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Theoretical investigation of semiconductor supported tunable terahertz dielectric loaded surface plasmons waveguides
Optics Communications 356 (2015) 64–69
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Theoretical investigation of semiconductor supported tunable terahertz dielectric loaded surface plasmons waveguides Chunlin Liu, Xiaoyong He n, Zhenyu Zhao, Hao Zhang, Wangzhou Shi Department of Physics, Mathematics & Science College, Shanghai Normal University, No. 100 Guilin Road, Shanghai 200234, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 16 April 2015 Received in revised form 12 July 2015 Accepted 21 July 2015
The tunable propagation properties of semiconductor-based dielectric loaded surface plasmons (DLSPs) structures have been theoretically investigated in the THz regime, including the effects of temperature, operation frequency, and the thermo-optic effect of dielectric stripe materials. The results show that the waveguide properties of DLSPs structure can be modulated in a wide range via changing the temperature. For instance, when the temperature is changed in the range of 300–600 K, the modulation depth of propagation length can reach more than 80%. With the increase of refractive index of the dielectric stripe, the modulation depth of the effective indices and propagation lengths increase. In addition, the propagation length and figure of the merit can be improved obviously with the hybrid dielectric stripe structure (by coating Si on the SiO2 layer). The results are very helpful to design novel waveguide devices, such as modulators, switchers, sensors and polarizers. & 2015 Elsevier B.V. All rights reserved.
Keywords: Terahertz DLSPs waveguides Tunable
1. Introduction The last two decades witness the rapid development of terahertz (THz) technology [1–5]. For its application in the fields of imaging, biological sensor, and wireless communication, the investigation of waveguide devices is vital important, which is closely related to the surface plasmons (SPs) [6]. SPs are two-dimensional (2D) electromagnetic waves confined at the metal–dielectric interfaces [7–10], offering the promise to control the electromagnetic waves at subwavelength scales [11–15]. Considerable efforts have been devoted to develop various SPs waveguide structures, such as metal wire waveguide [16], metal–dielectrics–metal structure [17], air–dielectrics–metal waveguides [18,19], the triangle grooves structure [20], and the hybrid surface plasmonic structure. The hybrid surface plasmonic waveguide displays the merits of strong mode confinement, low propagation loss, and high compatibility with semiconductor fabrication [21]. It mainly includes two kinds of typical structures, i.e. the dielectric fiber–dielectrics–metal waveguide structure and dielectric loaded surface plasmons (DLSPs) waveguides [22,23]. Consisting of a dielectric fiber (Si) with high refractive index separated from a metal layer by a low dielectric gap (SiO2), the dielectric fiber–dielectrics– metal structure can be applied to fabricate micro and nanolasers and plasmonic integrated circuits [24,25]. Additionally, surface plasmons waveguides have also been widely applied in many research fields, e.g. n
Corresponding author. E-mail address: [email protected] (X. He).
http://dx.doi.org/10.1016/j.optcom.2015.07.062 0030-4018/& 2015 Elsevier B.V. All rights reserved.
in the design of high power super-luminescent diodes (SLEDs). For instance, by using active multi-mode interferometer structures, the high power SLEDs has been put forward, which manifests high efficiency and obviously thermal resistance reduction [26,27]. By depositing dielectric stripe on the metallic substrate, the DLSPs waveguides can be realized, manifesting the merits of strong lateral confinement, low bending losses and high component integration density [28–30]. Recently, much experimental and theoretical research has been carried out in this aspect [31– 34]. For instance, Sorger et al. first experimental proved that the ultra-small propagating waves at visible wavelength, the mode size reduced down to 50 60 nm2 [28]. By inserting a thin gold strip between two high refractive indices Si spacer, Magno et al. proposed a kind of long-range propagation DLSPs structures [29], which displayed long propagation length and the application to fabricate high sensitivity optical sensors (the resolution can reach about 10 6–10 7). On the base of DLSPs structures, the micro-ring resonator filters and monitors have also been shown, the extinction ratios as high as 17 dB have been attained [31,32]. In 2013, by inserting the graphene layer into the DLSPs waveguides, a new type of electro-optical modulators have been proposed [33,34], the optical bandwidth exceeds 12 THz, the operation speed is about 500 GHz, which shows great potential applications in the fields of integrated on-chip devices and tunable waveguides device design. But for above mentioned DLSPs waveguides, it is very difficult to vary the permittivity of metal and modulate the propagation properties of hybrid modes, which severely degrades the quality of
C. Liu et al. / Optics Communications 356 (2015) 64–69
SPs resonance and limits the practical application of devices. Fortunately, some doped semiconductors (e.g. InSb) also show metallic characters in the THz regime, and the corresponding dielectric constants are similar to that of metal in the visible/UV regime. Furthermore, the plasmonic properties of semiconductor devices can be dynamically modulated by changing the temperature or doping. The tunable DLSPs waveguide structure has been proposed by replacing the metal layer with the semiconductor InSb layer, giving us much more freedom to control the optical properties. With the finite element method (FEM), the tunable propagation properties of semiconductor-supported DLSPs waveguide structure in the THz regime have been numerically investigated and compared with other kinds of waveguide structures, including the influence of operation frequency, different kinds of dielectric stripes, and temperature. Our investigation manifest that via controlling the temperature, the propagation properties of the suggested semiconductor-supported DLSPs structure can be modulated in a wide range, e.g. the modulation depth of propagation length can reach more than 80%.
