- Email: [email protected]
dielectric nonlinear metamaterial multilayers
Accepted Manuscript Title: Optical multistability in metal/dielectric nonlinear metamaterial multilayers Author: Zhaohong Li Huajun Zhao PII: DOI: Reference:
S0030-4026(16)30586-1 http://dx.doi.org/doi:10.1016/j.ijleo.2016.05.133 IJLEO 57759
To appear in: Received date: Accepted date:
29-3-2016 28-5-2016
Please cite this article as: Zhaohong Li, Huajun Zhao, Optical multistability in metal/dielectric nonlinear metamaterial multilayers, Optik - International Journal for Light and Electron Optics http://dx.doi.org/10.1016/j.ijleo.2016.05.133 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.
Optical multistability in metal/dielectric nonlinear metamaterial multilayers *
Zhaohong Li 1,2 ,Huajun Zhao 1
college of Electronic and Electrical Engineering, Chongqing University of Arts and Sciences, Chongqing 402160, China 2
Chongqing Key Laboratory of Micro/Nano Materials Engineering and Technology, Chongqing 402160, China Corresponding e-mail address :
[email protected]
The bistable properties are investigated in a composite optical configuration based on Fibonacci sequence and nonlinear material. Bistablility is realized near the wavelength of perfect transmission resonance. A small threshold of the switch is attained by introducing a bulk nonlinear defect in the matamaterial photonic bandgap system. Meanwhile, the small offset to the defect mode, the lower threshold of the switch is. When the thickness of the defect layer decreases, multistability is easy to appear if the localized electric field distributed in the form of several isolated centers. Some others, such as the transmission of the linear multilayer and the bistability of band edge states are also calculated by the transfer-matrix method.
Keywords: Multistability; Quasicrystals; Kerr effects.
1.Introduction Photonic crystals (PCs) have been considered as one of the most research topic since the pioneering works of Yablonovithc[1] and John[2] for the fascinating properties. Bistability is one of phenomena which can be realized in the nonlinear system. The nonlinear features between the input and output intensity which manifest a hysteresis loop is a basis of the optical switch device in an optical communication system. As a key component of integrated optics to control the photon transportation, optical switch has inspired considerable interest in recent years. In modern all-optical integrated circuits there is one main approach to realize the optical switch operation. One is to use nonlinear optical effects, such as Kerr effect, free-carrier effect, Black phosphorus (BP) nanosheet, thermo-optic effect, PC waveguide, and so on[3-9]. as far as the promising way to realize optical switching , there are two requirements to operate the nonlinear PCs to engender bistability. One is the strong nonlinearity to enhance the transmission, another is the spatially asymmetric to coupling between the resonance states. Hence, the structure is important to the performance of devices[10]. Quasicrystal is a intermediate stage between traditional periodic photonic and disorder systems [11], effectively containing both characteristic of bandgap and localization properties due to short-range disorder. Fibonacci is the typical quasicrystal and has earned more attention in the past years. The linear transmission spectrum of such photonic structures was shown to have a perfect transmission resonance [12]. Our interest in the Fibonacci multilayer is inspired by the fact that the linear transmission curve possesses rich variety of resonances ranging from very narrow to rather broad, furthermore, it is easier to have bistability in the condition of the narrower resonance [13]. In this paper, we present a composite optical configuration formed by two fifth generation Fibonacci sequence and a nonlinear defect. The perfect transmission resonance and bistability properties of are studied by the nonlinear transfer-matrix method. Though some progress in the transmission resonance optical properties have been made in symmetry[14] and asymmetry[15] systems, the bistability/multibistabiliy are hardly reported in metamatertial structures. Our purpose is to study the perfect transmission and bistability in the fifth generation Fibonacci nonlinear composite multilayer structure. 2.Structure and Theory The Fibonacci quasiperiodic structure is built of two types of arbitrary dielectric layers indicated with letters A and B. The Fibonacci sequence is a recurse by the rule sn sn 1sn 2 ,for n 2 ,with s1 A and s2 AB . Consequently, the fifth generation is ABAABABA . We stack two fifth generation Fibonacci
sequences with a nonlinear layer placed in center Fig.1, which a one-dimensional multilayer is stacked in the form of ABAABABACABAABABA . The background dielectric parameters are 0 0 1 . The permittivity and permeability of metamaterial layers A are assumed to be A 1.5 and
A -12.67 respectively. Layers B and C are the Kerr-nonlinear material. The permittivity of B and C are B 2.25 and C 5 with permeability B C 1 . The parameters mentioned above is invariant except the thickness of the C slab in this paper. The thickness of the three materials A , B and C are
0 / 4 A A
, 0.50 , 30 with constant 0 532 nm respectively. For simplicity, the
absorption of the nonlinear material is ignored.
