Generation and manipulation of chiral terahertz waves in the three-dimensional topological insulator Bi2Te3

Arbitrary manipulation of broadband terahertz waves with flexible polarization shaping at the source has great potential in expanding numerous applications, such as imaging, information encryption, and all-optical coherent control of terahertz nonlinear phenomena. Topological insulators featuring unique spinmomentum–locked surface state have already exhibited very promising prospects in terahertz emission, detection, and modulation, which may lay a foundation for future on-chip topological insulator-based terahertz systems. However, polarization-shaped terahertz emitters based on topological insulators with an arbitrarily manipulated temporal evolution of the amplitude and the electric-field vector direction have not yet been explored. We systematically investigated the terahertz radiation from topological insulator Bi2Te3 nanofilms driven by femtosecond laser pulses and successfully realized the generation of efficient chiral terahertz waves with controllable chirality, ellipticity, and principal axis. The convenient engineering of the chiral terahertz waves was interpreted by a photogalvanic effect (PGE)-induced photocurrent, while the linearly polarized terahertz waves originated from linear PGE-induced shift currents. Our work not only provides further understanding of femtosecond coherent control of ultrafast spin currents but also describes an effective way to generate spin-polarized terahertz waves at the source.


Introduction
Spin-polarized terahertz waves with twisted electric-and magnetic-field directions and temporal evolution of the amplitudes have been widely utilized for non-thermal and selective excitation of electron spins, resulting in the discovery of the spin-galvanic effect, 1 the manipulation of terahertz nonlinear phenomena, 2 and optical spin injection in novel materials. [3][4][5] Recently, Mashkovich et al. 6 found that linearly polarized terahertz waves can nonlinearly excite gigahertz spin waves in antiferromagnetic FeBO 3 via terahertz field-induced inverse Cotton-Mouton effect. However, the inverse Faraday effect excited by circularly polarized terahertz waves has not yet been experimentally demonstrated, despite many theoretical predictions reported. 7 Therefore, spin-polarized terahertz waves are essential for revealing electron spin-related linear and nonlinear ultrafast phenomena. In addition, spin-polarized terahertz waves naturally carrying chiral properties offer multifaceted spectroscopic capabilities for investigating low-energy macromolecular vibrations in biomaterials, understanding the mesoscale chiral architecture 8 and confidential communication. 9,10 However, the proliferation of terahertz circular dichroism spectroscopy and other applications is impeded by the lack of generation and manipulation of chiral terahertz waves.
Conventional polarization shapers working in the terahertz electromagnetic frequency range are mainly segmental waveplates and metasurfaces, 11,12 both of which can be well applied for narrow bandwidth applications and low tolerance of flexible tunability. Various scalar manipulations of terahertz waves have been demonstrated in poled nonlinear optical crystals 13,14 or even in water 15 and have synthesized different frequencies generated in spatially distributed sources. 16 However, it would be much more elegant if polarization shaping functionality could be vectorized and integrated into the sources. Currently, there are already several such kinds of vector-manipulated polarized terahertz sources. [17][18][19][20][21] For example, femtosecond laser amplifier pumped air plasma, when intensively tuning the phase delay and the pump pulse polarization between two different pump pulse colors, can enable the opportunity to obtain arbitrarily shaped terahertz polarizations. 17 Furthermore, terahertz polarization pulse shaping with arbitrary field control has also been demonstrated by optical rectification of a laser pulse whose instantaneous intensity and polarization state are flexibly controlled by an optical pulse shaper. 18 Recently, elliptically and circularly polarized terahertz waves have been realized in W/CoFeB/Pt heterostructures through deliberately engineering the applied magnetic-field distribution and the cascade emission method, respectively. 19,20 In a NiO single crystal, double-pulse excitation has also been used to vectorially control magnetization by radiating spin-polarized terahertz waves, 21 in which an electron spin-polarized wave (magnon oscillation) has been purposely vectorially controlled and manipulated. In addition, highly efficient chiral terahertz emission with elliptical polarization states has been obtained from the fascinating topological phase matter, bulk Weyl semimetal TaAs with reflection terahertz emission geometry. 22 Inspired by these pioneering breakthroughs, the question of whether we can utilize the spin freedom of electrons to generate and manipulate spin-polarized terahertz waves has become very interesting.
