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Abstract
Polarization-sensitive Tamm plasmons are investigated in a multi-layer photonic configuration where a monolayer black phosphorus (BP) is coated on a Bragg mirror separated by a dielectric. Owing to the in-plane anisotropy of BP, the Tamm plasmon can be excited selectively by tuning the BP carrier density. Cross-polarization conversion occurs when the armchair direction of BP makes an angle with the incident plan, i.e., ϕ≠0 or 90°. The BP-based Tamm device can be used as an intensity modulator with a modulation depth up to ∼100% and an insertion loss smaller than -0.55 dB. By analyzing the polarization evolution carefully, a multichannel polarization division multiplexing scheme is proposed and discussed. These findings open a new avenue for exploiting versatile tunable THz devices based on the monolayer of BP.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Polarization is an intrinsic property of electromagnetic waves. Active control and modulation of the polarization state of electromagnetic wave is of practical importance in many areas such as imaging, sensing and communications [1–3]. Many active materials with different tuning mechanisms have been proposed including electrically-tuned liquid crystals [4], mechanically-deformed elastic materials [5], and electrically-modulated graphene metamaterials [6,7]. Among them, graphene metamaterials show advantages owing to the its atomic thickness, high electrical modulation speed, and wide operation spectrum from sub-terahertz to near-infrared [8]. In the THz region, graphene metamaterials with subwavelength patterned structure have been designed to functioned as quarter wave plate [9], asymmetric transmission [6,10], polarization converter [11], etc. However, the electrical modulation of the patterned graphene is difficult. To address this problem, researchers assemble a complete large area graphene with anisotropic/chiral subwavelength structure made of metal or dielectric [12,13]. In 2017, Kim and co-workers combined a chiral metamaterial with a gated single-layer sheet of graphene [13]. They demonstrated that transmission of a terahertz wave with one circular polarization can be electrically controlled with large-intensity modulation depths (>99%) [13].
Black phosphorus (BP) has recently emerged as a strong competitor to graphene, due to its high carrier mobility and uniquely tunable band gap ranging from 0.3 to 2 eV [14–16]. The striking in-plane anisotropy of BP promises novel polarization-dependent and angle-resolved optoelectronic devices without the need of patterned structure or additional subwavelength metal/dielectric structure [17,18]. In 2016, the anisotropic property of BP has been visualized by a polarized optical microscopy [19]. The crystallographic orientation of BP can be precisely determined by a polarized reflectance measurement [17,19–21]. In 2017, Yang et al. compared the optical birefringence of different anisotropic 2D layered materials [22]. They found that birefringence in BP (∼0.245) is larger than those of anisotropic transition metal dichalcogenides such as ReS2 (∼0.037) and ReSe2 (∼0.047) and is larger than that of the current state of the art bulk materials (e.g., CaCO3) [22]. The layered BP can rotate polarization state with of ∼0.005° per atomic layer [22]. BP has found applications ranging from ultrafast photonic absorbers [23], multidisciplinary biomedical applications [24], vector soliton fiber lasers [25], to all-optical signal processing [26]. Recently, BP has expanded its applications to the THz region. In 2020, Guo et al. fabricated a BP-based THz photon detector with excellent sensitivity of 297 VW−1 and low noise equivalent power of 58 pW/Hz0.5, which can be used for THz imaging [27]. Taking the advantage of the anisotropic property of BP, unusual polarization-dependent perfect absorption has been demonstrated [28]. Using BP as saturable absorber, a switchable dual-wavelength Q-switched fiber laser was fabricated, which might be applied to effectively generate THz signal [29]. However, the instability of BP hinders the application of black phosphorus, and a breakthrough has been made in this field in recent years. Some recent work has proved how to improve the quality of monolayer black phosphorus to solve the problem of oxidation and instability [26].
Although the BP has strong anisotropy, the monolayer of BP is far from being able to manipulate the polarization state efficiently. Here, a BP-based Tamm (BPT) structure is designed with a monolayer BP siting on the top of the structure. The excitation of the Tamm plasmon enhances the interaction between BP and electromagnetic field, thus the polarization state of the reflected field can be modulated flexibly. Tamm plasmons are electromagnetic modes confined between a distributed Bragg reflector (DBR) and a metal (e.g. gold in the visible region, graphene in the THz region) [30]. Comparing to the surface-plasmons, the Tamm plasmons may be excited by both the TE and TM modes and present a relatively high quality factor [31]. Tamm plasmons therefore finds applications in refractive-index sensing [32,33], perfect absorption [34], and thermal emission [35].