Fig.1. (Color online) Schematic diagram of the asymmetry multilayer with the sequence of ABAABABACABAABABA. Orange represent negative refractive metamaterial layers and White represent the two Kerr-nonlinear lays B and C respectively. TE wave propagates from the negative z direction.
Let a TE polarized plane wave be normal incident from the left of the structure and propagating along with the Z direction (Fig.1). Therefore, we can numerically calculate the electric and magnetic fields by the characteristic matrix of each layer. The matrix of the metamaterial layers A in the form: [16] cos( K A d A ) MA i sin( K A d A )
i
sin( K A d A ) cos( K A d A )
(1)
Where the ratio coefficient is A / A and the wave vector K A is ( / c) A A in the media. The transfer matrix of the Kerr-nonlinear layers B and C are shown as:
MN
K0 Kz Kz
Kz Kz exp(iK z d N ) exp(iK z d N ) K exp(iK z d N ) K exp(iK z d N ) 0 0 Kz Kz Kz Kz [exp(iK z d N ) exp(iK z d N )] exp(iK z d N ) exp(iK z d N ) 2 K K K 0 0 0
(2)
where N is the number of the layer. K z and K z are the forward and backward propagating the wave vector. For a given transmitted intensity I t we calculated in the first nonlinear layer from the right end. In the first nonlinear slab the relation between I t and K z are defined by the following two expressions:
K1z K0 N (1 I1 2I1 )
I1 I 1
1 K 1 K 0
(3) 2
1 1 K1 M A I t 0 K 0
(4)
, where I t At .The and At are the Kerr-nonlinear coefficient and the amplitude of the 2
transmitted intensity. We can obtain the K z and nonlinear layer matrix M B by (3) and (4) using the Newton iterative methodfor a fixed transmitted intensity I t . The matrix of the first nonlinear slab is obtained by formula (2). Similarly, the I 2 in the second nonlinear layer which is on the right hand of the structure is expressed as: I 2 I 2
1 K 2 K 0
2
1 1 K 2 M A M B M A I t 0 K 0
(5)
At last we can calculate all the characteristic matrix of nonlinear layers in such way. Once the matrix of the individual layer is defined, the transfer matrix of the composite structure is obtained as: (6) M M A M B ...M A M C M A ...M B M A Hence, we can computed the transfer matrix elements and the transmittance is given by: 2n0 T (n0 M 22 M 21 ) (n0 M12 M11 )
2
(7)
Where n0 is the refractive index of the background defined by 0 0 . Once the transmittance is known, the corresponding incident intensity I in can be expressed I t / T . 3.Numerical Results The linear field profiles and transmission spectra calculated using the transfer-matrix method is shown in Fig.2. There are five distinct defect modes with high transmittance ranging from 550 nm to 850 nm. As we all know, bulk material can introduce a defect mode into the multilayer photonic bandgap system and hence an incident wave can pass through the structure if the incident frequency is tuned at the linear defect mode. The refractive index of Kerr nonlinear material is independent of field intensity, giving rise to self-phase modulation, therefore, the corresponding defect mode wavelength will shift with the local electric field. If the defect modes move to the incident wavelength, the optical bistability will appear. An S-shape bistable curve is shown in Fig.2 (b) with an incident wavelength 688 nm near the resonance mode 686.7 nm noted by point P. The switch-up and switch-down point are noted with U and D point, respectively. The corresponding incident intensities are 0.1693 and 0.0897 (a.u.). The electric field distribution of the perfect transmission resonance wavelength noted by arrowhead in Fig.2 (a) is shown by blue solid line in Fig.2 (b) and it is evident that electric intensity is mainly localized in the center of the structure.
Fig.2. (a)Transmission spectrum of the multilayer with the thickness dc 30 .The matamaterial refractive indices are
A -12.68
and
A 1.5 ,
wavelength of 688 nm.
(b) the bistable behavior of the transmission intensity verses the incident intensity at
An S-shape bistable curve is showed noted by the switch point U and D respectively. The inset in (b) is
the electric field distribution of resonance mode P in (a) taking no account of Kerr coefficient.
