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Effects of hydrostatic pressure, temperature and angle of incidence on the transmittance spectrum of TE mode in a 1D semiconductor photonic crystal
Optik 161 (2018) 64–69
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Original research article
Effects of hydrostatic pressure, temperature and angle of incidence on the transmittance spectrum of TE mode in a 1D semiconductor photonic crystal Francis Segovia-Chaves a,b,∗ , Herbert Vinck-Posada a a b
Departamento de Física, Universidad Nacional de Colombia, AA 055051 Bogotá, Colombia Departamento de Ciencias Naturales, Universidad Surcolombiana, AA 385 Neiva, Colombia
a r t i c l e
i n f o
Article history: Received 1 January 2018 Accepted 24 January 2018 Keywords: Photonic crystal Hydrostatic pressure Angle of incidence Transfer-matrix method
a b s t r a c t In this paper, we study the effects of hydrostatic pressure, temperature and angle of incidence on the transmittance spectrum of TE mode of a one-dimensional photonic crystal, using the transfer-matrix method. We consider that the crystal is formed by alternated layers of air and GaAs, with the dielectric constant of GaAs as a function of the temperature and pressure applied. We found that the spectrums dependence on temperature is negligible, the optical response of the system is due mainly to the pressure applied. When increasing the hydrostatic pressure, the spectrum shifts to a short-wavelength, which is caused by the decrease of the dielectric constant of GaAs. These results agree with the electromagnetic variational theorem. We also found if the angle of incidence of the modes increases, the band width increases compared to the normal incidence modes. © 2018 Elsevier GmbH. All rights reserved.
1. Introduction Light propagating on periodic one-dimensional structures dates back to 1887 when Lord Rayleigh discovered that when the angle of incidence varies [1], it is possible to obtain regions where the light is totally reflected. A century later, these regions were called photonic band gap (PBG). In 1987, the work of Eli Yablonovitch [2], who worked on inhibiting the spontaneous emission of electrons in semiconductors, and Sajeev Jhon [3], who studied the effects of localization of light in disordered systems, proposed making periodic structures named photonic crystals (PC), thus appearing the concept of photonic band gap in two and three dimensions. The principle of functionality in PCs is the periodic spatial variation of the dielectric constant, equivalent to a periodic potential in an atomic crystal [4]. The allowed and forbidden (PBG) states in PC are due to the Bragg diffraction, generated by the scatterers that form the crystal [5]. In PBGs no light mode with a frequency within the frequency range of PBG can propagate, regardless its polarization and angle of incidence. PCs are formally described by Maxwell’s electromagnetic theory [6]. It is possible to find the PBGs and the states of the electromagnetic field by solving the equations. To solve Maxwell’s equations there are the following methods: plane waves, finite-difference frequency and time domain, transfer-matrix, scattering-matrix, among others [7–10]. The possibility of tuning the PBG by modifying the optical response of the materials that form the PC using external parameters such as electrical and magnetic fields [11,12], temperature [13,14] and hydrostatic pressure [15], allows it to be
∗ Corresponding author at: Grupo de óptica e información cuántica, Universidad Nacional. Departamento de Ciencias Naturales, Universidad Surcolombiana. E-mail address: [email protected] (F. Segovia-Chaves). https://doi.org/10.1016/j.ijleo.2018.01.087 0030-4026/© 2018 Elsevier GmbH. All rights reserved.
F. Segovia-Chaves, H. Vinck-Posada / Optik 161 (2018) 64–69
65
Fig. 1. Photonic crystal one-dimensional Air/(HL)N H/Air.
