Terahertz time-domain magnetospectroscopy. The sample studied was a large (~ 0.8 × 20 × 30 mm3) crystal of Te-doped n-InSb with an electron density of 2.3 × 1014 cm− 3 and a 2 K mobility of 7.7 × 104 cm2 V− 1s− 1. At a temperature (T) of 40 K, the Fermi energy EF = 0.9 meV (or 0.21 THz), the plasma frequency ωp = 2πfp = 2π × 0.3 THz, and the scattering rate ν = 0.03 THz.16 We used a time-domain THz magneto-spectroscopy setup17,18 equipped with a commercial THz QWP and wire-grid polarizers to characterize the sample (see the Methods section). Figure 1 shows experimental and calculated transmittance contour maps and spectra for LCP and RCP modes at magnetic fields (B) of 0, 0.3, 0.6, and 0.9 T at T = 40 K. At B = 0 T, a sharp plasma edge exists at 0.3 THz for both the LCP and RCP modes. For the LCP mode (which is the cyclotron resonance active, or CRA, mode), with increasing B, the plasma edge moves towards higher frequencies, and a wide near-zero transmission region appears, whose center also moves towards higher frequencies with increasing B. However, for the RCP mode (which is the cyclotron resonance inactive, or CRI, mode), the plasma edge moves towards lower frequencies with increasing B and eventually becomes unobservable when it becomes lower than the low-frequency limit of our setup (∼ 0.15 THz). At B > 0.3 T, the transmittance spectrum for the RCP mode is a horizontal line without any spectral features, as shown in Fig. 1f and 1h.
These results indicate that a linearly polarized incident THz beam with a frequency inside the zero-transmission band, propagating along the direction of the applied B field, is fully converted into CPL, i.e., the RCP mode. Note that the roles of the two modes are interchanged when the direction of the B field is flipped. Namely, if a linearly polarized THz beam with a frequency inside the zero-transmission band propagates in the negative B direction, it will be fully converted into the LCP mode. Furthermore, the transmission of a circularly polarized THz beam would be nonreciprocal. A RCP THz beam will transmit through (be blocked by) the InSb plate if it propagates in the positive (negative) B direction. Therefore, the InSb plate in B also works as a broadband isolator for circularly polarized THz light.19–22 These concepts are schematically depicted in Fig. 2a.
We calculated transmittance spectra using classical transmission analysis of a THz pulse through a thick sample slab (see the Methods section). The obtained transmittance spectra are in good agreement with the experimental spectra, as shown in Fig. 1. The calculated isolation of the RCP mode, which is defined as\(Iso{\text{=10}} \cdot {\text{lg}}\left( {{{{T_{{\text{RCP}}}}} \mathord{\left/ {\vphantom {{{T_{{\text{RCP}}}}} {{T_{{\text{LCP}}}}}}} \right. \kern-0pt} {{T_{{\text{LCP}}}}}}} \right)\) where TRCP and TLCP are the transmittances of the RCP mode and LCP mode at fc, is as high as − 449 dB at 0.9 T. The isolation value obtained from the experimental results is about − 36 dB, limited only by the signal-to-noise ratio of the THz experimental setup.
