A tunable metal/magneto-optic plasmonic lens for terahertz isolator is demonstrated. Based on the magneto-optical effect of the semiconductor material and non-symmetrical structure, this plasmonic lens has not only the focusing feature but also nonreciprocal transmission property. Moreover, a transmission enhancement through this device greatly larger than that of the ordinary metallic slit arrays is contributed by the extraordinary optical transmission effect of the magneto surface plasmon polaritons. The results show that the proposed isolator has an isolation bandwidth of larger than 0.4THz and the maximum isolation of higher than 110dB, and its operating frequency also can be broadly tuned by changing the external magnetic field or temperature. This low-loss, high isolation, broadband tunable nonreciprocal terahertz transmission mechanism has a great potential for terahertz application systems.
©2013 Optical Society of America
Terahertz (THz) waves show great applications in THz sensing, imaging, spectroscopy, and communication. However, due to lack of feasible broadband low-loss nonreciprocal trans-mission (or called one-way transmission) devices in the THz regime, such as isolator  and circulator [2, 3], THz echoes of the reflection and scattering for system components bring some noise severely limiting the performance of these THz systems. In recent years, a number of nonreciprocal photonic devices have been investigated in the microwave and near-infrared regime [4–6]. Since the nonreciprocal materials responding at THz frequencies are very rare, until recently, some preliminary works for nonreciprocal terahertz transmission have been reported based on waveguide devices by Fan et.al [7, 8] and Hu et.al . However, a large coupling loss exists when the THz waves are coupled from the free space into these waveguide devices, which limit practical applications.
Plasmonic lens (PL) can focus light beyond diffraction limit by means of surface plasmon polarisons (SPPs), which has attracted much attention recently due to its unique feature of extraordinary enhanced transmission [10–12]. The different phase retardations can be achieved by structuring the PLs to manipulate the spatial distribution of the output beam. In the THz regime, doped semiconductors are often used as plasmonic materials to replace metals. When an external magnetic field (EMF) is applied, the cyclotron frequency of high electron mobility semiconductors such as InSb, HgTe and graphene locates in the THz regime [13, 14], which become magneto-optical materials, and of which dielectric properties can be tuned by the EMF. Hu et.al  proposed an active PL based on a symmetrical InSb-air-InSb waveguide to actively control the focusing position of THz waves. Moreover, the magneto surface plasmon polarisons (MSPPs) can be excited in these materials. MSPPs can support nonreciprocal transmission in a non-symmetrical system due to break the time reversal symmetry. However, so far, the PL with the property of one-way transmission in the THz regime has not been reported.
In this paper, according to the dispersion and gyrotropy properties of the InSb in the THz regime, we proposed a tunable metal/magneto-optic plasmonic lens (MMOPL) for non-reciprocal terahertz transmission. The dispersion relations as well as the transmission and isolation properties of this MMOPL are investigated. The operating frequency of this MMOPL can be broadly tuned by changing the external magnetic field or temperature. Moreover, we find a great transmission enhancement through this device for the contribution of the extraordinary optical transmission effects of the MSPPs.
2. Theoretical analysis
In this work, we use the Al as metal and the undoped InSb as magneto-optical semiconductor material. In the THz regime, the permittivity of the metal Al follows the dispersion relation . When an EMF along the z direction is applied, the semiconductor InSb shows a strong gyrotropy near the cyclotron frequency ωc, and becomes a nonreciprocal medium. The ωc is proportional to the EMF through, where B is the magnetic flux density, m* is the effective mass of the carrier, and e is the electron charge. m* = 0.015me for the InSb, and me is the mass of electron. In this case, the dielectric function of the InSb becomes a nonreciprocal tensor expressed as [8, 9]Eq. (1) follow17, 18]Eqs. (1-3) with the different EMF and temperature as shown in Fig. 1 .
The structure of the proposed MMOPL is shown in Figs. 2(a) and 2(b), which is composed by a slit array with a periodic arrangement of the metal and InSb grating in turn. The width of the slit in a unit cell is W = 20μm, the thickness of the device is H = 0.5mm, the total area of the device is larger than 3mm × 3mm. Compared with the previous single waveguide isolators [2–9], our MMOPL structure is an area array device. Since our structure is larger than the area of THz beam as shown in Fig. 2(a), we don’t need compressing THz beam to subwavelength scale to couple it into a single waveguide structure by using any couplers, so the MMOPL has a smaller insertion loss. Light propagation within the MMOPL can be seen as the propagation in a series of independent metal/insulator/magneto-optic/metal (MIMOM) waveguide as shown in Fig. 2(c), because the skin depth of THz wave in the metal is far less than the metal width of W. The THz waves transmit in this hybrid magneto-optical waveguide composed of the air and InSb with the width of 2W. For this MIMOM structure, we can express the electromagnetic field components Hz and Ey of the incident TM polarized THz wave in the four regions shown in Fig. 2(c) as follows:Fig. 3 by both analytical and numerical methods according to Eq. (4).
