Nonreciprocal Fabry-Perot effect and performance enhancement in a magneto-optical InSb-based Faraday terahertz isolator

Elise Keshock; Peisong Peng; Jiangfeng Zhou; Diyar Talbayev

doi:10.1364/OE.411581

1. Introduction

The Terahertz (THz) regime of electromagnetic waves refers to the frequency range between the far infrared and microwave wavelengths (0.1–10 THz). THz-frequency light has found unique applications in medical imaging, wireless communications, security screening, and nondestructive evaluation [1–3]. For continued advances towards large-scale deployment of THz technology, lost-cost and high-performance optical elements are needed as parts of THz systems. We focus on THz optical isolators in this work. An isolator is a passive optical element that allows the light to pass in one (forward) direction and efficiently blocks back-reflected light traveling in the opposite (backward) direction. Faraday optical isolators are common commercial devices at visible wavelengths used to protect laser sources from harmful backward reflections. For THz frequencies, the need in optical isolation will grow together with the output power of THz sources, especially those based on photonics [1,2]. Optical isolation typically relies on the presence of magnetic field or of a magnetized substance. The classic longitudinal magneto-optical configuration, in which the direction of the magnetic field is parallel to the direction of incident light, creates optical isolation due to the Faraday effect in the magneto-optical medium sandwiched between a pair of linear polarizers with a relative 45-degree orientation. A variety of materials has been investigated as candidate magneto-optical media for Faraday THz isolators, including high-mobility semiconductors [4–10], 2D van der Waals crystals [11–19], ferrofluids [20], and magnetic insulators [21]. On the other hand, studies of actual complete THz isolator devices are relatively rare [11,21]. A Faraday THz isolator created by Shalaby et al. relied on Faraday rotation in a ferrite and was distinguished by broadband operation [21]. A near-optimal graphene-based isolator designed by Tamagnone et al. reached 20 dB of isolation with −7.5 dB of insertion loss in magnetic field of 7 T [11].

We select InSb as our Faraday medium due to its unique electronic and THz optical properties. These include a small electron effective mass m* = 0.014m₀ [22], a small band gap (170 meV at room temperature [23]), and a high electron mobility [24]. The small bandgap allows the tuning of the intrinsic carrier density and plasma frequency by temperature [25]. The small electron effective mass is the key feature of InSb for application in Faraday THz isolators, as it allows the device to perform in low applied magnetic field. The high electron mobility is also essential to achieve a high isolator performance. The THz magneto-optical properties of InSb have been well documented [26–28]. For example, Wang et al. reported a magnetically-induced transparency of semiconductor plasma in InSb that arises from interference between left- and right-circular eigenpolarizations in magnetic field [26]. Due to its low effective mass and high-mobility, InSb also serves as an excellent material for THz plasmonic applications [29], including the theoretical proposals for magneto-plasmonic metamaterials that enable THz optical nonreciprocity [30–33]. Fan et al. considered computationally a magneto-plasmonic structure of InSb sandwiched between two wire grid polarizers oriented at 90 degrees and predicted an isolation performance of over 70 dB [34]. Fabry-Perot reflections were considered in their work and found to enhance transmittance symmetrically in both forward and backward directions at the same frequencies. We also note that the polarizers before and after the Faraday medium must be oriented at 45 degrees to create an optical isolator. Experimental studies of InSb-based THz magneto-plasmonic structures have also been reported [35,36].

In this article, we present the experimental and theoretical evaluation of a complete Faraday THz isolator that is built from InSb as the Faraday medium and gold wire grids as the linear polarizers oriented at 45 degrees to each other. Our main finding is an overlooked nonreciprocal Fabry-Perot effect that can enhance multi-fold the isolation performance of the device as well as lower the insertion loss. We observe that the Fabry-Perot reflections (echoes) enhance transmittance in the forward direction but suppress the transmittance in backward direction at the same set of frequencies.

