dielectric function's dependence on pressure and temperature [10]. In PC fibers, experiments show that both pressure and temperature reduce PBG width, change the state of polarization, and cause the transmission spectrum to drift toward longer wavelengths as temperature increases from 61 °C to 80 °C [11,12]. For a pressure sensor composed circular Si rods arranged in a square lattice embedded in air, the resonance wavelength of the sensor shift towards longer wavelengths when pressure increases from 0 to 7 GPa [13]. Segovia et al. [14,15] have studied the effects of pressure and temperature on the PBS in a two-dimensional photonic crystal (2D-PC), composed rods with circular cross sections arranged in square and hexagonal lattices. These authors report that the optical response of the structure is mainly prompted by the applied pressure, further causing the PBS to shift towards higher frequency regions as pressure increases. In addition, when a circular rod is removed from the 2D-PC, a confined mode may be observed within the PBG, which changes position as pressure increases. Recently, Segovia et al. have also reported on rods with triangular cross-sections arranged in a hexagonal lattice [16]. In
⁎
Corresponding author at: Grupo de Física Teórica Universidad Surcolombiana. E-mail address: [email protected] (F. Segovia-Chaves).
https://doi.org/10.1016/j.ijleo.2019.163595 Received 21 July 2019; Accepted 11 October 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 202 (2020) 163595
F. Segovia-Chaves, et al.
this study, they found that when the rotation angle of the triangular dispersers increases, PBG width also increases. In this work, we consider a waveguide in a 2D-PC composed rods with triangular cross-sections arranged in a honeycomb lattice and embedded in air.The rods are composed of GaAs and their dielectric function depends on both the applied hydrostatic pressure and temperature. Using the plane wave expansion (PWE) method and the supercell technique, we calculate the TM band diagram for the waveguide in a honeycomb lattice. The outline of the paper is as follows: Section 2 presents the PWE theoretical model, and Section 3 discusses the results obtained for the TM band diagram when the rotation angle of the triangular rods changes and pressure increases. Finally, Section 4 discusses the conclusions of this research study. 2. Theoretical model Maxwell's equations are the foundation of electromagnetic theory, which is used to study the propagation of light in material → media. In this work, we consider isotropic non-magnetic materials with a dielectric function of ϵ( r ) that changes only with its → → → → position. From the Fourier analysis, the electric (E ( r , t ) ) and magnetic (H ( r , t ) ) fields can be written as a superposition of har→ → → → → → → → monic modes with an angular frequency of ω: E ( r , t ) = E ( r ) e−iωt and H ( r , t ) = H ( r ) e−iωt . In the absence of sources, the → → → → equations that govern the propagation of E ( r ) and H ( r ) are given by
1 → → → → ω2 → ∇ × ∇ × E ( r ) = 2 E (→ r) → c ϵ( r )
(1)
→ ⎛ 1 → → →⎞ ω2 → ∇ × ⎜ → ∇ × H ( r )⎟ = 2 H (→ r) c ⎝ ϵ( r ) ⎠
(2)
where c is the velocity of light [1]. In this paper, we have selected a 2D-PC composed GaAs rods with triangular cross-sections, arranged in a honeycomb lattice embedded in background of air. This 2D-PC has infinite length in the z direction and presents a discrete translational symmetry in the → → → → xy plane. The dielectric function of the rods satisfies ϵ( r ) = ϵ( r + R ) for any R vector that is an integral multiple of a (lattice constant) [17]. In addition, the dielectric function of the GaAs rods depends on the hydrostatic pressure (P) and temperature (T),
ϵGaAs (P , T ) = (ϵ 0 + AeT / T0 P ) e−αP
(3)
where ϵ0 = 12.446, A = 0.21125, T0 = 240.7 K, and α = 0.00173 kbar
−1
[10]. → → → This work focuses on the TM band diagram, which is characterized by the H ( r ) field lying on the xy plane of the k wave vectors, → → → → with the E ( r ) field being perpendicular to said plane. According to Bloch's theorem, the E ( r ) field may be written as an expansion → → of Bloch modes in the z direction with a wave vector of k and a two-dimensional reciprocal lattice vector G [18]:
→ → E (r ) = z
→ → →
ˆ ∑ CGTM ei (G + k )· r
(4) → → are the Fourier coefficients. The inverse of the ϵ−1 ( r ) dielectric function may also be expanded into a Fourier series of G G
where CGTM
1 = r) ϵ(→
→→
∑ ηG eiG · r
(5)
G
where the ηG Fourier coefficients play a key role in determining the TM band diagram.