Three-dimensional topological insulator Bi 2 Te 3 single crystals grown on Al 2 O 3 substrates not only feature the spinmomentum-locked (topologically protected) surface states but also are adapted for various heterostructures with tunable quantum layers. 23 They are predicted to have the potential for achieving vectorial polarization shaping terahertz sources. The charge current and spin current inside topological insulators are simultaneously excited when being driven by femtosecond laser pulses. [24][25][26][27][28] However, topological insulator-based terahertz emission mechanisms are far more complicated when compared with conventional terahertz emitters. It is clarified that nonlinear effects play a predominant role in the ultrafast terahertz emission process. 29 Nonlinear effects include optical rectification, photogalvanic effect (PGE), and photon drag effect, among which PGE has a strong correlation to electron spin. Thus, there is the feasibility to generate a femtosecond laser helicity-dependent current component resulting in synthesizing chiral terahertz light.
In this work, we systematically studied terahertz emission from Bi 2 Te 3 nanofilms with various thicknesses grown on Al 2 O 3 substrates excited by femtosecond laser pulses. Through elaborately controlling the pump laser pulse polarization, its incident angle, and the sample azimuthal angle, we not only successfully realized efficient linear polarization tunable terahertz radiation but also obtained high-quality chiral terahertz waves, as shown in Fig. 1(a). In addition, we also demonstrated arbitrary manipulation of the ellipticity and chirality of the emitted terahertz waves from topological insulators. Based on the radiated terahertz characteristics, which originated from the helicitydependent photocurrent component featuring the spin selectivity in the surface state [ Fig. 1(b)] and other components in the bulk state 30 [ Fig. 1(c)], we provide a phenomenological PGE-based interpretation for the polarization tunable terahertz radiation mechanism. Such polarization-shaped topological insulatorbased terahertz source can be used for terahertz circular dichroism spectroscopy, terahertz secure wireless communication, and electron spin correlated coherent excitation and manipulation investigations.

Experimental Results
In this experiment, we employed a commercial Ti:sapphire laser oscillator delivering nJ magnitude pulse energy with a central wavelength of 800 nm, a pulse duration of 100 fs, and a repetition rate of 80 MHz. The pump laser polarization was varied by inserting either a half-wave plate or a quarter-wave plate. The radiated terahertz polarization was probed by a polarizationresolved electro-optic sampling detection method. The topological insulator samples were grown by molecular beam epitaxy (MBE), and their lattice structures and the characterizations  First, we examined the sample azimuthal angle-dependent terahertz radiation polarization properties to unveil the radiation mechanism in topological insulators driven by femtosecond laser pulses. For the p-polarized pump laser at nearly normal incidence, as shown in Fig. 2(a), the radiated terahertz signal from 10-nm-thick Bi 2 Te 3 was ∼1∕20 compared with that from 1-mm-thick ZnTe (see Supplementary Note 2 and Fig. S2 in the Supplementary Material) and with ∼100 dynamic range, which manifests its feasibility for terahertz spectroscopy. The radiated terahertz waves were always linearly polarized with a three-fold rotation angle depending on the sample azimuthal angle, as shown in Figs. 2(b) and 2(c). For the pump-fluence-dependence results [ Fig. 2(d)], the terahertz signals of different thickness all exhibited a linear increasing tendency, indicating that the terahertz radiation mechanism under the linear-polarization laser pump was predominated by a second-order nonlinear effect.