2. Experimental methods

The structure of our Faraday isolator device is shown in Fig. 1. The isolator consists of two wire-grid polarizers (WGPs) and a 0.5-mm-thick InSb layer between them. We used a commercial nominally undoped InSb wafer. The WGPs are oriented with their transmission axes at 45 degrees to each other. They are lithographically fabricated on 0.5-mm-thick fused silica substrates and consist of a gold wire grid with 40 µm period and 20-µm-wide wires. The thickness of gold is 200 nm. The gold wire grids are placed adjacent to the InSb layer in the assembled device and the WGPs are glued to InSb using PELCO Quickstick 135 Wax. The assembled Faraday isolator was placed in the field of a small NdFeB ring magnet with the measured field strength of 0.13 T.

Fig. 1. A schematic view of the Faraday isolator. The middle layer is semiconductor InSb. The outer layers are fused silica with lithographically deposited gold wire gratings. The thickness of the fused silica substrates and the InSb wafer is 0.5 mm. The gold gratings are on the inner fused silica surfaces adjacent to the InSb layer. The three layers are glued together using wax glue. The estimated thickness of the glue is in the 10–20 $\mu m$ range.

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Terahertz spectroscopic characterization was performed using a home built THz time-domain spectrometer based on a mode-locked femtosecond Ti:Sapphire laser and photoconductive antennas for THz emission and detection. The assembled Faraday isolator and the ring magnet were mounted on the cold finger of a liquid nitrogen cryostat to perform measurements at 180 K to allow the semiconductor InSb to reach transparency as the carrier density is lowered. The incident THz wave was plane-polarized along the transmission axis of the front WGP. We measured the full polarization state of the transmitted THz wave and confirmed that it was plane-polarized at 45 degrees to the incident wave. The backward transmittance was measured either by turning around the isolator-magnet assembly inside the cryostat or by turning around only the magnet. The two methods should provide identical results and indeed did so in our experiments.

3. Results and discussion

Figure 2(a) shows the measured forward (T_f) and backward (T_b) transmittance of our Faraday isolator. We define transmittance T as the ratio if the transmitted intensity I_t divided by the incident intensity I₀: T = I_t / I₀. The incident light is assumed to be polarized along the transmission axis of the first WGP. The isolator is not transparent in either direction below 0.4 THz, which is the plasma frequency of InSb at T=180 K. Above about 0.7 THz, the forward and backward transmittances are different, and the difference peaks close to 1.2 THz. Prominent Fabry-Perot interference fringes are observed in both forward and backward transmittance spectra. We define isolation as the ratio between the forward and backward transmittances (T_f / T_b) and find that isolation also displays a series of peaks, Fig. 2(b). A comparison with Fig. 1(a) shows that the highest points in isolation occur at the Fabry-Perot minima of the backward transmittance, where the isolation reaches 75 (or 18.8 dB) near 1.2 THz. Thus, Fig. 2(b) illustrates the main finding of our work – the Fabry-Perot effect can lead to a dramatic enhancement of Faraday isolator performance.

Fig. 2. (a,b) Measured transmittance and isolation of the Faraday isolator. Panel (a) shows forward and backward transmittances recorded using the full length of the THz time-domain scan. Panel (b) shows the corresponding isolation - the ratio of the forward and backward transmittances. The green line shows the isolation calculated using the full length of the THz time-domain scan that includes the Fabry-Perot echoes in the fused silica WGP substrates and in the InSb layer. The blue line shows the isolation calculated after the echoes were windowed out and the Fourier transform of only the main transmitted THz pulse was computed. (c,d) The calculated transmittance and isolation of the isolator. The plotted quantities and colors correspond to the measured quantities shown in the panels directly above in the first row.

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We now explore in more detail how this enhancement originates from the Fabry-Perot effect. In THz time-domain spectroscopy, the Fabry-Perot interference fringes can often be filtered out of the measured intensity by limiting the length of the time-domain THz scan [37,38]. This is because the incident THz pulse is shorter than the separation between the Fabry-Perot echoes in the time domain. We shortened our sample and reference time-domain spectra to include only the main transmitted THz pulse and computed the resulting transmittance and isolation spectra. The measured main-pulse isolation is shown in Fig. 2(b), blue line. The main-pulse isolation is much smoother and exhibits a broad peak centered around 1.2 THz and reaching 25. It is clear that the Fabry-Perot interference can strongly enhance or suppress the isolation. For narrowband or single-frequency THz sources often found in imaging and communications solutions, the actual isolation will include the Fabry-Perot fringes and the Faraday isolator can exhibit much enhanced performance.