The mode area can be defined as the ratio of the total mode energy and peak energy density, which can be given by [15]
Am =
Wm 1 = max {W (r ) } max {W (r ) }
W (r ) =
where ε1 ¼ 15.75, ω is the radiation frequency, T is the temperature, Γ is the phenomenological scattering rate, i.e.
Γ (T ) = 1/(m*μ (T )/e)
(2)
in which m* is the effective mass of the electron, m* ¼0.015 me, me ¼ 9.108 10 31 ㎏, e¼1.602 10 19 C, and μ is the carrier mobility, its value of is taken from Ref. [37]. The plasma frequency ωp can be expressed as
ωp =
(e2n)/(ε0 m*)
where
ε0 ¼8.854 10 12 F/m, n is the carrier concentration [38],
n = 2.9 × 1011(2400 − T )0.75 (1 + 2.7 × 10−4 T ) T 1.5 × ⎛ 0.129 − 1.5 × 10−4 T ⎞ exp ⎜ − ⎟ ⎝ kB T ⎠
(3)
(4)
kB is the Boltzmann constant.
Fig. 1. The sketch of DLSPs waveguide structure, a dielectric stripe is deposited on the InSb substrate layer, the width and length of dielectric stripe is w and h with the values of both 100 μm.
(6)
ef
The modulation depth (Mod) can be defined as
Mod =
xmax − xmin xmax
(7)
Here x stands for neff, L and Am/A0, respectively. The figure of merits (FoM) of the propagation mode is defined as
L
3. Results and discussions
(1)
(5)
A0 is the normalized diffraction-limited area and calculated by λ2/4. The normalized effective mode area is Am/A0. The effective index neff and propagation length L can be defined by neff ¼ Re(β)/k0 and L = 4π Imλ(N ) , respectively.
Fig. 1 shows that the sketch of the DLSPs waveguide structures. The dielectric stripe is deposited on the InSb substrate layer. The length and width of the dielectric stripe is h and w, respectively. The dielectric constant of the InSb is calculated by using the following formula [35,36]:
ω [ω + iΓ (T ) ]
W (r ) d2r
⎞ 1 ⎛ d (ε (r ) ω) ⎜ E (r ) 2 + μ 0 H (r ) 2 ⎟ ⎠ 2 ⎝ dω
FoM =
ϵ(ω, T ) = ϵ∞ −
+∞
∫−∞
where Wm and W(r) are the electromagnetic energy and energy density, respectively. W(r) is calculated by the formula:
2. Theoretic model and research method
ωp2 (T )
65
Am
(8)
The real part of the effective index (Re(neff)) and propagation length (Lspp) of the modes versus temperature for different kinds of dielectric stripes are shown in Fig. 2. The width and thickness of the dielectric stripe are both 100 μm. The numerical results have been obtained from the FEM software package-COMSOL MULTIPHYSICS 4.0. The dielectric stripe materials are SiO2, Al2O3, and Si, respectively. Because temperature changes in a wide range, i.e. 300–600 K, the refractive index of the dielectric stripe changes significantly. Therefore, the thermo-optic effect has been taken into account in the simulation. The thermo-optic coefficients of SiO2, Al2O3 and Si are 1 10 5/K, 4.7 10 5/K and 1.8 10 4/K, respectively [39,40]. It can be found from Fig. 2 that as the refractive index of dielectric stripe increases, the value of Re(neff) increases, the propagation length decreases, which means that the propagation modes can be better confined in the dielectric stripe. In addition, Fig. 2 also shows that the influences of temperature on the propagation properties. As the temperature increases, the value of Re(neff) decreases, the propagation length increases. The reasons are shown in the following. As the temperature increases, the carrier concentration of InSb increases, resulting into the enhancement of the dielectric constant of InSb at higher temperature. For instance, the dielectric constants of InSb are 8.25 101 þ2.62 101i, 1.18 103 þ 7.16 102i, and 3 3 2.10 10 þ 1.69 10 i at the temperatures of 300 K, 500 K, and 600 K, respectively. Thus, InSb layer shows better “metallic” properties at higher temperature. Consequently, much less modes penetrates into the substrate layer, leading into the value of the effective indices of propagation modes decreasing. Fig. 3 shows the modulation depth of the DLSPs waveguides versus temperature for different kinds of dielectric stripes. The modulation depth has been defined by using the Eq. (7), and x stands for neff. It can be found from Fig. 3 that the modulation depth of Re(neff) and LSPP increase with the increase of temperature, resulting from the value of the effective index of hybrid modes decrease at higher temperature. Furthermore, with the permittivity of the dielectric stripe increases, the modulation depth of the LSPP increases, i.e. the value of Si stripe is much larger
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C. Liu et al. / Optics Communications 356 (2015) 64–69
Fig. 2. The real part of the effective index (a) Re(neff) and (b) propagation length of the DLSPs waveguide modes versus temperature for different kinds of dielectric stripes. The operation frequency is 1.0 THz. The dielectric materials are SiO2, Al2O3 and Si, respectively. The value of Re(neff) and propagation length decreases and increases with the increases of temperature. The solid, dash, and dot lines represent SiO2, Al2O3and Si, respectively.