ηG = (2)
where S by
1 S (2)
→→
∫cell ϵ−1(→r ) e−iG · r d→r
(6)
is the unit cell area. For the GaAs triangular rods in a honeycomb lattice, the ηG coefficients in an air background are given
2 1 ⎧1 + L ⎛ − 1⎞ ⎪ 2a2 ⎝ ϵGaAs (P , T ) ⎠ ηG = ⎨ L2 1 − 1⎞ [I (Gx , Gy ) + I (−Gx , Gy )] ⎪ 2⎛ ⎠ ⎩ 2a ⎝ ϵGaAs (P , T ) ⎜
⎟
⎜
where
L
I (Gx , Gy ) =
2i 3 Gy L
⎟
represents
Exp
(
Gy L 2 3
−
equation is given by [14,19]
Gx L 4
the
) ⎡⎣Exp (
length
i 3 LG y 4
) sinc ( ω
→ G =0 → G ≠0 (7) of
the
Gx L − 3 G y L
) − sinc (
4
) ⎤⎦
of
the
triangles,
and
[16]. For the TM polarization, the eigenvalue
2
∑ ηG′−G [(Gx + k x )2 + (Gy + k y )2] CGTM = ⎛ c ⎞ CGTM ′ G
Gx L 4
sides
⎝ ⎠
(8)
When considering different rotation angles (θ) for the triangular rods in the 2D-PC, the Fourier transform ηG for the rotated x′ and y′ coordinate system is given by 2
Optik - International Journal for Light and Electron Optics 202 (2020) 163595
F. Segovia-Chaves, et al.
Fig. 1. (a) Two-dimensional photonic crystal waveguide with a line defect in a honeycomb lattice of GaAs triangles embedded in an air background. (b) TM band diagram at P = 0 kbar. (c) Electric field intensity |Ez|2 for kxa = 0.52 (top) and kxa = 1.05 (bottom). (d) TM band diagram at P = 70 kbar. The values used in the numerical simulations are L = 0.49a and T = 4 K. (For interpretation of the references to color in the text, the reader is referred to the web version of this article.)
η¯G =
1 S (2)
∫ ϵ−1(→r ′) e−i (G¯ x′+G¯ y′) d→r x
y
(9)
where G¯x are G¯ y are functions of the rotation angle of the rods; given by,
⎛G¯x ⎞ = ⎛ cos θ sin θ ⎞ ⎛Gx ⎞ ⎜ ¯ ⎟ − sin θ cos θ ⎠ ⎝Gy ⎠ ⎝Gy ⎠ ⎝ ⎜
⎟
(10)
3. Numerical results and discussion This section presents the numerical results of the TM band diagram for a waveguide in a honeycomb lattice comprising GaAs rods with triangular cross-sections embedded in a background of air. In our calculations, the waveguide is created by removing a row of triangular rods while holding the temperature (4 K), and the triangle side length (L = 0.49a) values constant. Fig. 1(a) displays the waveguide in a honeycomb lattice, where the dielectric constant of the GaAs is ϵGaAs = 12.66 at a pressure of 0 kbar. At P = 0 kbar, Fig. 1(b) shows a TM band diagram for the 2D-PC in the direction of the component of the normalized wave vector along the line defect (kxa). The gray area represents the continuum of TM modes which may be propagate within the 2D-PC at given x component values for the Bloch wave vector kx. In a regular 2D-PC (without defects), no mode is allowed to propagate within the PBG. However, when a row of triangles is removed from the 2D-PC, a defect band may be observed within the PBG. Fig. 1(c) denotes the calculation of the intensity profiles for the defect band's electric field |Ez|2 for kxa = 0.52 (top) and kxa = 1.05 (bottom). For kxa = 0.52, a confinement of defect mode may be observed with a higher concentration of intensity |Ez|2 around the defect. In addition to increasing kxa, the defect band approaches the regular crystal band, thus losing its confinement properties and becoming a 3
Optik - International Journal for Light and Electron Optics 202 (2020) 163595
F. Segovia-Chaves, et al.
Fig. 2. (a) Two-dimensional photonic crystal waveguide with a line defect in a honeycomb lattice of rotated triangles (b) TM band diagram. (c) Electric field intensity |Ez|2 for kxa = 0.52 (top) and kxa = 1.05 (bottom). The values used in the numerical simulations are L = 0.49a, θ = 30°, P = 0 kbar and T = 4 K.