Mechanism of Linearly Polarized Terahertz Emission
Nonlinear effect radiation mechanisms include optical rectification (OR), photon-drag effect (PDE), and PGE. To evaluate the relative importance of optical rectification in our experiment, we compared the emitted terahertz amplitude from the topological insulator sample with that from a ZnTe crystal, which is known for relatively strong optical rectification at the pump wavelength used here. We found that the terahertz signal from topological insulator was almost 3 orders of magnitude larger than that from ZnTe when we normalized the amplitudes by the emitter thicknesses (see Supplementary Note 2 and Fig. S2 in the Supplementary Material). Therefore, we can safely rule out optical rectification as the predominant radiation mechanism. 31 Linear photon-drag effect (LPDE) can be microscopically understood as the linear transfer from photon momentum to electron momentum, which means that when the opposite incident angle was employed, we would obtain opposite polarity of terahertz signals. However, when we varied the incident angles with opposite signs under the p-polarized laser pumping, which was a pure LPDE case, such behaviors neither for S x ðtÞ nor for S y ðtÞ were not observed, as described in Supplementary Note 4 and Fig. S4 in the Supplementary Material. Hence, we can further safely remove the LPDE as the primary radiation origin. Macroscopically, PGE is a second-order nonlinear process occurring in the noncentral symmetric structure. From the microscopic physical picture, PGE can be understood as optical transition on the surface states, 31,32 which can be divided into linear PGE (LPGE) and circular PGE (CPGE). The linear photogalvanic current, also named a shift current, can be explained in that the electron density distribution is spatially shifted from state 1 to state 2 along the Bi-Te bonds [see Fig. 2(e)] when the pump laser pulses illuminate the topological insulator, following a relaxation process such as asymmetric scattering due to the optical electric field. 31 For this mechanism underlying the terahertz emission scenario, we can quantify the behavior on both x and y directions by analyzing the symmetry of the terahertz signals. The three-fold rotational symmetry of signals is in coordination with the space group of the surface and bulk, such that we can write the two-dimensional waveform set Sðt; φÞ as a linear combination of three bases, 31 Sðt; φÞ ¼ AðtÞ þ BðtÞ sinð3φÞ þ CðtÞ cosð3φÞ: (1) Thus, these three bases can completely characterize Sðt; φÞ. Among the three coefficients, BðtÞ and CðtÞ represent the pure LPGE-dependent components, while AðtÞ represents a direct current component, which might be ascribed to a thermal current caused by the thermal potential gradient when the sample was under pumping. From the extracted coefficients in the y axis, as shown in Fig. 2(f) (coefficients in the x axis can be found in Supplementary Note 4 and Fig. S5 in the Supplementary Material), it is obvious that the thermal current contributes much less than the shift current. Therefore, the thermal effect in our Bi 2 Te 3 material under excitation can be safely ignored. We only need to consider the shift current in the generation process of the linearly polarized terahertz wave scenario.
As for a more profound insight about the shift current generated under the short-pulse excitation, this process results in a step-like charge displacement Δx sh ΘðtÞ, where Δx sh is the spatial displacement of the electron density, ΘðtÞ denotes the unit step function, whose temporal derivative is proportional to the shift current J sh . A phenomenological relaxation process such as scattering should also be included so that an exponential decay term with time constant τ sh has to be introduced. Therefore, we obtain the shape of the shift current as follows: The convolution (marked by *) with the laser intensity profile I p ðtÞ (normalized to unit) accounts for the influence under the process of excitation. This model illustrates that the profile of J sh initially follows the envelope of I p ðtÞ and subsequently becomes bipolar, as shown in Fig. 2(g) (blue line). Due to the relatively flat response function of ZnTe 33 within the lowfrequency range (<3 THz), we can semi-quantitatively retrieve the photocurrent (green line) inside the material [see Appendix (Sec. 5) and Supplementary Note 4 and Fig. S6 in the Supplementary Material]. Moreover, to fit the photocurrent through the theoretical model, I p ðtÞ exhibits an extension of its pulse duration from 140 to 170 fs (red line), and the retrieved photocurrent agrees well with the calculated results among the whole profile. According to the fitting process, we obtain the relaxation time τ sh ¼ 22 fs, which is consistent with that in the previous literature, 31 further proving the validity of the LPGE mechanism under this circumstance. It has been reported that the shift current in Bi 2 Te 3 originated from the transient electron transfer along the Bi-Te bonds involving the surface-statesrelated optical transitions. However, for ultrathin Bi 2 Te 3 nanofilms, the coupling between the top and bottom surface opens a gap in the surface state dispersion and, hence, suppresses the surface-states-related optical transitions. 23 Therefore, as shown in Fig. 2(d), the shift current as well as its resulting terahertz emission decreases considerably.

