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Analysis of the photonic band gap of plasma photonic crystals with filmy structure
Optik 125 (2014) 532–535
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Analysis of the photonic band gap of plasma photonic crystals with filmy structure Hong Wei Yang a,c,∗ , Ze Kun Yang b , Cheng-Ke Zhu a , Ai Ping Li a , Xiong You a a b c
Department of Physics, College of Science, Nanjing Agricultural University, Nanjing 210095, PR China School of Computer Science and Technology, Xidian University, Xi’an 710126, PR China State Key Laboratory of Millimeter Waves, Nanjing 210096, PR China
a r t i c l e
i n f o
Article history: Received 1 March 2013 Accepted 3 July 2013
Keywords: Plasma photonic crystals Finite-difference time-domain Photonic band gap
a b s t r a c t Plasma photonic crystals are artificially periodic structures, which are composed of plasmas and dielectric structures. In this paper, the Runge–Kutta exponential time differencing (RKETD) finite-difference time-domain (FDTD) method is applied to study the one dimension plasma photonic crystals. From the perspective of frequency-domain, the effects of the relative dielectric constant, the plasma collision frequency and the plasma frequency on the photonic band gap are analyzed and discussed. According to numerical simulation, the results show that the electromagnetic band gaps of unmagnetized plasma photonic crystals can be tuned by the parameters. These may provide some useful information for basic theoretical study and designing plasma photonic crystals devices. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction In 1987, Yablonovitch [1] and John [2] independently proposed the photonic crystal concept. Photonic crystals (PCs) are artificial materials with periodically dielectric modulated function, which are characterized by the particular ability to control the flow of light or electromagnetic waves within a frequency band called the photonic band gap (PBG) [3–5]. In 1999, the Science magazine of the United States made photonic crystals as one of the top ten hot topics, harbingering of 21st century is the century of a photon. Recently, the research of photonic crystal has became a common research focus in the fields of physics, optoelectronics, electromagnetic theory, materials science, nanotechnology research, and so on. By making use of its fascinating properties, various functional devices such as frequency filters, waveguides, time delay device, diode laser and microwave circuit components can be created [6,7]. The plasma photonic crystals [8] are artificially periodic array composed of alternating unmagnetized (or magnetized) plasmas and dielectric materials. It is well known that the unmagnetized plasma can be characterized by a complex frequency-dependent permittivity. On the one hand, the unmagnetized plasma is a frequency dispersive medium. On the other hand, the unmagnetized collisional plasma is a dissipative medium. So, comparing with
∗ Corresponding author at: Department of Physics, College of Science, Nanjing Agricultural University, Nanjing 210095, PR China. E-mail address: phd [email protected] (H.W. Yang). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.07.032
conventional photonic crystals, the plasma photonic crystals are expected to obtain many new particular characteristics, and it not only has unique characteristics of PBG and photon localization property, but also has the characteristics of plasma [9,10]. We can adjust to control and change parameters of plasma to realize the dynamic adjustment of photonic band gap structure. So, the study of the plasma photonic crystals is of a practical significance. In this paper, the RKETD-FDTD method [11,12] is used to simulate the one dimensional unmagnetized plasma photonic crystals. The effects of relative permittivity, plasma frequency and plasma collision frequency on photonic band gap are discussed and analyzed, respectively. 2. Numerical algorithms and physical model 2.1. RKETD-FDTD The finite-difference time-domain method is a simple and effective method, which has been widely used to simulate the transient solutions of electromagnetic wave propagation in various media including dispersive media [13,14]. Compared to the transfer matrix method, FDTD is an accurate numerical algorithm. Not only dose this method takes into account all the electromagnetic loss, but also it is more suitable for the electromagnetic simulation of complex dielectric structures [15]. In this paper, the Runge–Kutta exponential time differencing (RKETD) method, proposed in literature [10], is applied to simulation the plasma photonic crystals. The algorithm has high accuracy, and use less CPU time and compute memory.
