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Polarization properties of defect mode in one-dimensional magnetic photonic crystal
Accepted Manuscript Title: Polarization properties of defect mode in one-dimensional magnetic photonic crystal Authors: Guangbin Wu, Yanling Han, Yang Li, Hong Wang PII: DOI: Reference:
S0030-4026(17)30148-1 http://dx.doi.org/doi:10.1016/j.ijleo.2017.02.005 IJLEO 58821
To appear in: Received date: Revised date: Accepted date:
25-10-2015 3-2-2017 7-2-2017
Please cite this article as: Guangbin Wu, Yanling Han, Yang Li, Hong Wang, Polarization properties of defect mode in one-dimensional magnetic photonic crystal, Optik - International Journal for Light and Electron Optics http://dx.doi.org/10.1016/j.ijleo.2017.02.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Polarization properties of defect mode in one-dimensional magnetic photonic crystal
Guangbin Wua, Yanling Hand,Yang Lid, Hong Wangb,c
a
Science and Research Office, Shenzhen Polytechnic, Shenzhen, China b School of Automation, China University of Geosciences, Wuhan , China cHubei key Laboratory of Advanced Control and Intelligent Automation for Complex Systems, Wuhan , China d School of Mathematics and Physics, China University of Geosciences, Wuhan, China
Corresponding author E-mail address: [email protected]
Abstract Using 4 4 transfer matrix method, we investigate the polarization properties of defect mode in magnetic photonic crystal consisting of gyrotropic material. It is found that the position of the defect mode significant depend on the polarization of the incident light and the intensity of the applied magnetic field. As the intensity of the magnetic field increases, defect modes move toward lower frequency for right-circularly polarized light or higher frequency for left-circularly polarized light. Furthermore, we also find that position of the defect modes critically depend on the thickness of magnetized medium. Keywords magnetic photonic crystal; polarization of mode; Magneto-optical effect; 4 4 transfer
matrix method
1. Introduction Magnetic photonic crystals (MPCs) are spatially periodic composite structures with one of the components or even only a defect being a magnetic material, such as a ferromagnet or a ferrite [1, 2]. Over the past ten years, MPCs have been the objects of intense theoretical and experimental investigation of promising application in optoelectronics and telecommunication. Such structures exhibit very unique optical and magneto-optical properties, and many interesting properties, such as Enhanced Faraday Effect [3, 4], magnetic super-prism effect [5] and nonreciprocal or magnetically controllable photonic, are theoretically predicted [6, 7]. Among the properties of photonic crystals, the band gap is of particular interest because its properties would be fundamental problems of interpreting other optical properties. The band gap formation, local normal mode coupling and Bloch states in birefringent magneto photonic periodic stacks is studies [8-11]. Degenerate band gap periodic magneto-optic systems were analyzed [12]. Furthermore, the dynamic tuning of the band gap has also been explored. The magnetic tunability is examined by band gap analysis. The effect of dielectric permittivity or magnetic permeability on gap width, band gap center wavelength is investigated [13-18]. Such magneto photonic stop band manipulation is particularly interesting as it may have important technological implications for fast optical switching given the extremely fast magnetic
response of magneto-optic media. Recent results have shown 40fs magnetization switching effects [19]. In this paper, we discuss magnetically controllable defect modes in one-dimensional magnetic photonic crystal, which is a different mechanism of magnetic tuning particularly suited to one-dimensional magnetic photonic crystal and applicable to integrated on-chip devices. The essential cause of the effect can be traced to circular Bragg phenomenon (CBP). The possibility to control the spectra of such a structure by variation of the external magnetic field is demonstrated. It is shown that that the position of the defect mode significant depend on the polarization of the incident electromagnetic light and the intensity of the applied magnetic field. Specifically, it is shown that as the intensity of the magnetic field increases, defect modes move toward lower frequency for RCP wave or higher frequency for LCP wave. Furthermore, we find that position of the defect modes also depend on the thickness of magnetized medium. 