Rainbow trapping and releasing in InSb graded subwavelength grooves by thermal tuning at the terahertz range

Ruoying Kanyang; Fan Zhang; Genquan Han; Yan Liu; Yao Shao; Jincheng Zhang; Yue Hao

doi:10.1364/OME.8.002954

1. Introduction

Surface plasmon polaritons (SPPs) as a new field of science and technology are being explored for their potentials in constructing chip-based nanoscale buffers, spectrometers, filters, data processors, and quantum optical memories. SPPs are electromagnetic waves bounded to the interface between two different media [1, 2], which have opposite permittivity, such as conductor and dielectric. The ability of strong field confinement to the surface ensures the SPPs to widely apply in optical buffers, filters, and enhanced light-matter interactions [3–6]. It has been demonstrated that designed plasmonic waveguide or metamaterial could reduce the speed of light at different locations along the propagation direction [7]. This special phenomenon of rainbow trapping has paved the way for the applications of SPPs in improving the performance of optical switching or optical signal precession [8–12]. Tremendous research shows that rainbow trapping could be achieved by different structures such as concentric circular grooves, quantum-dot emitters, photonic crystals, hyperbolic metamaterial structure or waveguide trapper on metal-insulator-metal [13–16]. Among the above mentioned structures, photonic crystals are comparatively disadvantaged because of the delay-bandwidth resulted from the group delay from an optical resonance, and the delay is inversely proportional to the bandwidth within where the delay occurs. Therefore, the minimum group velocity is limited for a pulse with a given temporal duration and corresponding bandwidth [17]. For example, in a coupled resonator optical waveguides structure, the minimum velocity that can be reached is about 10⁻² c (c is the velocity of light). That’s why photonic crystals encounter the bottleneck in stopping light. In order to overcome the limitation, the graded grating structure has been studied extensively. It has been demonstrated that the group velocity could be reduced to nearly zero by using graded grating model [18].

Most of the existing researches focus on realizing rainbow trapping of electromagnetic waves by various approaches [19–21], but little attention is paid to releasing the trapped SPPs mode [22]. In order to effectively control the speed of surface waves, it is important to study the releasing of electromagnetic waves. As we all know, metal can excite SPPs, and it has been demonstrated that the graded grating structure with silver could realize rainbow trapping and releasing [22]. For achieving an efficient coupling between the incident waves and free charges, plasmonic effects require that the penetration of the electromagnetic radiation in the material is significant. The coupling becomes stronger with lower value of permittivity, $ϵ$ . And the $ϵ$ is determined by carrier density, which influences the value of plasma frequency $ω_{p}$ . At low frequency, InSb behaves like metal, which makes it possible to excite SPPs at terahertz (THz) frequency. However, with $| ε | \geq 10^{5}$ at THz frequency, a long SPP decay length results from the high plasma frequency of metal, which limits the applications of low-frequency plasmonics [1]. Compared to metal, the permittivities of some semiconductors, such as silicon and InSb, are much lower than metals because of smaller carrier density [23], which makes them more suitable for the applications of SPPs at THz frequency. And semiconductor material has been used in lots of optical applications [24–27]. Besides, the carrier concentration of InSb can be flexibly tuned by changing temperature or doping. Therefore, compared with metal, InSb plasmonic graded structure can be a better candidate for the applications of SPPs at low frequency. And InSb has been thoroughly investigated in the THz region, which can be applied in lots of plasmonic applications [28–32]. In recent years, people pay more and more attention to the application of thermo-optic materials in the THz region [33-34], and it has been demonstrated previously that the electromagnetic waves could be released by inserting the grating grooves with thermo-optic materials, whose refractive index would change with the temperature [35–37]. However, considering the complexity of manufacture process, it is significant to research the SPPs pulses propagating on the gradient-corrugated grating waveguide based on thermo-optic material InSb.

In this paper, the dispersive characteristics of the gradient grating waveguide along the intrinsic InSb surface in the THz frequency region are investigated. The propagation characteristics are simulated by computer simulation technology (CST) microwave studio. The trapping and releasing properties of InSb graded grating structure are thoroughly compared by tuning the temperature. Such a structure based on thermo-optic material InSb exhibits an excellent trapping and releasing capability, which is significant for future low frequency surface plasmonic optical applications.