scattered mode within the 2D-PC, as shown in Fig. 1(c) (bottom). The dependence of the dielectric constant on the applied hydrostatic pressure is given by Eq. (3). At P = 70 kbar, the dielectric constant of the GaAs decreases (ϵGaAs = 11.21), maintaining PBG width constant and causing the TM band diagram to shift into higher frequency regions (blue area) against the TM band diagram at P = 0 kbar (gray area), as shown in Fig. 1(d). In addition, the defect band tunes to higher frequencies within the PBG. The effect originated on the TM band diagram when changing the orientation of the triangular rods is determined by Eqs. (9) and (10). Fig. 2(a) displays the waveguide in a honeycomb lattice when the rods are rotated θ = 30°. Fig. 2(b) denotes an increase in PBG width accompanied by a small shift of the defect band with respect to the triangles that are not rotated (see Fig. 1(b)). Fig. 2(c) depicts the intensity |Ez|2 of the defect band for kxa = 0.52 (top) and kxa = 1.05 (bottom). As can be observed, for small values of kxa, 4
Optik - International Journal for Light and Electron Optics 202 (2020) 163595
F. Segovia-Chaves, et al.
Fig. 3. (a) TM band diagram. (c) Electric field intensity |Ez|2 for kxa = 0.52 (top) and kxa = 1.05 (bottom). The values used in the numerical simulations are L = 0.49a, θ = 30°, P = 30 kbar and T = 4 K.
the intensity |Ez|2 is greater than the intensity reported for the unrotated triangles (see Fig. 1(c)), evidencing a larger confinement around the line defect, as shown in Fig. 2(d) (top)). Fig. 2(d) (bottom) denotes intensity |Ez|2 for kxa = 1.05, where it may be observed that the scattering of |Ez|2 is less than the scattering reported in Fig. 1(c) (bottom). Fig. 3(a) displays the TM band diagram when pressure increases to 70 kbar at a given triangle rotation angle (θ = 30°). In this case, the behavior is similar to the one reported in Fig. 1(d), where the TM band diagram shift to higher frequencies. Fig. 3(b) (top), for kxa = 0.52, shows a guided mode which loses its confinement for kxa = 1.05 whenever the defect band is close to the regular crystal band, as shown in Fig. 3(b) (bottom). 4. Conclusions Through the PWE method and the supercell technique, we were able to calculate the TM band diagram for a waveguide in a honeycomb lattice composed triangular GaAs rods embedded in air. As part of the results, a defect band that propagates throughout the line defect is reported within the PBG. When rotating the triangular rods, the intensity of the electric field (|Ez|2) is greater for small component values of the normalized wave vector along the line defect. In addition, we report that the TM band diagram and the defect band are tuned to higher frequencies when increasing the hydrostatic pressure while holding PBG width constant. 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] J. Joannopoulos, S. Johnson, R. Meade, Photonic Crystals: Molding the Flow of Light, Princenton University Press, 2007. [2] E. Yablanovitch, Inhibited spontaneous emission in solid state physics and electronics, Phys. Rev. Lett. 58 (1987) 2059, https://doi.org/10.1103/PhysRevLett. 58.2059. [3] E. Yablanovitch, Photonic crystals: semiconductors of light, Sci. Am. 285 (2001) 46–55, https://doi.org/10.1038/scientificamerican1201-46. [4] M. Loncar, T. Yoshie, A. Scherer, Low-threshold photonic crystal laser, Appl. Phys. Lett. 81 (2002) 2680. [5] M. Imada, S. Noda, A. Chutinan, T. Tokuda, Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure, Appl. Phys. Lett. 75 (1999) 316. [6] A. Christ, S. Tikhodeev, N. Gippius, J. Kuhl, H. Giessen, Waveguide-plasmon polaritons: strong coupling of photonic and electronic resonances in a metallic photonic crystal slab, Phys. Rev. Lett. 91 (2003) 183901. [7] J. Foresi, P. Villeneuve, J. Ferrera, E. Thoen, G. Steinmeyer, S. Fan, J. Joannopoulos, L. Kimerling, H. Smith, E. Ippen, Photonic-bandgap microcavities in optical waveguides, Nature 390 (1997) 143–145.