2. Materials and theoretical model A schematic of the magnetic resonance cavity structure is illustrated in Fig. 1, which consists of alternating non-magnetic dielectric layers infiltrated with magnetic layer, i.e., AB m C ( AB) m . The indices A and B correspond to two different dielectric layers with relative permittivity 1 and 2 , respectively. The thickness of A, B and C are d A , d B and d C , respectively. In general, for magnetic materials both dielectric permittivity and relative permeability are tensor quantities. The permittivity tensors ij for a gyrotropic magnetic medium magnetized C along the z-axis can be represented as follows [20]: xx i xy 0 i xy xx 0 , 0 0 zz
(1)
where xx , yy and zz are real and xx yy . The xy defines the gyrotropy factor responsible for magneto-optic effect
(circular birefringence)
and
in
first
approximation linear with the magnetization vector, corresponding to either uniaxial or isotropic magnetic material placed in a magnetic field parallel to the z axis. In the optical wavelength regime, the relative permeability in most of the practice cases is reduced to a scalar which is the permeability of free space 0 . The 4 4 transfer matrix method is effective in describing the optical properties of multilayered structures, which has been widely applied to uniaxial, biaxial dielectric and magnetic material [21, 22]. Based on the Maxwell equations for a monochrome light propagating in the medium, we have d z i z z , dz c
(2)
where the complex amplitude of electric and magnetic field components are arranged in a column forms:
z E x ( z ), E y z , H x z , H y z T ,
and the superscript T denotes the
transpose operator, is the angular frequency of the incident wave and c is the light speed in vacuum.
z
is the function of permittivity, permeability and incident
wave vector, its elements for permittivity tensors given in (1) is 1 1 X 2 zz 0 0 xy xx 0 0 0 0 xx X 2 xy
0 0 , 1 0
(3)
where, X sin air and air is incidence angle. The total transfer matrix in the present configuration is P PA PB PC PA PB m
m
,
(4)
where PA , PB and PC are the propagation matrices for A, B and C layers, respectively. The propagation matrix within each homogeneous layer with thickness h can be obtained by direct calculation
P( z h) SGS 1 , where
(5)
ex1 e x 2 u y2 u S y1 e ey 2 y1 u x1 u x 2 eik 0 1h 0 G 0 0
ex 3 ex 4 u y3 u y 4 , ey 3 ey 4 u x 3 u x 4 0 e
0 0
ik 0 2 h
eik 0 3 h 0
0 0
(6)
. ik 0 4 h e 0 0 0
(7)
Here, the e , u and , 1,2,3,4 are elements of the eigenvector and eigenvalue of the matrix in Eq.(3), respectively[23]. With the above results, we can calculate the transmission coefficient T by following relation [24] E xt cos 2 air E yt
2
E xi cos air E yi
2
2
T
2
2
.
(8)
Here, incidence angle air 0 in our all simulation. Using the methods proposed above, various cases could be calculated. In this paper, we focus on calculating the transmission spectrum to check phenomenon of defect-modes splitting in magnetic photonic crystal.
Fig.1 3. Results and discussions To form the band structure, the dielectric permittivity is assumed to be 1 2 , and 1 9 , 2 3 .The corresponding thickness is d A 0.403m , d B 0.556m . For magneto-optic material, we assume the xx 3 , zz 9 and dC 2d A . In the present calculations, we use the values of material parameters Y 2.5Bi 0.5 Fe5O12 films which were given by Inoue et al. [25], the off-diagonal dielectric element xy varied from 0 to 0.1.
Before studying the polarization properties of defect mode in the presence of external magnetic fields, we start with the characterization of a layered structure without magnetization, i.e., xy 0 , xx 2 and zz 1 . We consider RCP and LCP wave propagating normally to the above layered structure. By compare, we easily find that the peaks for different polarized light wave are overlapping each other, and located in the middle of the photonic band gap. The resonant frequency of defect mode is determined as a function of the thickness and the relative permittivity of the defect layer.