H.W. Yang et al. / Optik 125 (2014) 532–535
533
Considering one dimensional plasma photonic crystals, The Maxwell’s equations and constitutive relation for non-magnetized collision plasma are given by ∂E +J ∂t
(1)
∂H ∂t
(2)
dJ + J = ε0 ωp2 E dt
(3)
∇ × H = ε0
∇ × E = −0
where E is the electric field, H is the magnetic field, J is the polarization current density, ε0 and 0 are the permittivity and permeability of free space, respectively. is the electron collision frequency, ωp is the plasma frequency. To derive the Runge–Kutta exponential time differencing (RKETD) scheme [13], multiplying Eq. (3) by et , letting, t = tn + , tn+1 = tn + t F(tn + ) = ε0 ωp2 Ex (tn + ), to give
t
Jxn+1 = e−t Jxn + e−t
e F(tn + )d
(4)
0
Applying the second RKETD method [10], the first step is taken to give K1 = e−t Jxn +
F(tn , Jx )(1 − e−t )
F(tn + ) = F(tn , Jx ) +
(5)
2 [F(tn + t, K1 ) − F(tn , Jx )] + o((t) ) t
(6)
Substitution of Eq. (6) in Eq. (4) yields, after some integral manipulation, the component of J at n + 1 time step can be written as n+1/2
Jx
(k) = e− t Jxn (k) +
(1 − e− t )ε0 ωp2
(e− t − 1 + t)ε0 ωp2
+
2 t
Exn (k)
1+
+ +
−1
ωp2
(e−t − 1 + t) 2
2v
(7)
2
1−
t 1 n+1/2 Hy k+ 2 ε0 z
n−1/2
− Hy
k−
n+1/2
k+
1 2
2v
(1 − e−t )
(e−t − 1 + t) Exn (k) 2
t − (1 + e−t )Jxn (k) 2ε0
Hy
tωp2
ωp2
n−1/2
= Hy
k−
1 2
plasma slab with a thickness of 1.5 cm. The incident wave used in the simulation is the derivative of the Gaussian pulse in order to eliminate the effect of the frequency at zero. The incident wave that is perpendicular to the plasma slab. The computational domain is subdivided into 500 cells, the plasma occupies 200 cells from 200 to 400, and other cells are free space, five cells perfectly matched layer (PML) medium are used at the terminations of the space to eliminate unwanted reflected wave. The mesh width is 75 m and time step is 0.125 ps. The plasma parameters are as follows: ωp = 2 × 28.7 × 109 rad/s, = 20 GHz. The simulation is allowed to run for 20,000 time steps and the result is convergent. Figs. 1 and 2 show the graph of the reflection coefficients and the transmission coefficient of electromagnetic wave through a nonmagnetized collision plasma slab. As is shown in the figures, the simulation results of the two methods agree with very well, and fully consistent with results of literature [11], so we can verify the correctness of the algorithm. 3. Numerical results and discussion
[Exn+1 (k) − Exn (k)]
Combining Eqs. (1) and (2), using the Yee grid and leap-frog integration, the update equations of H field and E field is given respectively by Exn+1 (k) =
Fig. 1. Comparison of the reflection coefficients.
1 2
The plasma photonic crystals are an artificially periodic array composed of alternating discharged plasma and other dielectric materials, including vacuum. Calculation model as shown in Fig. 3: the structure of the photonic crystal in this paper is (AB)n A, where A and B represent medium and plasma, respectively. N = 3 and N is the number of the repeating unit. Each cycle thickness is d = 3 cm, d = a + b. Thickness of each layer of plasma and medium thickness is equal, a = b. The mesh width is z = 1.5 × 10−3 m and the time step is t = 2.5 × 10−12 s, the plasma frequency is ωp = 4 × 109 rad/s, the plasma collision frequency is = 3 GHz. Five cells perfectly matched layer (PML) medium were used at the terminations of the space
(8)
−
t [E n (k + 1) − Exn (k)] (9) 0 z x
Here, z is the mesh width, t is the time step, literature [11] gives a detailed derivation of the formulation. 2.2. Validity and accuracy of the RKETD algorithm In order to check the correctness and the accuracy of the upper algorithm, we compute the reflection and transmission coefficients of electromagnetic wave through a non-magnetized collisional
Fig. 2. Comparison of the transmission coefficients.