Theoretical model. When a magnetic field is applied to an n-InSb sample along the light propagation direction (that is, in the Faraday geometry), a linearly polarized THz wave propagates in the sample as a superposition of the two transverse normal modes of the magnetoplasma: the LCP mode, also called the ‘extraordinary’ or CRA mode, and the RCP mode, also called the ‘ordinary’ or CRI mode. The “dielectric constants” (squares of refractive indices) for the LCP (+) and RCP (−) modes are given by23
$${\varepsilon _ \pm }={\varepsilon _l}\left( {1 - \frac{{\omega _{{\text{p}}}^{2}}}{{\omega \left( {\omega \mp {\omega _{\text{c}}}+i\nu } \right)}}} \right)$$
1
,
where \({\epsilon }_{\pm }= {\epsilon }_{xx}\pm i{\epsilon }_{xy}\), \({\varepsilon _l}\) = 16 is the lattice dielectric constant, ω is the angular frequency, \({\omega _{\text{p}}}=\sqrt {\frac{{n{e^2}}}{{{m^ * }{\varepsilon _l}{\varepsilon _0}}}}\) is plasma frequency, \({\omega _{\text{c}}}=\frac{{eB}}{{{m^ * }}}\) is the cyclotron frequency, ν is the electron scattering rate, n is the electron density, e is the electronic charge, ε0 is the vacuum permittivity, and m* is the effective mass of the electrons. For ν = 0 (neglecting scattering losses), the equation simplifies to a real dielectric constant
$${\varepsilon ^{\prime}_ \pm }={\varepsilon _l}\left( {1 - \frac{{\omega _{{\text{p}}}^{2}}}{{\omega \left( {\omega \mp {\omega _{\text{c}}}} \right)}}} \right)$$
2
.
Figure 2b shows the lines of \({\varepsilon ^{\prime}_ \pm }\) equal to 0, 1, and ∞. The shadowed area bounded by the 0 and ∞ lines represents the region of negative \({\varepsilon ^{\prime}_ \pm }\), where LCP mode is almost totally reflected (reflectivity ≈ 1). The line of \({\varepsilon ^{\prime}_ \pm }\) = 1 (the real part of the refractive index = 1) represents reflectivity = 0, while the \({\varepsilon ^{\prime}_ \pm }\) = ∞ line represents cyclotron resonance for the LCP mode or ω = 0 for the RCP mode. The \({\varepsilon ^{\prime}_ \pm }\) = 0 line represents the plasma edge, which splits into the two magnetoplasmon frequencies with increasing B, given by
$${\omega _ \pm } \cong \frac{1}{2}\left( {\sqrt {\omega _{{\text{c}}}^{2}+4\omega _{{\text{p}}}^{2}} \pm {\omega _{\text{c}}}} \right)$$
3
.
As B increases, the frequency \({\omega _{\text{+}}}\) asymptotically approaches the electron cyclotron frequency\({\omega _{\text{c}}}\), whereas \({\omega _ - }\) monotonically decreases and asymptotically approaches zero as \(B\to \infty\). The total-reflection region (shadowed area) becomes narrower for both the LCP and RCP modes with increasing B. The two modes are interchangeable by flipping the direction of the B field, as indicated by Eqs. 1 and 2. The nonreciprocal behavior of CPL in the zero-transmission band in Fig. 1 comes from the gyrotropic dielectric tensor, which results in the plasma-edge splitting of the InSb magnetoplasma.23 In this work, we concentrate on the zero-transmission region in the LCP mode, while the zero-transmission region in the RCP mode would work for microwave and millimeter waves.
Figures 3 show that the real and imaginary parts of the dielectric constants for both the LCP and RCP modes in the InSb sample at T = 40 K and B = 0.5 T. In Fig. 3, Im(ε+) shows a resonance peak at 1 THz and Re(ε+) shows a Lorentz dispersion shape due to cyclotron resonance. The lower shadow area represents the high reflectivity region where Re(ε+) is negative. The top shadow area also represents a high reflectivity region where Re(ε+) is large. Re(ε−) and Im(ε−) do not show any spectral feature at the frequency of cyclotron resonance. Therefore, under cyclotron resonance conditions, the LCP mode is strongly attenuated due to resonance absorption whereas the RCP mode propagates without any attenuation except for the interface reflection loss. In addition, the high reflectivity on both sides of the resonance frequency further attenuates the LCP mode.
By tuning the B field, near-perfect circularly polarized THz light with a tunable broad bandwidth is realized. The bandwidth of the zero-transmission region is about 0.45 THz for this sample, which is proportional to the electron scattering rate ν, as indicated in Eq. 1. The tunable frequency range of the zero-transmission region is from ƒp (∼ 0.3 THz) to the Reststrahlen absorption band starting at 160 cm− 1 (∼ 4.8 THz).24 By using other lightly doped semiconductors that have different Reststrahlen absorption bands, one would be able to generate CPL covering the whole THz frequency band (0.3−10 THz).