than those of SiO2 and Al2O3. For instance, when the temperature changes in the range of 300–600 K, for the dielectric stripe of SiO2 and Al2O3, the modulation depth of the Re(neff) are 6% and 7%, respectively. It is also confirmed by the 2D field distribution obtained from the FEM software, as shown in Fig. 5. Another point should be noted is that the modulation depth of Lspp is much larger that of the Re(neff). For example, for Si stripe, the modulation depth of the Lspp and Re(neff) are about 80% and 10% when the temperature changes in the range of 300–600 K. The mode confinement can be measured by the mode area, which is defined by Eq. (5). Fig. 4(a) shows the normalized mode area of DLSPs waveguides structure for different kinds of dielectric stripes. As temperature increases, the mode area increases, the confinement decreases. This is in accordance with the effective index of the modes decreases with the temperature. It can also be found from Fig. 4(a) that as the refractive index of the dielectric stripe increases, the mode area decreases, which means that the dielectric stripe with larger permittivity shows better confinement. The modulation depth of mode area of DLSPs waveguides vs. temperature can be found in Fig. 4(b). As the temperature increases, the modulation depth of mode area can be changed in a wide range. When the temperature is changed in the range of 300–600 K, the modulation depth of mode area can reach more than 60% for the dielectric stripe of Si. As the temperature decreases, the value of mode area decreases, leading to the modulation depth of mode area increase. In addition, as the refractive index of dielectric stripe increases, the modulation depth of modes area increases, i.e. the modulation depth of the Si is larger than
Fig. 3. The modulation depth of the (a) Re(neff) and (b) propagation length of the DLSPs waveguides modes versus temperature for different kinds of dielectric stripe. The operation frequency is 1.0 THz. The dielectric materials are SiO2, Al2O3 and Si, respectively. For the dielectric stripe of SiO2, Al2O3, and Si, the modulation depth of the Re(neff) are 6%, 7% and 10%, respectively, and the modulation depth of theLsppare 60%, 70% and 80%, respectively. The square, circle, and triangle symbols represent the dielectric stripes of SiO2, Al2O3 and Si, respectively.
that of SiO2. This means that the propagation modes of the dielectric stripe with larger refractive index are affected by the changes of temperature much more significantly. For instance, at the temperature of 600 K, the modulation depth of the mode area for the SiO2, Al2O3 and Si are 30.4%, 46.3% and 61.8%, respectively. The electric fields distribution of above several kinds of dielectric stripe can be found in Fig. 5. As the refractive index of the dielectric stripe increases, there are much more modes confined in the dielectric stripe, which means that the confinement of DLSPs waveguides increases, as shown in Fig. 5(b), (d) and (f). Additionally, as the temperature increases, more modes leak into the surrounding air, resulting into the value of the effective index decreasing. For instance, when the dielectric stripe material is SiO2, the value of the effective indices of the DLSPs modes are 1.72 and 1.61 at the temperatures of 300 K and 600 K, respectively. The waveguides properties can be improved by coating the DLSPs structure with a thin dielectric layer with larger refractive index, as the inset shown in Fig. 6(b). The thickness of the Si layer is 10 μm, and the length and thickness of SiO2 are both 90 μm. The real part of the effective index Re(neff) and propagation length versus temperature for the above mentioned waveguides structure are shown in Fig. 6(a) and (b), respectively. It can be found from Fig. 6 that by coating Si layer on the DLSPs waveguides, the value of Re(neff) increases, the propagation length is also much longer, especially at higher temperature. The possible reasons are shown in the following. For the modified DLSPs waveguides structure, it
C. Liu et al. / Optics Communications 356 (2015) 64–69
can be regarded as a similar capacitor like structure. With the increase of the refractive index of dielectric stripe, more modes energy can be stored, which means that the propagation modes
Fig. 4. (a) The normalized effective mode area Am/A0 and (b) the modulation depth of normalized effective mode area Mod(Am/A0) of the DLSPs waveguide modes versus temperature for different kinds of dielectric materials stripe. The operation frequency is 1.0 THz. The dielectric materials are SiO2, Al2O3 and Si, respectively. The mode area increases and the confinement decreases as temperature increases. In (a) the solid, dash, and dot lines represent SiO2, Al2O3 and Si, respectively. In (b) the square, circle, and triangle symbols represent the dielectric stripes of SiO2, Al2O3 and Si, respectively.