534
H.W. Yang et al. / Optik 125 (2014) 532–535
Fig. 3. Schematic diagram of one-dimensional plasma photonic crystals.
to eliminate unwanted reflections. The incident wave used in the simulation is a Gaussian-derivative pulsed plane wave. 3.1. Effect of relative dielectric constant In recent years, the photonic crystals in intelligent materials, chemical and biological sensors, new optical device, and information storage materials have achieved a rapid development. They mainly have two basic characteristics: one is photonic band gaps (PBG), the other one is localized modes at the defects, which can be applied to control and limit the propagation of electromagnetic waves. Recently, thin-film materials with structured photonic crystal structures used in solar cells have aroused widespread concern and research [16,17]. Relevant studies show that the special structure of photonic crystal film can increase the rate of absorption, photoelectric conversion performance enhancements. In 2007, the United States of Marian Florescu [18] put forward an efficient and reliable solar cells using photonic crystal technology. This structure can improve thermal radiation spectrum and angle characteristics, which can be used on spectrum and angle of the hot photovoltaic devices selective demand and increase the efficiency of solar cell components together. In 2009, Ye Hong analysis of photonic crystal as a filter in the field of solar photovoltaic applications, and improved the filter membrane structure. In 2011, the scientists at Stanford University have developed a new type of thin-film solar cells [19], combined with the plasma to improve the performance and economic feasibility. Visible, photonic crystals and plasma in a new generation of thin film solar cells in the research and development has important application value. Film as a kind of material is widely used in industry, agriculture and daily life. Commonly film raw materials: SiO2 , TiO2 , ZnO, MgF, PE. In this section, we study the effect of several common film raw materials on PBG of 1D-PPC. The calculation model is shown as in Table 1. In the case of keeping plasma parameters constant, only change the relative dielectric constant of the medium. Fig. 4 shows the results of RKETD-FDTD calculation program. When εr = 2.3, reflection coefficient frequency spectrum in 4 GHz appears the obvious forbidden band; when εr = 4.5, in 3 GHz and 6.2 GHz there are two obvious forbidden band, and the bandwidth of the EBG is broadening; when εr = 9.7, there are three forbidden band at 2.2 GHz, 4.5 GHz and 7.3 GHz, and the shock between the gaps is intensifying. In short, as dielectric constant increases, the number of PBG and depth increased, and the bandwidth of the PBG has been broadened, so in order to get more of the energy band structure, you should choose the relative dielectric constant greater media.
Fig. 4. Effects of different media relative dielectric on PBG.
3.2. Effect of the plasma collision frequency Calculation model is the same as Fig. 1: the relative dielectric constant is εr = 4, and the plasma frequency is ωp = 4 × 109 rad/s, collision frequency of plasma is taken respectively 2 GHz, 8 GHz, 16 GHz, 32 GHz. Other parameters are the same as above. The transformation of the PBG is shown in Fig. 6. S11 is used to represent reflection coefficient, and S21 is forward transmission coefficient in Fig. 5. It is obvious from Fig. 6 that: with the increase in plasma collision frequency, the peak of the PBG in PPC reduces, and the reflection coefficient has a remarkable reduction when the frequency of incident electromagnetic wave is lower than the plasma frequency. Total value of the reflection coefficient and transmission coefficient decreases with the increase of plasma collision frequency, it is because that the collisional plasma can efficiently absorb electromagnetic waves under some conditions,
Table 1 the relative dielectric constant under normal temperature. Material
PE
SiO2
Fe2 O3
MgO
Relative dielectric constant
2.3
4.5
6.5
9.7 Fig. 5. Effects of different plasma collision frequency on PBG.
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dielectric constant, plasma frequency, and plasma collision frequency. The results show that the number of PBG increases as relative dielectric constant increases; because the unmagnetized, collisional plasma is a dissipative medium, the energy of the EM reduce with the plasma collision frequency increasing, at the same time, the peaks (upwards) of reflection coefficient decrease, however, the positions of the PBG remain constant; what is more, plasma photonic crystals is of the characteristic of a high-pass filter, the change of plasma frequency has less influence on the PBG of high frequency than low frequency. The PBG of the plasma photonic crystals is of flexible adjustability, so it has high value of research and wide application prospect. Acknowledgements This research work is jointly supported by the Natural Science Foundation of China (No. 11171155), the Fundamental Research Funds for the Central Universities (No. Y0201100265), and Foundation of State Key Laboratory of Millimeter Waves (No. K201319). References
Fig. 6. Effects of different plasma frequency on PBG.