2. Model and theory
As shown in Fig. 1(a), the BPT structure consists of a monolayer of BP, a defect layer, and a distributed Bragg reflector (DBR). BP shows metallic property in the THz region. The DBR is composed of 15 alternating dielectric layers of silicon (Si) and silicon dioxide (SiO2). The refractive indices of Si and SiO2 are respectively 3.42 and 1.9 with quarter-wave layer thickness for a center frequency of 1 THz, namely, dSi=21.93 µm, dSiO2=39.47 µm. A Si defect layer of thickness ds is sandwiched by BP and DBR, which can tune the resonance dip.
Fig. 1. (a) Schematic diagram of BP-based Tamm structure, where ds is the thickness of the Si defect layer between BP and the DBR. (b) Modulus of reflection coefficients |rBP| (for a Si-air interface containing BP), |rDBR| (for a Si-DBR interface), and |rpp| (for THz wave incident normally onto a BP-based Tamm structure) and phase ${\mathop{\rm Arg}\nolimits} [{r_{BP}}{r_{DBR}}\exp (2i\delta )]$ changing with frequency. (c) A single BP layer surrounding by two dielectrics with refractive index of nj and nj+1. (d) A stack of dielectrics containing a BP layer. (e) A dissymmetric cavity, enclosed by a sheet of BP and a DBR.
Let us consider a monochromatic light beam illuminating onto BP-based Tamm structure. The photonic property of BP can be described by a surface conductivity [33]. Due to the in-plane anisotropic property, the conductivity is different along the armchair or zigzag crystalline directions. Under the Drude model, it is [36,37]
where Darm,zig=πe2n/marm,zig is the Drude weight. marm=ћ2/(2γ2/Δ+ηc), mzig=ћ2/2vc are respectively the electron mass along the armchair and zigzag directions, where η=10meV, Δ=2 eV, ηc=ћ2/0.4m0, vc= ћ2/1.4m0, γ=4a/π eVm, with the scale length of BP being a = 0.223 nm. The electron carrier density ρ can be changed by electric doping via a bias voltage [38,39].By rotating the BP, the armchair axis of BP crystal makes an angle of ϕ with the incident plane. Therefore, the conductance matrix connecting the surface current and electric light field can be given by σ=[σpp, σps; σsp, σss], where σpp=σarmcos2ϕ+σzigsin2ϕ, σss=σarmsin2ϕ+σzigcos2ϕ, and σsp=σps=(σzig-σarm)sinϕcosϕ [15,34]. The cross conductivity σsp vanishes for isotropic 2D materials such as graphene [40]. For BP, however, σsp is generally nonzero except when the BP angle ϕ=0 and 90°.
Modified transfer matrix has been derived in our previous work to describe the light propagation through layered media with BP layers [41]. In this work, a 4×4 matrix is employed since the cross conductivity of BP will induce a coupling between p and s waves. According to Ref. [41], the transmission matrix for a dielectric interface containing BP layer (see Fig. 1(b)) is
The effect of BP layer is excluded in the propagation matrix as the thicknesses of BP layers are neglected. The propagation matrix in a homogenous medium j with a thickness of dj is
According to Eq. (4), the Fresnel reflection coefficients can be obtained. By setting C0=BN+1=DN+1=0, we have
The BPT structure can be regarded as a dissymmetric cavity, enclosed by a sheet of BP on one side and a DBR on the other [34]. For the case of ϕ=0 and 90°, the resonance condition for the cavity with incident p (TE) and s (TM) plane wave can be given respectively by [31,32]
δ is the propagation phase within the deflection layer. Since $r_{\textrm{BP}}^{pp}$ and $r_{\textrm{BP}}^{\textrm{s}p}$ are different, the excitation conditions of Tamm plasmons for TE and TM modes are different, which is different from the Tamm plasmons based on graphene and metals. When ϕ≠0 or 90°, the TE and TM plane waves couple each other. In this situation, however, the cross polarization conversion effect can be enhanced.3. Results and discussions
Figure 1(b) shows the modulus and phase of reflection coefficients, when a p-polarized 1 THz wave is illuminated normally onto the BPT structure with BP angle ϕ=0, i.e., the polarization direction is along the armchair axis of BP. The characteristic stop band of DBR is clearly observed in the red line of Fig. 1(b), which is centered at 1 THz with a bandwidth of ∼0.36 THz. The reflection coefficient for a Si–air interface containing BP layer is shown in light yellow line, with a relatively constant modulus of ∼0.633 over the spectral range. This relatively small modulus results in low quality factor of asymmetric cavity [42]. The green curve presents for comparison the full reflectance of BPT structure |rpp|, the Tamm plasmon appears as a dip at 1 THz. The resonance condition expressed in Eq. (2) requires ${\mathop{\rm Arg}\nolimits} [{r_{BP}}{r_{DBR}}\exp (2i\delta )] = 0$. As shown by the green line in Fig. 1(b), this zero-crossing point is observed at 1 THz, and coincides with the reflectance dip for |rpp|.