67
can be better confined in the dielectric stripe. This is also confirmed by the field distribution shown in Fig. 8. Simultaneously, the existence of low loss SiO2 layer also results much more mode spreading into in the upper section of dielectric stripe (Si),
Fig. 6. The real part of the effective index (a) Re(neff) and (b) propagation length of DLSPs waveguides modes versus temperature for different kinds of dielectric stripes. The operation frequency is 1.0 THz. The dielectric materials are Si, SiO2 and SiO2–Si, respectively. The inset in (c) shows the sketch of the proposed hybrid dielectric stripe structure of DLSPPs waveguides structure. The Re(neff) decreases and propagation length increases as temperature increases. The solid line, dash line, and dot line represent SiO2, Si, and SiO2–Si, respectively.
Fig. 5. The electric field distribution of the DLSPs waveguides structure for different kinds dielectric stripe materials. The dielectric stripes are SiO2 (a, b), Al2O3 (c, d) and Si (e, f), respectively. The temperature are 300 K (a, c and e) and 600 K (b, d and f), respectively. The operation frequency is 1.0 THz. The width and thickness of SiO2, Al2O3 and Si are 100 μm.
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C. Liu et al. / Optics Communications 356 (2015) 64–69
Fig. 7. Comparison of the FoM versus temperature for different kinds of dielectric stripe materials. The dielectric materials are Si, SiO2 and SiO2–Si, respectively. The thickness of the coating Si layer on the SiO2 is 10 μm. The temperatures is changed in the range of 300–600 K. The solid line, dash line, and dot line represent SiO2, Si, and SiO2–Si, respectively.
value of the FoM increases. In addition, the value of FoM of the modified DLSPs waveguides is largest among the three waveguides structures. By coating a thin Si layer on the DLSPs waveguides, not only the confinement increases, but the loss (propagation length) also decreasing (increasing). Consequently, the value of FoM of the modified DLSPs structure increases, which means that the overall performance of the modified structure can be significantly increases by coating a dielectric layer. The electric field distribution of the fundamental mode for different kinds of waveguide structures has been shown in Fig. 8. We can found that much more modes have been confined in the modified dielectric stripe structure, as shown in Fig. 8(b) and (d). In addition, as the temperature of InSb layer increases, more modes leak into the surrounding air, resulting into the value of effective index decreasing. For instance, if the dielectric stripe material is SiO2, the values of effective indices of the DLSPs modes are 1.72 and 1.61 at the temperatures of 300 K and 600 K, respectively. While in the new type DLSPs waveguide, i.e. around the SiO2 stripe covered with 10 μm thick Si, there will be much more mode confined in the dielectric stripe, leading to the value of the effective index increasing. For instance, in the modified new structure, the values of the effective indices of the modified DLSPs mode are 1.86 and 1.75 at the temperature of 300 K and 600 K, respectively.
4. Conclusion
Fig. 8. The electric field distribution of the propagation mode for different kinds of waveguide structures. The waveguides are the original DLSPs structure (a, b) and the modified structure (c, d). The temperature are 300 K (a, c) and 600 K (b, d). The operation frequency is 1.0 THz. In the modified structure, the thickness of the Si layer is 10 μm, and the length and thickness of SiO2 are both 90 μm.
The tunable propagation properties of semiconductor-supported DLSPs waveguide structures have been theoretically investigated in the THz regime. The influences of temperature, operation frequency, and thermo-optic effect of dielectric stripe materials have been given and discussed. The results show that the waveguide properties of DLSPs structure can be modulated in a wide range by changing the temperature. As the temperature increases, the effective index of propagation mode decreases, while the propagation loss increases. When the temperature is changed in the range of 300–600 K, the modulation depth of propagation length can reach more than 80%. As the refractive index of the dielectric stripe increases, the modulation depth of propagation modes increases. Additionally, for the hybrid dielectric stripe structure, i.e. by coating Si on the SiO2 layer, the propagation length and figure of the merit can be improved obviously. Our results are very useful to have a better understanding of the propagation mechanism of the DLSPs structure and design novel waveguide devices, such as such as modulators, switchers, sensors and polarizer.
Acknowledgments reducing the interaction of modes with the high loss InSb substrate. Thus, the propagation losses also decrease as well. Furthermore, as the temperature increases, the carrier concentration of InSb layer increases, resulting into the enhancement of dielectric constant. Much less mode penetrates into the substrate layer at higher temperature, leading into the influence of the dielectric coating layer increasing. As given in Eq. (8), FoM is a suitable measurement for a tradeoff between mode confinement and loss, which has been adopted to quantitatively compare the proposed modified DLSPs waveguide with the original structure. Fig. 7 shows the relationship of FoM to temperature for different kinds of dielectric materials, i.e. Si, SiO2 and SiO2–Si structures. As the temperature increases, the value of FoM increases. This can be explained by the fact that as the temperature increases, the value of the dielectric constant of InSb increases, leading to the value of Re(neff) decrease, and the propagation loss increases. Because the influence of temperature on the propagation length is larger than that of the Re(neff), the
This work is supported by the Funding of Shanghai Pujiang Program under the Grant no. 15PJ1406500, the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, China, the Research Funding of Shanghai Normal University under the Grant no. SK201529, the Initial Funding of Scientific Rsearch for the Introduction of Talents of Shanghai Normal University, and the Funding of National Natural Science Foundation of China under the Grant no. 61307130, and the Funding of the Shanghai Municipal Education Commission under the Grant no. 14YZ077.