Figure 3 shows the main-pulse forward and backward transmitted intensities. We simulate this measurement by an optical model that includes three layers - two layers representing the WGPs on fused silica substrates and one layer for InSb. The refractive index and absorption of the fused silica layers with WGPs were measured separately (see the Supplement 1). An unknown component of this optical model is the precise thickness, refractive index, and absorption of the glue layers between the WGPs and InSb. We estimate the glue thickness to be in the 10–20 µm range. The glue is a good dielectric with a low refractive index and only a negligible fraction of light is absorbed by it. Therefore, we leave the glue layers out of our intensity transmittance model. We assume that an optical interface is formed directly between the WGPs and the InSb layer. The transmission and reflection coefficients at this interface are computed using the refractive indices of fused silica and InSb. We also introduce the amplitude transmission coefficients from air into fused silica and from fused silica into air:

(1)$$\; {t_{01}} = \frac{2}{{{n_1} + 1}}\; \; \textrm{and}\; \; {t_{10}} = \frac{{2{n_1}}}{{{n_1} + 1}},$$

where the subscripts 0 and 1 refer to air and fused silica, respectively, and ${n_1}$ is the complex refractive index of fused silica. The amplitude propagation coefficient through the fused silica layers if given by

(2)$${p_1}\; = \; \textrm{exp} \left( {\frac{{i\omega {d_1}{n_1}}}{c}} \right),$$

where ${d_1}$ = 0.5 mm is the thickness of the WGP substrates.

Fig. 3. Symbols show the measured forward and backward transmittance of the main THz pulse only. The F-P reflections have been windowed out in the time domain. Solid lines show a least-squares fit of the measured transmittances using Eqs. (1)–(12).

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To describe the optical properties of InSb, we use the Drude formula for the dielectric tensor of InSb in the magnetic field [27]

(3)$${\varepsilon _{xx}}\; = \; {\varepsilon _\infty }\; - \; \frac{{{\omega _p}^2\; ({{\omega^2} + i\gamma \omega } )}}{{({{\omega^2} + i\gamma \omega } )- {\omega ^2}{\omega _p}^2\; }}\; + \; {\varepsilon _{ph}},$$

(4)$${\varepsilon _{CRA}}\; = \; {\varepsilon _\infty } - \; \frac{{{\omega _p}^2}}{{\omega ({\omega + i\gamma - {\omega_c}} )}}\; + \; {\varepsilon _{ph}},$$

(5)$${\varepsilon _{CRI}}\; = \; {\varepsilon _\infty }\; - \; \frac{{{\omega _p}^2}}{{\omega ({\omega + i\gamma + {\omega_c}} )}}\; + \; {\varepsilon _{ph}},$$

where ${\omega _p} = \frac{{n{e^2}}}{{{m^\ast }{\varepsilon _0}}}$ is the plasma frequency that depends on the carrier density n and carrier effective mass ${m^\ast }$ and ${\omega _c}\; = \frac{{eB}}{{{m^\ast }}}$. is the electron cyclotron frequency proportional to the magnetic field B. ${\varepsilon _{ph}}$ is the contribution to the dielectric function from optical phonons

(6)$$\varepsilon _{ph}{\rm \; } = {\rm \; }\varepsilon _\infty \left( {\displaystyle{{\omega _t^2 -\omega _1^2 } \over {\omega _t^2 -\omega ^2-i\gamma _{ph}\omega }}} \right),$$

where ${\omega _t}/2\pi = 5.90$ THz and ${\omega _l}/2\pi = 5.54$ THz are the transverse and longitudinal phonon frequencies and ${\gamma _{ph}} = 3.77$ THz is the phonon scattering rate [27]. The magnetic field and its Lorentz force induce the cyclotron resonance trajectories for electrons that absorb energy from one circular THz wave polarization (cyclotron resonance active, CRA, Eq. (4)) and do not absorb energy from the other polarization (cyclotron resonance inactive, CRI, Eq. (5)) [5]. The CRA and CRI polarizations become the eigenpolarizations for the THz propagation through InSb along the z axis shown in Fig. 1 that is described by the propagation coefficient