Fig.2 Next, Let us study the nature of the corresponding defect mode in the presence of external magnetic fields i.e., xy 0 , xx 2 and zz 1 . We repeated the calculation of similar structures with different parameter value xy . The corresponding spectrum in Fig.3 exhibits a significant dependence of defect modes on the polarization state of a normally incident electromagnetic wave. It was found that the defect mode will shift in the opposite direction with increase of gyrotropy factor xy for different polarized incident light. Specifically, as the intensity of the magnetic field increases, defect modes move toward lower frequency for LCP wave or higher frequency for RCP wave.
Fig.3. In Fig.4, We give the dependence of the transmission peaks on gyrotropy factor
xy for RCP and LCP wave. It is found that resonance frequency of defect mode linearly increase with xy for right-circularly polarized light or linearly decrease with
xy for left-circularly polarized light.
Fig.4 The polarization properties of defect mode results from circular Bragg phenomenon (CBP) [22].The electromagnetic eigenmodes of the layered structure with permittivity tensors given in (1) are all circularly polarized. Generally, the Bragg phenomenon is independent of polarization of a normally incident electromagnetic wave. In this case, the RCP and LCP wave is degenerate, which implies defect modes are no independent of RCP and LCP wave. However, this conclusion holds only for isotropic material i.e. xy 0 . For Anisotropic material, i.e. xy 0 , incident electromagnetic plane waves of LCP and RCP waves are reflected and transmitted differently in the Bragg wavelength-regime, For LCP and RCP wave, in the presence of external magnetic fields, corresponding refractive indices is n2 xx xy , where n is for RCP wave and n for LCP wave. From this equation, we can see that n increases and n decrease in the presence of external magnetic fields. In such a case, defect-mode for RCP and LCP wave move toward lower and higher frequency, respectively, according to the fundamental Bragg condition 2n . In addition to gyrotropy factor xy , we have checked effect of thickness of magnetized material on the polarization of defect mode. If defect layer can be fractional magnetized, we define a parameter
f dm dC to describe the
magnetization of the defect layer. The f 1 indicates that defect layer is fully magnetized and f 0 is not magnetized. We repeated the calculation of similar structures with different value of f . The corresponding result in Fig. 5 indicates that
polarization of defect modes critically depends on thickness of magnetization. It was shown that the resonant frequency of defect modes moves toward lower frequency or higher frequency with increasing of parameter value f for RCP or LCP wave.
Fig. 5 4. Conclusion In summary, using 4 4 transfer matrix method, we investigate the polarization properties of defect mode in MPCs consisting of gyrotropic material. It is found that the position of the defect mode depend on the polarization of the incident light and the intensity of the applied magnetic field. Specifically, as the intensity of the magnetic field increases, defect modes move toward lower frequency for RCP wave or higher frequency for LCP wave. Furthermore, we also find that position of the defect modes critically depend on the thickness of magnetized medium. On other hand, the tunability is particularly useful for the fabrication of photonic integrated devices, such as tunable filters, optical switches and gates. Based on one-dimensional photonic crystal made of nematic liquid crystal (LC), the electric tunable filters has been proposed [26]. Comparing with Wang’s research [27], a similar conclusion can be obtained that it is possible to effectively control the spectrum of such a structure by an external magnetic field, but our results further confirm that the polarimetric characteristic of incident light is another factor to control the spectrum of magnetic photonic crystal. That is to say, our results present another approach to realize tunable filters by using MPCs. Acknowledgments This work was supported by Natural Science Foundation of Hubei Province (Grant N0:2008CHB413), National Natural Science Foundation of China (Grant No. 60878037)
References [1]
M. Inoue, R. Fujikawa, A. Baryshev, et al, Magnetophotonic crystals, J. Phys. D: Appl. Phys. 39 (2006) R151–R161
[2]
I. L. Lyubchanskii, N. N. Dadoenkova, M. I. Lyubchanskii, et al, Magnetic photonic crystals J. Phys. D: Appl. Phys. 36 (2003) R277–R287
[3]
M. Sadatgol, M. Rahman, E. Forati,et al, Enhanced Faraday rotation in hybrid magneto-optical metamaterial structure of bismuth-substituted-iron-garnet
with
embedded-gold-wires,
Article in Journal
of
Applied
Physics
119(2016)103105 -103110 [4]
R. Fujikawa, Alexander V Baryshev, Alexander Khanikaev et al, Enhancement of Faraday rotation in 3D/Bi:YIG/1D photonic heterostructures, Journal of Materials Science Materials in Electronics 209(2008)493-497
[5]
Alexander M Merzlikin，A. P. Vinogradov，M. Inoue，et al，Superprism effect in one-dimensional magnetic photonic crystals. Physics of the Solid State 50(2008)873-877.
