1. Introduction

While many improvements to technologies in the terahertz (THz) frequency regime have been devised and implemented over the past few decades [1–3], further advancements are in high demand and are necessary for methods utilizing the THz range to better compete with existing technologies. For example, in many applications that utilize THz pulses, such as those involving imaging, sensing, and communication, there is a strong call for increased data acquisition rate, signal-to-noise ratio (SNR), and bandwidth [4,5].

One way to address these needs is to improve the spatial control, and in particular the steering ability, of optically-induced, short-duration, THz-bandwidth pulses, both in terms of precision and speed. Throughout this work, steering is defined as changing the direction of THz propagation within the laboratory reference frame (e.g., relative to a fixed optical excitation beam). The earliest means of steering THz pulses was to change the angle of incidence of the pump pulses onto a thin emitter by rotating the emitter [6], a method which can only steer radiation that is emitted in the backward direction at a rate which is limited to the rate of mechanical rotation of the emitter. Subsequently, another technique employed spatial modulation of the THz signals generated from a photoconductive (PC) antenna array by spatially varying the bias voltage or applying a sinusoidal variation to the impinging pump pulse [7]. Both of these techniques produced four separate beams of multi-cycle THz transients with relatively narrow bandwidths.

While these techniques may have sufficed for the earliest applications of THz technologies, the need for improved steering has led to a recent surge of new methods, such as the use of difference-frequency generation from two separate spatially chirped optical pulses that impinge upon a thin emitter at different angles of incidence [8]. This technique also produces only narrow band THz radiation, and the rate of steering is limited by the rate at which the angles of incidence can be varied. Another technique uses an optically-programmed THz spatial light modulator (SLM) [9], which can be used as a THz diffraction grating to disperse a THz pulse angularly and thereby steer narrow band THz radiation. The SLM can also be used for pixel-by-pixel imaging, although it is limited by the fact that only one pixel of the THz pulse is used for each pixel of the image, thereby greatly reducing the SNR. Similarly, an electrically programmable THz diffraction grating that can focus THz pulses has been reported [10], but, again, this device can only steer narrow band THz radiation. Other techniques have achieved steering by using a THz liquid-crystal display [11] or coherently controlling two different types of current in GaAs by manipulating the polarization of the pump pulse [12]. However, the former method can only steer THz radiation between two discrete angles, while the latter requires a tightly focused pump beam and is therefore not scalable to higher THz-pulse energy levels.

In this proof-of-concept work we present a new method to steer single-cycle THz pulses generated from thin emitters that relies on excitation by large-area, tilted ultrafast pulse-fronts and is capable of being efficient, scalable, and rapid. To the best of our knowledge, this is also the first report of the use of tilted optical pulse-fronts to generate THz radiation from a thin emitter.

2. Theory

The propagation of optically generated THz pulses in a bulk material was first experimentally explored using an ultrafast optical pulse with a beam width w that satisfies w ≪ λ_thz, where λ_thz is the central wavelength of the broad THz-pulse spectrum in the material [13]. This spot-size condition permits the use of a model that treats the focused pump pulse as a point source of THz radiation traveling at a speed, v_s = v_g, where v_g is the group velocity of the pump. Following Huygens’ Principle, the THz point source will emit outgoing spherical THz waves that propagate with phase velocity v_p and coherently form a cone of radiation [14], as depicted in Fig. 1(a). Basic trigonometry reveals that

 

Fig. 1 The generation of coherent THz radiation by (a) a small-area optical pulse traveling through a bulk, THz-emitting material and (b) a large-area optical pulse impinging upon a thin, THz-emitting material (shown in gray). The gold ellipses are cross-sectional snapshots of the propagating optical pulse, and the dark red lines and arrows define the THz wavefronts and directions of propagation, respectively. The Huygens’ spheres are removed from (b) to avoid clutter.