In Fig. 3, β>0 indicates forward propagation and β<0 indicates backward propagation. When there is no EMF applied, as shown in Fig. 3(a), the dispersion curve is symmetric for the forward and backward propagations, which shows a reciprocal transmission of an ordinary waveguide. This dispersion curve also shows a bandgap originating from the SPPs at 1.5~1.65THz (pink region in Fig. 3(a)). When an EMF of 0.1T is applied as shown in Fig. 3(b), due to the gyrotropy of the InSb under the EMF and the non-symmetry structure, the SPPs split as two different MSPPs under the EMF, and these two MSPPs have opposite angular momentum in the x-y plane forming a right rotating and a left rotating mode with different dispersion relations , respectively. Thus the dispersion curve becomes non-symmetric, the branch of the forward propagation moves to a higher frequency, and the backward moves to a lower frequency. Correspondingly, the bandgaps supporting by the MSPPs of each branch also moves and they no longer overlap at the same frequency range. For the backward wave, its bandgap is located at 1.45~1.55THz (yellow region in Fig. 3(b)), while the forward wave still transmit at this frequency range. Therefore, this is an isolation region which permits the forward wave but forbids the backward wave. Likewise, the blue regions in Figs. 3(b) and 3(c) also have the isolation property only permitting the backward propagation. When a larger EMF of 0.5T is applied as shown in Fig. 3(c), the splitting of the two branches becomes larger, and the bandwidth of the isolation regions becomes wider at 1.1~1.4THz due to a stronger MSPP mode. As the description of Eq. (2), the dielectric tensor elements of the InSb depend on the EMF and temperature, and thus the MSPP mode in the MIMOM waveguide can be broadly tuned by the EMF or temperature.
3. Results and discussion
The functions of the proposed MMOPL are based on the property of the MIMOM waveguide discussed above. This MMOPL has two essential functions: the first one is focusing and collimating the light beam, which is the same as other ordinary lens; the second one is a unique feature that is isolating the backward waves as an isolator in the free space, so we will mainly discuss the isolation feature of this MMOPL. Its performance mainly depends on two aspects: one is the transmittance of the forward propagation, which determines the insertion loss of the isolator; the other is the isolation between the backward transmission Tback and forward transmission Tfor expressed as.
The transmission spectra of forward and backward waves through the MMOPL with the different EMF at T = 185K are simulated by the finite difference time domain (FDTD) method shown in Fig. 4 , and the corresponding isolation spectra are shown in Fig. 5 . As shown in Fig. 4(a), the transmission without an EMF (B = 0T) has a low average power flow of 0.1W/m and a band-gap in the range of 1.42~1.55THz, which well coincides with the description of the dispersion relation shown in Fig. 3(a). We also use the metal to replace the InSb grating to form an ordinary PL as comparison, and its average power flow is still 0.1 W/m level without any bandgaps and remarkable peaks (labeled by the brown color). When an EMF is applied, the SPP mode splits to be the MSPP modes. Take B = 0.5T spectral line for example, the transmission spectra are consistent with the dispersion relation shown in Fig. 3(c). For the forward wave, a passband with high transmittances locates in 1.1~1.45 THz, and the max-imum power flow is higher than 3W/m. For the backward waves, extremely low transmit-tances locate at 1.1~1.45 THz, that is the isolation region shown in Fig. 5. The device obtains the 30dB isolation bandwidth of larger than 300GHz and the maximum isolation of higher than 110dB at 1.24THz. Outside the isolation range, the forward and backward transmittances are similar to the B = 0T case. Therefore, this device achieves the one-way transmission in the range of 1.1-1.45THz when B = 0.5T and T = 185K.