[6]
A. Figotin, I. Vitebsky, Nonreciprocal magnetic photonic crystals, Phys.Rev. E, 63(2001) 066609-066625
[7]
D. Szaller, S. Bordacs, I. Kezsmarki, Symmetry conditions for nonreciprocal light propagation in magnetic crystals. Phys.Rev. B,87(2013) 014421-014426
[8]
M.Levy, A.A.Jalali, J.Opt. Soc.Am.B Band structure and Bloch states in birefringentone-dimensional magnetophotonic crystals:an analytical approach,24(2007)1603-1609
[9]
M.Levy, Normal modes and birefringent magnetophotonic crystals,J.Appl.Phys.99(2006),073104-073108
[10] M.Levy, A.A.Jalali,Z.Zhou,N.Dissanayake, Bandgap formation and selective suppression of Bloch states in birefringent gyrotropic Bragg waveguides,Opt.Express 16(2008),13421-13430 [11] A.A.Jalali, M.Levy , Local Normal Mode Coupling and Energy Band Splitting in Elliptically Birefringent 1D Magnetophotonic Crystals,J.Opt. Soc.Am.B 25(2008),119-133 [12] A.M.Merzlikin,M.Levy,A.A.Jalali,A.P.Vinogradov, Polarization degeneracy at Bragg reflectance in magnetized photonic crystals,Phys.Rev.B 79(2009),195103-195109 [13] F. Wang, A.Lakhtakia, controllable intra-Brillouin-zone band gaps in one-dimensional helicoidal magnetophotonic crystals Phys.Rev.B 79(2009),193102-193105 [14] A.Figotin, Y.A.Godin, I.Vitebskiy, Two-dimensional tunable photonic crystals,Phys.Rev.B 57(1998),2841-2848 [15] C.Xu,X.Hu,Z.Li,X.Liu,R.Fu,J.Zi, Semiconductor-based tunable photonic crystals by means of an external magnetic field,Phys.Rev.B 68(2003),193201-193204 [16] H.Chen,C.T.Chan,S.Liu,Z.Lin, A simple route to a tunable electromagnetic gateway,New J.Phys.11(2009),083012-083024 [17] I. L. Lyubchanskii, N. N. Dadoenkova, M. I. Lyubchanskii, One-dimensional bigyrotropic magnetic photonic crystals, Appl.Phys.Lett.85(2004)5932-5934 [18] W. Hong, W. Guangjun, H.Yanling, Polarization gaps in one-dimensional magnetic photonic crystal, Optics Communications,310(2014)199–203 [19] C.D.Stanciu,F.Hansteen,A.V.Rimel,A.Kirilyuk,A.Tsukamoto,A.Itoh,Th.Rasing, All-Optical Magnetic Recording with Circularly Polarized Light,Phys.Rev.Lett.99(2007),047601-047604 [20] H. Hajian, B. Rezaei, A. Soltani Vala, M. Kalafi, Tuning of switching by means of external magnetic fields, Physica B 405 (2010) 2996 -2999 [21] I. Abdulhalim, Analytic propagation matrix method for anisotropic magnetic-optic layered media, J.Opt. A 2 (2000) 557-564 [22] A. Lakhtakia, J.A. Reyes, Theory of electrically controlled exhibition of circular Bragg phenomenon by an obliquely excited structurally chiral material-Part 1: Axial de electric field, Optik 119 (2008) 253-259 [23] Y. Pochi, Electromagnetic propagation in birefringent layered media, J. Opt. 69 (1979) 742-756 [24] C. J. Chen, A. Lien, M. I. Nathan, 4 4 and 2 2 matrix formulations for the optics in stratified and biaxial media, Opt. A 14 (1997) 3125-3134
[25] M. Inoue, K. Isamoto, T. Yamamoto, T. Fujii, Magneto-optical Faraday effect of discontinuous magnetic media with a one-dimensional array structure, J. Appl. Phys. 79 (1996) 1611-1614 [26] C. J. Chen, A. Lien, Tunable band gap in a photonic crystal modulated by a nematic liquid crystal, Phys. Rev. B 72 (2005) 045133-045137 [27] F. Wang, A. Lakhtakia,Magnetically controllable intra-Brillouin-zone band gaps in one-dimensional helicoidal magnetophotonic crystals Phys. Rev. B 79(2009),193102-193105
List of Figure Captions [1]
[2]
