1. Introduction

Intensity modulators are fundamental components in optics with broad application spaces. Resonant free-space intensity modulators are a subclass of intensity modulators, allowing modulation of the intensity of free-space beams at a single frequency. The active area, modulation frequency, insertion loss, acceptance angle, and modulation efficiency are some of the key performance metrics for resonant free-space intensity modulators. Free-space resonant intensity modulators with centimeter-square scale area, megahertz modulation frequency, high modulation efficiency, and low insertion loss are crucial components for applications including wide-field lock-in detection [1–5] and phase shift time-of-flight (ToF) imaging relying on standard image sensors [6–9]. Higher modulation frequency translates to a higher ranging accuracy, and is therefore preferred for ToF imaging applications.

Many physical mechanisms exist that allow control over the intensity of free-space optical beams. The solid-state approaches can be broadly classified into three categories: electro-optic, liquid-crystal-based, and acousto-optic. Electro-optic approaches offer high speed (exceeding gigahertz frequencies), but face challenges in modulation efficiency or limited acceptance angle due to reliance on optical resonance to boost efficiency [10–18], whereas liquid crystals are limited in their modulation frequency to the kilohertz regime [19–23]. Acousto-optics is a promising pathway, especially for resonant modulators. Mechanically resonant devices can attain quality factors significantly greater than those of radio frequency (RF) resonators at room temperature, resulting in high modulation efficiencies for acousto-optic devices.

Acousto-optic intensity modulators typically consist of a piezoelectric transducer bonded to a suitable crystal. The piezoelectric transducer is excited with an RF signal, which launches acoustic waves into this crystal and modulates the incident light on the crystal. Existing acousto-optic modulators have historically been classified into Bragg cells (some of them are referred to as acousto-optic tunable filters) [24–28] and photoelastic modulators [29,30]. Bragg cells rely on the diffraction of light (usually into an orthogonal polarization) by the acoustic wave. This is a phase-matched interaction, resulting in the modulation of light only for a narrow range of angles of incidence. Typical photoelastic modulators are resonant devices, where the input aperture determines the resonant frequency of the device. Centimeter-square scale input apertures result in fundamental resonant frequencies in the kilohertz regime, thus sub-millimeter apertures are required to reach megahertz frequencies.

We have recently demonstrated a new type of acousto-optic modulator that we refer to as a longitudinal piezoelectric resonant photoelastic modulator [31]. This class of modulators offer a simple fabrication process: coating the top and bottom surfaces of a suitable piezoelectric material with transparent surface electrodes. These modulators have allowed centimeter-square areas, low insertion loss, operation at megahertz frequencies, and significantly higher modulation efficiency compared to state of the art. These modulators are also optically broadband, similar to typical acousto-optic modulators. The spectral bandwidth of operation is only limited by the optical transparency window of the photoelastic medium and the transparent electrodes, which exceed several hundred nanometers of optical wavelength for many materials. In contrast with typical photoelastic modulators, our longitudinal resonator’s resonant frequency is determined by its thickness, which lies along the optical axis. This arrangement decouples the input aperture from the resonant frequency, enabling designs with simultaneously large input apertures and high resonant frequencies. Additionally, the proposed modulator scatters light into the same polarization as the incident light (unlike acousto-optic tunable filters that scatter light into an orthogonal polarization), allowing the intensity of light to be modulated with the addition of polarizers. The operation of our modulator is therefore similar to a resonant electro-optic modulator placed between polarizers to achieve intensity modulation, but with the advantage of the significantly higher quality factor attainable for piezoelectric resonators compared to RF resonators at room temperature. This advance has allowed watt level power consumption modulation of light at megahertz frequencies with a centimeter-square aperture.

