In this work, we investigate the nonreciprocal circular dichroism for terahertz (THz) waves in magnetized InSb by the theoretical calculation and numerical simulation, which indicates that longitudinally magnetized InSb can be applied to the circular polarizer and nonreciprocal one-way transmission for the circular polarization THz waves. Furthermore, we propose a double-layer magnetoplasmonics based on the longitudinally magnetized InSb, and find two MO enhancement mechanisms in this device: the magneto surface plasmon resonance on the InSb-metal surface and Fabry–Pérot resonances between two orthogonal metallic gratings. These two resonance mechanisms enlarge the MO polarization rotation and greatly reduce the external magnetic field below 0.1T. The one-way transmission and perfect linear polarization conversion can be realized over 70dB, of which the transmittance can be modulated from 0 to 80% when the weak magnetic field changes from 0 to 0.1T under the low temperature around 200K. This magnetoplasmonic device has broad potential as a THz isolator, modulator, polarization convertor, and filter in the THz application systems.
© 2016 Optical Society of America
Terahertz (THz) radiation is electromagnetic radiation whose frequency lies 0.1 to 10 THz between the microwave and infrared regions of the spectrum. With great successful progress on THz science and technology, more and more applications in imaging, sensing and communication have been realized [1–3]. For further development of the THz application system, there is a high demand on efficient devices for guiding, modulating, and manipulating THz wave in its amplitude, phase, and polarization. In order to obtain high performance, novel artificial electromagnetic microstructures such as photonic crystals , metamaterial , and plasmonics  has been developed and utilized into the THz regime. The unique nonreciprocal effect and magnetic tunability of magneto-optical (MO) device make it play an irreplaceable role in the high performance isolator , polarization controller , MO modulator , tunable filter , and magnetic field sensor . However, due to the lack of high performance THz MO materials and the limitation of device fabrication, the improvement of THz MO devices is still in challenge.
MO material introduced into the artificial microstructure, such as magnetic photonic crystal  or magnetoplasmonics , has become a research hotspot in recent years [14, 15]. Through the reasonable design of device structure, MO effect can be significantly enhanced by plasmonic resonance or bandgap effect [16, 17]; conversely, MO effect leads to some new physical mechanism and phenomena such as the splitting of plasmonic resonance, nonreciprocal transmission, and enhanced Faraday rotation effect [18–21]. For examples, a magnetically induced THz transparency in the n-doped InSb was demonstrated, owing to the interference between left and right circularly polarized magnetoplasmon eigenmodes . The tunable THz magnetoplasmons and their MO splitting were observed in the graphene  and ferrofluid . In our previous works, we reported series of THz nonreciprocal isolators with magnetoplasmonic and metasurface structures based on the Voigt MO effect in InSb under a transverse biased magnetic field (i.e. the external magnetic field direction is orthogonal to the direction of light propagation, called as Voigt configuration), which can achieve very high isolation ratio of over 60dB and a low insertion loss of only 2dB [25–27].
However, applying longitudinal magnetic field is more convenient for the two-dimensional materials and devices. As a classic longitudinal MO effect (i.e. the external magnetic field direction is parallel to the direction of light propagation, called as Faraday configuration), Faraday effect can lead to a non-reciprocal rotation of the linear polarized light in the MO materials, which can be widely used as polarization rotator, isolator and MO modulators if a large Faraday rotation angle can be achieved. The Faraday rotations were observed in some high electron mobility semiconductors, for examples, InSb , HgTe , and graphene  in the THz regime. Shuvaev et.al. observed the first giant Faraday effect in the THz regime on epitaxial HgTe thin films at room temperature . The maximum Faraday rotation reached 0.25 rad at 0.35THz when B = 1T, which corresponded to a very large Verdet constant V = 3 × 106rad∙T−1∙m−1 with the 70 nm thickness. A. Fallahi et.al. also presented a graphene metasurfaces to manipulate giant Faraday rotation in the THz regime , of which rotation reaches 0.1 rad with a broad bandwidth of over 1THz, and this operating frequency band can be broadly tuned from 0.5 to 5THz by the different magnetic field from 1 to 7T. Tamagnone et. al.  reported a THz nonreciprocal isolator based on monolayer graphene under a strong biased magnetic field of 7T, which exhibits about 20 dB of isolation and only 7.5 dB of insertion loss at 2.9 THz. Although these materials have great Verdet constant, the Faraday rotation angle is limited due to their thin thickness relative to the THz wavelength, and requires an extremely high magnetic field.