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Rather than a focused pump propagating inside a bulk material, one could also consider the illumination of a thin material at an arbitrary angle of incidence, θ_i, by a pump beam with a large width, satisfying w ≫ λ_thz. Additionally, one could allow the pump pulse to have its pulse-front tilted at an angle γ and to travel through the surrounding material with a group velocity v_g, as in Fig. 1(b). It is also equally valid to consider the illumination of a thick (even semi-infinite) material as long as the optical penetration depth, and therefore the THz-generation region, is thin. All media having thin THz-generation regions will henceforth be regarded as thin THz emitters. Given a THz refractive index, n, and THz-generation region thickness, t, for the thin emitter such that

In this case, instead of a traveling THz point source, the THz radiation in the thin emitter geometry is now produced from an infinite series of stationary point sources distributed throughout the generation region - each one of which begins to radiate once the optical pulse-front reaches it. However, since the pulse-front is flat, the series of phased stationary point sources is directly analogous to a single point source traveling at a speed, v_s, which is in general no longer equal to v_g. More trigonometry reveals that the speed of the analogous point source is

From Eq. (4) it is clear that a THz pulse may be steered by controlling the tilt angle of the pulse-front of the optical pump beam. This is in contrast to the THz pulses generated in bulk media using tilted pulse-fronts [15], which cannot be steered and are velocity matched for only a small range of pulse-front tilt angles. The method of controlling the pulse-front tilt is completely arbitrary and could potentially be done electronically to facilitate rapid steering. Alternatively, steering could also be achieved by manipulating either the THz phase velocity or optical group velocity of the surrounding material. Lastly, both the forward (assuming γ ≠ 0) and backward emission can also be steered by rotating the thin emitter to change the angle of incidence. While it may not provide the fastest means of steering, this last method is significantly easier to implement experimentally and was chosen for the proof-of-concept experiment described in Sec. 3.

For the special case of zero pulse-front tilt, γ = 0, and a surrounding material yielding v_p = v_g (such as dry air), Eq. (4) reduces to θ_f,b = θ_i. Thus the THz pulse travels collinear to the pump beam’s reflection and transmission directions. This result is consistent with early experiments conducted with crystalline semiconductors [16], where due to their large near-infrared absorption coefficients [17], ${\alpha }_{\text{nir}}\gg {\lambda }_{\text{thz}}^{-1}$, the semiconductors (e.g., GaAs) acted as thin emitters regardless of their actual thickness for typical near-infrared pump pulses.

For γ ≠ 0, and a surrounding material yielding v_g = v_p, there is always a range of θ_i values such that v_s < v_p. Within this range, there is no constructive interference in any direction. This is the so-called subluminal regime in which the resulting radiation, if any, is no longer of the Cherenkov type.

An important aspect of this theory is that the direction of coherent emission does not depend on the mechanism of THz emission (recognizing that source elements such as dipoles produce a radiation pattern but do not change the direction in which the combined emission of THz radiation from many point sources is temporally coherent). However, the actual amount of energy radiated in the coherent direction will depend on the mechanism of generation (e.g., χ⁽²⁾) in addition to losses from interface reflections and THz absorption in each medium. Furthermore, Eq. (4) was derived assuming the transverse profile of the pump pulse was flat rather than the commonly encountered Gaussian shape. Thus, in practice several other factors must be taken into account to provide a more quantitative description of the angular power distribution of the THz radiation.

3. Experiment

To verify that Eq. (4) accurately describes the direction of coherent THz emission from thin emitters, a pulse-front-tilted pump beam was used to illuminate a (100)-cut, 700 μm thick GaAs wafer at a variety of angles of incidence, and the resulting angular distribution of THz pulse energy was measured using a standard time-domain PC-sampling scheme. The ∼ 3 mm 1/e² diameter pump beam nominally consisted of 100 fs pulses centered at 800 nm wavelength delivered at a repetition rate of 80 MHz from a Ti:Sapphire oscillator with an average power output of ∼ 600 mW. To create the pulse-front tilt, the pump beam was directed onto a diffraction grating. The m = −1 diffraction order beam was then imaged from the grating onto the GaAs wafer using a cylindrical 4-f imaging system (Fig. 2) similar to that employed for tilted-pulse-front pump-beam experiments conducted with bulk THz emitters [18].