(7)$${p_{CRA,CRI}}\; = \textrm{exp} \left( {\frac{{i\omega d{n_{CRA,CRI}}}}{c}} \right),$$

where ${n_{CRA,CRI}}\; = \; \sqrt {{\varepsilon _{CRA,CRI}}} $ . In our experiments, a plane-polarized wave is incident on a gold wire grid near the surface of InSb with THz electric field oscillating perpendicular to the wires. The proximity of the metal wires to the InSb surface forces the boundary condition ${E_{||}} = \; 0$ at the InSb surface, where ${E_{||}}$ is the electric field component along the wires of the WGP. In other words, in the immediate vicinity of the front and back InSb surfaces, we neglect the off-diagonal components of the dielectric tensor and use the diagonal component of the tensor and refractive index to compute the front and back surface amplitude transmission coefficients:

(8)$${t_{12}} = \frac{{2{n_1}}}{{{n_1} + {n_{xx}}}}\; \; \textrm{and}\; \; {t_{21}} = \frac{{2{n_{xx}}}}{{{n_{xx}} + {n_1}}}$$

with ${n_{xx}}\; = \; \sqrt {{\varepsilon _{xx}}} $ . The electric fields that impinge on the back InSb surface are (E₀ is the initial incident plane-polarized electric field)

(9)$$E_{CRA,CRI}^{bk} = {p_{CRA,CRI}}\cdot{t_{12}}\cdot{p_1}\cdot\; {t_{01}}\cdot{E_0}/\sqrt 2 , $$

(10)$$E_x^{bk} = ({E_{CRA}^{bk} + E_{CRI}^{bk}} )/\sqrt {2,\; } \; \; \; \; \; E_{y\; }^{bk} = ({E_{CRA}^{bk} - E_{CRI}^{bk}} )/i\sqrt 2 ,$$

(11)$$E_\parallel ^{bk} = ({E_x^{bk}\; - \; E_y^{bk}} )/\sqrt 2 ,\; \; \; \; \; \; E_ \bot ^{bk} = ({E_x^{bk}\; + \; E_y^{bk}} )/\sqrt 2 .$$

In Eq. (9), we compute the propagation of the circular eigenpolarizations of the THz wave through InSb to its back surface. In Eq. (10), we transform the CRA and CRI electric fields to the laboratory cartesian coordinate system, Fig. 1. And in Eq. (11), we project the cartesian x and y components onto the directions parallel and perpendicular to the gold wires of the second WGP. The parallel component $E_\parallel ^{bk}\; $ is blocked by the WGP and the perpendicular component $E_ \bot ^{bk}\; $ is allowed to pass through the second fused silica substrate and onto the THz receiver. Thus, the transmission coefficients of the Faraday isolator in the cartesian laboratory coordinates become

(12)$${t_{x,y\; }} = \; {t_{10}}\cdot{p_1}\cdot{t_{21}}\cdot(E_ \bot ^{bk}/{E_0})\; /\sqrt 2 . $$

The coefficients ${t_x}$ and ${t_y}$ are the same because of the 45-degree orientation of the second WGP. Intensity transmittance is computed as $T = {[{t_x^2 + t_y^2} ]^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}} \right.}\!\lower0.7ex\hbox{$2$}}}}$. We use this transmittance model to conduct a least-squares fit of the transmitted forward and backward intensities with ${\omega _p}$, $\gamma $, ${\varepsilon _\infty }$, and B as the fitting parameters. The solid lines in Fig. 3 show the best fit achieved with ${\omega _p}\; $ = 0.43 THz, $\gamma $ = 0.88 THz, ${\varepsilon _\infty }$ = 22.8, and B = 0.12 T. The blue line in Fig. 2(d) shows the corresponding computed isolation and can be compared to the measured isolation in Fig. 2(b), blue line. We see that the computed forward and backward intensities and isolation match well the measured ones.