In this work, we investigate the longitudinal MO effects of magnetized InSb in the THz regime, which show that THz waves makes its MO responses quite different from the typical Faraday effect in the visible and near infrared lights. By the theoretical derivation and numerical simulation, we confirm the nonreciprocal circular dichroism for THz waves in magnetized InSb. Based on the above, we propose a double-layer magnetoplasmonics to realize one-way transmission, magneto modulation and polarization conversion. The magneto surface plasmon resonance on the InSb-metal surface and Fabry–Pérot (F-P) resonances between two metallic gratings greatly enhance the MO effect under a weak external magnetic field to form high transmission peaks at resonance frequencies. Moreover, the dependences and tunability of this device on the external magnetic field, temperature and device structure parameters are also investigated.
2. Terahertz nonreciprocal circular dichroism in magnetized InSb
When an external magnetic field is applied, the semiconductor InSb shows a strong gyrotropy near the cyclotron frequency ωc. The ωc is proportional to the biased magnetic field by, where B is the magnetic flux density, m* is the effective mass of the carrier and m* = 0.014me for the InSb [22, 33–36], me is the mass of electron; e is the electron charge. When the biased magnetic field is along the z direction, the dielectric function of InSb becomes a nonreciprocal tensor, which is expressed as [25, 26, 37]:Eq. (1) can be written as [25, 26, 37]:33, 34], so the γ is also dependent on the temperature.
Moreover, the dielectric property of the InSb greatly depends on the N, and the N strongly depends on the temperature T, which follows [22, 33–36],
For a plane wave propagating as a Faraday configuration along z axis, the wave equation can be written as:
Such waves correspond to the following two circularly polarized eigenwaves
The eigenwave of Ey = jEx in Eq. (5) represents a left-handed wave (counterclockwise circularly polarized wave, CCW) along + z axis or a right-handed wave (clockwise circularly polarized wave, CW) along −z axis. εL = ε1−ε2 is the effective permittivity of the circularly polarized eigenwave of β1. On the contrary, the eigenwave of Ey = −jEx in Eq. (6) represents a right-handed wave along + z axis or a left-handed wave along −z. εR = ε1 + ε2 is the effective permittivity of the circularly polarized eigenwave of β2.
The εL and εR spectra in the THz regime are calculated according to Eqs. (1)-(6) under the different temperatures and magnetic field as shown in Fig. 1. Figures 1(a) and 1(c) show that the real part of εL Re(εL) is negative in the low frequency range and increases monotonously with the frequency as a Drude lineshape, and the imaginary part of εL Im(εL) is small and monotonously reduced, tend to be 0 in the higher frequency band. The frequency point of Re(εL) = 0 is defined as the effective plasma frequency of eigenwave β1, expressed asFig. 1(a), the ωp increases accordingly, so the ωp1 move to a higher frequency and the Re(εL) becomes smaller. As the biased magnetic field increasing shown in Fig. 1(c), the ωc increases accordingly, so the ωp1 moves to a lower frequency and the Re(εL) becomes larger.
Figures 1(b) and 1(d) show that the Re(εR) can be divided into three regions. Re(εR)>0 in the low frequency range, and it has a singularity at ω = ωc with a strong resonance as a Drude- Lorentzian lineshape. This is the first point of Re(εR) = 0. And for Im(εR), it is always larger than 0 and initially increase with the frequency. At this point of ω = ωc, Im(εR) reaches its peak, which is much larger than Im(εL). The second frequency point of Re(εR) = 0 at a higher frequency is defined as the effective plasma frequency of eigenwave β2 as follows:Fig. 1(b), the ωp increases accordingly but ωc is not changed, so the position of forbidden band keeps still, but both the ∆ωp2 and Re(εR) become larger. As the biased magnetic field increasing shown in Fig. 1(d), the ωc increases accordingly, so the position of forbidden band moves to a higher frequency, and the ∆ωp2 becomes smaller.