2. Device structure

To investigate the SPPs property of intrinsic InSb gradient-corrugated grating waveguide thoroughly, the structure we used is exhibited in Fig. 1. And we examine the dispersive relation by the eigenmode solver of commercial software, CST Microwave Studio. In this software, the dispersive relation can be obtained through parameters calculation, and the parameters includes collision frequency and plasma frequency. In the simulation, the thickness t of the intrinsic InSb strip along the z direction is 65 μm, the period is represented by p, the depth of the groove is represented by h, the groove width is represented by d, and the width of the space bar is represented by a. The width and height of the strip are marked as W and H, respectively. We can optimize the structure by properly choosing the values for a, d, h, p. The period p is 215 μm, the width of the space bar a is 150 μm, the groove width d is 65 μm, and the duty factor is 2.3. The height for the whole graded grating waveguide H is set to be 600 μm. The groove depth h changes from 100 to 500 μm gradually along the x direction. So, the whole length of the strip W is 8965 μm including 41 periods. The standard printed circuit board process could be used to fabricate the experiment sample on a flexible InSb clad laminate. The clad laminate is composed of a 65 um InSb layer and a dielectric clad sheet. The graded structure could be fabricated using the electron beam lithography (EBL) and inductively coupled plasma (ICP) from the Z direction shown in Fig. 1. And the simulation result shows that the dielectric substrate has no influence on the SPPs properties of the structure.”

Fig. 1 3D schematic of the designed plasmonic graded grating waveguide with intrinsic InSb.

Download Full Size | PDF

3. Results and discussion

In order to analyze the dispersion characteristics of SPPs in the intrinsic InSb plasmonic graded grating waveguide, firstly, we take a repeating unit of the grating structure. The calculated dispersion curves of the device are demonstrated in Fig. 2 (a). The permittivity of InSb that we used in the calculation is well described by the Drude model [38]:

ε(ω {)=ε}_{\infty} (1 - \frac{ω_{p}^{2}}{ω^{2} + τ^{- 2}} + \frac{i ω_{p}^{2} τ^{- 1}}{ω (ω^{2} + τ^{- 2})}),

where the high-frequency permittivity is represented by

ε_{\infty}

and

ε_{\infty} = 15.7

for InSb at room temperature. The average collision time τ is related to carrier mobility μ by

μ = τ e / m^{*}

, and InSb is a semiconductor with a very large electron mobility μ ≈

7.7 \times 10^{4}

cm²V⁻¹s⁻¹. And the plasma frequency is represented by

ω_{p} (= \sqrt{N e^{2} / ε_{\infty} ε_{0} m^{*}})

. Here N represents the charge carrier concentration of InSb, and N ≈10¹⁶ cm⁻³ for intrinsic InSb here at room temperature. And e represents the fundamental charge, while

ε_{0}

represents the vacuum permittivity and m* represents the charge carrier effective mass. The

m_{0}

represents the electron mass and the electron effective mass of InSb is

0.014 m_{0}

. As we can see, the upward trend of the dispersive curve gradually slows until it flattens out, and hence the cutoff frequency of the curve is corresponding to the point where the upward trend disappears. Moreover, the cutoff frequency decreases along with the deeper groove due to stronger confinement of the SPPs waves. It can be seen that the minimum value of the cutoff frequency is 0.084 THz, which is corresponding to the groove depth of 500 μm. And the maximum cutoff frequency is 0.326 THz, which is corresponding to the groove depth of 100 μm. To further demonstrate the propagation characteristics of SPPs, we compare the dispersive relation of one periodic unit of the designed spoof plasmonic graded grating waveguides based on doped InSb and doped silicon, the latter of which has been thoroughly analyzed before in reference [39]. The structure we used in doped silicon is similar to the structure used here. In contrast, the dispersive relations of highly n-doped InSb with a carrier density of 7 × 10¹⁸ cm⁻³ and highly p-doped silicon with a carrier density of 5 × 10¹⁹ cm⁻³ are depicted in Fig. 2(b) and (c). Similarly, the almost exactly same tendency is observed in these two figures. When the groove depth of silicon increases to 500 μm, there exists an unusual trend towards the higher frequency in Fig. 2(c), which is not shown here. As we can see, the curves of intrinsic InSb have more significant deviation from the light line. In general, intrinsic InSb has stronger confinement of light than doped InSb and silicon with deeper groove depth. For further explanation, the skin depths of different materials are compared and explored in Fig. 2(d), which can modify the effective size of the corrugate structures directly. According to the Drude model, the skin depth varies with the scattering rate, which is also influenced by carrier density and temperature. The skin depth of doped silicon with a carrier density of 5 × 10¹⁹ cm⁻³ and that of gold are indicated by a white dot and a yellow dot respectively. By varying the temperature from 225 K to 325 K and the doping concentration from 1 × 10¹⁶ to 7 × 10¹⁸ cm⁻³, the two regions of photoexcited InSb are circled by the green ellipse and the red ellipse respectively. From Fig. 2(d), it can be observed that the skin depth varies from 0.48 to 7.43 μm along with the carrier density decreasing, which can be further raised to 29.79 μm when the temperature drops to 225 K. This value is much larger than that of silicon and gold. Since the effective size of the groove increases with the larger skin depth, the coupling of the incident radiation can also be enhanced due to the larger effective refractive index for coupling [38]. For any InSb grating structure with the thickness of several microns, the THz radiation can penetrate completely into InSb and be dissipated effectively, which will be detailed analyzed in the latter section.

