1. Introduction

Since Arago’s [1] first observation of light polarization rotation in a quartz crystal in 1811, optical activity has been an impetus for progress in chemistry [2, 3], biology [4], optics [5, 6], and the study of fundamental symmetries in particle physics [7, 8]. The origin of optical activity arises from electromagnetic interaction with structures possessing chirality [2]. The principle behind chirality is mirror-asymmetry of a structure’s sub-wavelength micro-, nano-, or atomic-scale molecular arrangement. Of the many manifestations of chirality, the helical conformation is the most universally significant in understanding the origins of optical activity. The significance of the helical shape in affecting light polarization was demonstrated over a century ago [9, 10] in experiments showing polarization rotation of radiation transmitted through helices, which qualitatively established the structural similarities between macroscopic helices and microscopic optically active molecules [11,12]. Since then, extensive theoretical and experimental work has been performed on the electromagnetic properties of helices. Theoretical frequency-domain calculations [13-19] and experimental work in the visible and microwave range [20-26] have revealed that the optical activity of helices fundamentally arises from the anisotropic scattering of left- and right- circular light from the geometrical shape of the helix. To date, however, experimental investigations of helices have been restricted to only irradiance-based experiments where the helix is excited by continuous light waves [20-26]. While such continuous wave experiments have enabled the extraction of the complex material parameters of helices [see, for instance, Ref. 23], they do not permit investigation of the transient scattering mechanism which gives rise to optical rotation and polarization circularization. An alternative method to study optical activity from a helix is to excite the structure with a temporally localized electromagnetic pulse and examine the impulsive scattering response directly in the time-domain. Such time-domain, transient investigations permit a more direct study of optical activity since scattering mechanisms within the helix can be ascertained from the arrival time and the instantaneous polarization state of the scattered radiation. Moreover, transmitted waveforms can be parsed to separate the complex interactions occurring in the helix.

Here, we explore the optical activity of a helical structure by studying its time-domain, electromagnetic response to impulsive excitation. By dynamically accessing the electric field pulse propagated along the axis of a helix, we correlate different optical activity signatures with the arrival time and polarization of the scattered radiation. Our experimental results show that impulsive excitation along the axis of a helix gives rise to a scattered pulse consisting of two distinctive regimes: an initial, transient regime where polarization circularization increases in time and a subsequent regime of sustained polarization circularization. Calculations of the electric field and Poynting vector evolution show that optical activity in the latter regime is associated with multiply scattered radiation within the helix. To our knowledge, this is the first experimental time-resolved phase and amplitude investigation of optical activity signatures arising from impulsive excitation of the helical shape and may stimulate future studies of other optically active chiral elements such as spirals and gammadions.

2. Experimental setup

For the measurements, we employ far-infrared electric field excitation together with synchronized, polarization-sensitive detection. Far-infrared radiation in the form of a linearly polarized, single-cycle pulse is generated from a semi-insulated GaAs photo-conductive emitter excited by < 20 fs wide, 800 nm pulses supplied by a Ti:sapphire oscillator operating at a repetition rate of 80 MHz. The broadband pulses encompass a spectral range from 0.1 THz to 1.5 THz, which corresponds to a wavelength range from 3 mm to 200 μm. The model chiral structure studied is a Cu helix consisting of n right-handed circular revolutions having a pitch s = 120 ± 10 μm, a wire diameter a = 60 ± 2 μm and a helical centre-to-centre diameter D = 295 ± 10 μm. The helix is fabricated by carefully winding the thin Cu wire into helical conformation. In our work, a metallic structure is studied (as opposed to a dielectric structure) because any optical activity signatures in the metallic structure must arise from scattering (reflection) from the helix. At THz frequencies, metals are perfectly opaque due to their large negative permittivity. Thus, THz light incident on metallic structures is almost completely reflected. The situation is more complicated in a dielectric helical structure, where light can 1) pass directly through the structure, 2) get absorbed in the dielectric, and/or 3) scatter from the surface of the structure. Moreover, broadband excitation is ideal for probing the electromagnetic response of the helix in regimes where πD > λ, πD ~ λ, and πD < λ; in particular, the helix diameter is chosen such that over the wavelength range 0.32 < πD/λ < 4.79 and s < λ. As shown in Fig. 1, the THz pulse (red) and the femtosecond probe pulse (green) are co-focused by a 2.5 cm focal length gold-coated parabolic mirror along the axis of the helix. To preclude THz radiation leakage around the helical structure, a 300 μm diameter metallic aperture is placed directly in front of the helix entrance. The transmitted THz electric field pulse is measured via time-gated electro-optic detection in a 500 μm thick <111> ZnSe crystal. In this configuration, the gating optical femtosecond pulse is focussed through the helix structure without affecting the helix. The temporal resolution of our experimental measurements is 0.08 ps.