implemented in potential applications of modern photonics such as optical switches and tunable filters, very important in wavelength-division multiplexing [16–18]. In this work, we studied the effects of the hydrostatic pressure applied, temperature and angle of incidence on the transmittance spectrum for the TE mode of a one-dimensional photonic crystal (1DPC), within the Maxwell framework and using the transfer-matrix method (TMM). We considered a 1DPC consisting of alternate layers of air and GaAs, in which the dielectric properties of the semiconductor depend on pressure and temperature. The work is organized as follows: Section 2, we present the theoretical framework. Section 3, we present the numerical results and discussion related to the calculation of the transmittance spectrum of the 1DPC for different temperatures, hydrostatic pressures, and angles of incidence. Finally, conclusions are presented in Section 4. 2. Theoretical model In Fig. 1, we present a finite 1DPC surrounded by air and consisting of alternate layers of materials, with high H and low L dielectric constants, whose thickness are dH and dL respectively. The wave vector of the incident medium is k 0 and the angle of incidence is , the PC has a homogeneous pattern in the xy plane and a periodicity in z direction. The number of periods of HL layers is given by N. We consider a linearly polarized electromagnetic wave propagating through the (x, z) plane with a wave vector qx along the x-axis. For the TE modes, which will be our focus, the electric field is given by
E j (x, z) = ey Aj eikj,z z + Bj e−ikj,z z e−iqx x where kj,z =
ω/c
2
(1)
j − q2x and j , is the z component of the wave vector and the dielectric constant in the jth layer,
respectively. The transverse component of the wave vector is qx = k0 sin, Aj and Bj values are calculated by the continuity conditions of the tangential electric and magnetic field components. In the TMM, each 1DPC layer may be represented by a matrix [19], Mj = Dj Pj Dj−1
j = H, L
(2)
In Eq. (2) the propagation matrix is
Pj =
eiϕj
0
0
e−iϕj
(3)
with phase ϕj given by: ϕj = kj,z dj =
2dj
In Eq. (4) dj and nj = TE mode is given by
Dj =
(4)
nj cos j
j are the thickness and refractive index in the jth layer, respectively. The dynamic matrix for the
1
1
nj cos j
−nj cos j
(5)
The total transfer-matrix for the 1DPC Air/(HL)N H/Air is defined as
M=
M11
M12
M21
M22
= D0−1 (MH ML )N MH D0
(6)
66
F. Segovia-Chaves, H. Vinck-Posada / Optik 161 (2018) 64–69
Fig. 2. GaAs dielectric constant as a function of pressure and temperature. Table 1 Values of the dielectric constant of GaAs as a function of temperature for a pressure P = 40 kbar. Temperature (K)
Dielectric constant
0 50 200
11.80 11.85 12.02
Table 2 Edges L , R and band width for an angle of incidence = 0◦ and P = 40 kbar. Temperature (K)
Band edge
0 50 200
Band width)
L
R
723.78 723.48 722.48
1574.03 1575.43 1580.18
850.25 851.95 857.71
with D0 the dynamic matrix of air. Transmittance I is calculated with matrix elements M11 of Eq. (6),
1 2
M
I=
(7)
11
In this study, we will assume that layer H is GaAs and layer L is air; the effects on the transmittance spectrum due to hydrostatic pressure and temperature, are determined by the dielectric constant of GaAs [20], −3
GaAs (P, T ) = 12.74e−1.73×10
P 9.4×10−5 (T −75.6)
e
for T ≤ 200 K
(8)
with hydrostatic pressure P in kbar and temperature T in Kelvin, in Fig. 2, we present that dependence. The thickness of GaAs layers depends on hydrostatic pressure and is given by [21,22] d(P) = d0 [1 − (S11 + 2S12 )P]
(9) = 1.16 × 10−3
kbar−1
where d0 is the initial thickness under zero pressure. The elastic constants for GaAs are S11 and S12 =−3.7 × 10−4 kbar−1 . We will choose a quarter of wavelength as the initial thickness of the 1DPC layers, with a design wavelength of 0 of 1000 nm. 3. Numerical results and discussion The dielectric constant of GaAs is calculated by Eq. (8); as the temperature increases the value of GaAs also increases, as shown in Table 1. The numerical results of the transmittance spectrum of a 1DPC for the TE mode with normal incidence = 0◦ , for a pressure P = 40 kbar and a period N = 10 is shown in Fig. 3(a) and (b). The width of the PBG is determined by Bragg’s reflection theory. The left L and right R band edges are given by [23], (nH dH + nL dL ) , cos−1 (−(nH − nL /nH + nL ))