Fig. 2 (a) The dispersion relations of SPPs in one unit of the designed spoof plasmonic graded grating waveguides with different groove depths. Parameters of a, d, H, and t are fixed at 150, 65, 600, and 65 μm, respectively. (b) The dispersive relations of highly doped InSb. (c) The dispersive relations of highly doped silicon. (d) Skin depth as a function of the plasma frequency and of the scattering rate at the frequency of 0.9THz. The white dot corresponds to the value for doped silicon, and the yellow dot corresponds to the value for gold. The red ellipse corresponds to the region of doped InSb, and the green ellipse corresponds to the region of intrinsic InSb with different temperatures.

Download Full Size | PDF

The incident waves can cause the collective oscillation of electrons on the surface of InSb. And the coupling between the electromagnetic field and surface charge oscillation constitutes the SPPs, which means the SPPs are excited in the grating strip. To further interpret the dispersion characteristics of SPPs mode in the graded grating waveguide for InSb, as is demonstrated in Fig. 3, the electric field magnitude distribution of TM-mode is extracted using COMSOL Multiphysics. And we observe the simulated electric magnitude distribution on a plane that is 1 μm above the grating strip in the z direction. The length of the whole strip is 8965 μm with 41 periods and the depths of grooves change gradually from 100 to 500 μm at a step of 10 μm. As we can see, the propagation distances of the SPPs waves are shortened step by step along with the increasing frequency, which corresponds to the different cutoff frequencies in the dispersive relations shown in Fig. 2. The group velocity is obtained from the slope of the dispersive curves in Fig. 2(a). As can be seen from Fig. 3, the group velocity of the SPPs falls to nearly zero along the x direction at the position 7525 μm, 6235 μm, 4515 μm, 2795 μm, 1075 μm for the cutoff frequency 0.084 THz, 0.112THz, 0.151THz, 0.214THz, 0.326THz, respectively.

Fig. 3 2D electric field magnitude distributions of the designed spoof plasmonic graded grating waveguides with intrinsic InSb at different frequencies. (a) 0.084 THz, (b) 0.112 THz, (c) 0.151 THz, (d) 0.214 THz, (e) 0.326 THz.

Download Full Size | PDF

Then there is another question after trapping the THz rainbow at different positions along the gradient-corrugated grating waveguide: how to release the trapped waves. The reported methods are to cap the metal grating with dielectric material or inset the dielectric material into the metal grooves [35, 36]. Both methods base on the thermo-optic properties of the material. The refractive index of the material is temperature-tunable because of thermo-optic effect. When the temperature changes, the changed permittivity of the thermo-optic material will lead to the variation of the dispersive relation. Compared with normal material, the refractive index of InSb can be easily changed by changing temperature, and hence we take InSb as an example. It is well known that the Bragg scattering is influenced obviously by changing temperature. As shown in Fig. 4(a) and (b), we can see that the absolute value of the real part of the permittivity decreases along with the increasing frequency and approaches to 0 at higher frequency. And the imaginary part also decreases along with the increasing frequency. Besides, the absolute values of both parts decrease at lower temperature with fixed frequency. And comparing Fig. 4(a) and (b), the absolute value of the permittivity for InSb is much lower than that of metals at THz, leading to a lower plasma frequency. Consequently, thermal switching of intrinsic InSb plasmonic grating structure seems promising. Then, we investigate the thermal dependence of the SPPs band gap in the corrugated grooves. It suffices to characterize the first SPPs band gap by restricting the calculation in the first SPPs Bragg scattering. Following expressions are chosen to analyze the band edge for the first gap [40]:

\frac{ω_{\pm} (T)}{c} \approx \frac{G}{2} {1 - \frac{1}{ε (T)} {[1 + s_{0} (T) f_{0} \mp | s_{0} (T) f_{1} |]}^{2}}^{- \frac{1}{2}},

ε (ω, T) = ε_{\infty} - \frac{ω_{p}^{2} (T)}{ω [ω + i Γ (T)]},

s_{0} (T) \equiv \frac{1 - ε (T)}{δ (T) ε (T)} {1 - {[δ (T)]}^{2}}^{\frac{1}{2}},

where c is velocity of light,

G = 2 π / p

is the reciprocal lattice vector, p is the grating period, d represents the groove width, and h represents the groove depth as demonstrated in Fig. 1. All magnitudes of ω are evaluated by cG/2. The Eq. (2) explicitly incorporates the dependence of temperature T. In the Drude-type dielectric function Eq. (3), the plasma frequency ω_p (T) and collision rate Γ (T) also depend on temperature. Equation (4) exhibits the surface impedance of the grating structure, which is valid for small optical skin depth

δ (ω, T) = (\frac{c}{ω}) {[- ε (ω, T)]}^{- \frac{1}{2}}

. In Eq. (2), the two first Fourier terms of the series expansion of the surface profile function are

f_{0} = \frac{h d}{p}

,

f_{1} = \frac{h}{π} \sin (π \frac{d}{p})

, which are applied for rectangular grooves.

Fig. 4 The real part (a) and imaginary part (b) of the permittivity of InSb as a function of frequency at different temperatures. (c) Thermal dependence of the first gap of SPPs on InSb grooves. The lines represent the lower and higher frequencies for the upper and lower band edges (corresponding to ω⁺ and ω^- in the Eq. (2)). The grooves have a depth of h = 100 μm. Shaded areas indicate the spectral region where the thermally-induced shift of the lower band edge occurs.

Download Full Size | PDF

From the Eq. (2), it is sufficient to obtain the first SPPs band gap of InSb arrays for corrugated grooves of depth h = 100μm, as shown in Fig. 4(c). As above equations demonstrate, there lies two asymmetric frequency edges of the first SPPs bandgap and the gap almost extends as the temperature decreases except for a sudden drop at low temperature. The upper one refers to the leak mode of SPPs wave, which will be coupled to freely propagating electromagnetic radiation. For the lower one, the SPPs waves reflect at every single groove and interfere constructively, which enhances the establishment of SPPs. Consequently, only the low-frequency band edge is meaningful and constructive. When the temperature decreases, due to the reduction of free carrier density, the value of $| ε |$ becomes smaller, which yields a lower plasma frequency. Thus, both the confinement of SPPs mode to the surface and the scattering become stronger, which is corresponding to the wider gap observed in Fig. 4(c). What’s more, the leakage at the higher frequency increases along with the broadening of the bandgap. When the temperature changes from 325 K to 225 K, the lower band edge shifts by 0.05 THz, which is almost 10% of the gap width at room temperature. Therefore, the SPPs ultrabroadband thermal tuning can be achieved by exploiting the lower gap edge.