Fig.1 Schematic diagram of magnetic resonance cavity C sandwiched between a pair of identical periodic non-magnetic stacks (Bragg reflectors). Fig.2 Transmission spectrum for non-magnetic structure depicted in Fig. 1, for differently polarized incident light, d A d B , m 7 (a) RCP wave (b) LCP wave with thickness of defect layer dC 2d A
[3]
Fig.3 Transmission spectra of magnetic structure with different gyrotropy factors xy for LCP wave, (a) xy 0.01 , (b) xy 0.02 , (c) xy 0.03 and for RCP wave (d) xy 0.01 , (e) xy 0.02 , (f) xy 0.03 .
[4]
Fig.4 Dependence of the resonance transmission peaks on gyrotropy factor xy for RCP and LCP wave
[5]
Fig. 5 Dependence of the resonance transmission peaks on parameter f for RCP and LCP wave, and xy 0.02 .
Fig 1
Fig.2
Fig.3
Frequency (c)
0.46
0.44
LCP
0.42 RCP
0.40 0.00
0.02
0.04
xy
0.06
Fig.4
Frequency (c)
0.45 0.44
LCP
0.43 0.42 0.41
RCP
0.40 0.0
Fig. 5
0.2
0.4
0.6 f(%)
0.8
1.0
S0030-4026(17)30148-1 http://dx.doi.org/doi:10.1016/j.ijleo.2017.02.005 IJLEO 58821
To appear in: Received date: Revised date: Accepted date:
25-10-2015 3-2-2017 7-2-2017
Please cite this article as: Guangbin Wu, Yanling Han, Yang Li, Hong Wang, Polarization properties of defect mode in one-dimensional magnetic photonic crystal, Optik - International Journal for Light and Electron Optics http://dx.doi.org/10.1016/j.ijleo.2017.02.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Polarization properties of defect mode in one-dimensional magnetic photonic crystal
Guangbin Wua, Yanling Hand,Yang Lid, Hong Wangb,c
a
Science and Research Office, Shenzhen Polytechnic, Shenzhen, China b School of Automation, China University of Geosciences, Wuhan , China cHubei key Laboratory of Advanced Control and Intelligent Automation for Complex Systems, Wuhan , China d School of Mathematics and Physics, China University of Geosciences, Wuhan, China
Corresponding author E-mail address: [email protected]
Abstract Using 4 4 transfer matrix method, we investigate the polarization properties of defect mode in magnetic photonic crystal consisting of gyrotropic material. It is found that the position of the defect mode significant depend on the polarization of the incident light and the intensity of the applied magnetic field. As the intensity of the magnetic field increases, defect modes move toward lower frequency for right-circularly polarized light or higher frequency for left-circularly polarized light. Furthermore, we also find that position of the defect modes critically depend on the thickness of magnetized medium. Keywords magnetic photonic crystal; polarization of mode; Magneto-optical effect; 4 4 transfer
matrix method
1. Introduction Magnetic photonic crystals (MPCs) are spatially periodic composite structures with one of the components or even only a defect being a magnetic material, such as a ferromagnet or a ferrite [1, 2]. Over the past ten years, MPCs have been the objects of intense theoretical and experimental investigation of promising application in optoelectronics and telecommunication. Such structures exhibit very unique optical and magneto-optical properties, and many interesting properties, such as Enhanced Faraday Effect [3, 4], magnetic super-prism effect [5] and nonreciprocal or magnetically controllable photonic, are theoretically predicted [6, 7]. Among the properties of photonic crystals, the band gap is of particular interest because its properties would be fundamental problems of interpreting other optical properties. The band gap