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Optik - International Journal for Light and Electron Optics 202 (2020) 163595
F. Segovia-Chaves, et al.
[8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
J. Smajic, C. Hafner, D. Erni, On the design of photonic crystal multiplexers, Optics Express 6 (2003) 566–571. V. Liu, Y. Jiao, D. Miller, S. Fan, Design methodology for compact photonic-crystal-based wavelength division multiplexers, Optics Lett. 36 (2011) 591–593. G. Samara, Temperature and pressure dependences of the dielectric constants of semiconductors, Phys. Rev. B 27 (1983) 3494. M. Tefelska, M. Chychowski, A.R.D. Czapla, et al., Hydrostatic pressure effects in photonic liquid crystal fibers, Proc. SPIE 7120 (2008) 712008. T. Wolinski, A. Czapla, M. Tefelska, A. Domanski, J. Wojcik, E. Nowinowski-Kruszelnicki, R. Dabrowski, Photonic liquid crystal fibers for sensing applications, IEEE Trans. Instrum. Meas. 57 (2008) 1796–1802. K. Vijaya, S. Robinson, Two-dimensional photonic crystal based sensor for pressure sensing, Photonic Sens. 4 (2014) 248–253. F. Segovia-Chaves, H. Vinck-Posada, Dependence of photonic defect modes on hydrostatic pressure in a 2D hexagonal lattice, Physica E 104 (2018) 49–57. F. Segovia-Chaves, H. Vinck-Posada, Effects of hydrostatic pressure on the band structure in two-dimensional semiconductor square photonic lattice with defect, Physica B 545 (2018) 203–209. F. Segovia-Chaves, H. Vinck-Posada, E. Navarro-Baron, Dependence on hydrostatic pressure in two-dimensional triangular photonic crystals, Physica B 557 (2019) 88–94. M. Skorobogatiy, J. Yang, Fundamentals of Photonic Crystal Guiding, Cambridge University Press, 2009. N. Aschcroft, D. Mermin, D. Wei, Solid State Physics, Revised Edition, Cengage Learning Asia, 2016. K. Sakoda, Optical Properties of Photonic Crystals vol. 80, Springer Science and Business Media, 2004.
6
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
TM band diagram of a waveguide in a honeycomb photonic lattice composed by triangular-shaped rods
T
Francis Segovia-Chavesa,b,*, Herbert Vinck-Posadaa, Erik Navarro-Baróna 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 TM band diagram Honeycomb lattice Waveguide Plane wave expansion method
Using the plane wave expansion and the supercell technique, we calculate the TM band diagram of a waveguide in a honeycomb lattice that composed rods with triangular cross-sections embedded in air. For this study, we use GaAs rods with a dielectric function dependent on pressure and temperature. We report the existence of a defect band within the photonic band gap that presents large confinement for small wave vector values along the line defect. However, for large wave vector values, the defect band approaches the regular crystal bands, thus scattering a greater amount of energy. When the triangles are rotated, greater confinement and smaller scattering of energy is observed than that for unrotated triangles. Additionally, we report that the TM band diagram and the defect band may be tuned to higher frequencies by increasing the applied pressure.