(nH dH + nL dL ) (10) cos−1 (nH − nL /nH + nL ) √ with nH and nL , the refractive index of the semiconductor ( GaAs ) and air, respectively. The thickness of the GaAs layer is dH calculated by Eq. (9) and dL the thickness of the air layer. In Table 2, we present the edges values and with of the 1DPC band, calculated with Eq. (10) for different temperature values. The width of the PBG band increases with temperature, as shown in Fig. 3(a). In Fig. 3(b) we present the transmittance spectrum of TE mode for = 0◦ ; we increased the temperature from T = 0 K to T = 200 K, for a pressure of P = 40 kbar. When increasing the temperature, the spectrum shifted to a longer L =
R =
F. Segovia-Chaves, H. Vinck-Posada / Optik 161 (2018) 64–69
67
Fig. 3. (a) Contour plot of transmittance as function of the wavelength and temperature for the TE mode. (b) Transmittance spectrum for the TE mode for temperatures T = 0 K (black line) and T = 200 K (red line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)
Fig. 4. Contour plot of transmittance as a function of the wavelength and temperature for the TE mode with P = 40 kbar (a) = 45◦ and (b) = 65◦ . Table 3 Values of the dielectric constant of GaAs as a function of pressure for a temperature T = 200 K. Pressure (kbar)
Dielectric constant
0 10 40
12.88 12.66 12.02
wavelength. The results agree with the electromagnetic variational theorem; low frequency modes tend to concentrate a greater fraction of energy in the regions where the dielectric constant is greater. However, if we increase the temperature (at a constant pressure) and the angle of incidence, the width of the PBG band shifts to shorter wavelength regions, increasing the width of the PBG compared to that of the normal incidence, as shown in Fig. 4(a) and (b). In Fig. 5(a) and (b), we present the transmittance spectrum for the TE mode with an angle of incidence of = 45◦ and = 65◦ , respectively. When increasing the temperature from T = 0 K to T = 200 K, for a given pressure of P = 40 kbar, the spectrum shifts to long wavelength as in the case of normal incidence. Finally, we analyze the effects of hydrostatic pressure on the transmittance spectrum of 1DPC. We choose a temperature of T = 200 K and a period of N = 10. When increasing the pressure, the dielectric constant of GaAs decreases, as shown in Table 3. In Fig. 6(a), we present the numerical results of the transmittance spectrum of TE mode with normal incidence = 0◦ as a function of the wavelength and hydrostatic pressure. When increasing the pressure, the edges and band width increase too, this data is shown in Table 4. In Fig. 6(b), we observe a noticeable transmittance spectrum shifting to short-wavelength when increasing the pressure from P = 0 kbar to P = 40 kbar, due to the dielectric constant decrease. These results agree with
68
F. Segovia-Chaves, H. Vinck-Posada / Optik 161 (2018) 64–69
Fig. 5. Transmittance spectrum for the TE mode with P = 40 kbar and temperature T = 0 K (black line) and T = 200 K (red line) for (a) = 45◦ and (b) = 65◦ . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)
Fig. 6. (a) Contour plot of transmittance as a function of the wavelength and pressure for the TE mode. (b) Transmittance spectrum for the TE mode for pressure P = 0 kbar (black line) and P = 40 kbar (red line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.) Table 4 Edges L , R and band width for an angle of incidence = 0◦ and T = 200 K. Pressure (kbar)
0 10 40
Band edge (nm)
Band width (nm)
L
R
723.79 723.46 722.48
1617.08 1607.79 1580.18
893.29 884.33 857.71
the electromagnetic variational theorem. Compared to the results in Fig. 3(b), we observed that the optical response of 1DPC is mainly due to the hydrostatic pressure applied, instead of the temperature. In Fig. 7(a) and (b) we show the numerical results of the transmittance spectrum as a function of the pressure and wavelength for = 45◦ and = 65◦ , respectively. The angle and the hydrostatic pressure increase the PBG width, which shifts to short-wavelength, due to the decrease of the semiconductor dielectric constant. 4. Conclusions With the transfer-matrix method, we studied the dependence of the transmittance spectrum of a one-dimensional photonic crystal consisting of alternate layers of air and GaAs on temperature, hydrostatic pressure and angle of incidence. We found out that temperature does not affect significantly the shifting of the band; the effects are mainly due to the variation of the thickness and dielectric constant of GaAs with hydrostatic pressure. When the pressure increases, the value of the dielectric constant of GaAs decreases and shifts to short-wavelength. Considering the propagation outside the periodicity axis, the width of the PBG increases, compared to that of the normal incidence.