3. Experimental results

Optical activity in bulk media manifests as both polarization rotation (tilt in the polarization plane) and polarization circularization (transformation from linear polarization to elliptical polarization) of the scattered light. In our experiments, the scattering mechanisms responsible for optical activity of the helix are probed by measuring the temporal electric field pulse transmitted via axial propagation along helices having various numbers of turns, n. In the experiments, we measure the co-polarized, E _∥(t) (parallel to incident polarization), and cross-polarized, E _⊥(t) (perpendicular to incident polarization), components of the pulses emerging from helical structures having lengths, L, ranging from 0.50 ± 0.05 mm (n = 4) to 1.52 ± 0.05 n = 12). To explore the optical activity associated with the helix, we decompose the transmitted electric field vector into orthogonal left-handed (LH) and right-handed (RH) circularly polarized components. The LH [i.e. E _l(t)] and RH [i.e. E_r(t)] circular components are determined from the linearly polarized field components via the relations E_l(t) = 1/√2[E _∥(t)+iE _⊥(t)] and E_r(t) = 1/√2[E _∥(t)–iE _⊥(t)]. In Fig. 1, we show the time-domain left- and right- circular electric field pulses emerging from the helical structure. Optical activity manifests as a difference in the propagation time and/or the transmitted amplitude of the left- and right- circularly polarized fields. The transmitted pulse through the aperture does not show optical activity, since the left- and right- circular pulses arrive at similar times and with identical amplitudes. The circularly polarized components of the electric field pulse transmitted through the helices, however, show markedly different behaviour. For helices with n = 4 up to n = 15, there is distinctive, preferential transmission of E_r(t) relative to E_l(t). The difference in the amplitudes of the RH and LH circularly polarized pulses increases as the helix length increases, showing that this effect is cumulative over the length of the helical structure. Due to the greater transparency of the helix to RH light relative to LH light, the electric field vector, E(t) = E _∥(t)+E _⊥(t), transmitted through the helix shows significant polarization circularization, as shown in Fig. 2. In addition to the preferential attenuation of E_l(t) relative to E_r(t), the helical structure also introduces significant temporal dispersion in both the E_l(t) and E_r(t) waveforms. As n increases, the E_l(t) and E_r(t) waveform durations are progressively broadened, and the waveforms acquire more subsidiary oscillations. Such pronounced, resonating waveforms indicate a geometrically-enhanced frequency selectivity of the structure with increasing length.

 

Fig. 1. (above) A diagram of the setup used to characterize the far-infrared on-axis transmission through a sub-wavelength helix. (below) The measured right circular E_r(t) (solid line) and left circular E_l(t), (empty circles) electric field pulses through helices of various lengths, along with a reference pulse transmitted through the screening aperture. Note that E_r(t) and E_l(t) are displayed 180° out of phase for illustrative purposes and clarity.

Download Full Size | PDF

 

Fig. 2. Trajectories of the tip of the electric field vector, E(f) = E _∥(t) + E _⊥(t), for the transmission through the aperture, and helices having n = 4, 6, 8, 10, and 12.