There are abundant low transmitted states near the defect modes. In order to explore the optical bistable properties of these band edge modes, four wavelengths near the band edge modes are chosen to impinge on the multilayer. The mode behaviors of the incident wavelengths are not all the same. Wavelength 805nm, which is near a perfect transmission resonance, takes on a multistablility with the
increase in incident intensity shown with red solid line in Fig.3. The reason to this phenomenon is the interplay between the isolated localized district in the structure[17]. The other three wavelengths are also provided with bistable property. We can see that some band edge states still have the ability to shift the defect mode even though the transmittance is not high enough, but due to the low transmittance it is difficult to engender the multibistable behavior.
Fig.3. The bistable behavior of the transmission intensity verses the incident intensity at the wavelength 805 nm: red solid line, 796 nm: blue dashed line, 787 nm: blue dotted line, 778 nm: blue solid line, respectively.
Comparing the defined parameter of thickness d C and d B , we change the thickness d C to 0.50 to enhance the asymmetry of the multilayer to a certain degree while other parameters have no change , the transmission spectrum of several resonances ranging from 300 nm to 900 nm is showed in Fig.4 (a) taking no account of Kerr coefficient. There are three resonances
at wavelength 341.9, 386.8,
744.1nm, respectively. They are perfect transmission, whereas the width of the resonance mode differs. Obviously the linear defect mode 341.9 nm and 386.8 nm is narrower than the one at wavelength of 744.1 nm. First, we calculate the transmitted behavior at the wavelength of 342.4 nm (red solid line), 387.6 nm (blue solid line), and 745.5 nm (blue dashed line) respectively, which are located near the resonances. The variations of incident intensity with the transmitted intensity are shown in Fig.4 (b). Multibistabilty appears with the incident wavelength 342.4 nm and 387.6 nm. Multibistabilty occurs easily shown in Fig.4.(b). The thresholds of the lower incident wavelength 342.4 nm is smaller than the other two wavelength 387.6 nm and 745.5 nm, and the deviation from the linear resonance mode is 0.5nm ,0.8nm and 1.4nm respectively. A conclusion can be drawn that the smaller deviation from the linear defect mode and the narrower the resonance is, the easier it is to have bistability. it coincides with Ref. [18].
Fig.4. (a) The transmission spectrum for the linear multilayer with the thickness dc 0.50 . (b) The bistable / multibistable behavior of transmitted intensity versus the incident intensity for the three incident wavelengths 342.4 nm: red solid line, 387.6nm: blue solid line, 745.5nm: blue dashed line, respectively.
To better illustrated the field distribution the field profiles are demonstrated along with position of the linear multilayer in Fig.5. It is evident that the energy is scatted distributed with one more localization centers for the wavelength 342.4nm and 387.6nm, while it is not strong localized center for the mode wavelength 745.5 nm. The reason to the multistability in multilayers system is the interplay between these localization centers. In addition, the field of wavelength 342.4 nm is strongly localized and the enhancement is as large as an order magnitude.
Fig.5. The field profiles at the different resonant wavelength along with the linear multilayer, red solid line: 342.4 nm, blue solid line: 387.6nm, blue dashed line: 745.5nm.
4. Conclusion In summary, we have investigated the bistable behavior in a composite optical configuration based on Fibonacci sequence and nonlinear material. An S-shape bistability curve is shown which can be achieved near the wavelength of perfect transmission resonance. Only a small threshold of the switch by introducing a bulk nonlinear defect is needed if incident near the defect mode wavelength in this matamaterial photonic bandgap system. Moreover, we change the thickness of the bulk nonlinear defect layer to 0 / 2 to make the structure more asymmetrical, and we draw a conclude that the small offset to the defect mode is corresponding to a lower threshold of switch. Acknowledgements
This work was financially supported by the Talents Foundation of CQUAS (R20120Q06), by Laboratory topics of Chongqing Key Laboratory of Micro/Nano Materials Engineering and Technology (KFJJ1305), by the Foundation of Chongqing Education Committee (KJ1401115), and by the Foundation and Frontier Research Project of Chongqing (cstc2014jcyja00042).
Referrences
1.
E.Yablonovitch, Inhibited spontaneous emission in solid-state physics and electronics, Phys. Rev. Lett. 58 (1987) 2059-2062.
2.
S.John, Strong localization of photons in certain disordered dielectric super-lattices, Phys. Rev. Lett. 58 (1987) 2486-2489.
3.
E. N. Bulgakov, A. F. Sadreev, All-optical diode based on dipole modes of Kerr microcavity in asymmetric L-shaped photonic crystal waveguide, Opt. Lett. 39 (2014) 1787–1790 .
4.
H.J. Zhao, Z.H. Li, D.R. Yuan, W.J. Li ,Optical bistability with low intensity threshold (1.83 MW/cm2) inSPPs resonator multilayer anostructure, Optik 127 (2016) 3509-3512.