Since the ωp2 is always larger than the ωp1, when the ω>ωp2, the εR≠εL>0, so the typical Faraday rotation effect can be obtained. But when ω<ωp2, the nonreciprocal circular dichroism can be obtained. For instance, when a linearly polarized wave is incident into the longitudinally magnetized InSb along + z, the left-handed component can pass through the InSb, but the right-handed component is totally forbidden in the band of ωc<ω<ωp2, so the output wave is a left-handed wave, as shown in Fig. 2. When the incident wave is left-handed wave in the band of ωc<ω<ωp2, it can pass through the InSb along + z, but cannot pass along –z. The right-handed wave is just the opposite. Therefore, these are nonreciprocal one-way transmission for the left and right-handed waves in the longitudinally magnetized InSb.
Then, we use the frequency domain solver of CST software to simulate and verify the above theoretical analysis. An InSb layer with h = 100μm thickness is simulated, of which results are shown in Fig. 3 with different temperatures and external magnetic fields. As shown in Fig. 3(a), when the frequency is lower than the ωp1, which correspond to the frequency point of Re(εL) = 0 shown in Fig. 1(a) very well, the transmittances of left-handed wave drop down sharply. Therefore, the left-hand wave can transmit through the InSb with a high transmittances when ω>ωp1. The fluctuations in the passband of transmission spectra originate from the F-P interference effect between the two interfaces of InSb. There are transmission peaks and dips periodically arranged alternately in the frequency domain, and this circle is determined by the InSb thickness. At transmission peak point, nearly 100% wave can be transmitted through InSb, but over 50% energy will be reflected at F-P dips. With the increase of the temperature, the ωp1 and the passband move to a higher frequency. On the contrary, with the increase of the external magnetic field, the ωp1 and the passband move to a lower frequency as shown in Fig. 3(c), which also corresponds to Fig. 1(c).
As shown in Figs. 3(b) and 3(d), there is a forbidden band in the transmission spectrum of the right-handed waves. The frequency point of the falling edge of this band is the ωc, the rising edge is the ωp2 in Eq. (8), and the bandwidth of this band is ∆ωp2 in Eq. (9). The increase of temperature does not influence the position of the falling edge, but increase the bandwidth. The rise of the magnetic field makes the forbidden band shift to a higher frequency band. All the simulation results correspond to the results shown in Figs. 1(b) and 1(d). Therefore, the roles of longitudinally magnetized InSb can be seen as a high-pass filter for the left-handed wave and a band-stop filter for the right-handed wave. In our simulations, some frequency bands realize the nonreciprocal circular dichroism. For instance, when T = 200K and B = 0.3T, the left-handed wave can pass through the InSb but the right-handed wave is forbidden with only −90dB in the 0.55−0.9THz range, which are the blue lines in Figs. 3(c) and 3(d).
3. Double-layer magnetoplasmonics
3.1 Device structure and working principle
To apply the MO property of longitudinally magnetized InSb, we design a double-layer magnetoplasmonics to realize one-way transmission and linear polarization conversion, as shown in Fig. 4. There are two metallic gratings orthogonally coated on the two surfaces of InSb crystal. One is vertical grating along the y axis (defined as positive surface here), the other is horizontal grating along the x axis (i.e. negative surface). The InSb is h = 100μm thickness, the grating constant is a = 30μm, and the width of metallic grating grid is d = 28μm. The thickness of metallic grating is 200nm, and the gold is selected to form the grating grids.
Metallic grating can only transmit the TM mode, that is to say, only the linear polarized light that is orthogonal to the grating grid direction can pass through the metallic grating. So the elementary working principle of this device is as follows: when an X-linear polarized wave is incident into the vertical grating surface of double-layer magnetoplasmonics as shown in Fig. 5(a), this light can pass through this surface, and the MO effect of longitudinally magnetized InSb can change its polarization state into an elliptically polarized light. Only the Y components of this light can output through the horizontal grating surface, so this device can realize the polarization conversion from one linear polarized state to its orthogonal one. When an X-linear polarized wave is incident into the horizontal grating surface of double-layer magnetoplasmonics along the backward direction, no light can transmit through the device, which is equivalent to the result shown in Fig. 5(b). Therefore, one-way transmission for a specific linear polarized light can be realized in this device, but this is not a nonreciprocal transmission but a reciprocal one-way transmission.