 

Fig. 2 Simplified schematic diagram of the experimental design used to measure the time-domain THz waveforms for a variety of incidence, emission, and pulse-front tilt angles. BS = beam splitter, FC = fiber coupler, SMF = single-mode fiber, GL = GRIN lens, R = PC-receiver, C = chopper, DL = delay line, DG = diffraction grating, L1 = 10 cm cylindrical lens, L2 = 15 cm cylindrical lens, and S = sample (GaAs wafer).

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The tilted-pulse-fronts of the pump-beam induced THz-frequency currents in the GaAs wafer that were mechanized by optical rectification as well as transient current-surges resulting from the surface depletion electric field. To detect the subsequent THz radiation, ∼ 2% of the pump beam was redirected and coupled into a single mode optical fiber whose output was focused using a 0.29 pitch GRIN lens onto a PC dipole antenna mounted on a 12 mm diameter aspherical, collimating, high-resistivity, float-zone silicon lens. The antenna and lens were further mounted inside a rotating cage assembly so that the orientation of the PC receiver could be adjusted to be selective to any linear THz polarization. The cage assembly was fixed to a 3-axis translation stage, which itself was fixed to a rotating circular breadboard such that the PC receiver faced, and could revolve about, its center. The GaAs THz-emission wafer was mounted in the center of a rotation stage whose axis was collinear with that of the rotating breadboard. Thus, a standard THz generation/detection scheme could be used, with the additional benefits of THz polarization selectivity and of the freedom to revolve the detector about the GaAs-wafer THz source while maintaining appropriate pump-probe timing (Fig. 2). For this work, the pump beam was p-polarized and the detector was oriented to be sensitive to whichever polarization resulted in the largest signal - either p or s.

To test the forward emission, a pulse-front tilt angle, γ, of nominally 50° was achieved using an 1800 line/mm grating, first order diffraction angle of 36°, and magnification in the 4-f system of M = 3/2 (see [18] for details). To test the backward emission, and to increase the diversity of pulse-front tilt angles used, a gamma of nominally 18° was prepared using the same imaging system but with a 600 line/mm grating and a diffraction angle of 4°. The GaAs wafer was rotated to change the pump beam’s angle of incidence. Due to the finite size of the GaAs wafer and the Gaussian shape of the pump pulse, the emitted THz pulses were not diffraction limited. Thus, the angles θ_f and θ_b in Eq. (4) should be interpreted as the angles of maximum radiated power. Consequently, for each angle of incidence, the detector was revolved about the GaAs wafer in 2° increments, and the time-domain waveform of the THz pulse was recorded. Due to practical considerations, the range of accessible detector angles was limited in the backward direction.

Pulse energies (uncalibrated) were calculated by integrating the square of the time-domain waveform after applying a Gaussian apodization function to remove reflection artifacts. The uncertainty in the angles of incidence, diffraction, and detection, were all approximately 0.5°. To determine the peak emission angle, the data for each pump-beam angle of incidence was fit with a Gaussian function, as shown in Figs. 3(a) and 3(c). The peak emission angles of the fits were then plotted and compared with Eq. (4), which was evaluated at the appropriate γ value with v_p = 1 and v_g = 1. Excellent agreement was obtained for both forward and backward THz emission, as observed in Figs. 3(b) and 3(d), validating the expression for the steerability of pulsed THz beams excited by ultrafast optical pulses having tilted pulse-fronts.