5.
D.C. Chen, G.Z. Xie, X.Y. Cai, M. Liu, Y. Cao, S.J. Su, Fluorescent Organic Planar pn Heterojunction Light-Emitting Diodes with Simplified Structure, Extremely Low Driving Voltage, and High Efficiency, Adv. Mater. 28 (2016) 239–244.
6.
P. J. Jeon, Y. T. Lee, J. Y. Lim, J. S. Kim, D. K. Hwang, S. Im, Black Phosphorus–Zinc Oxide Nanomaterial Heterojunction for p–n Diode and Junction Field-Effect Transistor, Nano Lett. (2016) 16 1293–1298.
7.
K. Gallo, G. Assanto, K. R. Parameswaran, M. M. Fejer, Alloptical diode in a periodically poled lithium niobate waveguide, Appl. Phys. Lett. 79 (2001) 314–316.
8.
B. Liu, Y.F. Liu, S.J. Li, X.D. He, High efficiency all-optical diode based on photonic crystal waveguide, Optics Communications 368 ( 2016) 7–11.
9.
T. Sato, S. Makino, T. Fujisawa, K. Saitoh, Design of a reflection-suppressed all-optical diode based on asymmetric L-shaped nonlinear photonic crystal cavity, J. Opt. Soc. Am. B 33 (2016) 54–61.
10. G.H.Ma, S.H. Tang, J. Shen, Z.J.Zhang, Z.Y.Hua, Defect-mode dependence of two-photon- absorption enhancement in a one-dimensional photonic bandgap structure, Opt. Lett. 29 (2004) 1769-1771.
11. T.Fujiwara , T.Yokokawa, in: Quasicrystals, eds. T Fujiwara , T Ogawa, Springer Ser. Solid State Sci., Springer Verlag, Heidelberg, 1990, pp. 196-198.
12. R,Nava, J.Tagüeña-Martínez, J.A.del Río, G.G.Naumis, Perfect light transmission in Fibonacci arrays of dielectric multilayers, J. Phys.: Condens. Matter, 21 (2009) 155901.
13. G. S. Agarwal, S. D. Gupta, Exact results on optical bistability with surface plasmons in layered media, Phys.Rev.B 34 (1986) 5239-5243.
14. P. W. Mauriz, M. S. Vasconcelos, E. L. Albuquerque, Optical transmission spectra in symmetrical Fibonacci photonic multilayers, Phys.Lett. A 373 (2009) 496-500.
15. V. Grigoriev and F. Biancalana, Scattering on plasmonic nanostructures arrays modeled with a surface integral formulation, Photon. Nanostr. Fundam. Appl. 8 (2010) 285-290.
16. G. S. Agarwal and S. D. Gupta, Optical multistability in a nonlinear Fibonacci multilayer, Phys.Rev.B 38 (1988) 3628-3631.
17. V. Grigoriev, F. Biancalana, Bistability, multistability and nonreciprocal light propagation in Thue-Morse multilayers, Nonlinear Photonics, NME 8 (2010) 1-2.
18. R. Wang, J. Dong, D. Y. Xing, Dispersive optical bistability in one-dimensional doped photonic band gap structures, Phys. Rev. E 55 (1997) 6301-6304.
Fig.1. (Color online) Schematic diagram of the asymmetry multilayer with the sequence of ABAABABACABAABABA. Orange represent negative refractive metamaterial layers and White represent the two Kerr-nonlinear lays B and C respectively. TE wave propagates from the negative z direction. Fig.2. (a) Transmission spectrum of the multilayer with the thickness dc 30 .The matamaterial refractive indices are
A -12.68
and
A 1.5 ,(b)
the bistable behavior of the transmission intensity verses the incident intensity at
wavelength of 688 nm. An S-shape bistable curve is showed noted by the switch point U and D respectively. The inset in (b) is the electric field distribution of resonance mode P in (a) taking no account of Kerr coefficient. Fig.3.The bistable behavior of the transmission intensity verses the incident intensity at the wavelength 805 nm: red solid line, 796 nm: blue dashed line, 787 nm: blue dotted line, 778 nm: blue solid line, respectively. Fig.4. (a) The transmission spectrum for the linear multilayer with the thickness dc 0.50 . (b) The bistable / multibistable behavior of transmitted intensity versus the incident intensity for the three incident wavelengths 342.4 nm: red solid line, 387.6nm: blue solid line, 745.5nm: blue dashed line, respectively. Fig.5. The field profiles at the different resonant wavelength along with the linear multilayer, red solid line: 342.4 nm, blue solid line: 387.6nm, blue dashed line: 745.5nm.