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Fig. 7. Contour plot of transmittance as a function of the wavelength and pressure for the TE mode with T = 200 K (a) = 45◦ and (b) = 65◦ .
Acknowledgements 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”, “Exploración y modelación de la iridiscencia en especies Colombianas. Código 110156933525, CT 026-2013, HERMES 17432” and “Modelación teóricoestadística de la reflectividad en especies colombianas, HERMES 35767”. F.S.-Ch. also acknowledges to Vicerrectoría de Investigación, Universidad Surcolombiana Neiva-Huila. References [1] L. Rayleigh, On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure, Philos. Mag. Sci. 24 (1887) 145–159. [2] E. Yablanovitch, Inhibited spontaneous emission in solid state physics and electronics, Phys. Rev. Lett. 58 (1987) 2059. [3] S. John, Localization of light, Phys. Rev. Lett. 58 (1987) 2486. [4] R.H. Lipson, C. Lu, Photonic crystals: a unique partnership between light and matter, Eur. J. Phys. 30 (2009) S33. [5] J. Joannopoulos, S. Johnson, J. Winn, R. Meade, Photonic Crystals: Molding the Flow of Light, Princeton University Press, 2007. [6] M. Skorobogatiy, J. Yang, Fundamentals of Photonic Crystal Guiding, Cambridge University Press, 2009. [7] H. Sözüer, J. Haus, R. Inguva, Photonic bands: convergence problems with the plane-wave method, Phys. Rev. B 45 (1992) 13962. [8] L. Li, Use of Fourier series in the analysis of discontinuous periodic structures, J. Opt. Soc. Am. A 13 (1996) 1870. [9] A. Taflove, S. Hagness, Computational Electrodynamics: The Finite-difference Time-domain Method, Atech House Publishers, 2005. [10] A. Yariv, P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation, Wiley-Interscience, 2002. [11] C. Xu, X. Hu, Y. Li, X. Liu, R. Fu, J. Zi, Semiconductor-based tunable photonic crystals by means of an external magnetic field, Phys. Rev. B 68 (2003) 193201. [12] K. Busch, S. John, Liquid-crystal photonic-band-gap materials: the tunable electromagnetic vacuum, Phys. Rev. Lett. 83 (1999) 967. [13] F. Segovia-Chaves, H. Vinck-Posada, Temperature dependence of defect mode in band structures of the one-dimensional photonic crystal, Optik 154 (2018) 467. [14] V. Kumar, B. Suthar, A. Kumar, K. Singh, A. Bhargava, The effect of temperature and angle of incidence on photonic band gap in a dispersive Si-based one dimensional photonic crystal, Phys. B: Condens. Matter 416 (2013) 106. [15] N. Porras-Montenegro, C. Duque, Temperature and hydrostatic pressure effects on the photonic band structure of a 2D honeycomb lattice, Phys. E: Low-dimens. Syst. Nanostruct. 42 (2010) 1865. [16] H. Nemec, L. Duvillaret, F. Garet, P. Kuzel, P. Xavier, J. Richard, D. Rauly, Thermally tunable filter for terahertz range based on a one-dimensional photonic crystal with a defect, J. Appl. Phys. 96 (2004) 4072. [17] S. Eliahou-Niv, R. Dahan, G. Golan, Design and analysis of a novel tunable optical filter, Microelectron. J. 37 (2006) 302. [18] Z. Luo, H. Wen, H. Guo, M. Yang, A time-and wavelength-division multiplexing sensor network with ultra-weak fiber Bragg gratings, Opt. Express 21 (2013) 22799. [19] P. Yeh, Optical Waves in Layered Media, Wiley-Interscience, 2005. [20] I. Erdogan, O. Akankan, H. Akbas, Simultaneous effects of temperature, hydrostatic pressure and electric field on the self-polarization and electric field polarization in a GaAs/Ga0.7 Al0.3 As spherical quantum dot with a donor impurity, Superlattices Microstruct. 59 (2013) 13. [21] A. Elabsy, Hydrostatic pressure dependence of binding energies for donors in quantum well heterostructures, Phys. Scr. 48 (1993) 376. [22] G. Samara, Temperature and pressure dependences of the dielectric constants of semiconductors, Phys. Rev. B 27 (1983) 3494. [23] S. Orfanidis, Electromagnetic waves and antennas, NJ Rutgers University, New Brunswick, 2002.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Original research article