Download Full Size | PDF

In contrast to irradiance polarization-state measurements, experimental access to the time-domain response of the helix enables direct characterization of the helix’s transient optical activity. We quantify the helix’s transient optical activity by calculating the instantaneous degree of polarization circularization, β(t) = [I_l(t) - I_r(t)] / [I_l(t) + I_r(t)], where I_r,l (t) = E^*_r,l(t)E_r,l(t) are the instantaneous intensities of the RH and LH circularly polarized components. Shown in Fig. 3(a) is β(t) for various helix lengths (n = 4, 6, 8, and 12) along with β(t) for the bare circular aperture. As expected for the aperture transmission, β(t) ≈ 0 over the duration of the transmitted pulse. There is a small offset in β(t) from 0 due to the slight ellipticity of the pulse emitted from the THz photo-conductive source. By inserting a 4-turn helical structure into the THz beam path, |β(t)| increases dramatically from ~ 0 up to 0.5 over the pulse duration, indicating that the helical conformation introduces significant polarization transformation. Interestingly, the magnitude of the degree of polarization circularization, |β(t)|, is not constant, but increases approximately linearly with time over the duration of the pulse. This suggests that manifestations of optical activity in the transient scattered field are not instantaneous, but rather build up over several electric field cycles. The degree of circularization for n = 6 shows similar behaviour as that observed for n = 4. The parameter |β(t)| linearly increases over the duration of the pulse and peaks at a value of 0.6 at 3.5 ps. As shown in Fig. 3(a), polarization circularization for the 8- and 12-turn helices reveals interesting dynamics for temporal durations exceeding t ≈ 3.5 ps. For both n = 8 and n = 12, |β(t)| increases linearly from 0 to ~ 0.8 between t = 0 and t = 3.7 ps, consistent with the observed trends for n = 4 and n = 6. However, for t > 3.7 ps, the degree of polarization circularization reaches steady state values and remains approximately constant. Based on this data, we identify τ ≈ 3.7 ps as the time required for sustained polarization circularization to develop in the transmission through the helical structure. This result shows that the transient optical activity of a helix involves two regimes; an initial regime where polarization circularization increases over several electric field cycles and a latter regime of sustained polarization circularization.

To correlate the spectral selectivity of the helical structure to the temporal evolution of polarization circularization, we determine the time-dependent spectral contents of the pulse by calculated in the time-partitioned Fourier transform of E_r(t). As shown in Fig. 3(b), the scattered pulse arriving before t is centered at approximately 1.4±0.1 THz. In contrast, the spectral components of the pulse arriving at t > τ are narrowband and centered about ν₁ = 0.85 ± 0.07 THz. To quantify the frequency dependent optically active response of the helix in these two regimes, we calculate the effective imaginary refractive index difference between the LH and RH circular pulses propagated through the helix, α(ω) = ln[E_l(ω) / E_r(ω)] / kL [23], where k = 2π/λ is the wavevector. It should be noted that the refractive index generally applies to continuous, linear, and homogeneous media. However, the optical properties of media having sub-wavelength scale heterogeneity can often be described by an effective medium that is continuous, linear, and homogeneous, but possesses an effective refractive index which accounts for the electromagnetic response of the heterogeneous structure. As shown in Fig. 3(c), α(ω) for an n = 12 helix is determined from the scattered electric field pulse arriving at 0 < t < τ (labelled “transient”) and at t > τ (labelled “steady state”). As seen in Fig. 3(c), the effective index difference calculated from the initial portion of the transmitted pulse varies from 0.03 to 0.01 over spectral range from 0.9 THz up to 1.6 THz. The magnitude of α(ω) increases with decreasing frequency due to stronger scattering of the lower frequency pulse components where πD/λ approaches unity. The effective refractive index difference calculated from the scattered field arriving at t > τ shows significantly greater optical activity. Over the spectral range of the later arriving fields, α(ω) is between the range of 0.05 to 0.07. The scattered fields arriving at times t > τ show nearly double the optical activity relative to the initial portion of the scattered waveform where polarization circularization begins to develop. Interestingly, Fig. 3(c) reveals that over a frequency range from 0.9 THz to 1.1 THz, the effective index difference of the helix abruptly changes from approximately -0.03 to approximately -0.06. This “gap” reflects different scattering mechanisms operating within the helical structure in the transient and steady state regimes that give rise to the effective electromagnetic properties of the helix.