Chiral Terahertz Wave Emission and Manipulation
As shown in Fig. 3(a), when the left-handed circularly polarized (LCP) pump light was incident onto the sample at an incident angle around þ20 deg, and, when we rotated the sample azimuthal angle, elliptically polarized terahertz waves were obtained and showed no significant change when scanning the azimuthal angle φ in Fig. 3(b). To produce circularly polarized terahertz pulses, simultaneously tuning the pump laser polarization and the sample azimuthal angle was necessary. The experimental results are shown in Fig. 3(c). When the sample azimuthal angle was fixed, we can also obtain elliptical terahertz beams with various ellipticities and principal axes. Within expectations, we can manipulate the chirality of the emitted terahertz waves via varying the incident laser helicity. It is obvious that the polarity of emitted terahertz signal along the x direction reversed under illumination of the laser with different chirality [ Fig. 3(d)] and sustained the sign along the yz plane [ Fig. 3(e)]. Therefore, the generation of pump laser helicity-dependent photocurrents with continuous tuning of the magnitude and polarity is the primary reason for the production of elliptically and circularly polarized terahertz pulses.
To further investigate the helicity-dependent terahertz signal, a terahertz pulse with chirality was manipulated by rotating the quarter-wave plate with an angle α at a fixed incident angle around þ20 deg. During the α scanning, the terahertz peak values clearly exhibited a 180-deg period [ Fig. 4(a)], and the waveform shape along the x axis was very similar to the result in Ref. 24 in the Sb 2 Te 3 thin film system. It illustrates the analogies among the family of topological insulators. Therefore, based on the geometric structure of the incident laser that was modulated via the quarter-wave plate (Supplementary Note 6 and Fig. S8 in the Supplementary Material), we can write the signal in the following form: 25,26,34 Sðt; αÞ ¼ CðtÞ sinð2αÞ þ L 1 ðtÞ sinð4αÞ þ L 2 ðtÞ cosð4αÞ þ DðtÞ: The coefficient CðtÞ describes the helicity-dependent terahertz radiation that originates from CPGE. L 1 ðtÞ denotes the coefficient induced by linearly polarized light, and LPGE may be responsible for it. For terahertz emission in 40-nm Sb 2 Te 3 , Yu et al. 34 attributed the origin of L 2 ðtÞ to LPDE and experimentally verified that the trend of L 2 ðtÞ þ DðtÞ under the variation of the incidence angle θ was in accordance with the trend of pure LPDE for p-polarized laser excitation. However, as aforementioned, LPDE has already been eliminated, so it is not suitable for elucidating this phenomenon. In terms of DðtÞ, there is no uniform statement, and it is still subject to controversial debate. LPGE, drift current out of plane, or photo-Dember effect may be the candidates for this coefficient.
Accordingly, we extracted all coefficients along the x axis and yz plane (see Sec. 2). All data were offset to ensure great clarity, as shown in Figs. 4(b) and 4(c), which show the timedomain trace for every component. The main characteristics are summarized as follows. (i) In magnitude, along the x axis, C x , L 1x , and D x dominate the main components, while L 2x is negligible. They are sorted from the largest to smallest L 1x > D x > C x . Along the yz plane, L 2yz and D yz dominate the main components, while L 1yz and C yz are negligible. In addition, L 2yz and D yz have almost the same magnitude.
(ii) C x shows similar characteristics to those for L 1x . In fast Fourier transformed spectra of C x and L 1x [Figs. 4(e) and 4(f)], they both presented the same spectral shapes, strongly indicating that both C x and L 1x shared the same physical origin and are consistent with previous observation. 25,26,[35][36][37][38] However, the D x , L 2yz , and D yz terms also presented similar shapes in the frequency domain, and very few works pay attention to the underlying mechanism about these three terms. Ours is the first observation that all five terms resemble each other in nature, and we elucidate this phenomenon in a macroscopic phenomenological PGE-induced photocurrent framework.