The above transmittance model does not include the Fabry-Perot echoes occurring in the WGP substrates or in the InSb layer. To account for the echoes, we explicitly include each of them in the transmission of both WGPs and of the InSb layer. We write the full transmitted electric field as

(13)$$\begin{array}{c} E({full} )= E({0,0,0} )+ E({1,0,0} )+ E({2,0,0} )+ E({0,0,1} )\\ + E({0,0,2} )+ E({1,0,1} )+ E({0,1,0} )\end{array}$$

In this sum, the first term represents the THz main pulse transmitted through each of the three layers and computed using Eq. (12). The numbers in the parentheses, $E({i,j,\; k} ),$ represent the number of back-reflections in each of the layers – zero reflections in each of the layers representing the main transmitted pulse. The first number ($i$) in the parentheses is the number of reflections in the first WGP substrate, the second ($j$) is the number of reflections in the InSb layer, and the third ($k$) – the number of reflections in the second WGP substrate. The second term, $E({1,0,0} ),$ represents the first echo in the first WGP that then traverses the InSb and the second WGP without back-reflections. The third term, $E({2,0,0} )$, accounts for the THz pulse that was back-reflected twice in the first layer and then traverses the rest of the structure without reflections, and so on. Because of the limited length of the time-domain spectra in our measurement, we include at most two echoes in the WGP substrates and only one echo in the InSb layer. InSb has a higher refractive index than the fused silica substrates of the WGPs, thus making it sufficient to include a single echo in the InSb layer.

To compute the various terms in Eq. (13), we combine the transmission and propagation coefficients of Eqs. (1)–(2) and Eqs. (7)–(8) with the following reflection coefficients:

(14)$${r_{12}} = \frac{{{n_1} - {n_{xx}}}}{{{n_1} + {n_{xx}}}},\; \; \; \; \; \; \; \; \; \; \; \; \; {r_{21}} = \frac{{{n_{xx}} - {n_1}}}{{{n_{xx}} + {n_1}}},$$

for the InSb-fused silica interfaces and

(15)$${r_{10}} = \frac{{{n_1} - 1}}{{{n_1} + 1}}$$

for the fused silica-air interfaces. We model the Fabry-Perot echoes ${E_x}({0,1,0} )$ and ${E_y}({0,1,0} )$ in InSb in Eq. (13) by computing the reflected field at the second InSb interface. Equations (9)–(11) describe the electric field incident on the interface. The component $E_\parallel ^{bk}$ is along the wires of the WGP and is fully reflected with a phase change of π. The perpendicular component $E_ \bot ^{bk}$ is reflected with the amplitude determined by the reflection coefficient ${r_{21}}$ from Eq. (14). We write the reflected fields at the back interface as

(16)$$E_\parallel ^{bk,r} = E_\parallel ^{bk}({ - 1} ),\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; E_ \bot ^{bk,r} = E_ \bot ^{bk}\cdot{r_{21}},$$

(17)$$E_x^{bk,r} = ({E_ \bot^{bk,r} + E_\parallel^{bk,r}} )/\sqrt 2 ,\; \; \; \; \; \; \; \; \; \; \; \; \; E_y^{bk,r} = ({E_ \bot^{bk,r} - E_\parallel^{bk,r}} )/\sqrt 2 ,$$

(18)$$E_{CRA}^{bk,r} = ({E_x^{bk,r} + iE_y^{bk,r}} )/\sqrt 2 ,\; \; \; \; \; \; \; \; \; \; \; E_{CRI}^{bk,r} = ({E_x^{bk,r} - iE_y^{bk,r}} )/\sqrt 2 .$$

The propagation of reflected fields $E_{CRA}^{bk,r}$ and $E_{CRI}^{bk,r}$ toward the front InSb interface is computed by multiplying them by the corresponding propagation coefficient in Eq. (7). The fields incident on the front interface are

(19)$$E_x^{fr} = ({E_{CRA}^{bk,r}\cdot{p_{CRA}} + \; E_{CRI}^{bk,r}\; \cdot{p_{CRI}}} )/\sqrt 2 , $$