 

Fig. 3. (a). Degree of polarization circularization for the transmitted pulse through the aperture and helices having n = 4, 6, 8, and 12. (b). depicts the normalized time-partitioned Fourier spectra of the right-circularly polarized transmission through the 12 turn helix, using a Fourier window of 3.2 ps. The experimental transmission spectra show a cut off frequency ν_c = 0.74 ± 0.05 THz. This cut off behaviour can be understood by considering the guided modes in the helix. Approximating the helix as an infinite cylindrical waveguide, the helix has a cut off frequency of ν_c = 1.841c/(πd), where d is the inner diameter of the helix [27]. Using an inner helical diameter of 235 μm, ν_c = 0.75 THz is estimated, in excellent agreement with the experimental data. (c) The frequency-dependent imaginary refractive index difference between the RH and LH fields propagated through the helix calculated over the duration 0 < t < τ (labeled “transient”) and over the duration t > τ (labeled “steady state”).

Download Full Size | PDF

4. Calculation results

We employ a 3D finite difference time-domain (3D FDTD) technique [28] to model the propagation of a linearly polarized, 1 ps far-infrared pulse through the helical structure. The Cu helix used in the calculations physically matches that employed in the experiments. For the simulations, the optical response of the Cu metal is described by the Drude model where the plasma and collision frequencies are given by ω_p = 1.9 × 10¹⁵ rad/s and ω_c = 8.3 × 10¹² rad/s, respectively, and the surrounding medium is vacuum. Figure 4 shows snapshots of the calculated vector electric field at the centre of the helix at various times ranging from t = 0 ps to 20 ps. As the electric field pulse is guided along the helical axis, it develops rotational sense evident by the cycling electric field vector and emerges with a right-handed sense of rotation matching that of the helix. To confirm that the calculations offer a true representation of the electric field and polarization dynamics, we compare the time-domain behaviour of the transmitted fields with our experiments. As depicted in Fig. 5(a), the 3D FDTD model determines that the helical structure introduces a group delay of 0.38 ± 0.05 ps/revolution, which is in good agreement with the measured value of 0.30 ± 0.05 ps/revolution. The calculated time-domain electric field pulse transmitted through an n = 12 helix is shown in Fig. 5(b), along with the experimentally measured pulse. In accord with the experiments, the calculations show that E_r(t) is preferentially transmitted through the helix relative to E_l(t). In addition, E_l(t) and E_r(t) consist of several temporal electric field oscillations. Polarization circularization is evident in the calculated time-domain evolution of the electric field vector [inset of Fig. 5(c)], which demonstrates polarization transformation of the pulse transmitted through a 12 turn helix that is strikingly similar to the experimental plot for n = 12 in Fig. 2. The model also predicts the experimentally observed spectral response of the helix associated the pulse components arriving at t > τ. The calculated peak at ν_1calc = 0.85 THz agrees well with the sharp spectral peak experimentally measured at ν₁ = 0.86 ± 0.07 THz for an n = 12 helical structure. In addition, the calculation results show cutoff behaviour for ν< 0.75 THz similar to the experiments. We chart in Fig. 5(d) the parameter β(t) over the pulse duration of the experimental and calculated pulses shown in Fig. 5(b). The calculated β(t) shows similar trends as the experiment; the magnitude of the degree of polarization circularization increases rapidly for the first ~ 3 ps of the pulse arrival. Afterward, β(t) saturates and remains approximately constant over the pulse duration. Thus, the 3D FDTD model concurrently predicts, with accuracy, the group delays, the resonance frequency, the cut off behaviour (ν_c = 0.74 ± 0.05 THz), and the transient polarization circularization of the measured pulses.

 

Fig. 4. A vector plot of the 3D FDTD-calculated electric field vector along the helical axis at times 0 ps, 4 ps, 7 ps, 14 ps, and 20 ps. The images include a cross-sectional view of the 15 turn helical structure employed in the simulation. The size parameters of the helix used in the simulations physically match those of the helix used in the experiments.