Macroscopically, as a second-order nonlinear optical process, the PGE-induced photocurrent vanishes in the bulk region where inversion symmetry exists. Only on the surface where inversion symmetry is broken, the photocurrent induced by PGE can occur. Thus, taking the C 3v symmetry into account, in our specific case in which the incident pulse shines on the yz plane, the arbitary polarized laser pulse-induced photocurrent can be separated to CPGE-and LPGE-induced components (see the derivation in Supplementary Note 6 in the Supplementary Material): The LPGE-induced photocurrent can be written in a compact way, j LPGE ¼ 0 @ j x j y j z 1 A , and all components are written as j x ¼ −2C 2 ðη 1 cos θ þ η 2 sin θÞ sin 4α; where the incident angle is θ, the rotated angle of the quarterwave plate is α, and γ and η denote the independent components for CPGE and LPGE tensor elements, respectively. From the above four PGE-induced photocurrent equations, we can obtain four conclusions. (i) For the incident laser with arbitrary polarization and directions, in addition to the CPGE-induced term j CPGE ∝ sin 2α, the LPGE-induced photocurrent can be written in a form of j LPGE ¼ j LPGE;1 sin 4αþ j LPGE;2 cos 4α þ j LPGE;3 . The combination of these two photocurrents explains why we can write the emitted terahertz signals in a form of Eq. (3). (ii) We can readily find that when the incident laser pulse changes chirality from left-handed (α ¼ π∕4) to right-handed (α ¼ 3π∕4), the CPGE-induced photocurrent will change signs. Furthermore, the CPGE-induced photocurrent only flows along the direction perpendicular to the incident plane, which is consistent with the results in Fig. 3(d).
(iii) From these four equations, we can qualitatively analyze the magnitude relationship of the photocurrent components. As in the x direction, from Eqs. (4) and (5), only CPGE-induced C x and LPGE-induced L 1x components contribute to the photocurrent. Because the incident angle is relatively small, and L 1x contains the cos θ term, L 1x should be significantly larger than C x . And there is no L 2x contribution in the x direction. In the yz plane, C yz and L 1yz are both 0, and the magnitudes of L 2yz and D yz are approximately identical. The magnitudes of these current components determined by the crystal symmetry in the x axis and yz plane are in good agreement with the relative magnitudes of the various components extracted in our experiments except for D x . There is a relatively large D x , observed in experiment, and the physical origin of this component may come from a dark current, 30 which needs further investigation. (iv) It is worth noting that the PGE theoretical framework illustrates that photocurrent components all originate from the same PGE mechanism, which is exactly the evidence reflected by their frequency spectral shapes. Therefore, we can conclude that the phenomenological PGE mechanism framework determined by the symmetry of the crystal surface is adequate for explaining most of the salient features of the resulting photocurrents.
To acquire spin-polarized terahertz pulses with higher quality, we can also readily gain inspiration from the above framework. When the incident laser pulse is left-handed (α ¼ π∕4), j CPGE;x ¼ −2γC 2 sin θ, j LPGE;x ¼ 0, j LPGE;yz ¼ fðθÞ. When the incident laser pulse is right-handed (α ¼ 3π∕4), j CPGE;x ¼ 2γC 2 sin θ, j LPGE;x ¼ 0, j LPGE;yz ¼ fðθÞ. It indicates that when we change the chirality of the laser pulse, we can manipulate the polarity of the CPGE-induced photocurrent, keeping the orthogonal component invariant. Based on this statement, we can further analyze the manipulation of phase and amplitude for terahertz radiation. As for the phase between the x axis and yz plane, two different physical processes actually exist. The physical process in LPGE represents a shift current along the Bi-Te bond. As for the process in CPGE, the laser with certain chirality will selectively excite a certain spin alongside the direction of incidence due to the spin texture of the surface states [ Fig. 4(d)]. Subsequently, the asymmetry distribution of spin would result in a photocurrent that is perpendicular to the direction of the spin because of the spin-momentum locking. 26 As long as a phase difference between CPGE-and LPGE-induced photocurrents exists, which has already been assured by the above two different physical processes, spin-polarized terahertz radiation is guaranteed. Regarding the amplitude, randomly choosing incident angle θ cannot warrant indentical amplitude along the x axis and yz plane. This explains why we cannot achieve optimal ellipticity when employing circularly polarized pulses. Therefore, to optimize the ellipticity, we can increase the incident angle and look forward to achieving higher ellipticity in a wider spectrum.
In short, we systematically analyzed the features of L 1 , L 2 , and D via the symmetry on the surface of Bi 2 Te 3 under a phenomenological PGE framework. We realized that the invariant L 2 , D components, and polarity-controllable C component were all prerequisites for the spin-polarized terahertz emission. Furthermore, this framework can summarize most features of extracted components except for D x and gives hints for achieving optimal spin-polarized terahertz emission, which owns significant guidance.