(20)$$E_y^{fr} = ({E_{CRA}^{bk,r}\cdot{p_{CRA}} - \; E_{CRI}^{bk,r}\; \cdot{p_{CRI}}} )/i\sqrt 2 . $$

Since the wires of the front WGP are vertical (its transmission axis is horizontal), the reflected fields at the front interface are given by

(21)$$E_x^{fr,r} = E_x^{fr}\cdot{r_{21}},\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; E_y^{fr,r} = E_y^{fr}\cdot({ - 1} ),$$

(22)$$E_{CRA}^{fr,r} = ({E_x^{fr,r} + iE_y^{fr,r}} )/\sqrt 2 ,\; \; \; \; \; \; \; \; \; \; \; \; E_{CRI}^{fr,r} = ({E_x^{fr,r} - \; iE_y^{fr,r}} )/\sqrt 2 .$$

The fields $E_{CRA}^{fr,r}\; $ and $E_{CRI}^{fr,r}\; $ propagate toward the back InSb interface, where we decompose them into plane polarized parallel and perpendicular components relative to the wires of the second WGP:

(23)$$E_{CRA}^{bk,1} = E_{CRA}^{fr,r}\cdot{p_{CRA}},\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; E_{CRI}^{bk,1} = E_{CRI}^{fr,r}\cdot{p_{CRI}},$$

(24)$$E_x^{bk,1} = ({E_{CRA}^{bk,1} + E_{CRI}^{bk,1}} )/\sqrt 2 ,\; \; \; \; \; \; \; \; \; \; \; \; E_y^{bk,1} = ({E_{CRA}^{bk,1} - E_{CRI}^{bk,1}} )/i\sqrt 2 ,$$

(25)$$E_\parallel ^{bk,1} = ({E_x^{bk,1} - E_y^{bk,1}} )/\sqrt 2 ,\; \; \; \; \; \; \; \; \; \; \; \; E_ \bot ^{bk,1} = ({E_x^{bk,1} + E_y^{bk,1}} )/\sqrt 2 .$$

The parallel component $E_\parallel ^{bk,1}\; $ is fully reflected by the WGP. The perpendicular component $E_ \bot ^{bk,1}\; $ is transmitted and the electric field transmission coefficients due to the first echo in the InSb layer (corresponding to the $E({0,1,0} )$ term in Eq. (13)) become

(26)$${t_{x,y}}({0,1,0} )= {t_{10}}\cdot{p_1}\cdot{t_{21}}\cdot(E_ \bot ^{bk,1}/{E_0})/\sqrt 2 . $$

The structure of Eq. (26) is similar to that of Eq. (12).

To compute the electric field of echoes $E({0,0,1} )$ and $E({0,0,2} )$, we use the following formulas:

(27)$${E_{x,y}}({0,0,1} )= {t_{10}}\cdot{p_1}\cdot{r_{12}}\cdot{p_1}\cdot{r_{10}}\cdot{p_1}\cdot{t_{21}}\cdot{E}_ \bot ^{\; bk}/\sqrt 2 ,$$

(28)$${E_{x,y}}({0,0,2} )= {t_{10}}\cdot{({{p_1}\cdot{r_{12}}\cdot{p_1}\cdot{r_{10}}} )^2}\cdot{p_1}\cdot{t_{21}}\cdot{E}_ \bot ^{\; bk}/\sqrt 2 ,$$

where $E_ \bot ^{\; bk}$ is defined in Eq. (11). To compute the electric field of echoes $E({1,0,0} )$ and $E({2,0,0} )$, we use the following electric field that impinges on the back InSb surface:

(29)$$E_{CRA,CRI}^{bk}({1,0,0} )= {p_{CRA,CRI}}\cdot{t_{12}}\cdot{p_1}\cdot{r_{10}}\cdot{p_1}\cdot{r_{12}}\cdot{p_1}\cdot{t_{01}}\cdot{E_0}/\sqrt 2 ,. $$

(30)$$E_{CRA,CRI}^{bk}({2,0,0} )= {p_{CRA,CRI}}\cdot{t_{12}}\cdot{({{p_1}\cdot{r_{10}}\cdot{p_1}\cdot{r_{12}}} )^2}\cdot{p_1}\cdot{t_{01}}\cdot{E_0}/\sqrt 2 .$$

We use these equations in place of Eq. (9), followed by Eqs. (10)–(12), to compute transmission coefficients ${t_{x,y}}({1,0,0} )$ and ${t_{x,y}}({2,0,0} )$ corresponding to the first and second Fabry-Perot echoes in the first WGP subsate.