formation, local normal mode coupling and Bloch states in birefringent magneto photonic periodic stacks is studies [8-11]. Degenerate band gap periodic magneto-optic systems were analyzed [12]. Furthermore, the dynamic tuning of the band gap has also been explored. The magnetic tunability is examined by band gap analysis. The effect of dielectric permittivity or magnetic permeability on gap width, band gap center wavelength is investigated [13-18]. Such magneto photonic stop band manipulation is particularly interesting as it may have important technological implications for fast optical switching given the extremely fast magnetic
response of magneto-optic media. Recent results have shown 40fs magnetization switching effects [19]. In this paper, we discuss magnetically controllable defect modes in one-dimensional magnetic photonic crystal, which is a different mechanism of magnetic tuning particularly suited to one-dimensional magnetic photonic crystal and applicable to integrated on-chip devices. The essential cause of the effect can be traced to circular Bragg phenomenon (CBP). The possibility to control the spectra of such a structure by variation of the external magnetic field is demonstrated. It is shown that that the position of the defect mode significant depend on the polarization of the incident electromagnetic light and the intensity of the applied magnetic field. Specifically, it is shown that as the intensity of the magnetic field increases, defect modes move toward lower frequency for RCP wave or higher frequency for LCP wave. Furthermore, we find that position of the defect modes also depend on the thickness of magnetized medium. 2. Materials and theoretical model A schematic of the magnetic resonance cavity structure is illustrated in Fig. 1, which consists of alternating non-magnetic dielectric layers infiltrated with magnetic layer, i.e., AB m C ( AB) m . The indices A and B correspond to two different dielectric layers with relative permittivity 1 and 2 , respectively. The thickness of A, B and C are d A , d B and d C , respectively. In general, for magnetic materials both dielectric permittivity and relative permeability are tensor quantities. The permittivity tensors ij for a gyrotropic magnetic medium magnetized C along the z-axis can be represented as follows [20]: xx i xy 0 i xy xx 0 , 0 0 zz
(1)
where xx , yy and zz are real and xx yy . The xy defines the gyrotropy factor responsible for magneto-optic effect
(circular birefringence)
and
in
first
approximation linear with the magnetization vector, corresponding to either uniaxial or isotropic magnetic material placed in a magnetic field parallel to the z axis. In the optical wavelength regime, the relative permeability in most of the practice cases is reduced to a scalar which is the permeability of free space 0 . The 4 4 transfer matrix method is effective in describing the optical properties of multilayered structures, which has been widely applied to uniaxial, biaxial dielectric and magnetic material [21, 22]. Based on the Maxwell equations for a monochrome light propagating in the medium, we have d z i z z , dz c
(2)
where the complex amplitude of electric and magnetic field components are arranged in a column forms:
z E x ( z ), E y z , H x z , H y z T ,
and the superscript T denotes the
transpose operator, is the angular frequency of the incident wave and c is the light speed in vacuum.