S0030-4026(16)30586-1 http://dx.doi.org/doi:10.1016/j.ijleo.2016.05.133 IJLEO 57759
To appear in: Received date: Accepted date:
29-3-2016 28-5-2016
Please cite this article as: Zhaohong Li, Huajun Zhao, Optical multistability in metal/dielectric nonlinear metamaterial multilayers, Optik - International Journal for Light and Electron Optics http://dx.doi.org/10.1016/j.ijleo.2016.05.133 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.
Optical multistability in metal/dielectric nonlinear metamaterial multilayers *
Zhaohong Li 1,2 ,Huajun Zhao 1
college of Electronic and Electrical Engineering, Chongqing University of Arts and Sciences, Chongqing 402160, China 2
Chongqing Key Laboratory of Micro/Nano Materials Engineering and Technology, Chongqing 402160, China Corresponding e-mail address :
[email protected]
The bistable properties are investigated in a composite optical configuration based on Fibonacci sequence and nonlinear material. Bistablility is realized near the wavelength of perfect transmission resonance. A small threshold of the switch is attained by introducing a bulk nonlinear defect in the matamaterial photonic bandgap system. Meanwhile, the small offset to the defect mode, the lower threshold of the switch is. When the thickness of the defect layer decreases, multistability is easy to appear if the localized electric field distributed in the form of several isolated centers. Some others, such as the transmission of the linear multilayer and the bistability of band edge states are also calculated by the transfer-matrix method.
Keywords: Multistability; Quasicrystals; Kerr effects.
1.Introduction Photonic crystals (PCs) have been considered as one of the most research topic since the pioneering works of Yablonovithc[1] and John[2] for the fascinating properties. Bistability is one of phenomena which can be realized in the nonlinear system. The nonlinear features between the input and output intensity which manifest a hysteresis loop is a basis of the optical switch device in an optical communication system. As a key component of integrated optics to control the photon transportation, optical switch has inspired considerable interest in recent years. In modern all-optical integrated circuits there is one main approach to realize the optical switch operation. One is to use nonlinear optical effects, such as Kerr effect, free-carrier effect, Black phosphorus (BP) nanosheet, thermo-optic effect, PC waveguide, and so on[3-9]. as far as the promising way to realize optical switching , there are two requirements to operate the nonlinear PCs to engender bistability. One is the strong nonlinearity to enhance the transmission, another is the spatially asymmetric to coupling between the resonance states. Hence, the structure is important to the performance of devices[10]. Quasicrystal is a intermediate stage between traditional periodic photonic and disorder systems [11], effectively containing both characteristic of bandgap and localization properties due to short-range disorder. Fibonacci is the typical quasicrystal and has earned more attention in the past years. The linear transmission spectrum of such photonic structures was shown to have a perfect transmission resonance [12]. Our interest in the Fibonacci multilayer is inspired by the fact that the linear transmission curve possesses rich variety of resonances ranging from very narrow to rather broad, furthermore, it is easier to have bistability in the condition of the narrower resonance [13]. In this paper, we present a composite optical configuration formed by two fifth generation Fibonacci sequence and a nonlinear defect. The perfect transmission resonance and bistability properties of are studied by the nonlinear transfer-matrix method. Though some progress in the transmission resonance optical properties have been made in symmetry[14] and asymmetry[15] systems, the bistability/multibistabiliy are hardly reported in metamatertial structures. Our purpose is to study the perfect transmission and bistability in the fifth generation Fibonacci nonlinear composite multilayer structure. 2.Structure and Theory The Fibonacci quasiperiodic structure is built of two types of arbitrary dielectric layers indicated with letters A and B. The Fibonacci sequence is a recurse by the rule sn sn 1sn 2 ,for n 2 ,with s1 A and s2 AB . Consequently, the fifth generation is ABAABABA . We stack two fifth generation Fibonacci
sequences with a nonlinear layer placed in center Fig.1, which a one-dimensional multilayer is stacked in the form of ABAABABACABAABABA . The background dielectric parameters are 0 0 1 . The permittivity and permeability of metamaterial layers A are assumed to be A 1.5 and
A -12.67 respectively. Layers B and C are the Kerr-nonlinear material. The permittivity of B and C are B 2.25 and C 5 with permeability B C 1 . The parameters mentioned above is invariant except the thickness of the C slab in this paper. The thickness of the three materials A , B and C are
0 / 4 A A
, 0.50 , 30 with constant 0 532 nm respectively. For simplicity, the
absorption of the nonlinear material is ignored.