To compute the fields ${E_{x,y}}({1,0,1} )$ and the corresponding transmission coefficients, we use Eq. (29) in place of Eq. (9), then carry out Eqs. (10)–(11) to determine $E_ \bot ^{\; bk}$, which we use in Eq. (27) to find ${t_{x,y}}({1,0,1} )$.

Figure 2(c) shows the transmitted forward and backward intensities calculated by taking into account the Fabry-Perot echoes in the WGP substrates and the InSb layer. Our calculation reproduces well all qualitative features of the measured intensities in Fig. 2(a). The forward transmittance features a strong slow Fabry-Perot oscillation with a superimposed weaker and faster oscillation. The slower oscillation is due to Fabry-Perot echoes in the WGP substrate and the faster one is due to the echoes in InSb. The backward transmittance is dominated by the fast oscillation in both the measured and simulated intensities, Figs. 2(a), (c). Even though the WGP substrates and InSb have similar thicknesses, their refractive indices differ by about a factor of 2, which makes the Fabry-Perot oscillation period appear so different in the forward and backward transmittances. The fast oscillation dominates the backward transmittance because the backward-traveling THz wave is polarized mostly along the wires of the second WGP, thus increasing dramatically the reflected amplitude at that interface. We also calculate the isolation including the described Fabry-Perot echoes, which is shown by the dark green line in Fig. 2(d). We find that our calculation describes the qualitative behavior of the measured isolation in Fig. 2(b) very well, including its dramatic enhancement due to the Fabry-Perot effect.

While the Fabry-Perot effect explains the oscillatory behavior of transmittance and isolation, we do not achieve a close quantitative agreement. For example, the fast oscillation in the measured forward intensity, Fig. 2(a), is more muted compared to the fast oscillation in the calculated forward intensity, Fig. 2(c). Our calculated isolation also overestimates the measured one. Some isolation peaks that are off the scale in Fig. 2(d) reach the values exceeding 1000. Such high values are not reached in the measured isolation. These discrepancies occur partly because our optical transmittance model simplifies the structure of our Faraday isolator by leaving out the glue layers between the WGPs and InSb. Nonetheless, our model correctly illuminates the physics of the isolation enhancement due to the Fabry-Perot effect in the InSb layer.

Our model also correctly predicts that the Fabry-Perot echoes in the WGP substrates alone do not lead to the interference fringes in the isolation, even though the fringes appear in the individual forward and backward transmittances. Figure 4(a) shows the calculated transmittance of the Faraday isolator that includes two Fabry-Perot echoes in both wire grid polarizers but does not include any echoes in the InSb layer. We observe the Fabry-Perot interference fringes in both forward and backward transmittances. The computed isolation, the ratio of the forward and backward transmittances, is shown in Fig. 4(b) and does not exhibit any Fabry-Perot fringes. We conclude that the Fabry-Perot interference fringes in isolation are the result of the echoes inside the InSb layer and not the echoes inside the WGP substrates.

Fig. 4. (a) Calculated forward and backward intensity transmittance of the Faraday isolator. The calculation includes the Fabry-Perot echoes in the substrates of the wire grid polarizers but does not include the echoes in the magneto-optical InSb layer. (b) Isolation – the ratio of the forward and backward transmittances from panel (a).