Download Full Size | PDF

To visual the dynamical fields within the helical structure, we show a snapshot of the electric field intensity within the helical structure after the passage of the initial pulse wavefront (corresponding to the regime of sustained polarization circularization) in Fig. 6. The most pronounced effect evident in Fig. 6 is the spiralling behaviour of the electric field intensity along the helical structure. Over four helical turns (n = 8 - 11), the electric field vector progressively re-orients and completes a single electric field rotation. The representative Poynting vector distribution (yellow arrows) shown at the 12^th turn reveals that the field has acquired a non-zero Poynting vector component perpendicular to the propagation axis. In contrast to purely circularly polarized light that has a Poynting vector parallel to the propagation direction, remarkably, the electromagnetic field emerging through a macroscopic helix not only carries intrinsic spin momentum (circular polarization), but also carries extrinsic orbital angular momentum evident by the gyrating Poynting vector. This trend is further illustrated in Fig. 7 depicting the Poynting vector distributions perpendicular to the propagation axis over four helical turns (n = 8 - 11). At the helix centre located at (300 μm, 300 μm), there is minimal electromagnetic energy flow within the plane perpendicular to the propagation direction. Near the inner walls of the helical structure, the Poynting vector distributions are characterized by circuitous pattern with a right-handed sense matching the handedness of the helix. The simulations suggest that electromagnetic energy does not propagate in a straight-line path directly through the helical structure. Rather, electromagnetic energy is scattered and re-directed between points in the helix.

 

Fig. 5. (a). The experimental and FDTD-calculated group delay is plotted versus n. (b). displays the experimental and calculated right-circular (blue) and left-circular (red) electric field pulses transmitted through an n = 12 helix. (c). The experimentally measured and 3D FDTD-calculated transmission power spectra for n = 12 are compared. The inset in (c) shows the calculated trajectory of the tip of the electric field vector for n = 12. (d) Calculated and experimental degree of polarization circularization for an n = 12 helix.

Download Full Size | PDF

 

Fig. 6. A plot of the electric field intensity at t = 14 ps along planes cutting through turn number 8, 9, 10, 11, and 12. The white arrows superimposed on the intensity plots indicate the orientation of the electric field vector on a plane. A representative plot of the Poynting vector at the plane cutting through turn number 12 shows the cycling behaviour of the electromagnetic energy flow in the helix.

Download Full Size | PDF

 

Fig. 7. (2.1MB) Movie of the calculated Poynting vector distributions from t = 5 ps to t = 12.5 ps within four planes perpendicular to the helical axis intersecting turns (a) 8, (b) 9, (c) 10, and (d) 11 of the helix. The helix is centred at (300 μm, 300 μm). The distributions are depicted from a viewpoint of an observer facing the wave propagation direction. The still frame shows the Poynting vector distributions at time t = 8.6 ps. [Media 1]