which leads to the reduction of total energy. However, the positions of the PBG remain constant. So, the collision frequency has less influence on PBG. 3.3. Effect of the plasma frequency Next, we compute the effects of plasma frequency on the PBG of unmagnetized plasma photonic crystals. Calculation model is the same as Fig. 1. The plasma collision frequency is chosen to be 2 GHz, which is fixed in this simulation. The relative dielectric constant is εr = 4, the plasma frequency is taken: ωp = 2 × 2 × 109 rad/s, ωp = 2 × 4 × 109 rad/s, ωp = 2 × 6 × 109 rad/s, respectively. Other parameters are the same as above. From Fig. 6, the following conclusions may be obtained. The effect of plasma frequency on plasma photonic crystal bandgap structure mainly reflects in the low frequency parts. When the plasma frequency exceeds EM wave frequency, the peaks of reflection coefficients decrease sharply with the increasing plasma frequency, especially in low EM wave frequency range, and the PBG is getting larger. The gap regions shift toward the higher frequency side appreciably with the increase of the plasma frequency. Physically, this can be explained by the fact that the isotropic plasma can act as a high-pass filter. EM wave frequencies above the plasma frequency constitute a pass band, and frequencies below the plasma frequency constitute a stop band. When the EM wave frequency is less than the plasma frequency, the EM wave is totally reflected. 4. Conclusions In summary, this paper adopts RKETD-FDTD algorithm to simulate the plasma photonic crystals structures. The reflection and transmission coefficients in frequency domain are computed, and the results of the PBG effect were discussed in terms of relative
[1] E. Yablonovitch, Inhibited spontaneous emission in solid-state physics and electronics, Phys. Rev. Lett. 58 (20) (1987) 2059–2062. [2] S. John, Strong localization of photons in certain disordered dielectric superlattices, Phys. Rev. Lett. 58 (23) (1987) 2486–2489. [3] J.D. Joannopoulos, P.R. Villeneuve, S. Fan, Photonic crystals: putting a new twist on light, Nature 386 (1997) 143–149. [4] K.M. Ho, C.T. Chen, C.M. Soukoulis, Existence of a photonic gap in periodic dielectric structures, Phys. Rev. Lett. 65 (25) (1990) 3152–3155. [5] E. Yablonovitch, Photonic band-gap structures, J. Opt. Soc. Am. B 10 (2) (1993) 283–295. [6] M. Bayindir, B. Temelkuran, E. Ozbay, Photonic-crystal-based beam splitters, Appl. Phys. Lett. 77 (24) (2000) 3902–3904. [7] M. Loncar, D. Nedeljkovic, T. Doll, J. Vuckovic, A. Scherer, T.P. Pearsall, Waveguiding in planar photonic crystals, Appl. Phys. Lett. 77 (13) (2000) 1937–1939. [8] H. Hitoshi, M. Atsushi, Dispersion relation of electromagnetic waves in onedimensional plasma photonic crystals, J. Plasma Fusion Res. 80 (2) (2004) 89–90. [9] Y. Liu, H.W. Yang, Shift operator method and Runge–Kutta exponential time differencing method for plasma, Optik Int. J. Light Electron Optik. 122 (23) (2011) 2086–2089. [10] L. Song, Z. Shuangying, L. Sanqiu, A study of properties of the photonic band gap of unmagnetized plasma photonic crystal, Plasma Sci. Technol. 11 (1) (2009) 14–17. [11] H.W. Yang, Y. Liu, Runge–Kutta exponential time differencing method analysis of non-magnetized plasma stealth, J. Infrared Millimeter Terahertz Waves 31 (9) (2010) 1075–1080. [12] S. Liu, S.-y. Zhong, S. Liu, Finite-difference time-domain algorithm for dispersive media based on Runge–Kutta exponential time differencing method, Int. J. Infrared Millimeter Waves 29 (3) (2008) 323–328. [13] D.M. Sullivan, Electromagnetic Simulation Using the FDTD Method, IEEE Press, New York, 2000. [14] A. Taflove, S.C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Artech House, Norwood, MA, 2005. [15] L. Shao-Bin, Z. Chuan-Xi, Y. Nai-Chang, FDTD simulation for plasma photonic crystals (in Chinese), Acta Phys. Sin. 54 (6) (2005) 2804–2808. [16] D.M. Sullivan, Frequency-dependent FDTD methods using Z transforms, IEEE Trans. Antennas Propag. 40 (10) (1992) 1223–1230. [17] E.W. McFarland, J. Tang, A photovoltaic device structure based on internal electron emission, Nature 421 (2003) 616–618. [18] A.O. Govorov, J. Lee, N.A. Kotov, Theory of plasmon-enhanced Forster energy transfer in optically excited semiconductor and metal nanoparticles, Phys. Rev. B 76 (12) (2007) 125308–125316. [19] M. Florescu, H. Lee, I. Puscasu, M. Pralle, L. Florescu, D.Z. Ting, J.P. Dowling, Improving solar cell efficiency using photonic band-gap materials, Sol. Energy Mater. Sol. Cells 91 (17) (2007) 1599–1610.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Analysis of the photonic band gap of plasma photonic crystals with filmy structure Hong Wei Yang a,c,∗ , Ze Kun Yang b , Cheng-Ke Zhu a , Ai Ping Li a , Xiong You a a b c