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Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Theoretical investigation of semiconductor supported tunable terahertz dielectric loaded surface plasmons waveguides Chunlin Liu, Xiaoyong He n, Zhenyu Zhao, Hao Zhang, Wangzhou Shi Department of Physics, Mathematics & Science College, Shanghai Normal University, No. 100 Guilin Road, Shanghai 200234, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 16 April 2015 Received in revised form 12 July 2015 Accepted 21 July 2015
The tunable propagation properties of semiconductor-based dielectric loaded surface plasmons (DLSPs) structures have been theoretically investigated in the THz regime, including the effects of temperature, operation frequency, and the thermo-optic effect of dielectric stripe materials. The results show that the waveguide properties of DLSPs structure can be modulated in a wide range via changing the temperature. For instance, when the temperature is changed in the range of 300–600 K, the modulation depth of propagation length can reach more than 80%. With the increase of refractive index of the dielectric stripe, the modulation depth of the effective indices and propagation lengths increase. In addition, the propagation length and figure of the merit can be improved obviously with the hybrid dielectric stripe structure (by coating Si on the SiO2 layer). The results are very helpful to design novel waveguide devices, such as modulators, switchers, sensors and polarizers. & 2015 Elsevier B.V. All rights reserved.
Keywords: Terahertz DLSPs waveguides Tunable
1. Introduction The last two decades witness the rapid development of terahertz (THz) technology [1–5]. For its application in the fields of imaging, biological sensor, and wireless communication, the investigation of waveguide devices is vital important, which is closely related to the surface plasmons (SPs) [6]. SPs are two-dimensional (2D) electromagnetic waves confined at the metal–dielectric interfaces [7–10], offering the promise to control the electromagnetic waves at subwavelength scales [11–15]. Considerable efforts have been devoted to develop various SPs waveguide structures, such as metal wire waveguide [16], metal–dielectrics–metal structure [17], air–dielectrics–metal waveguides [18,19], the triangle grooves structure [20], and the hybrid surface plasmonic structure. The hybrid surface plasmonic waveguide displays the merits of strong mode confinement, low propagation loss, and high compatibility with semiconductor fabrication [21]. It mainly includes two kinds of typical structures, i.e. the dielectric fiber–dielectrics–metal waveguide structure and dielectric loaded surface plasmons (DLSPs) waveguides [22,23]. Consisting of a dielectric fiber (Si) with high refractive index separated from a metal layer by a low dielectric gap (SiO2), the dielectric fiber–dielectrics– metal structure can be applied to fabricate micro and nanolasers and plasmonic integrated circuits [24,25]. Additionally, surface plasmons waveguides have also been widely applied in many research fields, e.g. n
Corresponding author. E-mail address: [email protected] (X. He).
http://dx.doi.org/10.1016/j.optcom.2015.07.062 0030-4018/& 2015 Elsevier B.V. All rights reserved.
in the design of high power super-luminescent diodes (SLEDs). For instance, by using active multi-mode interferometer structures, the high power SLEDs has been put forward, which manifests high efficiency and obviously thermal resistance reduction [26,27]. By depositing dielectric stripe on the metallic substrate, the DLSPs waveguides can be realized, manifesting the merits of strong lateral confinement, low bending losses and high component integration density [28–30]. Recently, much experimental and theoretical research has been carried out in this aspect [31– 34]. For instance, Sorger et al. first experimental proved that the ultra-small propagating waves at visible wavelength, the mode size reduced down to 50 60 nm2 [28]. By inserting a thin gold strip between two high refractive indices Si spacer, Magno et al. proposed a kind of long-range propagation DLSPs structures [29], which displayed long propagation length and the application to fabricate high sensitivity optical sensors (the resolution can reach about 10 6–10 7). On the base of DLSPs structures, the micro-ring resonator filters and monitors have also been shown, the extinction ratios as high as 17 dB have been attained [31,32]. In 2013, by inserting the graphene layer into the DLSPs waveguides, a new type of electro-optical modulators have been proposed [33,34], the optical bandwidth exceeds 12 THz, the operation speed is about 500 GHz, which shows great potential applications in the fields of integrated on-chip devices and tunable waveguides device design. But for above mentioned DLSPs waveguides, it is very difficult to vary the permittivity of metal and modulate the propagation properties of hybrid modes, which severely degrades the quality of
C. Liu et al. / Optics Communications 356 (2015) 64–69
SPs resonance and limits the practical application of devices. Fortunately, some doped semiconductors (e.g. InSb) also show metallic characters in the THz regime, and the corresponding dielectric constants are similar to that of metal in the visible/UV regime. Furthermore, the plasmonic properties of semiconductor devices can be dynamically modulated by changing the temperature or doping. The tunable DLSPs waveguide structure has been proposed by replacing the metal layer with the semiconductor InSb layer, giving us much more freedom to control the optical properties. With the finite element method (FEM), the tunable propagation properties of semiconductor-supported DLSPs waveguide structure in the THz regime have been numerically investigated and compared with other kinds of waveguide structures, including the influence of operation frequency, different kinds of dielectric stripes, and temperature. Our investigation manifest that via controlling the temperature, the propagation properties of the suggested semiconductor-supported DLSPs structure can be modulated in a wide range, e.g. the modulation depth of propagation length can reach more than 80%.