1. Introduction Photonic crystals (PC) are artificial structures with spatial periodic variation between high and low dielectric constant regions [1]. A fundamental property of PCs is the existence of frequency ranges known as photonic band gaps (PBGs), wherein light cannot propagate throughout the crystal's structure [2,3]. Among the most important applications of PBGs are cavities, which are formed when the translational symmetry of the PC breaks. In fact, important confined modes originate within PBGs in defective PCs (viz. point and line defects) in lasers [4,5], waveguides [6,7], and multiplexers [8,9]. A series of works have reported on PBG tuning as well as on photonic band structures (PBSs) through the application of external agents such as temperature and hydrostatic pressure. PBGs can be tuned based on the dielectric function's dependence on pressure and temperature [10]. In PC fibers, experiments show that both pressure and temperature reduce PBG width, change the state of polarization, and cause the transmission spectrum to drift toward longer wavelengths as temperature increases from 61 °C to 80 °C [11,12]. For a pressure sensor composed circular Si rods arranged in a square lattice embedded in air, the resonance wavelength of the sensor shift towards longer wavelengths when pressure increases from 0 to 7 GPa [13]. Segovia et al. [14,15] have studied the effects of pressure and temperature on the PBS in a two-dimensional photonic crystal (2D-PC), composed rods with circular cross sections arranged in square and hexagonal lattices. These authors report that the optical response of the structure is mainly prompted by the applied pressure, further causing the PBS to shift towards higher frequency regions as pressure increases. In addition, when a circular rod is removed from the 2D-PC, a confined mode may be observed within the PBG, which changes position as pressure increases. Recently, Segovia et al. have also reported on rods with triangular cross-sections arranged in a hexagonal lattice [16]. In
⁎
Corresponding author at: Grupo de Física Teórica Universidad Surcolombiana. E-mail address: [email protected] (F. Segovia-Chaves).
https://doi.org/10.1016/j.ijleo.2019.163595 Received 21 July 2019; Accepted 11 October 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 202 (2020) 163595
F. Segovia-Chaves, et al.
this study, they found that when the rotation angle of the triangular dispersers increases, PBG width also increases. In this work, we consider a waveguide in a 2D-PC composed rods with triangular cross-sections arranged in a honeycomb lattice and embedded in air.The rods are composed of GaAs and their dielectric function depends on both the applied hydrostatic pressure and temperature. Using the plane wave expansion (PWE) method and the supercell technique, we calculate the TM band diagram for the waveguide in a honeycomb lattice. The outline of the paper is as follows: Section 2 presents the PWE theoretical model, and Section 3 discusses the results obtained for the TM band diagram when the rotation angle of the triangular rods changes and pressure increases. Finally, Section 4 discusses the conclusions of this research study. 2. Theoretical model Maxwell's equations are the foundation of electromagnetic theory, which is used to study the propagation of light in material → media. In this work, we consider isotropic non-magnetic materials with a dielectric function of ϵ( r ) that changes only with its → → → → position. From the Fourier analysis, the electric (E ( r , t ) ) and magnetic (H ( r , t ) ) fields can be written as a superposition of har→ → → → → → → → monic modes with an angular frequency of ω: E ( r , t ) = E ( r ) e−iωt and H ( r , t ) = H ( r ) e−iωt . In the absence of sources, the → → → → equations that govern the propagation of E ( r ) and H ( r ) are given by
1 → → → → ω2 → ∇ × ∇ × E ( r ) = 2 E (→ r) → c ϵ( r )
(1)
→ ⎛ 1 → → →⎞ ω2 → ∇ × ⎜ → ∇ × H ( r )⎟ = 2 H (→ r) c ⎝ ϵ( r ) ⎠
(2)
where c is the velocity of light [1]. In this paper, we have selected a 2D-PC composed GaAs rods with triangular cross-sections, arranged in a honeycomb lattice embedded in background of air. This 2D-PC has infinite length in the z direction and presents a discrete translational symmetry in the → → → → xy plane. The dielectric function of the rods satisfies ϵ( r ) = ϵ( r + R ) for any R vector that is an integral multiple of a (lattice constant) [17]. In addition, the dielectric function of the GaAs rods depends on the hydrostatic pressure (P) and temperature (T),
ϵGaAs (P , T ) = (ϵ 0 + AeT / T0 P ) e−αP
(3)
where ϵ0 = 12.446, A = 0.21125, T0 = 240.7 K, and α = 0.00173 kbar
−1
[10]. → → → This work focuses on the TM band diagram, which is characterized by the H ( r ) field lying on the xy plane of the k wave vectors, → → → → with the E ( r ) field being perpendicular to said plane. According to Bloch's theorem, the E ( r ) field may be written as an expansion → → of Bloch modes in the z direction with a wave vector of k and a two-dimensional reciprocal lattice vector G [18]:
→ → E (r ) = z
→ → →
ˆ ∑ CGTM ei (G + k )· r
(4) → → are the Fourier coefficients. The inverse of the ϵ−1 ( r ) dielectric function may also be expanded into a Fourier series of G G
where CGTM
1 = r) ϵ(→
→→
∑ ηG eiG · r
(5)
G
where the ηG Fourier coefficients play a key role in determining the TM band diagram.