Fig.1. (Color online) Schematic diagram of the asymmetry multilayer with the sequence of ABAABABACABAABABA. Orange represent negative refractive metamaterial layers and White represent the two Kerr-nonlinear lays B and C respectively. TE wave propagates from the negative z direction.
Let a TE polarized plane wave be normal incident from the left of the structure and propagating along with the Z direction (Fig.1). Therefore, we can numerically calculate the electric and magnetic fields by the characteristic matrix of each layer. The matrix of the metamaterial layers A in the form: [16] cos( K A d A ) MA i sin( K A d A )
i
sin( K A d A ) cos( K A d A )
(1)
Where the ratio coefficient is A / A and the wave vector K A is ( / c) A A in the media. The transfer matrix of the Kerr-nonlinear layers B and C are shown as:
MN
K0 Kz Kz
Kz Kz exp(iK z d N ) exp(iK z d N ) K exp(iK z d N ) K exp(iK z d N ) 0 0 Kz Kz Kz Kz [exp(iK z d N ) exp(iK z d N )] exp(iK z d N ) exp(iK z d N ) 2 K K K 0 0 0
(2)
where N is the number of the layer. K z and K z are the forward and backward propagating the wave vector. For a given transmitted intensity I t we calculated in the first nonlinear layer from the right end. In the first nonlinear slab the relation between I t and K z are defined by the following two expressions:
K1z K0 N (1 I1 2I1 )
I1 I 1
1 K 1 K 0
(3) 2
1 1 K1 M A I t 0 K 0
(4)
, where I t At .The and At are the Kerr-nonlinear coefficient and the amplitude of the 2
transmitted intensity. We can obtain the K z and nonlinear layer matrix M B by (3) and (4) using the Newton iterative methodfor a fixed transmitted intensity I t . The matrix of the first nonlinear slab is obtained by formula (2). Similarly, the I 2 in the second nonlinear layer which is on the right hand of the structure is expressed as: I 2 I 2
1 K 2 K 0
2
1 1 K 2 M A M B M A I t 0 K 0
(5)
At last we can calculate all the characteristic matrix of nonlinear layers in such way. Once the matrix of the individual layer is defined, the transfer matrix of the composite structure is obtained as: (6) M M A M B ...M A M C M A ...M B M A Hence, we can computed the transfer matrix elements and the transmittance is given by: 2n0 T (n0 M 22 M 21 ) (n0 M12 M11 )
2
(7)
Where n0 is the refractive index of the background defined by 0 0 . Once the transmittance is known, the corresponding incident intensity I in can be expressed I t / T . 3.Numerical Results The linear field profiles and transmission spectra calculated using the transfer-matrix method is shown in Fig.2. There are five distinct defect modes with high transmittance ranging from 550 nm to 850 nm. As we all know, bulk material can introduce a defect mode into the multilayer photonic bandgap system and hence an incident wave can pass through the structure if the incident frequency is tuned at the linear defect mode. The refractive index of Kerr nonlinear material is independent of field intensity, giving rise to self-phase modulation, therefore, the corresponding defect mode wavelength will shift with the local electric field. If the defect modes move to the incident wavelength, the optical bistability will appear. An S-shape bistable curve is shown in Fig.2 (b) with an incident wavelength 688 nm near the resonance mode 686.7 nm noted by point P. The switch-up and switch-down point are noted with U and D point, respectively. The corresponding incident intensities are 0.1693 and 0.0897 (a.u.). The electric field distribution of the perfect transmission resonance wavelength noted by arrowhead in Fig.2 (a) is shown by blue solid line in Fig.2 (b) and it is evident that electric intensity is mainly localized in the center of the structure.
Fig.2. (a)Transmission spectrum of the multilayer with the thickness dc 30 .The matamaterial refractive indices are
A -12.68
and
A 1.5 ,
wavelength of 688 nm.
(b) the bistable behavior of the transmission intensity verses the incident intensity at
An S-shape bistable curve is showed noted by the switch point U and D respectively. The inset in (b) is
the electric field distribution of resonance mode P in (a) taking no account of Kerr coefficient.
There are abundant low transmitted states near the defect modes. In order to explore the optical bistable properties of these band edge modes, four wavelengths near the band edge modes are chosen to impinge on the multilayer. The mode behaviors of the incident wavelengths are not all the same. Wavelength 805nm, which is near a perfect transmission resonance, takes on a multistablility with the
increase in incident intensity shown with red solid line in Fig.3. The reason to this phenomenon is the interplay between the isolated localized district in the structure[17]. The other three wavelengths are also provided with bistable property. We can see that some band edge states still have the ability to shift the defect mode even though the transmittance is not high enough, but due to the low transmittance it is difficult to engender the multibistable behavior.