Operating at high modulation frequencies while requiring low power to operate the modulator is critical for many applications, including wide-field lock-in detection and ToF imaging using standard image sensors. The ranging accuracy of ToF imaging relying on standard image sensors is directly proportional to the modulation frequency, and lower drive power (i.e., higher modulation efficiency) greatly expands the application space (due to power constraints for many applications). Previous demonstrations of free-space intensity modulators are unable to reach high modulation frequencies and high modulation efficiencies simultaneously, including our recent demonstration of a longitudinal piezoelectric resonant photoelastic modulator using Y-cut lithium niobate (LN) [31]. Significant improvements in device performance are required, both in modulation frequency and modulation efficiency, to enable ToF imaging relying on standard image sensors.

The key feature missing in existing free-space intensity modulators is simultaneous high modulation efficiency and high frequency of operation. In this work, we introduce Y-Z cut lithium niobate longitudinal piezoelectric resonant photoelastic modulators to address this fundamental problem. Using this modulator, we demonstrate record-high modulation efficiency of $1.5~\text {W}/\text {cm}^2$ for 100% modulation at 7 MHz for 532 nm wavelength light. This is an improvement in modulation efficiency by more than a factor of 17 compared to our previous work and other free-space intensity modulators that operate at megahertz frequencies. Using this modulator, we demonstrate 4 megapixel ToF imaging on diffuse reflectors using a standard CMOS image sensor. Our demonstration sets a record in modulation efficiency and opens a promising path forward for the design of low-power, large area resonant intensity modulators operating in the megahertz frequency regime that can find use for a plethora of applications, especially for high spatial resolution ToF imaging.

2. Modulation principle

We provide an overview of the modulation principle for the modulator in this section. The modulator consists of a lithium niobate wafer coated on top and bottom surfaces with transparent electrodes. To operate the modulator, RF signal with frequency corresponding to the fundamental resonant frequency ($f_r$) of the modulator is applied to the surface electrodes to excite the fundamental shear resonance mode in the electrode region. Light propagating through the electrode region of the modulator is polarization modulated at the applied RF frequency, which can be expressed in terms of the static phase $\phi _s$ and the dynamic phase change $\phi _D(x',y')$ (between the excited ordinary and extraordinary waves in the wafer). These phase terms are expressed in Eq. (1) and Eq. (2), where $L$ is the modulator thickness, $r$ is the radius of the active region of the modulator, $S'_{yz}(x',y',z')$ the excited strain amplitude in the rotated coordinate frame as a function of spatial location (see Fig. 1(b)), $n_o$ is the ordinary refractive index of LN, $n'_y$ the refractive index experienced by the extraordinary wave, $\lambda$ the wavelength of light in vacuum, $p'_{14}$ and $p'_{24}$ the rotated photoelastic tensor components, $(x',y',z')$ spatial coordinates in the rotated frame.

 

Fig. 1. Modulator design and electromechanical characterization. (a) A lithium niobate wafer having a diameter of 50.8 mm and a thickness of 255 $\mu$m is coated on top and bottom surfaces with transparent electrodes of diameter 12.7 mm. The top and bottom electrodes are connected to an RF power supply via an impedance matching transformer. (b) Side view of the lithium niobate wafer shown in (a). ITO is used for the transparent top and bottom electrodes, and aluminum strips on top and bottom surfaces are used to carry the RF power from the source to the ITO coated region of the wafer. The material coordinate system is shown with $(x,y,z)$, and the rotated coordinate system with primed notation ($x'$, $y'$, $z'$), where $\beta$ is the rotation angle along the yz axis. (c) The simulated distribution of the dominant strain component amplitude ($S'_{yz}$) in the primed coordinate frame when the wafer is excited at 6.926 MHz with 2Vpp applied to the surface electrodes is shown for the center of the wafer. $\beta = 5^{\circ }$ and a quality factor of $6 \times 10^3$ is used for the simulation to match experimental results. (d) The fabricated modulator mounted and wirebonded to a PCB is shown. (e) Simulated (blue) and experimental (red) of the device scattering parameter $|s_{11}|$ is shown when no impedance matching is used. The desired mode is highlighted in gray. (f) Simulated (blue) and experimental (red) of the device scattering parameter $|s_{11}|$ is shown when a transformer with 37 turns in the primary, and 18 turns in the secondary is used. The desired mode is highlighted in gray. (g) Simulated equivalent series resistance is plotted alongside model results as a function of $\beta$ with $Q = 6 \times 10^3$. (h) Model and simulation results for the required RF power to drive the modulator at 100% optical intensity modulation are plotted against the cut angle $\beta$ (where $\phi _D = 1.2$) with $Q = 6 \times 10^3$.