3.2 Results and discussions
To verify the above analysis, we simulate the transmission property of this double-layer magnetoplasmonics by the frequency domain solver of the CST software. Firstly, an X-linear polarized plane wave is normally incident into the positive surface of the device. Two pairs of periodic boundary conditions are set to the simulation model of one unit grating cell. All the material parameters are set as the data calculated in the Section 2. The transmission spectra are detected by a monitor, and all the output components are Y-linear polarized waves. The results under the different magnetic fields are shown in Fig. 6. There are four transmission peaks P0~P3 and three resonance dips in the 0.1~1.5THz range. With the increase of magnetic field from 0.001T to 0.08T, the transmittances of peaks gradually rise up from 0 to 0.78 as shown in Fig. 6(a). When the magnetic field continues to increase from 0.08 to 0.3T as shown in Figs. 6(b) and 6(d), the transmittances of peaks no longer increase but the bandwidth of P1~P3 become larger, while the first peak P0 gradually splits as two peaks and one resonance dip.
We also simulate the distributions of power flow density in Fig. 7 and the electirc field distributions in x-z cutting plane of the double-layer magnetoplasmonics in Fig. 8. We can see that the first peak P0 is quite different from the later ones P1, P2, P3…There is no field localization in the InSb for P0, and most of the energy is located at the two interfaces of InSb and metallic grating. But for P1~P3, there are resonance modes in the InSb. They are two different resonance peaks with different mechanisms in this double-layer magnetoplasmonics. The P0 originates from the magneto surface plasmon resonance (MSPR) at the interface between the magnetized InSb and metallic grating structures [13–16]. Figure 8(b) shows that surface plasmons locate and resonant at the metal-InSb interface, especially at the metallic grid gap. The frequency position of P0 can be qualitatively described as [35, 36], where nspp is the effective refractive index of surface plasmonic mode and g is the geometry factor of the device, so this frequency peak is affected by two factors: one is the optical property of InSb, which is mainly determined by the carrier concentration dependent on the temperature T as well as the cyclotron resonance property dependent on the magnetic field B; another one is the geometry of metallic grating. The value of g is determined by grating period a, grating width d, and InSb thickness h. If the a, d or h increases, g will increase. Although it is hard to write a simple analytical expression, we can get the dependence of resonant frequency on these geometric parameters by numerical simulation in the following discussion.
The P1~P3 are the first to third order F-P resonances. The two metallic gratings form a resonance cavity to generate the F-P resonances, which follows:Figs. 1(a) and 1(c).
Moreover, from the direction of electric vector in Fig. 8(b), we can notice that the electric field is resonating along the X-axis at the input plane. After it transmits through the InSb, the electric field becomes along the Y-axis at the output plane. Therefore, the X-linear polarized THz wave can be converted efficiently to the Y-linear polarized one at the certain frequency points, and these Y-linear polarized transmission peaks can be modulated from 0 to 80% sensitively by the external magnetic field from 0 to 0.08T, so this double-layer magnetoplasmonics can be used as a perfect polarization converter and sensitive MO modulator with a good filtering output characteristic under a very weak magnetic field.
How do these MSPR and F-P resonances generate such high transmission peaks with orthogonal polarization to the incident one even under a weak magnetic field? In Section 3.1, the simple principle of this device is introduced; here we do a deeper analysis: As the ωc is located at low frequency under a weak magnetic field, the MO effect of InSb is weak in the THz regime. In spite of the weak MO effect, the polarization state of incident X-polarized wave can be changed in the InSb and generate some Y components to output through the horizontal grating surface. MSPR can localize the light on the interface between the InSb and metallic grating [13–16]. This is a quasi-static resonance field, of which group velocity is very slow so that the local oscillating fields in the InSb surface greatly interact with the MO medium, so more Y components are converted form X components at the frequency of MSPR though MO effect of the material itself is very weak. For F-P resonance, though the Y component is very small one time, the wave at the F-P resonance frequency can be reflected repeatedly between two metal gratings. Every after 2h distance, Y components are outputted one time. Since the Q value of this F-P cavity is very high, the transmission of Y component is greatly enhanced after oscillating many times. In general, both the MSPR and F-P resonances increase the effective MO interaction distance, and enhance the MO rotation in the device under a limited MO effect in the MO material. Obviously, increasing the external magnetic field can improve the MO effect of material, and thereby significantly increase the conversion of the Y components as shown in Figs. 6(a) and 6(c). However, outside the resonance frequency, the conversion is always very low even if the magnetic field increases since there are no MO enhancements existed at non-resonant frequencies.