Department of Physics, College of Science, Nanjing Agricultural University, Nanjing 210095, PR China School of Computer Science and Technology, Xidian University, Xi’an 710126, PR China State Key Laboratory of Millimeter Waves, Nanjing 210096, PR China
a r t i c l e
i n f o
Article history: Received 1 March 2013 Accepted 3 July 2013
Keywords: Plasma photonic crystals Finite-difference time-domain Photonic band gap
a b s t r a c t Plasma photonic crystals are artificially periodic structures, which are composed of plasmas and dielectric structures. In this paper, the Runge–Kutta exponential time differencing (RKETD) finite-difference time-domain (FDTD) method is applied to study the one dimension plasma photonic crystals. From the perspective of frequency-domain, the effects of the relative dielectric constant, the plasma collision frequency and the plasma frequency on the photonic band gap are analyzed and discussed. According to numerical simulation, the results show that the electromagnetic band gaps of unmagnetized plasma photonic crystals can be tuned by the parameters. These may provide some useful information for basic theoretical study and designing plasma photonic crystals devices. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction In 1987, Yablonovitch [1] and John [2] independently proposed the photonic crystal concept. Photonic crystals (PCs) are artificial materials with periodically dielectric modulated function, which are characterized by the particular ability to control the flow of light or electromagnetic waves within a frequency band called the photonic band gap (PBG) [3–5]. In 1999, the Science magazine of the United States made photonic crystals as one of the top ten hot topics, harbingering of 21st century is the century of a photon. Recently, the research of photonic crystal has became a common research focus in the fields of physics, optoelectronics, electromagnetic theory, materials science, nanotechnology research, and so on. By making use of its fascinating properties, various functional devices such as frequency filters, waveguides, time delay device, diode laser and microwave circuit components can be created [6,7]. The plasma photonic crystals [8] are artificially periodic array composed of alternating unmagnetized (or magnetized) plasmas and dielectric materials. It is well known that the unmagnetized plasma can be characterized by a complex frequency-dependent permittivity. On the one hand, the unmagnetized plasma is a frequency dispersive medium. On the other hand, the unmagnetized collisional plasma is a dissipative medium. So, comparing with
∗ Corresponding author at: Department of Physics, College of Science, Nanjing Agricultural University, Nanjing 210095, PR China. E-mail address: phd [email protected] (H.W. Yang). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.07.032
conventional photonic crystals, the plasma photonic crystals are expected to obtain many new particular characteristics, and it not only has unique characteristics of PBG and photon localization property, but also has the characteristics of plasma [9,10]. We can adjust to control and change parameters of plasma to realize the dynamic adjustment of photonic band gap structure. So, the study of the plasma photonic crystals is of a practical significance. In this paper, the RKETD-FDTD method [11,12] is used to simulate the one dimensional unmagnetized plasma photonic crystals. The effects of relative permittivity, plasma frequency and plasma collision frequency on photonic band gap are discussed and analyzed, respectively. 2. Numerical algorithms and physical model 2.1. RKETD-FDTD The finite-difference time-domain method is a simple and effective method, which has been widely used to simulate the transient solutions of electromagnetic wave propagation in various media including dispersive media [13,14]. Compared to the transfer matrix method, FDTD is an accurate numerical algorithm. Not only dose this method takes into account all the electromagnetic loss, but also it is more suitable for the electromagnetic simulation of complex dielectric structures [15]. In this paper, the Runge–Kutta exponential time differencing (RKETD) method, proposed in literature [10], is applied to simulation the plasma photonic crystals. The algorithm has high accuracy, and use less CPU time and compute memory.