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Intensity modulation is achieved by placing the modulator between polarizers. The incoming unpolarized laser beam having perpendicular incidence to the wafer surface with intensity profile $I_{0}(x',y')$ is intensity modulated. The intensity of the modulated laser beam is represented as $I(x',y',t)$, where $t$ stands for time, and is expressed in Eq. (3), where $J_0$ and $J_1$ are the zeroth and first order Bessel functions of the first kind, respectively; HOH stands for the higher order harmonics.

3. Modulator design and fabrication

3.1 Design

We begin the design of the modulator by choosing the active area for the modulator (radius $r$) and the desired intensity modulation frequency ($f_r$). The two parameters that need to be determined are the modulator thickness ($L$) and the cut angle from Z to Y axis of lithium niobate, which we denote as $\beta$. The cut angle is a critical parameter that controls both the piezoelectric and photoelastic couplings. Due to large aspect ratio (area to thickness) and anisotropic acoustic propagation, a pure mode cannot be excited with circular electrodes. This makes it difficult to accurately calculate the resonant frequency with dominant $S'_{yz}$. However, we observe that the stiffness coefficient coupling to $S'_{yz}$ does not vary strongly as a function of $\beta$ (see Supplement 1 Fig. S4). Therefore, the phase velocity for $S'_{yz}$ can be approximated through $\sqrt {\frac {c_{44}^E}{\rho }}$, where $c_{44}^E$ is the stiffness coefficient, and $\rho$ the density of LN. A thickness of $L_i = \frac {\sqrt {\frac {c_{44}^E}{\rho }}}{2f_r}$ is initially chosen. Using this approximate thickness, we next determine the cut angle for the wafer.

The cut angle of Y-Z LN is critical for controlling the mechanical resistance of our resonator, which should be significantly greater than the electrode resistance in order to operate efficiently. Electrode resistance is a challenge in our device due to the use of transparent electrodes, which are essential to the propagation of light through the device but are high resistivity compared to non-transparent conductors. The mechanical resistance of our device is inversely proportional to the product of the resonant frequency squared, quality factor ($Q$), and the active area. Simultaneous high operating frequency and large $Q$ (to achieve high modulation efficiency) result in a mechanical resistance that is orders of magnitude smaller than the electrode resistance. This makes it impossible to increase the resonant frequency and the $Q$ at the same time because the electrode resistance dominates the mechanical resistance. To overcome this problem, we derive a BVD model of the piezoelectric resonator using rotated material parameters. We observe that the BVD model mechanical resistance can be effectively controlled by adjusting the cut angle $\beta$ (see Supplement 1 section 3).

The sum of the mechanical resistance ($R_m$) and the electrode resistance ($R_s)$ is approximated in Eq. (4) in terms of the modulator parameters ($R_t = R_m + R_s$), where $e'_{34}$ is the rotated piezoelectric stress constant, $c'_{44}$ the rotated stiffness coefficient, and the phase velocity for the acoustic wave is $v' = 2 L f_r$. The primed notation indicates that we are in the rotated coordinate frame (see Fig. 1). We note that the phase velocity $v'$, and therefore the resonant frequency, is relatively stable for small variations of $\beta$ around 0 degrees (Z-cut), whereas $R_t$ varies quite considerably (therefore $R_m$ varies considerably, since $R_s$ is not affected by $\beta$). Consequently, controlling $\beta$ serves as a powerful control knob over $R_m$.