The dispersive relations of intrinsic InSb in one repeating unit of the designed gradient-corrugated grating waveguide with different groove depths are shown in Fig. 5, and the temperature varies from 270 to 325 K. The parameters used in simulation are the same with those in Fig. 2. As can be seen from the Fig. 5, the cutoff frequency of the dispersive curves increases along with the increasing temperature. When h = 100μm, the cutoff frequency is 0.249 THz at the temperature of 295 K, and the SPPs mode at this frequency is trapped. As the temperature increases to 325 K, the cutoff frequency increases to 0.28 THz, which implies a weaker confinement for SPPs mode at 325K. The cutoff frequency increases with increasing temperature at fixed depth. And according to the conclusion shown in Fig. 2, the deeper the depth, the lower the cutoff frequency of the groove. To trap the surface mode at the same frequency, a groove with deeper depth is required when the temperature is higher. In other word, the surface mode will be trapped at the groove with deeper depth when the temperature is higher, which corresponds to the realization of releasing. Compared with 295K, the electromagnetic wave is released at 325K. When the temperature decreases to 270 K, the cutoff frequency decreases to 0.165 THz. Compared with 295K, the electromagnetic wave is tightly trapped at 270K, which means the surface mode can be trapped by the grating groove with a depth smaller than 100 μm. When the temperature decreases from 325 K to 270 K, the shift of frequency is different from the result in Fig. 4(c) due to the insufficient finite array number N [40]. The similar ability of thermal switching is also observed in Fig. 5(b). For h = 500 μm, the frequency range is from 0.18 to 0.29 THz. The electromagnetic wave is more strongly trapped at 270K and released at 325K. Therefore, tuning temperature can release or more tightly trap SPPs mode in such InSb spoof plasmonic graded grating waveguides.

Fig. 5 The dispersion relations of SPPs in one unit of the designed spoof plasmonic graded grating waveguides with different groove depths at different temperatures.

Download Full Size | PDF

To get a better picture of the thermo-optic property of InSb, 2D electric field distributions of the designed gradient-corrugated grating waveguide are simulated in Fig. 6. We observe them on the plane that is 1 μm above the grating structure along the z direction. For comparing the propagation properties of InSb at different temperatures, the fixed frequency 0.214 THz is chosen for the convenience of covering all the temperature range. The propagation lengths of SPPs are 3225 μm, 2795 μm, 2150 μm, 1935 μm, 1505 μm, 430 μm, at the temperatures of 325 K, 295 K, 270 K, 255 K, 240 K, 225 K, respectively. It can be seen that the propagation distances are shortened step by step along with the decreasing temperature. Because the groove can trap the electromagnetic waves with frequencies higher than its cutoff frequency. This trend corresponds to a decrease in the cutoff frequency as the temperature decreases, so that the electromagnetic wave can’t pass through the groove that could have passed, which is the same with the tendency shown in Fig. 5. By comparing with the 2D electric field distributions shown in Fig. 3, the trapped mode is released at 325 K, and tightly trapped below 295 K, which permits the meaningful plasmonic applications for speed control of surface waves. At the temperature of 295 K, the group velocity of the SPPs falls to nearly zero at the position 2795 μm for the cutoff frequency of 0.214 THz. At 325K, the propagation distance is longer than 2795 μm, which corresponds to the releasing. When the temperature is below 295K, the propagation distance is less than 2795 μm, which corresponds to the trapping. Therefore, the 2D electric field distributions clearly indicate that the rainbow can be trapped and released by tuning temperature in the InSb graded grating structure. Changing temperature can easily change the permittivity of semiconductor and thus affect the propagation lengths of SPPs. A simple model of the flat interface structure between the conductor and air is used to qualitatively analyze the InSb grating. Certainly, the corrugation of the surface will yield effective impedance, which is furtherly beneficial to the establishment of the SPPs mode [41]. If the absolute value of $ε^{'}$ is larger than $ε^{''}$ , the propagation length of SPPs on a flat interface between the conductor and air approximately is [38]:

δ_{s p p} \approx 2 \frac{c_{0}}{ω} {(\frac{ε^{'} + 1}{ε^{'}})}^{\frac{3}{2}} \frac{ε^{'}^{2}}{ε^{''}},

the angular frequency is ω, the speed of light in vacuum is represented by

c_{0}

, and the complex permittivity of the conductor is

ε = ε^{'} + i ε^{''}

. As we can see, the propagation length decreases as the

ε^{'}

reduces or the

ε^{''}

increases. And the skin depth of the conductor approximately is given by

δ_{I n S b} \approx \frac{c_{0}}{ω} {(\frac{ε^{'} + 1}{ε^{' 2}})}^{\frac{1}{2}},

when the

ε^{'}

is smaller, the skin depth into the conductor is larger. When the temperature of the InSb grating is lowered, the permittivity decreases and the effective refractive index of the air-conductor model increases, which yields a more tightly coupling at low frequency. The absolute values of

ε^{'}

and

ε^{''}

decrease as the temperature gets lower, which results in a larger skin depth and a shorter

δ_{S P P}

. For InSb, the permittivity can be easily influenced by changing temperature. At 325 K, the absolute value of

ε^{'}

is 200 and

ε^{''}

is 110. And

| ε^{'} |

decreases to 10 while

ε^{''}

decreases to 6 at 225 K. According to the above equation, the decrease of InSb’s permittivity results in a larger skin depth and a smaller propagation length. As a consequence of the increasing skin depth of InSb, the dissipation of SPPs is more effective, which leads to a shorter

δ_{S P P}

as demonstrated in Fig. 6.