Effects of hydrostatic pressure, temperature and angle of incidence on the transmittance spectrum of TE mode in a 1D semiconductor photonic crystal Francis Segovia-Chaves a,b,∗ , Herbert Vinck-Posada a a b
Departamento de Física, Universidad Nacional de Colombia, AA 055051 Bogotá, Colombia Departamento de Ciencias Naturales, Universidad Surcolombiana, AA 385 Neiva, Colombia
a r t i c l e
i n f o
Article history: Received 1 January 2018 Accepted 24 January 2018 Keywords: Photonic crystal Hydrostatic pressure Angle of incidence Transfer-matrix method
a b s t r a c t In this paper, we study the effects of hydrostatic pressure, temperature and angle of incidence on the transmittance spectrum of TE mode of a one-dimensional photonic crystal, using the transfer-matrix method. We consider that the crystal is formed by alternated layers of air and GaAs, with the dielectric constant of GaAs as a function of the temperature and pressure applied. We found that the spectrums dependence on temperature is negligible, the optical response of the system is due mainly to the pressure applied. When increasing the hydrostatic pressure, the spectrum shifts to a short-wavelength, which is caused by the decrease of the dielectric constant of GaAs. These results agree with the electromagnetic variational theorem. We also found if the angle of incidence of the modes increases, the band width increases compared to the normal incidence modes. © 2018 Elsevier GmbH. All rights reserved.
1. Introduction Light propagating on periodic one-dimensional structures dates back to 1887 when Lord Rayleigh discovered that when the angle of incidence varies [1], it is possible to obtain regions where the light is totally reflected. A century later, these regions were called photonic band gap (PBG). In 1987, the work of Eli Yablonovitch [2], who worked on inhibiting the spontaneous emission of electrons in semiconductors, and Sajeev Jhon [3], who studied the effects of localization of light in disordered systems, proposed making periodic structures named photonic crystals (PC), thus appearing the concept of photonic band gap in two and three dimensions. The principle of functionality in PCs is the periodic spatial variation of the dielectric constant, equivalent to a periodic potential in an atomic crystal [4]. The allowed and forbidden (PBG) states in PC are due to the Bragg diffraction, generated by the scatterers that form the crystal [5]. In PBGs no light mode with a frequency within the frequency range of PBG can propagate, regardless its polarization and angle of incidence. PCs are formally described by Maxwell’s electromagnetic theory [6]. It is possible to find the PBGs and the states of the electromagnetic field by solving the equations. To solve Maxwell’s equations there are the following methods: plane waves, finite-difference frequency and time domain, transfer-matrix, scattering-matrix, among others [7–10]. The possibility of tuning the PBG by modifying the optical response of the materials that form the PC using external parameters such as electrical and magnetic fields [11,12], temperature [13,14] and hydrostatic pressure [15], allows it to be
∗ Corresponding author at: Grupo de óptica e información cuántica, Universidad Nacional. Departamento de Ciencias Naturales, Universidad Surcolombiana. E-mail address: [email protected] (F. Segovia-Chaves). https://doi.org/10.1016/j.ijleo.2018.01.087 0030-4026/© 2018 Elsevier GmbH. All rights reserved.
F. Segovia-Chaves, H. Vinck-Posada / Optik 161 (2018) 64–69
65
Fig. 1. Photonic crystal one-dimensional Air/(HL)N H/Air.