For our design, we choose an active area radius of $r = 5~\text {mm}$ with a desired operation frequency of approximately 7 MHz, which is the highest frequency we can achieve for the $S'_{yz}$ mode based on wafer availability ($250~\mu \text {m}$ is the thinnest wafer we could obtain with customized cut angle). We chose $r = 5~\text {mm}$, since an important application space for these modulators is ToF imaging, with the intention of placing the modulator in front of the image sensor pixel array or the lens of a camera. Standard image sensors offering megapixel resolution have centimeter-square scale areas, necessitating similar dimensions for the active area of the optical modulator to make use of all the pixels while also having a large acceptance angle. Similarly, standard image sensors have lenses with centimeter-square scale areas, requiring similar dimensions for the active area of the modulator to have high light collection efficiency.

In addition to $R_t$, another critical metric of our modulator is the RF power required ($P_{RF}$) to operate the device at some root mean square (rms) $S'_{yz}$, expressed in Eq. (5) (see Supplement 1 section 4 for derivation). To achieve high modulation efficiency, $P_{RF}$ should be made as small as possible when operating the device at 100% optical intensity modulation (rms $S'_{yz}$ to make $\phi _D = 1.2$). Therefore, in the design of the modulator, both $R_m$ and $P_{RF}$ should be taken into consideration when choosing $\beta$.

Our initial estimate of the quality factor is $Q = 1 \times 10^3$ for this modulator (based on our previous device). To achieve $R_t \approx 60~\Omega$ (to ensure that $R_m$ > $R_s$) while also attaining high modulation efficiency (i.e., small $P_{RF}$), we choose $\beta = 5^\circ$. We want to keep our mechanical resistance larger than the electrode resistance to operate the device efficiently (i.e., have high $Q$).

Choosing the right electrode thickness is important for achieving high modulation efficiency for these modulators. Thicker electrodes have lower resistance, and therefore allow smaller loss of RF power to the electrodes in the form as heat [32,33]. However, thicker electrodes also lead to larger acoustic attenuation and therefore lower $Q$, since acoustic waves are attenuated heavily in metals and alloys.

With the chosen cut, the device is simulated using finite element modeling software (COMSOL Multiphysics) [34] with the initial thickness $L_i$. The resonant frequency ($f_{i}$) is identified corresponding to $S'_{yz}$ by inspecting the admittance as a function of frequency in simulation. The final thickness can now be determined using scale invariance as: $L = L_i\frac {f_{i}}{f_r}$ to reach the desired modulation frequency of $f_r$.

The dominant strain component in our desired resonant mode is confined to the electrode region according to COMSOL simulation, as seen in Fig. 1(c). The other strain components in the wafer are shown in Supplement 1 Fig. S1. These other strain components have a significantly smaller power than the dominant $S'_{yz}$ component. In order to simplify the working principle in the preceding design equations, we assumed that only $S'_{yz}$ is excited in the wafer.

3.2 Fabrication

We coat the top and bottom surfaces of a double-side polished lithium niobate wafer having a thickness of 255 $\mu$m and 5.08 cm diameter with 210 nm thick indium tin oxide (ITO) electrodes (to limit $R_s$ to several ohms) of radius 6.35 mm. The orientation for the lithium niobate wafer is $\beta = 5^\circ$, with rotational symbol $(YXl)$ $5^\circ$ (see Fig. 1(b)). The ITO was deposited in a load locked chamber using sputter coating. The initial sheet resistance of the ITO electrodes was $43~\Omega /\text {sq}$, and this was improved by heating at room pressure. Since the LN wafer with orientation $\beta = 5^\circ$ is highly pyroelectric (charge accumulation along the top and bottom surfaces when heated), the heating of the wafer was performed while shorting the top and bottom surfaces of the wafer with an aluminum foil. The wafer covered with the aluminum foil shorting the top and bottom surfaces was heated on a hot plate; the temperature was raised by $5^\circ \text {C}$ every two minutes starting from $20^\circ \text {C}$ until $230^\circ \text {C}$ was reached. The temperature was then brought back to $20^\circ \text {C}$ by reducing the temperature by $5^\circ \text {C}$ every two minutes until $20^\circ \text {C}$ was reached. The final sheet resistance of the ITO electrodes was $22~\Omega /\text {sq}$.