z
is the function of permittivity, permeability and incident
wave vector, its elements for permittivity tensors given in (1) is 1 1 X 2 zz 0 0 xy xx 0 0 0 0 xx X 2 xy
0 0 , 1 0
(3)
where, X sin air and air is incidence angle. The total transfer matrix in the present configuration is P PA PB PC PA PB m
m
,
(4)
where PA , PB and PC are the propagation matrices for A, B and C layers, respectively. The propagation matrix within each homogeneous layer with thickness h can be obtained by direct calculation
P( z h) SGS 1 , where
(5)
ex1 e x 2 u y2 u S y1 e ey 2 y1 u x1 u x 2 eik 0 1h 0 G 0 0
ex 3 ex 4 u y3 u y 4 , ey 3 ey 4 u x 3 u x 4 0 e
0 0
ik 0 2 h
eik 0 3 h 0
0 0
(6)
. ik 0 4 h e 0 0 0
(7)
Here, the e , u and , 1,2,3,4 are elements of the eigenvector and eigenvalue of the matrix in Eq.(3), respectively[23]. With the above results, we can calculate the transmission coefficient T by following relation [24] E xt cos 2 air E yt
2
E xi cos air E yi
2
2
T
2
2
.
(8)
Here, incidence angle air 0 in our all simulation. Using the methods proposed above, various cases could be calculated. In this paper, we focus on calculating the transmission spectrum to check phenomenon of defect-modes splitting in magnetic photonic crystal.
Fig.1 3. Results and discussions To form the band structure, the dielectric permittivity is assumed to be 1 2 , and 1 9 , 2 3 .The corresponding thickness is d A 0.403m , d B 0.556m . For magneto-optic material, we assume the xx 3 , zz 9 and dC 2d A . In the present calculations, we use the values of material parameters Y 2.5Bi 0.5 Fe5O12 films which were given by Inoue et al. [25], the off-diagonal dielectric element xy varied from 0 to 0.1.
Before studying the polarization properties of defect mode in the presence of external magnetic fields, we start with the characterization of a layered structure without magnetization, i.e., xy 0 , xx 2 and zz 1 . We consider RCP and LCP wave propagating normally to the above layered structure. By compare, we easily find that the peaks for different polarized light wave are overlapping each other, and located in the middle of the photonic band gap. The resonant frequency of defect mode is determined as a function of the thickness and the relative permittivity of the defect layer.
Fig.2 Next, Let us study the nature of the corresponding defect mode in the presence of external magnetic fields i.e., xy 0 , xx 2 and zz 1 . We repeated the calculation of similar structures with different parameter value xy . The corresponding spectrum in Fig.3 exhibits a significant dependence of defect modes on the polarization state of a normally incident electromagnetic wave. It was found that the defect mode will shift in the opposite direction with increase of gyrotropy factor xy for different polarized incident light. Specifically, as the intensity of the magnetic field increases, defect modes move toward lower frequency for LCP wave or higher frequency for RCP wave.
Fig.3. In Fig.4, We give the dependence of the transmission peaks on gyrotropy factor
xy for RCP and LCP wave. It is found that resonance frequency of defect mode linearly increase with xy for right-circularly polarized light or linearly decrease with
xy for left-circularly polarized light.
Fig.4 The polarization properties of defect mode results from circular Bragg phenomenon (CBP) [22].The electromagnetic eigenmodes of the layered structure with permittivity tensors given in (1) are all circularly polarized. Generally, the Bragg phenomenon is independent of polarization of a normally incident electromagnetic wave. In this case, the RCP and LCP wave is degenerate, which implies defect modes are no independent of RCP and LCP wave. However, this conclusion holds only for isotropic material i.e. xy 0 . For Anisotropic material, i.e. xy 0 , incident electromagnetic plane waves of LCP and RCP waves are reflected and transmitted differently in the Bragg wavelength-regime, For LCP and RCP wave, in the presence of external magnetic fields, corresponding refractive indices is n2 xx xy , where n is for RCP wave and n for LCP wave. From this equation, we can see that n increases and n decrease in the presence of external magnetic fields. In such a case, defect-mode for RCP and LCP wave move toward lower and higher frequency, respectively, according to the fundamental Bragg condition 2n . In addition to gyrotropy factor xy , we have checked effect of thickness of magnetized material on the polarization of defect mode. If defect layer can be fractional magnetized, we define a parameter
f dm dC to describe the
magnetization of the defect layer. The f 1 indicates that defect layer is fully magnetized and f 0 is not magnetized. We repeated the calculation of similar structures with different value of f . The corresponding result in Fig. 5 indicates that
polarization of defect modes critically depends on thickness of magnetization. It was shown that the resonant frequency of defect modes moves toward lower frequency or higher frequency with increasing of parameter value f for RCP or LCP wave.