H.W. Yang et al. / Optik 125 (2014) 532–535
533
Considering one dimensional plasma photonic crystals, The Maxwell’s equations and constitutive relation for non-magnetized collision plasma are given by ∂E +J ∂t
(1)
∂H ∂t
(2)
dJ + J = ε0 ωp2 E dt
(3)
∇ × H = ε0
∇ × E = −0
where E is the electric field, H is the magnetic field, J is the polarization current density, ε0 and 0 are the permittivity and permeability of free space, respectively. is the electron collision frequency, ωp is the plasma frequency. To derive the Runge–Kutta exponential time differencing (RKETD) scheme [13], multiplying Eq. (3) by et , letting, t = tn + , tn+1 = tn + t F(tn + ) = ε0 ωp2 Ex (tn + ), to give
t
Jxn+1 = e−t Jxn + e−t
e F(tn + )d
(4)
0
Applying the second RKETD method [10], the first step is taken to give K1 = e−t Jxn +
F(tn , Jx )(1 − e−t )
F(tn + ) = F(tn , Jx ) +
(5)
2 [F(tn + t, K1 ) − F(tn , Jx )] + o((t) ) t
(6)
Substitution of Eq. (6) in Eq. (4) yields, after some integral manipulation, the component of J at n + 1 time step can be written as n+1/2
Jx
(k) = e− t Jxn (k) +
(1 − e− t )ε0 ωp2
(e− t − 1 + t)ε0 ωp2
+
2 t
Exn (k)
1+
+ +
−1
ωp2
(e−t − 1 + t) 2
2v
(7)
2
1−
t 1 n+1/2 Hy k+ 2 ε0 z
n−1/2
− Hy
k−
n+1/2
k+
1 2
2v
(1 − e−t )
(e−t − 1 + t) Exn (k) 2
t − (1 + e−t )Jxn (k) 2ε0
Hy
tωp2
ωp2
n−1/2
= Hy
k−
1 2
plasma slab with a thickness of 1.5 cm. The incident wave used in the simulation is the derivative of the Gaussian pulse in order to eliminate the effect of the frequency at zero. The incident wave that is perpendicular to the plasma slab. The computational domain is subdivided into 500 cells, the plasma occupies 200 cells from 200 to 400, and other cells are free space, five cells perfectly matched layer (PML) medium are used at the terminations of the space to eliminate unwanted reflected wave. The mesh width is 75 m and time step is 0.125 ps. The plasma parameters are as follows: ωp = 2 × 28.7 × 109 rad/s, = 20 GHz. The simulation is allowed to run for 20,000 time steps and the result is convergent. Figs. 1 and 2 show the graph of the reflection coefficients and the transmission coefficient of electromagnetic wave through a nonmagnetized collision plasma slab. As is shown in the figures, the simulation results of the two methods agree with very well, and fully consistent with results of literature [11], so we can verify the correctness of the algorithm. 3. Numerical results and discussion
[Exn+1 (k) − Exn (k)]
Combining Eqs. (1) and (2), using the Yee grid and leap-frog integration, the update equations of H field and E field is given respectively by Exn+1 (k) =
Fig. 1. Comparison of the reflection coefficients.
1 2
The plasma photonic crystals are an artificially periodic array composed of alternating discharged plasma and other dielectric materials, including vacuum. Calculation model as shown in Fig. 3: the structure of the photonic crystal in this paper is (AB)n A, where A and B represent medium and plasma, respectively. N = 3 and N is the number of the repeating unit. Each cycle thickness is d = 3 cm, d = a + b. Thickness of each layer of plasma and medium thickness is equal, a = b. The mesh width is z = 1.5 × 10−3 m and the time step is t = 2.5 × 10−12 s, the plasma frequency is ωp = 4 × 109 rad/s, the plasma collision frequency is = 3 GHz. Five cells perfectly matched layer (PML) medium were used at the terminations of the space
(8)
−
t [E n (k + 1) − Exn (k)] (9) 0 z x
Here, z is the mesh width, t is the time step, literature [11] gives a detailed derivation of the formulation. 2.2. Validity and accuracy of the RKETD algorithm In order to check the correctness and the accuracy of the upper algorithm, we compute the reflection and transmission coefficients of electromagnetic wave through a non-magnetized collisional
Fig. 2. Comparison of the transmission coefficients.
534
H.W. Yang et al. / Optik 125 (2014) 532–535
Fig. 3. Schematic diagram of one-dimensional plasma photonic crystals.