Based on the above theoretical analysis, we summarize the terahertz emission experimental results in Fig. 5. From these results, we can see that, for a fixed laser incidence angle, linearly polarized pump light can only produce linearly polarized terahertz waves [Figs. 2 and 5(c)], circularly polarized light can produce chiral terahertz waves, and its chirality is consistent with that of the pump laser [Figs. 5(a) and 5(b)]. To make the terahertz generation and manipulation more advanced, we can successfully realize various complex terahertz wave polarization shaping in topological insulators.

Conclusion
We systematically studied terahertz emission from topological insulator Bi 2 Te 3 nanofilms driven by femtosecond laser oscillator pulses. Through experimentally investigating the sample thickness, azimuthal angle, and pump laser polarization-dependent terahertz radiation properties, we clarified that PGE played a predominant radiation mechanism in this light-matter ultrafast interaction process and produced efficient terahertz waves. Borrowing the spin-momentum-locked state-induced helicitydependent spin-polarized current generation and combing it with the helicity-independent photocurrent, we not only realized circularly polarized terahertz beam generation but also demonstrated arbitrary manipulation of their polarization shaping in topological insulators via simultaneously adjusting the pump laser polarization states and sample azimuthal angles. Our method may have the feasibility to be extended to a variety of topological insulator materials and other related Dirac quantum materials. The topological insulator nanofilms can be grown with high quality and large size possible for high-field terahertz sources at transmission emission geometry, which may also provide very promising applications in fundamental nonlinear terahertz investigations and other real applications such as terahertz circular dichroism spectroscopy, polarization-based imaging, and terahertz secure communication.

Appendix A: Sample Preparation and Characterization
Normal three-dimensional topological insulators are V to VI component semiconductors with a hexagonal crystal structure, in which the building block is a quintuple layer (QL). Our topological insulator nanofilms with 10-, 8-, and 5-nm thicknesses grown on 0.5-mm-thick sapphire are Bi 2 Te 3 prepared by molecular beam epitaxy. As shown in Supplementary Note 1 and Fig. S1(a) in the Supplementary Material, each QL of Bi 2 Te 3 consists of five atomic layers, terminating by Te atoms without dangling bonds, and a weak interaction called van der Waals force binds two neighboring QLs. This peculiar gap makes Bi 2 Te 3 able to overcome the large lattice mismatch with sapphire substrates (∼8%). Due to the weak adhesion between Bi 2 Te 3 and sapphire, we employed a two-step deposition procedure. The initial 1 to 2 QLs of Bi 2 Te 3 were deposited at a low temperature of 160°C. The lower temperature diminished the material quality, which promoted the stacking of atoms to the substrates. After finishing this first deposition, the substrate temperature slowly increased to 230°C. Based on these seed layers, the rest of the film could be deposited with a good quality.

Appendix B: Terahertz Emission Spectroscopy
In our experiment, we used a home-made terahertz time-domain spectrometer to carry out all of the measurements. The femtosecond laser pulses were from a commercial Ti:sapphire femtosecond laser oscillator with a central wavelength of 800 nm, a pulse duration of 100 fs, and a repetition rate of 80 MHz. About 90% of the laser energy was used for terahertz generation in topological insulators, and the residual was employed for electro-optic sampling in 1-mm-thick ZnTe detection crystal. Either a half-wave plate or a quarter-wave plate was inserted into the pumping beam before it illuminated onto the sample to vary the pump laser polarization states. The radiated terahertz pulses went through four parabolic mirrors and finally were focused together with the probing beam into the ZnTe detector.
Between the second and third parabolic mirrors, there were two terahertz polarizers inserted into the optical path to resolve the terahertz polarization. 19 The optical path from the terahertz emitter to the detector was sealed and pumped to eliminate the influence of water vapor. All of the experiments were carried out at room temperature.

Appendix D: Retrieving Femtosecond Photocurrents
The measured electro-optic signal SðtÞ and the terahertz electric field EðtÞ near the sample surface have a linear relationship in the frequency domain as the following equation: SðωÞ ¼ HðωÞEðωÞ; where HðωÞ is the transfer function. After getting frequency domain EðωÞ, Ohm's law can be used to obtain the photocurrent JðtÞ on the surface of Bi 2 Te 3 as J yz ðωÞ ¼ − n 2 cos θ þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 2 − sin 2 θ p Z 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 2 − sin 2 θ p E yz ðωÞ; where θ is the incident angle, n is the refractive index of