Fig. 6 2D electric field magnitude distributions of the designed spoof plasmonic graded grating waveguides along intrinsic InSb at different temperatures.

Download Full Size | PDF

In order to investigate the feasibility for some meaningful applications, the lifetime of the intrinsic InSb plasmonic mode is estimated by the expression $τ = 1 / α v_{g}$ . Here $v_{g}$ is the group velocity and $α$ is the propagation loss coefficient, and $α$ is impacted by both internal absorption and scattering loss. We extract the propagation loss $α$ from the intensity distribution |E| which is 1μm above the grating surface at 0.084 THz, 0.112THz and 0.151THz, respectively. The grating strips we used in our calculation have uniform groove depths which are fixed at 100 μm, 200 μm, 300 μm, 400 μm, and 500 μm, respectively. The propagation decay coefficient $α$ decreases gradually with the deeper grooves, as revealed in Fig. 7(a). As shown in Fig. 7(b), the SPPs lifetime is estimated at 0.084 THz. The blue dots are the extracted data from different depths and the red line is fitted to guide the eyes. When the fixed groove depth increases, the lifetime of SPPs also increases. The maximum lifetime of the InSb SPPs mode at 0.084 THz reaches approximately 35 ps. The propagation decay coefficient and lifetime at 0.112 THz and 0.151 THz are shown in Fig. 7(c) and (d), respectively. And the blue and black dots are the extracted data from different depths, the red and pink lines are fitted to guide the eyes. The lifetimes at the same depth increase with the increasing cutoff frequency. At 0.086 THz, the lifetime is 35 ps. And the lifetime increases to 182 ps at 0.151 THz. The lifetime of the metal plasmonic grating waveguide reported recently is approximately 3ps [22], and the lifetime of the highly doped silicon grating structure reaches approximately 1200ps [39]. Therefore, the maximum absorption of metal leads to the minimum lifetime of silver grating waveguide. As for intrinsic InSb grating structure, the less absorption of InSb than that of metal leads to a longer lifetime. Although the lifetime of InSb SPPs is not as good as silicon, such a lifetime may be long enough for the future on-a-chip optical communication.

Fig. 7 (a) The relationship between propagation decay coefficient α and the constant depth of grating. 1/α increases with the depth of grating increases. (b) Estimation of the SPPs lifetime, τ, along the grating surfaces for various depths. The blue dots are extracted data for various depths, the red line is an exponential growth fitted to guide the eyes. (c) The relationship between propagation decay coefficient α and the constant depth of grating at 0.112 THz and 0.151 THz. (d) Estimation of the SPPs lifetime at 0.112 THz and 0.151 THz.

Download Full Size | PDF

4. Conclusion

In this work, SPPs propagating at the surface of grating grooves based on InSb achieves rainbow trapping in the THz region successfully. The propagation characteristics of the InSb grating grooves are thoroughly analyzed by the dispersive relation curves, electric field magnitude distribution, the propagation loss and the lifetime of the plasmonic mode. The electric field magnitude distributions of the gradient InSb grating waveguide in a fixed frequency at different temperatures are compared, which proves that the InSb grating structure possesses excellent ability to trap and release SPPs at terahertz range. Therefore, InSb material is a splendid candidate for future plasmonic integrated devices.

Funding

National Natural Science Foundation of China (61534004, 61604112, and 61622405).