to eliminate unwanted reflections. The incident wave used in the simulation is a Gaussian-derivative pulsed plane wave. 3.1. Effect of relative dielectric constant In recent years, the photonic crystals in intelligent materials, chemical and biological sensors, new optical device, and information storage materials have achieved a rapid development. They mainly have two basic characteristics: one is photonic band gaps (PBG), the other one is localized modes at the defects, which can be applied to control and limit the propagation of electromagnetic waves. Recently, thin-film materials with structured photonic crystal structures used in solar cells have aroused widespread concern and research [16,17]. Relevant studies show that the special structure of photonic crystal film can increase the rate of absorption, photoelectric conversion performance enhancements. In 2007, the United States of Marian Florescu [18] put forward an efficient and reliable solar cells using photonic crystal technology. This structure can improve thermal radiation spectrum and angle characteristics, which can be used on spectrum and angle of the hot photovoltaic devices selective demand and increase the efficiency of solar cell components together. In 2009, Ye Hong analysis of photonic crystal as a filter in the field of solar photovoltaic applications, and improved the filter membrane structure. In 2011, the scientists at Stanford University have developed a new type of thin-film solar cells [19], combined with the plasma to improve the performance and economic feasibility. Visible, photonic crystals and plasma in a new generation of thin film solar cells in the research and development has important application value. Film as a kind of material is widely used in industry, agriculture and daily life. Commonly film raw materials: SiO2 , TiO2 , ZnO, MgF, PE. In this section, we study the effect of several common film raw materials on PBG of 1D-PPC. The calculation model is shown as in Table 1. In the case of keeping plasma parameters constant, only change the relative dielectric constant of the medium. Fig. 4 shows the results of RKETD-FDTD calculation program. When εr = 2.3, reflection coefficient frequency spectrum in 4 GHz appears the obvious forbidden band; when εr = 4.5, in 3 GHz and 6.2 GHz there are two obvious forbidden band, and the bandwidth of the EBG is broadening; when εr = 9.7, there are three forbidden band at 2.2 GHz, 4.5 GHz and 7.3 GHz, and the shock between the gaps is intensifying. In short, as dielectric constant increases, the number of PBG and depth increased, and the bandwidth of the PBG has been broadened, so in order to get more of the energy band structure, you should choose the relative dielectric constant greater media.
Fig. 4. Effects of different media relative dielectric on PBG.
3.2. Effect of the plasma collision frequency Calculation model is the same as Fig. 1: the relative dielectric constant is εr = 4, and the plasma frequency is ωp = 4 × 109 rad/s, collision frequency of plasma is taken respectively 2 GHz, 8 GHz, 16 GHz, 32 GHz. Other parameters are the same as above. The transformation of the PBG is shown in Fig. 6. S11 is used to represent reflection coefficient, and S21 is forward transmission coefficient in Fig. 5. It is obvious from Fig. 6 that: with the increase in plasma collision frequency, the peak of the PBG in PPC reduces, and the reflection coefficient has a remarkable reduction when the frequency of incident electromagnetic wave is lower than the plasma frequency. Total value of the reflection coefficient and transmission coefficient decreases with the increase of plasma collision frequency, it is because that the collisional plasma can efficiently absorb electromagnetic waves under some conditions,
Table 1 the relative dielectric constant under normal temperature. Material
PE
SiO2
Fe2 O3
MgO
Relative dielectric constant
2.3
4.5
6.5
9.7 Fig. 5. Effects of different plasma collision frequency on PBG.
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dielectric constant, plasma frequency, and plasma collision frequency. The results show that the number of PBG increases as relative dielectric constant increases; because the unmagnetized, collisional plasma is a dissipative medium, the energy of the EM reduce with the plasma collision frequency increasing, at the same time, the peaks (upwards) of reflection coefficient decrease, however, the positions of the PBG remain constant; what is more, plasma photonic crystals is of the characteristic of a high-pass filter, the change of plasma frequency has less influence on the PBG of high frequency than low frequency. The PBG of the plasma photonic crystals is of flexible adjustability, so it has high value of research and wide application prospect. Acknowledgements This research work is jointly supported by the Natural Science Foundation of China (No. 11171155), the Fundamental Research Funds for the Central Universities (No. Y0201100265), and Foundation of State Key Laboratory of Millimeter Waves (No. K201319). References
Fig. 6. Effects of different plasma frequency on PBG.