With the EMF increasing, the passband of forward wave moves to a lower frequencies and the maximum transmittance becomes higher; while the forbidden band of backward wave also moves to the same lower frequencies and the bandwidth becomes larger. Accordingly, the isolation peak moves to the lower frequency, and its bandwidth becomes larger as shown in Fig. 5. However, when the EMF continues increasing, the isolation peak value dramatically decreases. The reason is that the cyclotron frequency ωc is gradually far away from the THz frequency range and the gyrotropy of the InSb becomes weak, leading to a lower nonreciprocal transmission. The central operating frequency of this isolator can be broadly tuned from 1.5THz to 1.1THz with the EMF increased form 0T to 0.9T.
We further discuss the influence of the EMF and temperature on the device. Firstly, the isolation dependences on the EMF at T = 185K with 1, 1.2 and 1.4THz wave are shown in Fig. 6(a) . The isolation of each frequency reaches a peak value with the different EMF. When the EMF continues to increase, the isolation drops with the cyclotron frequency ωc far from the operating frequency according to Figs. 1(a) and 1(b). The peak value is induced under a higher EMF at a lower operating frequency. Secondly, the isolation dependence on the temperature under 0.5T is shown in Fig. 6(b). According to the results in Figs. 1(c) and 1(d), the MSPPs can be induced by a feasible EMF in the low-temperature range of 160~220K. When the temperature is close to room temperature, an extremely large EMF should be applied. A lower temperature corresponds to a smaller EMF reaching the peak of the isolation, so using a lower temperature can decrease the EMF applied for the device. As a conclusion, both the EMF and temperature strongly impact on the one-way transmission of the MMOPL, so the operating frequency band of this isolator can be feasibly tuned by the EMF and temperature.
At last, we focus on the transmission and spatial distribution of the forward wave through the MMOPL. The power flow distributions normally incident into the MMOPL for the for-ward and backward propagations at 1.25THz under 0.5T and 185K are shown in Figs. 7(a) and 7(b), respectively, and their corresponding magnetic field distributions in the MIMOM are shown in Fig. 7(c). The THz wave of 1.25THz realizes one-way transmission, and the forward wave is focused at 1mm away from the back surface of the PL. Figure. 7(d) exhibits a rapid damping process of the backward wave at the interface between the InSb and metal. Outside the isolation frequency region, both forward and backward propagations at 0.8THz can transmit through the MMOPL the same as an ordinary metal slit PL focusing at 0.7mm as shown in Fig. 7(e).
The spatial distributions of the forward waves along the x direction at focusing position with different EMF and temperature conditions are shown in Fig. 8(a) for 1.25THz and Fig. 8(b) for 0.8THz, respectively. According to the discussions above, among all the conditions shown in Fig. 8, only the 0.5T 185K for 1.25THz case exhibits a strong isolation, and the other cases have no isolating property. Significantly, Fig. 8 reveals that only the 0.5T 185K case has an extremely high transmission, other cases without one-way transmission are all in a similar low transmittance level. The reason for this high transmission of the nonreciprocal transmission case is that all the backward waves are forbidden in the waveguide and all the reflecting waves in any cases are transmit through the MMOPL as the forward waves. With this propagation process, the MSPP waves in the InSb are gradually enhanced just as a reverse process shown in Fig. 7(d). Finally, very strong MSPPs are coupled into the free space to form a stronger extraordinary transmission effect. Therefore, due to the contribution of the extraordinary optical transmission of the MSPPs within the nonreciprocal transmission frequency range, a transmission enhancement through this MMOPL is about 30 times larger than that of the ordinary metallic or semiconductor slit arrays as shown in Figs. 8(a) and 4(a), which greatly reduces the transmission loss of the forward wave in this MMOPL.
In conclusion, based on the dispersion and magneto-optical property of the InSb, the non-symmetrical dispersion relation and one-way transmission property of the proposed MMOPL are investigated in the THz regime. The numerical results show that the proposed isolator has an isolation bandwidth of larger than 0.4THz and the maximum isolation of higher than 110dB at the temperature of 185K. The operating frequency of this device also can be broadly tuned by changing the external magnetic field or temperature. The further discussions show that a transmission enhancement through this MMOPL is about 30 times larger than that of the ordinary metallic or semiconductor slit arrays. This low-loss, high isolation, broadband tunable nonreciprocal THz transmission mechanism has a great potential application in promoting the performances of the THz application systems.
This work is supported by the National High Technology Research and Development Program of China (863 Program) (Grant No. 2011AA010205), the National Natural Science Foundation of China (Grant No. 61171027), and the Natural Science Foundation of Tianjin of China (Grant No. 10JCZDJC15200).
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