References and links

1. J. Gómez Rivas, M. Kuttge, H. Kurz, P. Haring Bolivar, and J. A. Sánchez-Gil, “Low-frequency active surface plasmon optics on semiconductors,” Appl. Phys. Lett. 88(8), 082106 (2006). [CrossRef]

2. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999). [CrossRef]

3. F. Xia, L. Sekaric, and Y. Vlasov, “optical buffers on a silicon chip,” Nat. Photonics 1, 65–71 (2007). [CrossRef]

4. J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, “Transmission and group delay of microring coupled-resonator optical waveguides,” Opt. Lett. 31(4), 456–458 (2006). [CrossRef] [PubMed]

5. M. SoljaČiĆ and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3, 211–219 (2004). [CrossRef] [PubMed]

6. Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438(7064), 65–69 (2005). [CrossRef] [PubMed]

7. L. Thevenaz, “Slow and fast light in optical fibres,” Nat. Photonics 2(8), 474–481 (2008). [CrossRef]

8. Z. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored light in an optical fiber via stimulated brillouin scattering,” Science 318(5857), 1748–1750 (2007). [CrossRef] [PubMed]

9. H. Nakamura, Y. Sugimoto, and K. Asakawa, “Ultra-fast photonic crystal/quantum dot all-optical switch for future photonic networks,” Opt. Express 12(26), 6606–6614 (2004). [CrossRef]

10. D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Ultracompact and low-power optical switch based on silicon photonic crystals,” Opt. Lett. 33(2), 147–149 (2008). [CrossRef] [PubMed]

11. Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. U.S.A. 108(13), 5169–5173 (2011). [CrossRef] [PubMed]

12. J. Park, K.-Y. Kim, I.-M. Lee, H. Na, S.-Y. Lee, and B. Lee, “Trapping light in plasmonic waveguides,” Opt. Express 18(2), 598–623 (2010). [CrossRef] [PubMed]

13. S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics 1(8), 449–458 (2007). [CrossRef]

14. T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, and T. W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. 26(24), 1972–1974 (2001). [CrossRef] [PubMed]

15. M. S. Jang and H. Atwater, “Plasmonic rainbow trapping structures for light localization and spectrum splitting,” Phys. Rev. Lett. 107(20), 207401 (2011). [CrossRef] [PubMed]

16. C. Winnewisser, F. Lewen, and H. Helm, “Transmission characteristics of dichroic filters measured by THz time-domain spectroscopy,” Appl. Phys., A Mater. Sci. Process. 66(6), 593–598 (1998). [CrossRef]

17. M. F. Yanik and S. Fan, “Stopping light all optically,” Phys. Rev. Lett. 92(8), 083901 (2004). [CrossRef] [PubMed]

18. L. Chen, P. W. Guo, Q. Gan, and F. J. Bartoli, “Trapping of surface-plasmon polaritons in a graded Bragg structure: Frequency-dependent spatially separated localization of the visible spectrum modes,” Phys. Rev. B 80(16), 161106 (2009). [CrossRef]

19. H. F. Ma, X. P. Shen, Q. Cheng, W. X. Jiang, and T. J. Cui, “Broadband and high-efficiency conversion from guided waves to spoof surface plasmon polaritons,” Laser Photonics Rev. 8(1), 146–151 (2014). [CrossRef]

20. B. C. Pan, Z. Liao, J. Zhao, and T. J. Cui, “Controlling rejections of spoof surface plasmon polaritons using metamaterial particles,” Opt. Express 22(11), 13940–13950 (2014). [CrossRef] [PubMed]

21. Z. Liao, J. Zhao, B. C. Pan, X. P. Shen, and T. J. Cui, “Broadband transition between microstrip line and conformal surface plasmon waveguide,” J. Phys. D. 47(31), 315103 (2014). [CrossRef]

22. Q. Gan, Y. J. Ding, and F. J. Bartoli, ““Rainbow” trapping and releasing at telecommunication wavelengths,” Phys. Rev. Lett. 102(5), 056801 (2009). [CrossRef] [PubMed]

23. A. Berrier, R. Ulbricht, M. Bonn, and J. G. Rivas, “Ultrafast active control of localized surface plasmon resonances in silicon bowtie antennas,” Opt. Express 18(22), 23226–23235 (2010). [CrossRef] [PubMed]

24. M. T. Nouman, H. W. Kim, J. M. Woo, J. H. Hwang, D. Kim, and J. H. Jang, “Terahertz modulator based on metamaterials integrated with metal-semiconductor-metal varactors,” Sci. Rep. 6(1), 26452 (2016). [CrossRef] [PubMed]

25. Z. Vafapour, “Slowing down light using terahertz semiconductor metamaterial for dual-band thermally tunable modulator applications,” Appl. Opt. 57(4), 722–729 (2018). [CrossRef] [PubMed]