Fig.3. The bistable behavior of the transmission intensity verses the incident intensity at the wavelength 805 nm: red solid line, 796 nm: blue dashed line, 787 nm: blue dotted line, 778 nm: blue solid line, respectively.
Comparing the defined parameter of thickness d C and d B , we change the thickness d C to 0.50 to enhance the asymmetry of the multilayer to a certain degree while other parameters have no change , the transmission spectrum of several resonances ranging from 300 nm to 900 nm is showed in Fig.4 (a) taking no account of Kerr coefficient. There are three resonances
at wavelength 341.9, 386.8,
744.1nm, respectively. They are perfect transmission, whereas the width of the resonance mode differs. Obviously the linear defect mode 341.9 nm and 386.8 nm is narrower than the one at wavelength of 744.1 nm. First, we calculate the transmitted behavior at the wavelength of 342.4 nm (red solid line), 387.6 nm (blue solid line), and 745.5 nm (blue dashed line) respectively, which are located near the resonances. The variations of incident intensity with the transmitted intensity are shown in Fig.4 (b). Multibistabilty appears with the incident wavelength 342.4 nm and 387.6 nm. Multibistabilty occurs easily shown in Fig.4.(b). The thresholds of the lower incident wavelength 342.4 nm is smaller than the other two wavelength 387.6 nm and 745.5 nm, and the deviation from the linear resonance mode is 0.5nm ,0.8nm and 1.4nm respectively. A conclusion can be drawn that the smaller deviation from the linear defect mode and the narrower the resonance is, the easier it is to have bistability. it coincides with Ref. [18].
Fig.4. (a) The transmission spectrum for the linear multilayer with the thickness dc 0.50 . (b) The bistable / multibistable behavior of transmitted intensity versus the incident intensity for the three incident wavelengths 342.4 nm: red solid line, 387.6nm: blue solid line, 745.5nm: blue dashed line, respectively.
To better illustrated the field distribution the field profiles are demonstrated along with position of the linear multilayer in Fig.5. It is evident that the energy is scatted distributed with one more localization centers for the wavelength 342.4nm and 387.6nm, while it is not strong localized center for the mode wavelength 745.5 nm. The reason to the multistability in multilayers system is the interplay between these localization centers. In addition, the field of wavelength 342.4 nm is strongly localized and the enhancement is as large as an order magnitude.
Fig.5. The field profiles at the different resonant wavelength along with the linear multilayer, red solid line: 342.4 nm, blue solid line: 387.6nm, blue dashed line: 745.5nm.
4. Conclusion In summary, we have investigated the bistable behavior in a composite optical configuration based on Fibonacci sequence and nonlinear material. An S-shape bistability curve is shown which can be achieved near the wavelength of perfect transmission resonance. Only a small threshold of the switch by introducing a bulk nonlinear defect is needed if incident near the defect mode wavelength in this matamaterial photonic bandgap system. Moreover, we change the thickness of the bulk nonlinear defect layer to 0 / 2 to make the structure more asymmetrical, and we draw a conclude that the small offset to the defect mode is corresponding to a lower threshold of switch. Acknowledgements
This work was financially supported by the Talents Foundation of CQUAS (R20120Q06), by Laboratory topics of Chongqing Key Laboratory of Micro/Nano Materials Engineering and Technology (KFJJ1305), by the Foundation of Chongqing Education Committee (KJ1401115), and by the Foundation and Frontier Research Project of Chongqing (cstc2014jcyja00042).
Referrences
1.
E.Yablonovitch, Inhibited spontaneous emission in solid-state physics and electronics, Phys. Rev. Lett. 58 (1987) 2059-2062.
2.
S.John, Strong localization of photons in certain disordered dielectric super-lattices, Phys. Rev. Lett. 58 (1987) 2486-2489.
3.
E. N. Bulgakov, A. F. Sadreev, All-optical diode based on dipole modes of Kerr microcavity in asymmetric L-shaped photonic crystal waveguide, Opt. Lett. 39 (2014) 1787–1790 .
4.
H.J. Zhao, Z.H. Li, D.R. Yuan, W.J. Li ,Optical bistability with low intensity threshold (1.83 MW/cm2) inSPPs resonator multilayer anostructure, Optik 127 (2016) 3509-3512.
5.