implemented in potential applications of modern photonics such as optical switches and tunable filters, very important in wavelength-division multiplexing [16–18]. In this work, we studied the effects of the hydrostatic pressure applied, temperature and angle of incidence on the transmittance spectrum for the TE mode of a one-dimensional photonic crystal (1DPC), within the Maxwell framework and using the transfer-matrix method (TMM). We considered a 1DPC consisting of alternate layers of air and GaAs, in which the dielectric properties of the semiconductor depend on pressure and temperature. The work is organized as follows: Section 2, we present the theoretical framework. Section 3, we present the numerical results and discussion related to the calculation of the transmittance spectrum of the 1DPC for different temperatures, hydrostatic pressures, and angles of incidence. Finally, conclusions are presented in Section 4. 2. Theoretical model In Fig. 1, we present a finite 1DPC surrounded by air and consisting of alternate layers of materials, with high H and low L dielectric constants, whose thickness are dH and dL respectively. The wave vector of the incident medium is k 0 and the angle of incidence is , the PC has a homogeneous pattern in the xy plane and a periodicity in z direction. The number of periods of HL layers is given by N. We consider a linearly polarized electromagnetic wave propagating through the (x, z) plane with a wave vector qx along the x-axis. For the TE modes, which will be our focus, the electric field is given by
E j (x, z) = ey Aj eikj,z z + Bj e−ikj,z z e−iqx x where kj,z =
ω/c
2
(1)
j − q2x and j , is the z component of the wave vector and the dielectric constant in the jth layer,
respectively. The transverse component of the wave vector is qx = k0 sin, Aj and Bj values are calculated by the continuity conditions of the tangential electric and magnetic field components. In the TMM, each 1DPC layer may be represented by a matrix [19], Mj = Dj Pj Dj−1
j = H, L
(2)
In Eq. (2) the propagation matrix is
Pj =
eiϕj
0
0
e−iϕj
(3)
with phase ϕj given by: ϕj = kj,z dj =
2dj
In Eq. (4) dj and nj = TE mode is given by
Dj =
(4)
nj cos j
j are the thickness and refractive index in the jth layer, respectively. The dynamic matrix for the
1
1
nj cos j
−nj cos j
(5)
The total transfer-matrix for the 1DPC Air/(HL)N H/Air is defined as
M=
M11
M12
M21
M22
= D0−1 (MH ML )N MH D0
(6)
66
F. Segovia-Chaves, H. Vinck-Posada / Optik 161 (2018) 64–69
Fig. 2. GaAs dielectric constant as a function of pressure and temperature. Table 1 Values of the dielectric constant of GaAs as a function of temperature for a pressure P = 40 kbar. Temperature (K)
Dielectric constant
0 50 200
11.80 11.85 12.02
Table 2 Edges L , R and band width for an angle of incidence = 0◦ and P = 40 kbar. Temperature (K)
Band edge
0 50 200
Band width)
L
R
723.78 723.48 722.48
1574.03 1575.43 1580.18
850.25 851.95 857.71
with D0 the dynamic matrix of air. Transmittance I is calculated with matrix elements M11 of Eq. (6),
1 2
M
I=
(7)
11
In this study, we will assume that layer H is GaAs and layer L is air; the effects on the transmittance spectrum due to hydrostatic pressure and temperature, are determined by the dielectric constant of GaAs [20], −3
GaAs (P, T ) = 12.74e−1.73×10
P 9.4×10−5 (T −75.6)
e
for T ≤ 200 K
(8)
with hydrostatic pressure P in kbar and temperature T in Kelvin, in Fig. 2, we present that dependence. The thickness of GaAs layers depends on hydrostatic pressure and is given by [21,22] d(P) = d0 [1 − (S11 + 2S12 )P]
(9) = 1.16 × 10−3
kbar−1
where d0 is the initial thickness under zero pressure. The elastic constants for GaAs are S11 and S12 =−3.7 × 10−4 kbar−1 . We will choose a quarter of wavelength as the initial thickness of the 1DPC layers, with a design wavelength of 0 of 1000 nm. 3. Numerical results and discussion The dielectric constant of GaAs is calculated by Eq. (8); as the temperature increases the value of GaAs also increases, as shown in Table 1. The numerical results of the transmittance spectrum of a 1DPC for the TE mode with normal incidence = 0◦ , for a pressure P = 40 kbar and a period N = 10 is shown in Fig. 3(a) and (b). The width of the PBG is determined by Bragg’s reflection theory. The left L and right R band edges are given by [23], (nH dH + nL dL ) , cos−1 (−(nH − nL /nH + nL ))
(nH dH + nL dL ) (10) cos−1 (nH − nL /nH + nL ) √ with nH and nL , the refractive index of the semiconductor ( GaAs ) and air, respectively. The thickness of the GaAs layer is dH calculated by Eq. (9) and dL the thickness of the air layer. In Table 2, we present the edges values and with of the 1DPC band, calculated with Eq. (10) for different temperature values. The width of the PBG band increases with temperature, as shown in Fig. 3(a). In Fig. 3(b) we present the transmittance spectrum of TE mode for = 0◦ ; we increased the temperature from T = 0 K to T = 200 K, for a pressure of P = 40 kbar. When increasing the temperature, the spectrum shifted to a longer L =