The Bragg structure acts as a perfect reflector. Si-SiO2 period numbers will affect the performance of Bragg structure. We have calculated the reflection coefficients of |rDBR| and |rpp| for different period numbers. As shown by Fig. 2, the |rDBR| increases with the period number, resulting in the decrease of the dip of |rpp|. Although the side lobes of |rpp| vary with the period number T, the main lobes are almost identical when T > 7. Throughout the paper, the period number is fixed to T = 15.
Fig. 2. The reflection coefficients of |rDBR| (a) and |rpp| (b) vs frequency for different Si-SiO2 period T in Bragg structure.
The reflectance dip varies with the thickness of defect layer ds. The reflection coefficients |rpp| and |rss| as functions of frequency f at normal incidence for ds = 130, 133 and 136 µm are shown respectively in Fig. 3(a), where the vanishing |rps| is not shown since the BP angle is ϕ=0. The dip positions for |rpp| and |rss| have a little different owing to the anisotropic of BP layer. The dip position for |rpp| moves from 0.98 to 1.02 THz when ds increases from 130 to 136 µm. Further increase of ds can produce multiple reflection dips within the stop band between 0.81 ∼ 1.18THz (see Fig. 6(b) and 6(c)).
Fig. 3. (a) Reflection coefficients as functions of frequency f at normal incidence for different ds (a) and different carrier density (b). In (a), θ=0, ϕ=0°; θ=30°, ϕ=45° in (b).
When the BP angle is set as ϕ=45°, the p and s plane waves couple each other, resulting in nonzero non-diagonal reflection coefficients, rps=rsp≠0. As shown by Fig. 3(b), the |rps| is up to 0.24 at f = 1 THz, when ρ=8.55×1017 m-2. At this carrier density, |rpp| has a dip at 1 THz, which is down to 4.55 × 10−3, while the dip of |rss| is 0.148. However, without BP, ρ=0, there is not reflectance dip within the stopband, and rps vanishes.
To achieve a change in carrier density, it is usually necessary to apply a bias voltage. To this end, a several tenths of nanometer (∼10−4 wavelength) insulating layer is inserted between the BP and top silicon layers, as shown by Fig. 4(a). The inserted insulating layer will hardly affect the Fresnel reflection coefficients of the BP-based Tamm structure. According to Ref. [18,43], the carrier density of BP can be electrically modulated from a minimum of 0.17×1017 m-2 to a maximum of 12×1017 m-2.
Fig. 4. (a) The schematic of modulation models for reflection coefficients. (b) The Fresnel reflection coefficient as function of carrier density ρ. (c,d) The reflection coefficient as function of frequency for ρ=4.68×1017 m-2 (blue star in (b)) and 11.68×1017 m-2 (red srar in (b)).