Download Full Size | PDF

5. Discussions and interpretation

Although the 3D FDTD calculations provide excellent agreement with experiments, it is our premise that a description of polarization rotation based on the electromagnetic energy flow in the sub-wavelength helix can give a visual picture of optical activity. We show a graphical illustration in Fig. 8(a) of the Poynting vector evolution spanning n = 1 to n = 7 captured at a time t = 10.6 ps, after the passage of the initial pulse wavefront. As seen in the Fig. 8(a), the Poynting vector streamlines, which describe the spatial orientation of the instantaneous energy flow, show that electromagnetic energy is erratically scattered near the helix entrance and most of the electromagnetic energy escapes from the helix. When viewed along the axis [Fig. 8(b)], the Poynting vectors show random scattering of the electromagnetic energy from the inner helix walls where the electromagnetic energy flow abruptly changes direction. In comparison, we plot four representative Poynting vector streamlines spanning n = 8 to n = 12 captured at a a similar time of t = 10.6 ps. The calculated Poynting vectors near the exit end of the helix show that most of the electromagnetic wave gyrates through the helical structure via successive scattering from the inner helix walls. The intricate scattering patterns in Fig. 8(c) show complex energy flow where the Poynting vector streamlines interweave along the helical axis. Physically, these streamlines represent the propagation path of the delayed, resonant pulse components giving rise to sustained polarization circularization. Interestingly, when viewed along the helical axis as shown in Fig. 8(d), the streamlines show a well-defined right-handed helical pattern corresponding to a distinctive propagation mode where a significant portion of the electromagnetic energy is confined in and propagates along the helical structure via successive scattering from the inner helical walls. The enantiometric left-handed helical pattern is not supported in the right-handed helix and cannot propagate, which is the origin of the polarization circularization arising from the helix. From the calculated Poynting vector streamlines, we construct a 3D propagation mode consisting of three points A, B, and C defining the locations where the Poynting vector associated with the mode is scattered [the blue streamline in Fig. 8(c) and changes direction [Fig. 8(e)]. A, B, and C are extracted from the Poynting vector representations and correspond to spatial locations where the Poynting vector streamlines form apexes of the helical pattern. Geometrically, A, B, and C are displaced along the helical axis by 1.33 turns and azimuthally translated relative to each other by approximately 120°, forming the triangular pattern. Given point D, which corresponds to point A displaced 4 turns away, ABCD defines the fundamental propagation mode of the helix and has an optical path length, ξ= 760 μm. Interestingly, for the criterion ξ = mλ, where m is an integer, the electromagnetic wave completes an integer number of oscillations upon propagating via ABCD = ξ To ascertain the geometrical interpretation of this eigenmode, we compare the resonance frequencies observed in the experiment (ν₁ = 0.86 ± 0.07 THz) and calculations (ν_1calc = 0.85 THz) with the criterion ν= mc/ξ. Given ξ= 760 μm and m = 2, we obtain ν = 0.8 THz, reasonably agreeing with both the experimental and 3D FDTD-calculated values. It should be noted that the m = 1 mode is not present since it is below the cutoff of the helical structure, while higher order m > 2 modes are above the bandwidth of the experimental pulse (although they can be predicted via 3D FDTD). Furthermore, the geometrical interpretation of the optically active mode in the structure is valid for only a sub-wavelength helix.

 

Fig. 8. (a). several representative Poynting vector streamlines spanning n = 1 to n = 7 captured at an arbitrary time t = 10.6 ps calculated from the 3D FDTD simulations. Nearly all of the streamlines are scattered outside the helix. The width of the streamlines is proportional to the time rate of change of the energy density. The streamlines are depicted with a cross sectional view of the helical structure employed in the simulations. (b). A head-on perspective of the same vector streamlines shown in (a). (c) Shows four representative Poynting vector streamlines spanning n = 8 to n = 12 captured at t = 10.6 ps calculated from the 3D FDTD simulations. The yellow and red Poynting vector streamlines are scattered outside the helix after the third turn, while the green and blue streamlines are confined within the helix throughout the 4 turns. (d) Shows a head-on perspective of the same vector streamlines shown in (c). (e) Based on the spatial locations where energy flow abruptly changes direction, we construct the fundamental mode consisting of four points A, B, C, and D, coinciding with the locations where the Poynting vector changes direction.

Download Full Size | PDF

6. Conclusion

We demonstrate the first experimental time-domain characterization of optical activity associated with axial propagation along a sub-wavelength chiral structure. Via time-resolved measurements of the polarization rotation of few-cycle THz pulses propagated through a helical structure, we study transient optical activity arising from the helix. The central finding of our study is that optical activity is not instantaneous; rather, circularization of the electric field polarization gradually increases over several picoseconds until reaching sustained values. Using a 3D FDTD model, we provide visualization of the internal electric field and Poynting vector dynamics that lead to the steady state polarization circularization of light exiting from the helix. In particular, we show that steady state polarization circulation is associated with the formation of helical propagation modes within the helical structure. The results not only affirm the established picture that optical activity arises from multiple scattering in the helix, but also show that this process requires several electric field cycles to fully establish.