R =
F. Segovia-Chaves, H. Vinck-Posada / Optik 161 (2018) 64–69
67
Fig. 3. (a) Contour plot of transmittance as function of the wavelength and temperature for the TE mode. (b) Transmittance spectrum for the TE mode for temperatures T = 0 K (black line) and T = 200 K (red line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)
Fig. 4. Contour plot of transmittance as a function of the wavelength and temperature for the TE mode with P = 40 kbar (a) = 45◦ and (b) = 65◦ . Table 3 Values of the dielectric constant of GaAs as a function of pressure for a temperature T = 200 K. Pressure (kbar)
Dielectric constant
0 10 40
12.88 12.66 12.02
wavelength. The results agree with the electromagnetic variational theorem; low frequency modes tend to concentrate a greater fraction of energy in the regions where the dielectric constant is greater. However, if we increase the temperature (at a constant pressure) and the angle of incidence, the width of the PBG band shifts to shorter wavelength regions, increasing the width of the PBG compared to that of the normal incidence, as shown in Fig. 4(a) and (b). In Fig. 5(a) and (b), we present the transmittance spectrum for the TE mode with an angle of incidence of = 45◦ and = 65◦ , respectively. When increasing the temperature from T = 0 K to T = 200 K, for a given pressure of P = 40 kbar, the spectrum shifts to long wavelength as in the case of normal incidence. Finally, we analyze the effects of hydrostatic pressure on the transmittance spectrum of 1DPC. We choose a temperature of T = 200 K and a period of N = 10. When increasing the pressure, the dielectric constant of GaAs decreases, as shown in Table 3. In Fig. 6(a), we present the numerical results of the transmittance spectrum of TE mode with normal incidence = 0◦ as a function of the wavelength and hydrostatic pressure. When increasing the pressure, the edges and band width increase too, this data is shown in Table 4. In Fig. 6(b), we observe a noticeable transmittance spectrum shifting to short-wavelength when increasing the pressure from P = 0 kbar to P = 40 kbar, due to the dielectric constant decrease. These results agree with
68
F. Segovia-Chaves, H. Vinck-Posada / Optik 161 (2018) 64–69
Fig. 5. Transmittance spectrum for the TE mode with P = 40 kbar and temperature T = 0 K (black line) and T = 200 K (red line) for (a) = 45◦ and (b) = 65◦ . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)
Fig. 6. (a) Contour plot of transmittance as a function of the wavelength and pressure for the TE mode. (b) Transmittance spectrum for the TE mode for pressure P = 0 kbar (black line) and P = 40 kbar (red line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.) Table 4 Edges L , R and band width for an angle of incidence = 0◦ and T = 200 K. Pressure (kbar)
0 10 40
Band edge (nm)
Band width (nm)
L
R
723.79 723.46 722.48
1617.08 1607.79 1580.18
893.29 884.33 857.71
the electromagnetic variational theorem. Compared to the results in Fig. 3(b), we observed that the optical response of 1DPC is mainly due to the hydrostatic pressure applied, instead of the temperature. In Fig. 7(a) and (b) we show the numerical results of the transmittance spectrum as a function of the pressure and wavelength for = 45◦ and = 65◦ , respectively. The angle and the hydrostatic pressure increase the PBG width, which shifts to short-wavelength, due to the decrease of the semiconductor dielectric constant. 4. Conclusions With the transfer-matrix method, we studied the dependence of the transmittance spectrum of a one-dimensional photonic crystal consisting of alternate layers of air and GaAs on temperature, hydrostatic pressure and angle of incidence. We found out that temperature does not affect significantly the shifting of the band; the effects are mainly due to the variation of the thickness and dielectric constant of GaAs with hydrostatic pressure. When the pressure increases, the value of the dielectric constant of GaAs decreases and shifts to short-wavelength. Considering the propagation outside the periodicity axis, the width of the PBG increases, compared to that of the normal incidence.
F. Segovia-Chaves, H. Vinck-Posada / Optik 161 (2018) 64–69
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Fig. 7. Contour plot of transmittance as a function of the wavelength and pressure for the TE mode with T = 200 K (a) = 45◦ and (b) = 65◦ .