26. J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A. Maier, Z. Tian, A. K. Azad, H. T. Chen, A. J. Taylor, J. Han, and W. Zhang, “Active control of electromagnetically induced transparency analogue in terahertz metamaterials,” Nat. Commun. 3(1), 1151 (2012). [CrossRef] [PubMed]

27. Z. Vafapour and H. Ghahraloud, “Semiconductor-based far-infrared biosensor by optical control of light propagation using THz metamaterial,” JOSA B 35(5), 1192–1199 (2018). [CrossRef]

28. T. H. Isaac, W. L. Barnes, and E. Hendry, “Determining the terahertz optical properties of subwavelength films using semiconductor surface plasmons,” Appl. Phys. Lett. 93(24), 241115 (2008). [CrossRef]

29. T. H. Isaac, J. Gómez Rivas, J. R. Sambles, W. L. Barnes, and E. Hendry, “Surface plasmon mediated transmission of subwavelength slits at THz frequencies,” Phys. Rev. B 77(11), 113411 (2008). [CrossRef]

30. S. M. Hanham, A. I. Fernández-Domínguez, J. H. Teng, S. S. Ang, K. P. Lim, S. F. Yoon, C. Y. Ngo, N. Klein, J. B. Pendry, and S. A. Maier, “Broadband Terahertz Plasmonic Response of Touching InSb Disks,” Adv. Mater. 24(35), OP226–OP230 (2012). [CrossRef] [PubMed]

31. S. Lin, K. Bhattarai, J. Zhou, and D. Talbayev, “Thin InSb layers with metallic gratings: a novel platform for spectrally-selective THz plasmonic sensing,” Opt. Express 24(17), 19448–19457 (2016). [CrossRef] [PubMed]

32. S. Lin, S. Silva, J. Zhou, and D. Talbayev, “A One‐Way Mirror: High‐Performance Terahertz Optical Isolator Based on Magnetoplasmonics,” Advanced Optical Materials, 1800572 (2018). [CrossRef]

33. W. Li, Q. Meng, R. Huang, Z. Zhong, and B. Zhang, “Thermally tunable broadband terahertz metamaterials with negative refractive index,” Opt. Commun. 412, 85–89 (2018). [CrossRef]

34. A. Keshavarz and A. Zakery, “A Novel Terahertz Semiconductor Metamaterial for Slow Light Device and Dual-Band Modulator Applications,” Plasmonics 13(2), 459–466 (2018). [CrossRef]

35. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003). [CrossRef] [PubMed]

36. E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311(5758), 189–193 (2006). [CrossRef] [PubMed]

37. Y. Liu, Y. Wang, G. Han, Y. Shao, C. Fang, S. Zhang, Y. Huang, J. Zhang, and Y. Hao, “Engineering rainbow trapping and releasing in ultrathin THz plasmonic graded metallic grating strip with thermo-optic material,” Opt. Express 25(2), 1278–1287 (2017). [CrossRef] [PubMed]

38. J. Gómez Rivas, C. Janke, P. Bolivar, and H. Kurz, “Transmission of THz radiation through InSb gratings of subwavelength apertures,” Opt. Express 13(3), 847–859 (2005). [CrossRef] [PubMed]

39. Y. Liu, R. Kanyang, G. Han, Y. Shao, C. Fang, Y. Huang, S. Zhang, J. Zhang, and Y. Hao, “Rainbow trapping in highly doped silicon graded grating strip at the terahertz range,” photon Journal. 10, 5700309 (2018).

40. J. A. Sánchez-Gil and J. G. Rivas, “Thermal switching of the scattering coefficients of terahertz surface plasmon polaritons impinging on a finite array of subwavelength grooves on semiconductor surfaces,” Phys. Rev. B 73(20), 205410 (2006). [CrossRef]

41. J. Gómez Rivas, C. Schotsch, P. Haring Bolivar, and H. Kurz, “Enhanced transmission of THz radiation through subwavelength holes,” Phys. Rev. B 68(20), 201306 (2003). [CrossRef]

Rainbow trapping and releasing in InSb graded subwavelength grooves by thermal tuning at the terahertz range

Abstract

1. Introduction

2. Device structure

3. Results and discussion

4. Conclusion

Funding

References and links

Cited By

Figures (7)

Equations (6)

Optical Materials Express