ηG = (2)
where S by
1 S (2)
→→
∫cell ϵ−1(→r ) e−iG · r d→r
(6)
is the unit cell area. For the GaAs triangular rods in a honeycomb lattice, the ηG coefficients in an air background are given
2 1 ⎧1 + L ⎛ − 1⎞ ⎪ 2a2 ⎝ ϵGaAs (P , T ) ⎠ ηG = ⎨ L2 1 − 1⎞ [I (Gx , Gy ) + I (−Gx , Gy )] ⎪ 2⎛ ⎠ ⎩ 2a ⎝ ϵGaAs (P , T ) ⎜
⎟
⎜
where
L
I (Gx , Gy ) =
2i 3 Gy L
⎟
represents
Exp
(
Gy L 2 3
−
equation is given by [14,19]
Gx L 4
the
) ⎡⎣Exp (
length
i 3 LG y 4
) sinc ( ω
→ G =0 → G ≠0 (7) of
the
Gx L − 3 G y L
) − sinc (
4
) ⎤⎦
of
the
triangles,
and
[16]. For the TM polarization, the eigenvalue
2
∑ ηG′−G [(Gx + k x )2 + (Gy + k y )2] CGTM = ⎛ c ⎞ CGTM ′ G
Gx L 4
sides
⎝ ⎠
(8)
When considering different rotation angles (θ) for the triangular rods in the 2D-PC, the Fourier transform ηG for the rotated x′ and y′ coordinate system is given by 2
Optik - International Journal for Light and Electron Optics 202 (2020) 163595
F. Segovia-Chaves, et al.
Fig. 1. (a) Two-dimensional photonic crystal waveguide with a line defect in a honeycomb lattice of GaAs triangles embedded in an air background. (b) TM band diagram at P = 0 kbar. (c) Electric field intensity |Ez|2 for kxa = 0.52 (top) and kxa = 1.05 (bottom). (d) TM band diagram at P = 70 kbar. The values used in the numerical simulations are L = 0.49a and T = 4 K. (For interpretation of the references to color in the text, the reader is referred to the web version of this article.)
η¯G =
1 S (2)
∫ ϵ−1(→r ′) e−i (G¯ x′+G¯ y′) d→r x
y
(9)
where G¯x are G¯ y are functions of the rotation angle of the rods; given by,
⎛G¯x ⎞ = ⎛ cos θ sin θ ⎞ ⎛Gx ⎞ ⎜ ¯ ⎟ − sin θ cos θ ⎠ ⎝Gy ⎠ ⎝Gy ⎠ ⎝ ⎜
⎟
(10)
3. Numerical results and discussion This section presents the numerical results of the TM band diagram for a waveguide in a honeycomb lattice comprising GaAs rods with triangular cross-sections embedded in a background of air. In our calculations, the waveguide is created by removing a row of triangular rods while holding the temperature (4 K), and the triangle side length (L = 0.49a) values constant. Fig. 1(a) displays the waveguide in a honeycomb lattice, where the dielectric constant of the GaAs is ϵGaAs = 12.66 at a pressure of 0 kbar. At P = 0 kbar, Fig. 1(b) shows a TM band diagram for the 2D-PC in the direction of the component of the normalized wave vector along the line defect (kxa). The gray area represents the continuum of TM modes which may be propagate within the 2D-PC at given x component values for the Bloch wave vector kx. In a regular 2D-PC (without defects), no mode is allowed to propagate within the PBG. However, when a row of triangles is removed from the 2D-PC, a defect band may be observed within the PBG. Fig. 1(c) denotes the calculation of the intensity profiles for the defect band's electric field |Ez|2 for kxa = 0.52 (top) and kxa = 1.05 (bottom). For kxa = 0.52, a confinement of defect mode may be observed with a higher concentration of intensity |Ez|2 around the defect. In addition to increasing kxa, the defect band approaches the regular crystal band, thus losing its confinement properties and becoming a 3
Optik - International Journal for Light and Electron Optics 202 (2020) 163595
F. Segovia-Chaves, et al.
Fig. 2. (a) Two-dimensional photonic crystal waveguide with a line defect in a honeycomb lattice of rotated triangles (b) TM band diagram. (c) Electric field intensity |Ez|2 for kxa = 0.52 (top) and kxa = 1.05 (bottom). The values used in the numerical simulations are L = 0.49a, θ = 30°, P = 0 kbar and T = 4 K.