Acknowledgements 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”, “Exploración y modelación de la iridiscencia en especies Colombianas. Código 110156933525, CT 026-2013, HERMES 17432” and “Modelación teóricoestadística de la reflectividad en especies colombianas, HERMES 35767”. F.S.-Ch. also acknowledges to Vicerrectoría de Investigación, Universidad Surcolombiana Neiva-Huila. References [1] L. Rayleigh, On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure, Philos. Mag. Sci. 24 (1887) 145–159. [2] E. Yablanovitch, Inhibited spontaneous emission in solid state physics and electronics, Phys. Rev. Lett. 58 (1987) 2059. [3] S. John, Localization of light, Phys. Rev. Lett. 58 (1987) 2486. [4] R.H. Lipson, C. Lu, Photonic crystals: a unique partnership between light and matter, Eur. J. Phys. 30 (2009) S33. [5] J. Joannopoulos, S. Johnson, J. Winn, R. Meade, Photonic Crystals: Molding the Flow of Light, Princeton University Press, 2007. [6] M. Skorobogatiy, J. Yang, Fundamentals of Photonic Crystal Guiding, Cambridge University Press, 2009. [7] H. Sözüer, J. Haus, R. Inguva, Photonic bands: convergence problems with the plane-wave method, Phys. Rev. B 45 (1992) 13962. [8] L. Li, Use of Fourier series in the analysis of discontinuous periodic structures, J. Opt. Soc. Am. A 13 (1996) 1870. [9] A. Taflove, S. Hagness, Computational Electrodynamics: The Finite-difference Time-domain Method, Atech House Publishers, 2005. [10] A. Yariv, P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation, Wiley-Interscience, 2002. [11] C. Xu, X. Hu, Y. Li, X. Liu, R. Fu, J. Zi, Semiconductor-based tunable photonic crystals by means of an external magnetic field, Phys. Rev. B 68 (2003) 193201. [12] K. Busch, S. John, Liquid-crystal photonic-band-gap materials: the tunable electromagnetic vacuum, Phys. Rev. Lett. 83 (1999) 967. [13] F. Segovia-Chaves, H. Vinck-Posada, Temperature dependence of defect mode in band structures of the one-dimensional photonic crystal, Optik 154 (2018) 467. [14] V. Kumar, B. Suthar, A. Kumar, K. Singh, A. Bhargava, The effect of temperature and angle of incidence on photonic band gap in a dispersive Si-based one dimensional photonic crystal, Phys. B: Condens. Matter 416 (2013) 106. [15] N. Porras-Montenegro, C. Duque, Temperature and hydrostatic pressure effects on the photonic band structure of a 2D honeycomb lattice, Phys. E: Low-dimens. Syst. Nanostruct. 42 (2010) 1865. [16] H. Nemec, L. Duvillaret, F. Garet, P. Kuzel, P. Xavier, J. Richard, D. Rauly, Thermally tunable filter for terahertz range based on a one-dimensional photonic crystal with a defect, J. Appl. Phys. 96 (2004) 4072. [17] S. Eliahou-Niv, R. Dahan, G. Golan, Design and analysis of a novel tunable optical filter, Microelectron. J. 37 (2006) 302. [18] Z. Luo, H. Wen, H. Guo, M. Yang, A time-and wavelength-division multiplexing sensor network with ultra-weak fiber Bragg gratings, Opt. Express 21 (2013) 22799. [19] P. Yeh, Optical Waves in Layered Media, Wiley-Interscience, 2005. [20] I. Erdogan, O. Akankan, H. Akbas, Simultaneous effects of temperature, hydrostatic pressure and electric field on the self-polarization and electric field polarization in a GaAs/Ga0.7 Al0.3 As spherical quantum dot with a donor impurity, Superlattices Microstruct. 59 (2013) 13. [21] A. Elabsy, Hydrostatic pressure dependence of binding energies for donors in quantum well heterostructures, Phys. Scr. 48 (1993) 376. [22] G. Samara, Temperature and pressure dependences of the dielectric constants of semiconductors, Phys. Rev. B 27 (1983) 3494. [23] S. Orfanidis, Electromagnetic waves and antennas, NJ Rutgers University, New Brunswick, 2002.