1. Introduction

Slow light structures have many potential applications such as optical delay lines, nonlinear enhancements, etc [1]. There are many methods/structures with slow light effects including electromagnetically induced transparency [2], directly coupled resonators [3], stimulated Raman scattering [4], photonic crystal [5], surface plasmon polaritons [6] and so on. Recently, several slow light LHM waveguides, which is composed of LHM layer and normal dielectric medium layers, have been proposed [7–9]. Since the power in LHM layer flows in the opposite direction of that in normal dielectric layers, when they are almost equal, the guided wave will have very low group velocity. K. L. Tsakmakidis et al. went further and proposed a linear slowly tapered waveguide to achieve so-called “trapped rainbow” phenomena [1]. Such structure is able to slow down the light over a broad band at the room temperature and is more valuable in applications. However, the low-lossy uniform bulk of LHM is very difficult to realize now. Besides, The loss and the non-adiabatic tapered structure for a realistic LHM waveguide will severely limit the performance of slow-light propagation [10,11]. Though alternative structures of slow light waveguide based on negative-refractive-index photonic crystal or stack of alternating layer of silver and silica have been proposed [11–13], these tapered structures are all difficult to fabricate too.

Actually, LHMs or periodic structures are not necessary to realize opposite power flows in different layers of a waveguide. It has been found in previous work [14–17] that such opposite power flows can be obtained in high-permittivity-dielectric-loaded metallic waveguides. Thus, the dielectric-loaded metallic waveguide may also support the slow light propagation, as the LHM waveguide. However, previous papers [14–17] mainly focus on the backward waves or the abnormal dispersion relation for the dielectric loaded metallic waveguide.

In this article, we will show that the “trapped rainbow” (broadband slow light) can be obtained in simple and realizable structures based on GaAs-rod-loaded metallic waveguide. We investigate the dispersion relation of the mode in GaAs-rod-loaded circular metallic waveguides in terahertz frequency region. And, the existing conditions for slow light in the GaAs-rod-loaded metallic waveguide of various parameters are analyzed. Then, the realizable tapered waveguide structures are designed accordingly to achieve the “trapped rainbow”. We also show that the slow light with low loss can be realized in a practical GaAs-rod-loaded circular metallic waveguide.

2. Calculation and analysis

The waveguide we consider is shown in the inset of Fig. 1(a) . A metallic cylindrical hollow waveguide (with inner radius of R ₀) is coaxially inserted with a GaAs rod (with radius of R ₁). The GaAs is chosen for its high permittivity and low loss in terahertz region. The real part of permittivity for GaAs is assumed to be 12.9 [18]. Figure 1(a) shows the dispersion curve of HE₁₁ mode in an ideal lossless waveguide with parameters of R ₀ = 2R ₁ = 0.4mm in the wavelength range from 1.73mm to 2.51mm. The curve is quite similar to those for LHM waveguides [7–10,19]. Forward mode, backward mode and complex mode can all exist in the waveguide and are degenerated at the “critical point” [corresponding to the asterisk point in Fig. 1(a)]. It is easy to get that, at the “critical point”, the mode is of zero group velocity ($d\omega /d\beta$), and near the “critical point”, the forward- and backward- modes all have low group velocities. Thus, such a simple uniform waveguide without LHMs or periodic structures inside can also support slow light propagation.

 

Fig. 1 (a) Dispersion curve for the ideal GaAs-rod-loaded cylindrical metallic waveguide as shown in the inset. For the waveguide, R ₀ = 2R ₁ = 0.4mm. The permittivity of GaAs is 12.9, and the dielectric between the rod and hollow metallic waveguide is air. The dashed line corresponds to the complex mode. (b) The time average Poynting vector along the waveguide for the forward mode corresponding to circle point A in Fig. 1(a). (c) The time average Poynting vector for the backward mode B.

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The time average Poynting vectors of forward mode and backward mode are calculated, and illustrated in Fig. 1(b) and (c), respectively [corresponding to the modes pointed with circles A and B in Fig. 1(a)]. One can see that the power flow in the air gap of the waveguide is negative (the direction of the propagating vector is assumed to be positive) and anti-parallel to the direction of power flow in the rod. The reason is that the normal components of most magnetic fields in the rod are reverse to those in the air gap [17]. And, for the guided mode, the air gap is just like the LHM layer in an LHM waveguide. Therefore, the dispersion relation of the GaAs-rod-loaded metallic waveguide is quite similar to that of LHM waveguides. The mode will be backward (/forward) if its energy mainly flows in the air gap (/GaAs rod), and will have a zero group velocity if the absolute power flow in the air gap exactly equals to that in rod.

The change of the parameters R ₁ or R ₀ may vary the ratio of power flow in air gap and in the GaAs rod, resulting in the change of the group velocity of the guided wave, as shown in Fig. 2(a) . The group velocity of the guided forward wave decreases as the R ₁ of waveguide becomes small. So, for a forward wave propagating along a waveguide of which the rod is slowly tapered, it will slow down and its energy will accumulate. At the “critical point”, the wave will be “trapped” and have the strongest field [12,20,21]. Waves with different wavelength have different waveguide parameters at the “critical point”, as shown in Fig. 2(b) and 2(c). When a broad-band wave propagates along the waveguide of which the dielectric rod or hollow metallic shell is slowly tapered, the different wavelength components of the wave will be partly trapped in different location. Thus, they will be “separated”, and so-called “trapped rainbow” phenomenon is realized.