scattered mode within the 2D-PC, as shown in Fig. 1(c) (bottom). The dependence of the dielectric constant on the applied hydrostatic pressure is given by Eq. (3). At P = 70 kbar, the dielectric constant of the GaAs decreases (ϵGaAs = 11.21), maintaining PBG width constant and causing the TM band diagram to shift into higher frequency regions (blue area) against the TM band diagram at P = 0 kbar (gray area), as shown in Fig. 1(d). In addition, the defect band tunes to higher frequencies within the PBG. The effect originated on the TM band diagram when changing the orientation of the triangular rods is determined by Eqs. (9) and (10). Fig. 2(a) displays the waveguide in a honeycomb lattice when the rods are rotated θ = 30°. Fig. 2(b) denotes an increase in PBG width accompanied by a small shift of the defect band with respect to the triangles that are not rotated (see Fig. 1(b)). Fig. 2(c) depicts the intensity |Ez|2 of the defect band for kxa = 0.52 (top) and kxa = 1.05 (bottom). As can be observed, for small values of kxa, 4
Optik - International Journal for Light and Electron Optics 202 (2020) 163595
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Fig. 3. (a) TM band diagram. (c) Electric field intensity |Ez|2 for kxa = 0.52 (top) and kxa = 1.05 (bottom). The values used in the numerical simulations are L = 0.49a, θ = 30°, P = 30 kbar and T = 4 K.
the intensity |Ez|2 is greater than the intensity reported for the unrotated triangles (see Fig. 1(c)), evidencing a larger confinement around the line defect, as shown in Fig. 2(d) (top)). Fig. 2(d) (bottom) denotes intensity |Ez|2 for kxa = 1.05, where it may be observed that the scattering of |Ez|2 is less than the scattering reported in Fig. 1(c) (bottom). Fig. 3(a) displays the TM band diagram when pressure increases to 70 kbar at a given triangle rotation angle (θ = 30°). In this case, the behavior is similar to the one reported in Fig. 1(d), where the TM band diagram shift to higher frequencies. Fig. 3(b) (top), for kxa = 0.52, shows a guided mode which loses its confinement for kxa = 1.05 whenever the defect band is close to the regular crystal band, as shown in Fig. 3(b) (bottom). 4. Conclusions Through the PWE method and the supercell technique, we were able to calculate the TM band diagram for a waveguide in a honeycomb lattice composed triangular GaAs rods embedded in air. As part of the results, a defect band that propagates throughout the line defect is reported within the PBG. When rotating the triangular rods, the intensity of the electric field (|Ez|2) is greater for small component values of the normalized wave vector along the line defect. In addition, we report that the TM band diagram and the defect band are tuned to higher frequencies when increasing the hydrostatic pressure while holding PBG width constant. 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] J. Joannopoulos, S. Johnson, R. Meade, Photonic Crystals: Molding the Flow of Light, Princenton University Press, 2007. [2] E. Yablanovitch, Inhibited spontaneous emission in solid state physics and electronics, Phys. Rev. Lett. 58 (1987) 2059, https://doi.org/10.1103/PhysRevLett. 58.2059. [3] E. Yablanovitch, Photonic crystals: semiconductors of light, Sci. Am. 285 (2001) 46–55, https://doi.org/10.1038/scientificamerican1201-46. [4] M. Loncar, T. Yoshie, A. Scherer, Low-threshold photonic crystal laser, Appl. Phys. Lett. 81 (2002) 2680. [5] M. Imada, S. Noda, A. Chutinan, T. Tokuda, Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure, Appl. Phys. Lett. 75 (1999) 316. [6] A. Christ, S. Tikhodeev, N. Gippius, J. Kuhl, H. Giessen, Waveguide-plasmon polaritons: strong coupling of photonic and electronic resonances in a metallic photonic crystal slab, Phys. Rev. Lett. 91 (2003) 183901. [7] J. Foresi, P. Villeneuve, J. Ferrera, E. Thoen, G. Steinmeyer, S. Fan, J. Joannopoulos, L. Kimerling, H. Smith, E. Ippen, Photonic-bandgap microcavities in optical waveguides, Nature 390 (1997) 143–145.
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