which leads to the reduction of total energy. However, the positions of the PBG remain constant. So, the collision frequency has less influence on PBG. 3.3. Effect of the plasma frequency Next, we compute the effects of plasma frequency on the PBG of unmagnetized plasma photonic crystals. Calculation model is the same as Fig. 1. The plasma collision frequency is chosen to be 2 GHz, which is fixed in this simulation. The relative dielectric constant is εr = 4, the plasma frequency is taken: ωp = 2 × 2 × 109 rad/s, ωp = 2 × 4 × 109 rad/s, ωp = 2 × 6 × 109 rad/s, respectively. Other parameters are the same as above. From Fig. 6, the following conclusions may be obtained. The effect of plasma frequency on plasma photonic crystal bandgap structure mainly reflects in the low frequency parts. When the plasma frequency exceeds EM wave frequency, the peaks of reflection coefficients decrease sharply with the increasing plasma frequency, especially in low EM wave frequency range, and the PBG is getting larger. The gap regions shift toward the higher frequency side appreciably with the increase of the plasma frequency. Physically, this can be explained by the fact that the isotropic plasma can act as a high-pass filter. EM wave frequencies above the plasma frequency constitute a pass band, and frequencies below the plasma frequency constitute a stop band. When the EM wave frequency is less than the plasma frequency, the EM wave is totally reflected. 4. Conclusions In summary, this paper adopts RKETD-FDTD algorithm to simulate the plasma photonic crystals structures. The reflection and transmission coefficients in frequency domain are computed, and the results of the PBG effect were discussed in terms of relative
[1] E. Yablonovitch, Inhibited spontaneous emission in solid-state physics and electronics, Phys. Rev. Lett. 58 (20) (1987) 2059–2062. [2] S. John, Strong localization of photons in certain disordered dielectric superlattices, Phys. Rev. Lett. 58 (23) (1987) 2486–2489. [3] J.D. Joannopoulos, P.R. Villeneuve, S. Fan, Photonic crystals: putting a new twist on light, Nature 386 (1997) 143–149. [4] K.M. Ho, C.T. Chen, C.M. Soukoulis, Existence of a photonic gap in periodic dielectric structures, Phys. Rev. Lett. 65 (25) (1990) 3152–3155. [5] E. Yablonovitch, Photonic band-gap structures, J. Opt. Soc. Am. B 10 (2) (1993) 283–295. [6] M. Bayindir, B. Temelkuran, E. Ozbay, Photonic-crystal-based beam splitters, Appl. Phys. Lett. 77 (24) (2000) 3902–3904. [7] M. Loncar, D. Nedeljkovic, T. Doll, J. Vuckovic, A. Scherer, T.P. Pearsall, Waveguiding in planar photonic crystals, Appl. Phys. Lett. 77 (13) (2000) 1937–1939. [8] H. Hitoshi, M. Atsushi, Dispersion relation of electromagnetic waves in onedimensional plasma photonic crystals, J. Plasma Fusion Res. 80 (2) (2004) 89–90. [9] Y. Liu, H.W. Yang, Shift operator method and Runge–Kutta exponential time differencing method for plasma, Optik Int. J. Light Electron Optik. 122 (23) (2011) 2086–2089. [10] L. Song, Z. Shuangying, L. Sanqiu, A study of properties of the photonic band gap of unmagnetized plasma photonic crystal, Plasma Sci. Technol. 11 (1) (2009) 14–17. [11] H.W. Yang, Y. Liu, Runge–Kutta exponential time differencing method analysis of non-magnetized plasma stealth, J. Infrared Millimeter Terahertz Waves 31 (9) (2010) 1075–1080. [12] S. Liu, S.-y. Zhong, S. Liu, Finite-difference time-domain algorithm for dispersive media based on Runge–Kutta exponential time differencing method, Int. J. Infrared Millimeter Waves 29 (3) (2008) 323–328. [13] D.M. Sullivan, Electromagnetic Simulation Using the FDTD Method, IEEE Press, New York, 2000. [14] A. Taflove, S.C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Artech House, Norwood, MA, 2005. [15] L. Shao-Bin, Z. Chuan-Xi, Y. Nai-Chang, FDTD simulation for plasma photonic crystals (in Chinese), Acta Phys. Sin. 54 (6) (2005) 2804–2808. [16] D.M. Sullivan, Frequency-dependent FDTD methods using Z transforms, IEEE Trans. Antennas Propag. 40 (10) (1992) 1223–1230. [17] E.W. McFarland, J. Tang, A photovoltaic device structure based on internal electron emission, Nature 421 (2003) 616–618. [18] A.O. Govorov, J. Lee, N.A. Kotov, Theory of plasmon-enhanced Forster energy transfer in optically excited semiconductor and metal nanoparticles, Phys. Rev. B 76 (12) (2007) 125308–125316. [19] M. Florescu, H. Lee, I. Puscasu, M. Pralle, L. Florescu, D.Z. Ting, J.P. Dowling, Improving solar cell efficiency using photonic band-gap materials, Sol. Energy Mater. Sol. Cells 91 (17) (2007) 1599–1610.