When the magnetic field is further strengthen, the conversion of the Y components reaches saturation. The bandwidth of F-P resonance peaks becomes larger. The MSPR peak gradually splits into two peaks, because the surface plasmon splits as a pair of CW and CCW magneto surface plasmon modes with different effective refractive indexes under the strong external magnetic field [21–24]. Therefore, a small magnetic field of 0.05~0.1T is enough to support the good work of this device.
Next, we also calculate the power transmission spectra for forward and backward X-linear polarized waves at different temperatures under 0.05T as shown in Fig. 9(a). With the temperature increasing, the transmission peaks move to a higher frequency due to the rise of εL and εR with the increase of carrier density in InSb. All the transmittances of backward waves are lower than −60dB, so the one-way transmission for linear polarized waves is realized in this device. The isolation is defined as Iso = TP−TN, where TP and TN are the power transmittances in dB for forward and backward waves, respectively. The isolation spectra are shown in Fig. 9(b). We can find that the isolation peaks just correspond to the forward transmission peaks in Fig. 6(a), so the MSPR and F-P resonances also lead to the high isolation due to their enhancement on the MO effect in this double-layer magnetoplasmonics. The MSPR peak has the maximum isolation of over 75dB, while the F-P peaks have 60~70dB, lower than that of MSPR peak. This indicates that the MO enhancement effect of MSPR is more remarkable than that of F-P resonances.
Finally, we discuss the influences of device geometries on the device performance. Frist, the grating grid width d is changed from 28 to 16μm in Fig. 10(a). The F-P peaks significantly drop down with the d decreasing. Because the Q value of F-P cavity is reduced gradually in this case, the F-P resonances become weaken and the MO rotation output at these frequency points are damped down. On the contrary, the MSPR peak gets slightly higher since the change of grating grid width d has no impact on resonance intensity and frequency position of MSPR. Second, the grating constant a is changed from 20 to 50μm in Fig. 10(b), which shows that all the resonance peaks move to a lower frequency with the same frequency shift. Third, the InSb length h between the two metallic gratings is changed from 100 to 50μm in Fig. 10(c). Since the cavity length is shorter, the F-P resonances proportionally move to higher frequencies according to Eq. (10). Only the MSPR peak shifts a little because the InSb length has only a small impact on the MSPR, which is mainly determined by the MO material properties on the InSb-metal surface. If these is only one metallic grating, the conversion will be lower than that of double-layer metallic gratings, especially there will be no F-P resonance peaks since there is no high Q F-P cavity, but the MSPR will still exist.
On summary, we investigate the longitudinal MO effects of magnetized InSb as a Faraday configuration in the THz regime. By the theoretical derivation, we find the nonreciprocal circular dichroism for THz waves in magnetized InSb since both the cyclotron frequency and plasma frequency of InSb are just located in the THz frequency band, which indicate that the special frequency position in the electromagnetic spectrum of THz waves makes its MO responses quite different from the typical Faraday effect in the visible and near infrared lights. The numerical simulation results fit well with the theoretical calculations, which show that longitudinally magnetized InSb can be applied to the circular polarizer and nonreciprocal one-way transmission for the circular polarized waves.
Furthermore, we propose a double-layer magnetoplasmonics to realize one-way transmission and linear polarization conversion, and we find several transmission peaks in its transmission spectrum. By studying the cause of these transmission peaks, we find two MO enhancement mechanisms in this device: MSPR on the InSb-metal surface and F-P resonances between two metallic gratings. Moreover, the dependences and tunability of this device on the external magnetic field, temperature, and geometry parameters are also investigated, which further prove the existence of two resonance mechanisms and their corresponding influence factors. The simulation results show that the one-way transmission of over 70dB are realized in this devices, and it also can be used as a perfect polarization converter and sensitive MO modulator from 0 to 80% transmittance with a good filtering output characteristic under the weak magnetic field from 0 to 0.1T. This magnetoplasmonic device has broadly potentials for THz isolator, modulator, polarization convertor, and filter in the THz application systems.
National Basic Research Program of China (Program 973) (2014CB339800); National Natural Science Foundation of China (NSFC)(61378005, 61505088); Natural Science Foundation of Tianjin (15JCQNJC02100); Science and Technology Program of Tianjin (13RCGFGX01127); and Open project funds of Tianjin Key Laboratory and Key Laboratory of Ministry of Education.
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