Fig. 5 4. Conclusion In summary, using 4 4 transfer matrix method, we investigate the polarization properties of defect mode in MPCs consisting of gyrotropic material. It is found that the position of the defect mode depend on the polarization of the incident light and the intensity of the applied magnetic field. Specifically, as the intensity of the magnetic field increases, defect modes move toward lower frequency for RCP wave or higher frequency for LCP wave. Furthermore, we also find that position of the defect modes critically depend on the thickness of magnetized medium. On other hand, the tunability is particularly useful for the fabrication of photonic integrated devices, such as tunable filters, optical switches and gates. Based on one-dimensional photonic crystal made of nematic liquid crystal (LC), the electric tunable filters has been proposed [26]. Comparing with Wang’s research [27], a similar conclusion can be obtained that it is possible to effectively control the spectrum of such a structure by an external magnetic field, but our results further confirm that the polarimetric characteristic of incident light is another factor to control the spectrum of magnetic photonic crystal. That is to say, our results present another approach to realize tunable filters by using MPCs. Acknowledgments This work was supported by Natural Science Foundation of Hubei Province (Grant N0:2008CHB413), National Natural Science Foundation of China (Grant No. 60878037)
References [1]
M. Inoue, R. Fujikawa, A. Baryshev, et al, Magnetophotonic crystals, J. Phys. D: Appl. Phys. 39 (2006) R151–R161
[2]
I. L. Lyubchanskii, N. N. Dadoenkova, M. I. Lyubchanskii, et al, Magnetic photonic crystals J. Phys. D: Appl. Phys. 36 (2003) R277–R287
[3]
M. Sadatgol, M. Rahman, E. Forati,et al, Enhanced Faraday rotation in hybrid magneto-optical metamaterial structure of bismuth-substituted-iron-garnet
with
embedded-gold-wires,
Article in Journal
of
Applied
Physics
119(2016)103105 -103110 [4]
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List of Figure Captions [1]
[2]
Fig.1 Schematic diagram of magnetic resonance cavity C sandwiched between a pair of identical periodic non-magnetic stacks (Bragg reflectors). Fig.2 Transmission spectrum for non-magnetic structure depicted in Fig. 1, for differently polarized incident light, d A d B , m 7 (a) RCP wave (b) LCP wave with thickness of defect layer dC 2d A
[3]
Fig.3 Transmission spectra of magnetic structure with different gyrotropy factors xy for LCP wave, (a) xy 0.01 , (b) xy 0.02 , (c) xy 0.03 and for RCP wave (d) xy 0.01 , (e) xy 0.02 , (f) xy 0.03 .
[4]
Fig.4 Dependence of the resonance transmission peaks on gyrotropy factor xy for RCP and LCP wave
[5]
Fig. 5 Dependence of the resonance transmission peaks on parameter f for RCP and LCP wave, and xy 0.02 .
Fig 1
Fig.2
Fig.3
Frequency (c)
0.46
0.44
LCP
0.42 RCP
0.40 0.00
0.02
0.04
xy
0.06
Fig.4
Frequency (c)
0.45 0.44
LCP
0.43 0.42 0.41
RCP
0.40 0.0
Fig. 5
0.2
0.4
0.6 f(%)
0.8
1.0