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We now use our model to consider the performance of a simpler isolator structure where the gold wires are lithographically deposited directly on the InSb surfaces, which eliminates the fused silica substrates of the WGPs. This case is described by setting ${n_1} = \; 1$ in all of the above calculations. The computed forward and backward transmittances and the resulting isolation are shown in Fig. 5, where we calculated these quantities with and without the Fabry-Perot echoes in the InSb layer. The Fabry-Perot oscillations in the forward and backward transmittance are nearly out of phase and the minimum transmittance in the backward direction reaches nearly zero in the 0.8–1.3 THz range, which accounts for the Fabry-Perot isolation enhancement, Figs. 5(a), (b). In other words, the Fabry-Perot echoes enhance the transmittance in the forward direction, and they suppress the transmittance in the backward direction at the same frequencies within the 0.8–1.3 THz range. The Fabry-Perot effect is nonreciprocal. The forward and backward Fabry-Perot fringes slowly become more in-phase as the frequency approaches 2 THz, Fig. 5(b). This slow shift results from the differences in the phase of reflections at InSb surfaces. In the forward transmittance, this phase is determined by the reflection coefficient of the InSb – air interface. In the backward transmittance, this phase is determined by the reflection coefficient of the InSb – gold interface, because the wave incident on the second grating is largely polarized along the gold wires in this case. The forward transmittance of our isolator at the isolation peak near 1.2 THz corresponds to the insertion loss of −12.6 dB, Figs. 2(a), (b). The forward transmittance of the model isolator without the fused silica substrates at the isolation peak just above 1.2 THz corresponds to the insertion loss of −11.0 dB. The highest measured isolation is 18.8 dB, Fig. 2(b). Our analysis shows that the device design including the WGP substrates does not significantly affect the performance of the device, although placing the wire grids directly on InSb and eliminating the substrates does offer a small performance improvement.

Fig. 5. (a) Calculated forward and backward transmittance of the InSb Faraday isolator without the fused silica substrates for WGPs. The gold wires are deposited directly on the InSb surface. The transmittance with and without the F-P reflections is shown. (b) Calculated isolation of the Faraday isolator without the fused silica substrates. (c) The dependence of the calculated isolation peak near 1.2 THz from panel (b) on the applied magnetic field.

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Figure 5(c) shows how the isolation changes with the applied magnetic field. We selected the isolation peak just above 1.2 THz from Fig. 5(b) and calculated the isolation for the magnetic field in the 0.10–0.14 T range (the calculation in Fig. 5(b) is done with B = 0.12 T). We find that the magnetic field can enhance the isolation several-fold while having only a minor effect on the frequency.

4. Conclusion

We present our study of the Faraday THz isolator based on the magneto-optical response of semiconductor InSb resulting from the cyclotron resonance of conduction electrons. Our main finding is the enhancement of the optical isolation functionality by the nonreciprocal Fabry-Perot effect inside the InSb layer between the polarizing wire grids. We confirm this conclusion experimentally and computationally by comparing the forward and backward transmittance and isolation with and without the Fabry-Perot echoes. We find that the Fabry-Perot echoes add constructively in the forward direction and destructively in the backward direction at the same frequencies. This leads to a dramatic three-fold enhancement of isolation in the experiment with peak isolation reaching 18.8 dB, Fig. 2(b). Our modeling shows that this performance can be significantly improved by tuning the strength of the applied magnetic field. The observed experimental insertion loss was found to be −12.6 dB. The nonreciprocal Fabry-Perot effect confines the enhanced isolation to a sequence of narrow bands. This narrow-band effect makes this isolator a practical solution for applications that utilize continuous-wave THz sources, such as security and bio-medical imaging and wireless communications. We also predict that similar Fabry-Perot effects could be used for performance enhancement in alternative magneto-optical THz isolator architectures.

Funding

Newcomb College Institute; National Science Foundation (DMR-1554866, DMR-1919944); AFOSR/AOARD (FA2386-18-1-4104).

Disclosures

The authors declare no conflicts of interest.

See Supplement 1 for supporting content.

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Nonreciprocal Fabry-Perot effect and performance enhancement in a magneto-optical InSb-based Faraday terahertz isolator

Abstract

1. Introduction

2. Experimental methods

3. Results and discussion

4. Conclusion

Funding

Disclosures

References

Supplementary Material (1)

Cited By

Figures (5)

Equations (30)

Optics Express