A scheme for THz wave modulation is shown by Fig. 4(a), where both the incidence angle and BP angle are zero. By adjusting the BP carrier density, the amplitude changes dominate over the phase changes in the reflection coefficients because the monolayer BP is too thin for effective phase accumulation. When the carrier density ρ increases from 0, both |rpp| and |rss| decreases from 1 gradually. |rpp| reaches a minimum of 1.35×10−3 when ρ=4.71×1017 m-2, while the minimum of |rss| is 1.28×10−3 at ρ=11.73×1017 m-2, as shown by Fig. 4(b). Figure 4(c) shows the reflection coefficient as function of frequency for ρ=4.68×1017 m-2 (blue star in (b)) and 11.68×1017 m-2 (red star in (b)). One can conclude that, the Tamm plasmons for TE and TM wave can be selectively excited by controlling the carrier density of BP. This unique phenomenon originates from the anisotropic property of BP, cannot be found in isotropic materials such as metals and graphene. For isotropic materials, the reflection of TE and TM waves are identical at normal incidence. The intensity modulation depth for TE and TM waves can be defined as [44]
where $r_{pp,ss}^{\max }$ are the baselines defined as the maximum reachable reflection values of ${r_{pp,ss}}$. Assume that 0.17×1017 m-2 is the minimum carrier density of BP [18,43]. Thus, the maximum reachable reflection values of ${r_{pp,ss}}$ are 0.939 and 0.975, respectively. The intensity modulation depth for TE and TM waves are both 100%, with the insertion loss 10lg$\textrm{|}r_{pp,ss}^{\max }{\textrm{|}^\textrm{2}}$ of -0.55dB, -0.22dB, respectively. Comparing to the graphene-based THz modulator demonstrated by Chen et al. [44], our BP-based device can operate for both TE and TM modes with much lower insertion loss.Figure 5(a) shows the reflection coefficients when THz wave illuminated obliquely with θ=60°. As shown by Fig. 5(a), the |rss| can be effectively modulated by the BP carrier density effectively. However, the |rpp| can be hardly changed by the carrier density. |rpp| increases a litter from 0.3 to 0.33, when ρ increases from 0 to 6×1017 m-2. In Fig. 5(b), |rps| is no longer zero, when the BP angle is 45°. The incident angle and thickness of defected layer are optimized respectively to be θ=60° and ds=354.9 µm to maximize the polarization rotation angle. |rps| increases gradually with ρ, and approach to an asymptotic value of 0.10. Therefore, the reflected THz wave can be flexibly modulated by tuning the BP carrier density.
Fig. 5. (a) Reflection coefficients (a,b) and polarization angles (c,d) as functions of BP carrier density for (a,c) θ=60°, ϕ=90°; (b,d) θ=60°, ϕ=45°.
Fig. 6. (a,b) Reflection coefficients |rps| (b) and |rss| (c) as functions of frequency and carrier density. (b) Stokes parameters of the reflected beam changing with ρ; (c) The evolution of polarization states in Poincaré sphere. The red, blue and green lines in (b,c) represent f = 0.83, 1.00, and 1.17 THz, respectively.
In the following, we focus on attention on the polarization modulation by the carrier density. For the case of ϕ=90°, we assume that the incident polarization is diagonal polarization, i.e. [1,1]T. The polarization of the reflected wave is therefore [rpp,rss]T. As shown by Fig. 5(c), the reflected wave is in elliptical polarization, which varies with the carrier density. The polarization angle is defined as
S1, S2, and S3 are the Stokes parameters of the reflected waves. When the carrier density increases from 0 to 6×1017 m-2, the polarization angle changes from 75° to -58.5° gradually.In the case of ϕ=45°, the incident wave is assumed to be TE mode, namely, [0,1]T. The polarization of the reflected wave is [rps,rss]T. In this case, the reflected wave is linearly polarized. The polarization angle, increases from -90° with the carrier densy and approaches gradually to 81° with an angle change up to 171°. The insets show four special polarization states at ρ=0, 0.90×1017, 1.01×1017, 1.15×1017 and 2×1017 m-2, corresponding polarization angle corresponding polarization angles are -90°, -45°, 0°, 45°, 76°, respectively.