D.C. Chen, G.Z. Xie, X.Y. Cai, M. Liu, Y. Cao, S.J. Su, Fluorescent Organic Planar pn Heterojunction Light-Emitting Diodes with Simplified Structure, Extremely Low Driving Voltage, and High Efficiency, Adv. Mater. 28 (2016) 239–244.
6.
P. J. Jeon, Y. T. Lee, J. Y. Lim, J. S. Kim, D. K. Hwang, S. Im, Black Phosphorus–Zinc Oxide Nanomaterial Heterojunction for p–n Diode and Junction Field-Effect Transistor, Nano Lett. (2016) 16 1293–1298.
7.
K. Gallo, G. Assanto, K. R. Parameswaran, M. M. Fejer, Alloptical diode in a periodically poled lithium niobate waveguide, Appl. Phys. Lett. 79 (2001) 314–316.
8.
B. Liu, Y.F. Liu, S.J. Li, X.D. He, High efficiency all-optical diode based on photonic crystal waveguide, Optics Communications 368 ( 2016) 7–11.
9.
T. Sato, S. Makino, T. Fujisawa, K. Saitoh, Design of a reflection-suppressed all-optical diode based on asymmetric L-shaped nonlinear photonic crystal cavity, J. Opt. Soc. Am. B 33 (2016) 54–61.
10. G.H.Ma, S.H. Tang, J. Shen, Z.J.Zhang, Z.Y.Hua, Defect-mode dependence of two-photon- absorption enhancement in a one-dimensional photonic bandgap structure, Opt. Lett. 29 (2004) 1769-1771.
11. T.Fujiwara , T.Yokokawa, in: Quasicrystals, eds. T Fujiwara , T Ogawa, Springer Ser. Solid State Sci., Springer Verlag, Heidelberg, 1990, pp. 196-198.
12. R,Nava, J.Tagüeña-Martínez, J.A.del Río, G.G.Naumis, Perfect light transmission in Fibonacci arrays of dielectric multilayers, J. Phys.: Condens. Matter, 21 (2009) 155901.
13. G. S. Agarwal, S. D. Gupta, Exact results on optical bistability with surface plasmons in layered media, Phys.Rev.B 34 (1986) 5239-5243.
14. P. W. Mauriz, M. S. Vasconcelos, E. L. Albuquerque, Optical transmission spectra in symmetrical Fibonacci photonic multilayers, Phys.Lett. A 373 (2009) 496-500.
15. V. Grigoriev and F. Biancalana, Scattering on plasmonic nanostructures arrays modeled with a surface integral formulation, Photon. Nanostr. Fundam. Appl. 8 (2010) 285-290.
16. G. S. Agarwal and S. D. Gupta, Optical multistability in a nonlinear Fibonacci multilayer, Phys.Rev.B 38 (1988) 3628-3631.
17. V. Grigoriev, F. Biancalana, Bistability, multistability and nonreciprocal light propagation in Thue-Morse multilayers, Nonlinear Photonics, NME 8 (2010) 1-2.
18. R. Wang, J. Dong, D. Y. Xing, Dispersive optical bistability in one-dimensional doped photonic band gap structures, Phys. Rev. E 55 (1997) 6301-6304.
Fig.1. (Color online) Schematic diagram of the asymmetry multilayer with the sequence of ABAABABACABAABABA. Orange represent negative refractive metamaterial layers and White represent the two Kerr-nonlinear lays B and C respectively. TE wave propagates from the negative z direction. Fig.2. (a) Transmission spectrum of the multilayer with the thickness dc 30 .The matamaterial refractive indices are
A -12.68
and
A 1.5 ,(b)
the bistable behavior of the transmission intensity verses the incident intensity at
wavelength of 688 nm. An S-shape bistable curve is showed noted by the switch point U and D respectively. The inset in (b) is the electric field distribution of resonance mode P in (a) taking no account of Kerr coefficient. Fig.3.The bistable behavior of the transmission intensity verses the incident intensity at the wavelength 805 nm: red solid line, 796 nm: blue dashed line, 787 nm: blue dotted line, 778 nm: blue solid line, respectively. Fig.4. (a) The transmission spectrum for the linear multilayer with the thickness dc 0.50 . (b) The bistable / multibistable behavior of transmitted intensity versus the incident intensity for the three incident wavelengths 342.4 nm: red solid line, 387.6nm: blue solid line, 745.5nm: blue dashed line, respectively. Fig.5. The field profiles at the different resonant wavelength along with the linear multilayer, red solid line: 342.4 nm, blue solid line: 387.6nm, blue dashed line: 745.5nm.