 

Fig. 3 Distribution of THz pulse energy versus angle of emission, (a) in the forward direction with γ = 50° and (c) in the backward direction with γ = 18°, for different optical-pulse incidence angles. The circles are calculated THz-pulse energies from measured time-domain waveforms and the lines are Gaussian fits. The peak emission angles obtained from the Gaussian fits are plotted versus optical-pulse incidence angle in (b) and (d). The lines are from the theoretical expression in Eq. (4) evaluated at the corresponding value of γ.

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4. Discussion

From Fig. 3(b) and Eq. (4) it is theoretically predicted that for γ = 50°, only θ_i > 10° should produce coherent THz radiation. However, even with this limited range of incidence angles, the THz radiation can be steered continuously from −90° to 90° relative to a line normal to the thin emitter in both the forward and backward directions. In addition, the forward propagating THz pulse can be steered from 0° to 100° relative to the pump beam’s direction of propagation. This is significantly different from the result for γ = 0, as in [16], where only the backward beam can be steered.

For both of the geometries depicted in Fig. 1, the problem of determining the direction of coherent radiation is directly analogous to that of the infinite collinear (or planar) phased dipole array. Generally speaking, the sine of the emission angle (relative to the array normal) is equal to the spatial derivative of the phase of the array divided by the spatial derivative of the phase of the radiation [19]. This leads exactly to Eq. (1). Thus, it can be interpreted that the incident angle, group velocity, and pulse-front tilt angle of the optical pulse (Fig. 1(b)) directly modify the spatial derivative of the phase of the array and subsequently the direction of coherent emission.

The connection to phased arrays suggests that control over other characteristics of the emitted THz pulse is possible. For example, in addition to steering, it should also be possible to change the focus of the radiated THz pulses in a manner similar to that used in standard ultrasonic phased array technology [20]. For a pulse-front tilt angle that is constant across the optical pulse (i.e., a linear transverse group delay profile), like that considered in this work, the emitted THz beam should be well collimated. However, if the pulse-front is curved appropriately in addition to being tilted (i.e., a nonlinear transverse group delay profile) then complete control over the focus and steering of the THz pulse should exist. Generally speaking, any features enabled by a planar phased array should also be accessible by modifying the optical transverse group delay profile in some way.

5. Conclusion

A new method of steering THz pulses radiated from thin emitters has been proposed and verified. In the proof-of-concept experiment, an optical pulse with a pulse-front tilt angle of γ = 50° was used to steer the forward propagating THz beam over an angular range of ∼ 60° by varying the angle of incidence of the pump beam, in good agreement with theory. The technique treats the thin emitter as an infinite planar phased array and allows the relative phase between the elements to be altered via the angle of incidence, group velocity, and pulse-front tilt of the optical pump beam. With appropriate fast control over the optical pulse-front tilt, this technique could be employed for the rapid steering of THz beams. Potentially this could be accomplished by electrically controlling the combination of spatial chirp and dispersion [21]. Alternatively, this technique also permits the relatively slow steering of both the forward and backward THz beams by rotation of the thin emitter to change the angle of incidence of the pump beam.

The ability to steer THz pulses rapidly could have an immediate impact on THz imaging applications that require pixel-by-pixel scans, such as time-domain, focal-plane imaging [22]. For instance, these imaging systems currently suffer from long acquisition times that are limited by the rate at which the sample can be mechanically raster scanned [23].

Furthermore, the connection between the theory presented in Sec. 2 and planar infinite phased arrays suggests that, with appropriate control over the transverse group delay profile of the optical pulse, it should be possible to exert complete control over the propagation of the THz beam (e.g., direction, focus, etc.) using concepts from ultrasonic phased array technology [20]. This could open up new possibilities in the field of THz tomography [23].

Conversely, by measuring the output THz beam direction and structure, the pulse-front tilt of an optical beam or, more generally, its transverse group delay profile, could be measured using this method. However, a more detailed theoretical approach would be needed to accomplish this task and will be the subject of future work.