Multi-channel modulations can be achieved by increasing the thickness of the defect layer ds. Figures 5(a) and (b) shows the reflection coefficients for ϕ=0 at normal incidence, when ds=220.8 µm. As shown by Fig. 6(a), three dips can be found with the frequency in |rpp|, which are respectively f = 0.83, 1.00 and 1.17 THz at ρ=4.7×1017m-2. If the incident beam is circularly polarized, i.e., [1,i]T. The polarization state of the reflected wave is [rpp,irss]T. The Stokes parameters of the reflected beam S1, S2, and S3 can be calculated numerically, and the results are shown in Fig. 6(c). With the increase of the carrier density, parameter S3 increases gradually from -1 to + 1, and then decreases back to ∼0.5. Parameter |S2| is smaller than 0.2 for an arbitrary density. The evolution of polarization state on Poincaré sphere with the change of carrier density are shown in Fig. 6(d). The curves for three channels are almost overlapped. Therefore, one can control the polarizations of the reflected waves at three different frequency channels simultaneously.
As shown by Fig. 6(d), the reflected wave is in horizontal [1,0]T and vertical [0,1]T polarization states when ρ=4.71×1017 and 11.73×1017 m-2, respectively. When ρ=7.44×1017m-2, the polarization state is [1,ei(-π/4)]T. This elliptical polarization state can be considered as a superposition of horizontal and vertical polarization states with equal amplitude but a phase difference.
As demonstrated above, the BPT device can flexibly modulate the polarization state of the reflected light into two orthogonal polarization states as well as their superposition state by tuning the BP carrier density via bias voltage. Therefore, the proposed device can be employed for polarization-division multiplexing (PDM). PDM is a crucial technique that can significantly increase the transmission capacity of a single physical channel [13].
Figure 7 shows the multi-channel PDM scheme based on the BP-based Tamm structure. Three frequency channels are modulated simultaneously by the carrier density. As shown in Fig. 7, two independent information signals can be, respectively, encoded into two orthogonally polarized light beams with a binary format. Three different carrier densities of ρ1, ρ2, and ρ3 correspond to the horizontal, elliptical, and vertical polarization states, which associate to “10”, “11”, “01”, respectively. By tuning the carrier density, the reflected beams with a particular sequence of polarization states can be generated directly from the proposed structure without the need to couple multiple sources.
Fig. 7. The schematic of the multi-channel PDM technology, according to the superposition principle of the polarization state.
The proposed BPT device can be fabricated by stacking the 21.93 µm Si and 39.47 µm SiO2 slices alternatively [45]. And then the Si defected layer with thickness ds>100 µm is stacked on the top. Before transferring BP layer, a nanoscale thickness insulator layer (SiO2 or Al2O3) is deposited on the defect layer by atomic layer deposition. Finally, an Au electrode is thermally deposited on the BP layer. A voltage is applied via the Si defected layer and Au electrode to tune the carrier density of BP. The Bragg reflector can be also made of alternating air spacers and SiO2 layers, which is simpler in fabrication [45].
Before experimental measurement, it is necessary to determine crystalline axes of BP. As illustrated by Ref. [21], the armchair and zigzag axes of BP are associated respectively with maximum and minimum brightness of R channel, when analyzing the reflection of normal incident visible light with a polarizer and a CCD.
4. Conclusions
A novel multichannel THz polarization modulation scheme was proposed based on the BP-based Tamm structure. The excitation of Tamm plasmons enables an efficient and flexible modulation of THz waves. When the BP angle ϕ=0 or 90°, a perfect intensity modulator for TM and/or TE modes with a modulation depth up to ∼100% and an insertion loss smaller than -0.55 dB is demonstrated. When ϕ=45°, the polarization of the reflected THz wave rotates when the BP carrier density changes. The maximum polarization rotation angle is up to 171°. The polarization state changing with BP carrier density are visually shown in Poincare sphere for different frequency channels, based on which the multi-channel PDM is achieved. We believe the proposed versatile BPT device has potential applications in THz imaging, sensing and communications.
Funding
Science & Technology Project of Guangzhou (201604040005, 201605030002, 201704030105, 201707010396); Natural Science Foundation of Guangdong Province (2016A030311019, 2016A030313079, 2016TQ03X962, 2017A010102006, 2017A030313375); National Natural Science Foundation of China (61475066, 61505069, 61705086); College Students' Innovative Entrepreneurial Training Plan Program (201910559054).
Disclosures
The authors declare no conflicts of interest.
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