 

Fig. 2 (a) The group velocities of modes in the GaAs-rod-loaded metallic waveguide with various R ₁. The R ₀ is 0.4mm and wavelength of modes is 0.214mm. (b) The wavelength, λ_cp, of mode at the critical point for different R ₁ of waveguide. R ₀ = 0.4mm. (c) The λ_cp for different R ₀ of waveguide. R ₁ = 0.2mm.

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According to Fig. 2(a) and (b), a uniform metallic waveguide of R ₀ = 0.4mm with a tapered GaAs rod inserted is designed to obtain the broadband slow light, as shown in Fig. 3(a) . The radii of GaAs rod at two ports are 0.21mm and 0.23mm, respectively. The length of the waveguide is 12mm. The incident wave is coupled into the left port and propagates along to the right, and its polarization is shown in Fig. 3(b). Figure 3(c)-(f) show the finite element method simulation results of the absolute electric field (|E|) distribution for four different wavelengths, i.e. 2.16mm, 2.14mm, 2.12mm and 2.10mm. The four waves are separated due to being accumulated at different positions. The electric fields can be enhanced more if the waveguide has a larger R ₁ at the left port and longer length (to ensure that the waveguide is tapered enough slowly).

 

Fig. 3 (a) Schematic diagram of a tapered waveguide with R ₀ = 0.4mm, R ₁ = 0.23mm, R ₂ = 0.21mm and L = 12mm. (b) The electric field distribution of the wave coupled from left port of the waveguide. (c)-(f) the absolute electric field (|E|) distribution at the plane of y = 0 for four different incident waves in the tapered waveguide, respectively. (c) λ = 2.16mm, (d) λ = 2.14mm, (e) λ = 2.12mm, and (f) λ = 2.10mm.

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Besides R ₁ and R ₀, the wavelength of the “critical point” varies with the permittivity of dielectric rod, as illustrated in Fig. 4 . The wavelength of critical point changes from 2.147mm to 2.17mm when the permittivity of rod increases from 12.6 to 13. The permittivity of GaAs rod can be changed via thermo-optic effect [22]. So, in a tapered waveguide as shown in Fig. 3(a), the wavelength of the “trapped” wave in a particular position can be tuned by changing the temperature. As the temperature of the dielectric rod increases gradually, various wavelength components of the wave will be enhanced one by one. Such a novel structure may have potential applications in sensing or communications. Moreover, the “trapped rainbow” can also be obtained in a uniform structure of the waveguide, if the permittivity of the rod changes gradually along the waveguide (e.g. by placing a heater and a cooler at two ports of waveguide, the temperature gradually changes along the rod). Such waveguide may be easier to realize and more useful than tapered waveguides.

 

Fig. 4 The wavelength at the critical point varies with permittivity of dielectric rod of the waveguide (R ₀ = 2R ₁ = 0.4mm).

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Although the metal is nearly perfect conductor in low frequency of terahertz region, the loss of GaAs is much larger than that in infrared and is not negligible for the propagation of slow light. So, for the practical waveguide operating in THz range, we only consider the loss of GaAs, and the complex permittivity of GaAs is assumed to be 12.9 + 0.02i [18]. Figure 5(a) shows the guided modes vary with R ₁ at the fixed wavelength of λ = 2.14mm. In a lossy waveguide, the modes at the “critical point” do not degenerate, but instead split into a forward mode and a backward one. Therefore the mode with zero group velocity at the “critical point” cannot be obtained. However, the guided wave near the “critical point” is still with low group velocity and low loss, the field of the wave can be enhanced much at corresponding location in the tapered waveguide as well [20,21]. Figure 5(b) shows the |E| distribution for the wave of λ = 2.14mm propagating in a lossy tapered waveguide with the same structure as in Fig. 3(a). It can be seen that the wave will still be enhanced in a realistic lossy slow light waveguide, though the maximum field is a bit lower than that in an ideal waveguide [as shown in Fig. 3(e)]. If the dielectric rod is replaced with a gain medium, such as terahertz quantum cascade lasers [23], the loss of the wave can be compensated or the wave even can be amplified [13].

 

Fig. 5 (a) The propagation constants of modes as R ₁ of lossy and lossless waveguide varies. The wavelength of the guided mode equals to 2.14mm. For both waveguides, R ₀ = 0.4mm. The black and gray solid curves correspond to the lossy case, and the green thick lines and red dotted lines correspond to the lossless case. (b) The |E| distribution at plane of y = 0 for incident wave of λ = 2.14mm in the lossy waveguide as shown in Fig. 3(a).

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3. Conclusion

In summary, a simple GaAs-rod-loaded metallic waveguide with uniform or non-uniform structure for slow light operation was proposed and analyzed. It was shown that the so-called “trapped rainbow” can be realized in a tapered waveguide or a uniform waveguide with the permittivity of rod properly modulated. We also analyzed the practical lossy tapered waveguide in terahertz region. The results show that the slow light with low loss can still be obtained in a realistic waveguide. Besides, it is worth noting that the dielectric rod can be get by a similar method to fabricate the semiconductor core optical fiber [24], and that the waveguides we proposed can be simplified by replacing half of the structure with a metal plane, due to the symmetric modes in the circular waveguide [17]. And our other numerical analyses show that slow light can also be obtained in such semicircular GaAs-rod-loaded waveguides. This kind of waveguides can be fabricated more easily, since the dielectric rod in the waveguide is sustained by the metal plane.