The mode area can be defined as the ratio of the total mode energy and peak energy density, which can be given by [15]
Am =
Wm 1 = max {W (r ) } max {W (r ) }
W (r ) =
where ε1 ¼ 15.75, ω is the radiation frequency, T is the temperature, Γ is the phenomenological scattering rate, i.e.
Γ (T ) = 1/(m*μ (T )/e)
(2)
in which m* is the effective mass of the electron, m* ¼0.015 me, me ¼ 9.108 10 31 ㎏, e¼1.602 10 19 C, and μ is the carrier mobility, its value of is taken from Ref. [37]. The plasma frequency ωp can be expressed as
ωp =
(e2n)/(ε0 m*)
where
ε0 ¼8.854 10 12 F/m, n is the carrier concentration [38],
n = 2.9 × 1011(2400 − T )0.75 (1 + 2.7 × 10−4 T ) T 1.5 × ⎛ 0.129 − 1.5 × 10−4 T ⎞ exp ⎜ − ⎟ ⎝ kB T ⎠
(3)
(4)
kB is the Boltzmann constant.
Fig. 1. The sketch of DLSPs waveguide structure, a dielectric stripe is deposited on the InSb substrate layer, the width and length of dielectric stripe is w and h with the values of both 100 μm.
(6)
ef
The modulation depth (Mod) can be defined as
Mod =
xmax − xmin xmax
(7)
Here x stands for neff, L and Am/A0, respectively. The figure of merits (FoM) of the propagation mode is defined as
L
3. Results and discussions
(1)
(5)
A0 is the normalized diffraction-limited area and calculated by λ2/4. The normalized effective mode area is Am/A0. The effective index neff and propagation length L can be defined by neff ¼ Re(β)/k0 and L = 4π Imλ(N ) , respectively.
Fig. 1 shows that the sketch of the DLSPs waveguide structures. The dielectric stripe is deposited on the InSb substrate layer. The length and width of the dielectric stripe is h and w, respectively. The dielectric constant of the InSb is calculated by using the following formula [35,36]:
ω [ω + iΓ (T ) ]
W (r ) d2r
⎞ 1 ⎛ d (ε (r ) ω) ⎜ E (r ) 2 + μ 0 H (r ) 2 ⎟ ⎠ 2 ⎝ dω
FoM =
ϵ(ω, T ) = ϵ∞ −
+∞
∫−∞
where Wm and W(r) are the electromagnetic energy and energy density, respectively. W(r) is calculated by the formula:
2. Theoretic model and research method
ωp2 (T )
65
Am
(8)
The real part of the effective index (Re(neff)) and propagation length (Lspp) of the modes versus temperature for different kinds of dielectric stripes are shown in Fig. 2. The width and thickness of the dielectric stripe are both 100 μm. The numerical results have been obtained from the FEM software package-COMSOL MULTIPHYSICS 4.0. The dielectric stripe materials are SiO2, Al2O3, and Si, respectively. Because temperature changes in a wide range, i.e. 300–600 K, the refractive index of the dielectric stripe changes significantly. Therefore, the thermo-optic effect has been taken into account in the simulation. The thermo-optic coefficients of SiO2, Al2O3 and Si are 1 10 5/K, 4.7 10 5/K and 1.8 10 4/K, respectively [39,40]. It can be found from Fig. 2 that as the refractive index of dielectric stripe increases, the value of Re(neff) increases, the propagation length decreases, which means that the propagation modes can be better confined in the dielectric stripe. In addition, Fig. 2 also shows that the influences of temperature on the propagation properties. As the temperature increases, the value of Re(neff) decreases, the propagation length increases. The reasons are shown in the following. As the temperature increases, the carrier concentration of InSb increases, resulting into the enhancement of the dielectric constant of InSb at higher temperature. For instance, the dielectric constants of InSb are 8.25 101 þ2.62 101i, 1.18 103 þ 7.16 102i, and 3 3 2.10 10 þ 1.69 10 i at the temperatures of 300 K, 500 K, and 600 K, respectively. Thus, InSb layer shows better “metallic” properties at higher temperature. Consequently, much less modes penetrates into the substrate layer, leading into the value of the effective indices of propagation modes decreasing. Fig. 3 shows the modulation depth of the DLSPs waveguides versus temperature for different kinds of dielectric stripes. The modulation depth has been defined by using the Eq. (7), and x stands for neff. It can be found from Fig. 3 that the modulation depth of Re(neff) and LSPP increase with the increase of temperature, resulting from the value of the effective index of hybrid modes decrease at higher temperature. Furthermore, with the permittivity of the dielectric stripe increases, the modulation depth of the LSPP increases, i.e. the value of Si stripe is much larger
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Fig. 2. The real part of the effective index (a) Re(neff) and (b) propagation length of the DLSPs waveguide modes versus temperature for different kinds of dielectric stripes. The operation frequency is 1.0 THz. The dielectric materials are SiO2, Al2O3 and Si, respectively. The value of Re(neff) and propagation length decreases and increases with the increases of temperature. The solid, dash, and dot lines represent SiO2, Al2O3and Si, respectively.