Demonstration of a near-perfect circularly polarized THz beam. To further illustrate the one-way nonreciprocal propagation of circularly polarized THz light, we placed a THz narrow bandpass filter25 before the sample (see the Methods section), in order to obtain a circularly polarized THz wave after the sample. The filter has a central frequency right in the middle of the zero-transmission region of the InSb sample at B = ±0.5 T. We reconstructed the time evolution of the transmitted CPL field vector by measuring both the x and y components of the transmitted electric field. The red curve in Fig. 4a shows the RCP THz electric field E(t) at B = + 0.5 T; each point on the trace represents the tip of the THz electric field vector. The amplitude E(t) decays gradually as it rotates in the form of a clockwise helix looking from the positive z-direction. The black curve is the projection of E(t) onto the Ex − t plane, that is, the measured electric field Ex(t), while the green curve is the projection of E(t) onto the Ey − t plane, that is, the measured electric field Ey(t). The orange curve is the projection of E (t) onto the Ex − Ey plane.
At B = −0.5 T, the transmitted THz wave is LCP. Traces in Fig. 4b can be understood in the same way as in Fig. 4a. We plotted Ey(t) for both B = + 0.5 T and B = −0.5 T in Fig. 4c. The two traces are mirror-symmetric with respect to the horizontal line y = 0, and have an exact π phase shift, indicating the facile conversion of right/left CPL by switching the B field direction. In order to quantify the purity of the CPL, we calculated the ellipticity \(\eta \left( \omega \right)\) using the following formulas26–27
$$\eta \left( \omega \right)=\frac{{\left| {{{\tilde {T}}_ - }\left( \omega \right)} \right| - \left| {{{\tilde {T}}_+}\left( \omega \right)} \right|}}{{\left| {{{\tilde {T}}_ - }\left( \omega \right)} \right|+\left| {{{\tilde {T}}_+}\left( \omega \right)} \right|}}$$
4
,
where \({\tilde {T}_+}\left( \omega \right)\) and \({\tilde {T}_ - }\left( \omega \right)\) are the complex transmission coefficients of the LCP and RCP modes (see the Methods section), respectively. The calculated and experimental ellipticities are shown in Fig. 4d, showing a value of 1 in the zero-transmission region, indicating near-perfect CPL output.
Material scientists are seeking elusive lossless metals,28 that is, materials with a purely real and negative dielectric constant (\(\varepsilon ^{\prime\prime}\) ≈ 0 and \(\varepsilon ^{\prime}\) < 0), which would be ideal candidates to replace metals in plasmonic and metamaterial devices. Doped InSb represents a unique low-density metal in the THz spectral region, which has a low loss due to low carrier density and a negative dielectric constant for the RCP mode as shown in Fig. 2b and 3. In addition, a chiral plasma medium consisting of chiral objects embedded in a magnetized plasma has been proposed to enable a negative refractive index (both ε and µ are negative) in a broad frequency range.29
It is well known that nonreciprocal devices for microwaves are made using antiferromagnetic resonance in ferrites.30 However, great difficulties exist in using ferrites for millimeter and submillimeter (or THz) waves, such as the necessity for a strong B field and an increase in the forward loss. These difficulties can be overcome by using magnetoplasmas in doped semiconductors. To achieve efficient operation of semiconductor nonreciprocal devices based on a magnetoplasma, it is necessary to satisfy the condition\({\omega _c}\tau \gg 1\), where \(\tau\)= 1/ν is the momentum relaxation time. For n-InSb, the condition can be easily satisfied with a small magnetic field.16 In addition, our device can work at liquid nitrogen temperature and using permanent magnets to fulfil the above conditions below 2.5 THz.17