than those of SiO2 and Al2O3. For instance, when the temperature changes in the range of 300–600 K, for the dielectric stripe of SiO2 and Al2O3, the modulation depth of the Re(neff) are 6% and 7%, respectively. It is also confirmed by the 2D field distribution obtained from the FEM software, as shown in Fig. 5. Another point should be noted is that the modulation depth of Lspp is much larger that of the Re(neff). For example, for Si stripe, the modulation depth of the Lspp and Re(neff) are about 80% and 10% when the temperature changes in the range of 300–600 K. The mode confinement can be measured by the mode area, which is defined by Eq. (5). Fig. 4(a) shows the normalized mode area of DLSPs waveguides structure for different kinds of dielectric stripes. As temperature increases, the mode area increases, the confinement decreases. This is in accordance with the effective index of the modes decreases with the temperature. It can also be found from Fig. 4(a) that as the refractive index of the dielectric stripe increases, the mode area decreases, which means that the dielectric stripe with larger permittivity shows better confinement. The modulation depth of mode area of DLSPs waveguides vs. temperature can be found in Fig. 4(b). As the temperature increases, the modulation depth of mode area can be changed in a wide range. When the temperature is changed in the range of 300–600 K, the modulation depth of mode area can reach more than 60% for the dielectric stripe of Si. As the temperature decreases, the value of mode area decreases, leading to the modulation depth of mode area increase. In addition, as the refractive index of dielectric stripe increases, the modulation depth of modes area increases, i.e. the modulation depth of the Si is larger than
Fig. 3. The modulation depth of the (a) Re(neff) and (b) propagation length of the DLSPs waveguides modes versus temperature for different kinds of dielectric stripe. The operation frequency is 1.0 THz. The dielectric materials are SiO2, Al2O3 and Si, respectively. For the dielectric stripe of SiO2, Al2O3, and Si, the modulation depth of the Re(neff) are 6%, 7% and 10%, respectively, and the modulation depth of theLsppare 60%, 70% and 80%, respectively. The square, circle, and triangle symbols represent the dielectric stripes of SiO2, Al2O3 and Si, respectively.
that of SiO2. This means that the propagation modes of the dielectric stripe with larger refractive index are affected by the changes of temperature much more significantly. For instance, at the temperature of 600 K, the modulation depth of the mode area for the SiO2, Al2O3 and Si are 30.4%, 46.3% and 61.8%, respectively. The electric fields distribution of above several kinds of dielectric stripe can be found in Fig. 5. As the refractive index of the dielectric stripe increases, there are much more modes confined in the dielectric stripe, which means that the confinement of DLSPs waveguides increases, as shown in Fig. 5(b), (d) and (f). Additionally, as the temperature increases, more modes leak into the surrounding air, resulting into the value of the effective index decreasing. For instance, when the dielectric stripe material is SiO2, the value of the effective indices of the DLSPs modes are 1.72 and 1.61 at the temperatures of 300 K and 600 K, respectively. The waveguides properties can be improved by coating the DLSPs structure with a thin dielectric layer with larger refractive index, as the inset shown in Fig. 6(b). The thickness of the Si layer is 10 μm, and the length and thickness of SiO2 are both 90 μm. The real part of the effective index Re(neff) and propagation length versus temperature for the above mentioned waveguides structure are shown in Fig. 6(a) and (b), respectively. It can be found from Fig. 6 that by coating Si layer on the DLSPs waveguides, the value of Re(neff) increases, the propagation length is also much longer, especially at higher temperature. The possible reasons are shown in the following. For the modified DLSPs waveguides structure, it
C. Liu et al. / Optics Communications 356 (2015) 64–69
can be regarded as a similar capacitor like structure. With the increase of the refractive index of dielectric stripe, more modes energy can be stored, which means that the propagation modes
Fig. 4. (a) The normalized effective mode area Am/A0 and (b) the modulation depth of normalized effective mode area Mod(Am/A0) of the DLSPs waveguide modes versus temperature for different kinds of dielectric materials stripe. The operation frequency is 1.0 THz. The dielectric materials are SiO2, Al2O3 and Si, respectively. The mode area increases and the confinement decreases as temperature increases. In (a) the solid, dash, and dot lines represent SiO2, Al2O3 and Si, respectively. In (b) the square, circle, and triangle symbols represent the dielectric stripes of SiO2, Al2O3 and Si, respectively.
67
can be better confined in the dielectric stripe. This is also confirmed by the field distribution shown in Fig. 8. Simultaneously, the existence of low loss SiO2 layer also results much more mode spreading into in the upper section of dielectric stripe (Si),
Fig. 6. The real part of the effective index (a) Re(neff) and (b) propagation length of DLSPs waveguides modes versus temperature for different kinds of dielectric stripes. The operation frequency is 1.0 THz. The dielectric materials are Si, SiO2 and SiO2–Si, respectively. The inset in (c) shows the sketch of the proposed hybrid dielectric stripe structure of DLSPPs waveguides structure. The Re(neff) decreases and propagation length increases as temperature increases. The solid line, dash line, and dot line represent SiO2, Si, and SiO2–Si, respectively.
Fig. 5. The electric field distribution of the DLSPs waveguides structure for different kinds dielectric stripe materials. The dielectric stripes are SiO2 (a, b), Al2O3 (c, d) and Si (e, f), respectively. The temperature are 300 K (a, c and e) and 600 K (b, d and f), respectively. The operation frequency is 1.0 THz. The width and thickness of SiO2, Al2O3 and Si are 100 μm.
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C. Liu et al. / Optics Communications 356 (2015) 64–69
Fig. 7. Comparison of the FoM versus temperature for different kinds of dielectric stripe materials. The dielectric materials are Si, SiO2 and SiO2–Si, respectively. The thickness of the coating Si layer on the SiO2 is 10 μm. The temperatures is changed in the range of 300–600 K. The solid line, dash line, and dot line represent SiO2, Si, and SiO2–Si, respectively.
value of the FoM increases. In addition, the value of FoM of the modified DLSPs waveguides is largest among the three waveguides structures. By coating a thin Si layer on the DLSPs waveguides, not only the confinement increases, but the loss (propagation length) also decreasing (increasing). Consequently, the value of FoM of the modified DLSPs structure increases, which means that the overall performance of the modified structure can be significantly increases by coating a dielectric layer. The electric field distribution of the fundamental mode for different kinds of waveguide structures has been shown in Fig. 8. We can found that much more modes have been confined in the modified dielectric stripe structure, as shown in Fig. 8(b) and (d). In addition, as the temperature of InSb layer increases, more modes leak into the surrounding air, resulting into the value of effective index decreasing. For instance, if the dielectric stripe material is SiO2, the values of effective indices of the DLSPs modes are 1.72 and 1.61 at the temperatures of 300 K and 600 K, respectively. While in the new type DLSPs waveguide, i.e. around the SiO2 stripe covered with 10 μm thick Si, there will be much more mode confined in the dielectric stripe, leading to the value of the effective index increasing. For instance, in the modified new structure, the values of the effective indices of the modified DLSPs mode are 1.86 and 1.75 at the temperature of 300 K and 600 K, respectively.
4. Conclusion
Fig. 8. The electric field distribution of the propagation mode for different kinds of waveguide structures. The waveguides are the original DLSPs structure (a, b) and the modified structure (c, d). The temperature are 300 K (a, c) and 600 K (b, d). The operation frequency is 1.0 THz. In the modified structure, the thickness of the Si layer is 10 μm, and the length and thickness of SiO2 are both 90 μm.
The tunable propagation properties of semiconductor-supported DLSPs waveguide structures have been theoretically investigated in the THz regime. The influences of temperature, operation frequency, and thermo-optic effect of dielectric stripe materials have been given and discussed. The results show that the waveguide properties of DLSPs structure can be modulated in a wide range by changing the temperature. As the temperature increases, the effective index of propagation mode decreases, while the propagation loss increases. When the temperature is changed in the range of 300–600 K, the modulation depth of propagation length can reach more than 80%. As the refractive index of the dielectric stripe increases, the modulation depth of propagation modes increases. Additionally, for the hybrid dielectric stripe structure, i.e. by coating Si on the SiO2 layer, the propagation length and figure of the merit can be improved obviously. Our results are very useful to have a better understanding of the propagation mechanism of the DLSPs structure and design novel waveguide devices, such as such as modulators, switchers, sensors and polarizer.
Acknowledgments reducing the interaction of modes with the high loss InSb substrate. Thus, the propagation losses also decrease as well. Furthermore, as the temperature increases, the carrier concentration of InSb layer increases, resulting into the enhancement of dielectric constant. Much less mode penetrates into the substrate layer at higher temperature, leading into the influence of the dielectric coating layer increasing. As given in Eq. (8), FoM is a suitable measurement for a tradeoff between mode confinement and loss, which has been adopted to quantitatively compare the proposed modified DLSPs waveguide with the original structure. Fig. 7 shows the relationship of FoM to temperature for different kinds of dielectric materials, i.e. Si, SiO2 and SiO2–Si structures. As the temperature increases, the value of FoM increases. This can be explained by the fact that as the temperature increases, the value of the dielectric constant of InSb increases, leading to the value of Re(neff) decrease, and the propagation loss increases. Because the influence of temperature on the propagation length is larger than that of the Re(neff), the
This work is supported by the Funding of Shanghai Pujiang Program under the Grant no. 15PJ1406500, the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, China, the Research Funding of Shanghai Normal University under the Grant no. SK201529, the Initial Funding of Scientific Rsearch for the Introduction of Talents of Shanghai Normal University, and the Funding of National Natural Science Foundation of China under the Grant no. 61307130, and the Funding of the Shanghai Municipal Education Commission under the Grant no. 14YZ077.
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