To apply RF signal to the electrodes, for both the top and bottom surfaces of the wafer, we evaporate a 1 mm wide aluminum region around the edge of the ITO circular electrodes as well as a microstrip extension with width 1 mm to the edge of the wafer for wirebonding. 150 nm thick aluminum was coated on the wafer through evaporation in a load locked chamber. The aluminum allows the wirebonding and mechanical supports to be sufficiently far away from the acoustic mode which has a negligible strain profile outside the center 6.35 mm radius of the device, and therefore allows us to limit anchor losses. Additionally, the aluminum ring makes the active aperture of the modulator approximately 5 mm, since the aluminum at the edge of the circular ITO region is opaque.

4. Modulator characterization

4.1 Electromechanical characterization

We first characterize the fabricated modulator electromechanically. To perform this characterization, we mount the double sided electrode coated wafer on a printed circuit board (PCB), and wirebond the microstrip region on the top surface to the signal line of the PCB and the microstrip region on the bottom surface to the ground line of the PCB. We measure the $s_{11}$ reflection scattering parameter with respect to 50 $\Omega$ using a vector network analyzer (VNA) with excitation power of 0 dBm and bandwidth of 50 Hz. From this measurement, the $Q$ and $L$ (by comparing with COMSOL) of the modulator is extracted. Using this $Q$ and $L$, the device is simulated again in COMSOL. The overlaid simulation and experiment $|s_{11}|$ is shown in Fig. 1(e), showing a good match. The extracted $Q$ is $6 \times 10^3$ (higher than we expected during the design phase of the modulator), with extracted $R_t = 11.5~\Omega$. To impedance match the resonator to the input source having a characteristic impedance of $50~\Omega$, we use a transformer. We connect a toroidal core transformer with 38 turns in the primary, and 17 turns on the secondary. We measure the $s_{11}$ of the modulator again using a VNA with the impedance matching transformer. Figure 1(f) shows the overlaid simulation and experiment $|s_{11}|$. The impedance matching transformer allows us to couple more than 99.9% of the RF power from the source to the modulator (including electrode resistance $R_s$). The fabricated device mounted on a PCB with the impedance matching transformer is shown in Fig. 1(d).

4.2 Optical characterization

To estimate the spatial strain profile in the wafer and to measure the modulation efficiency we perform optical measurements. To reconstruct the dominant shear strain profile in the wafer ($S'_{yz}$), and to determine the modulation efficiency (RF power required to reach a certain volume rms strain level), we measure the intensity modulation imparted on a laser beam that propagates through the electrode region of the modulator with perpendicular incidence. Our setup is similar to that used in the characterization of our previous device [31]. A laser beam of wavelength 532 nm and with output power 10 mW is intensity modulated using an acousto-optic modulator (AOM) at $f_r(t) + 4$ Hz. We take the first order diffracted beam as the modulated output. Modulation is achieved by amplitude modulation of an 80 MHz carrier RF waveform that drives the AOM (see Fig. 2(a)). We measure a depth of modulation of 48% for the intensity modulated laser beam using a fast photodetector. This intensity modulated beam is adjusted using a 1 cm diameter aperture to match the laser beam area to the active area of the LN modulator. The intensity modulated laser beam passes through the active region of the LN modulator and is captured via heterodyne detection with a standard CMOS image sensor having 4 megapixel resolution.

 

Fig. 2. Optical characterization. (a) Schematic of the characterization setup. The setup includes a laser (L) with a wavelength of 532 nm that is intensity modulated at $f_r(t) + 4~\text {Hz}$ via an AOM (by modulating the carrier frequency $f_c = 80~\text {MHz}$), aperture (A) with a diameter of 1 cm, neutral density filter (N), two polarizers (P1) and (P2) with transmission axis $ {\hat {t} = (\hat {a}'_x + \hat {a}'_y)}/\sqrt {2}$, the modulator (W), and a standard CMOS camera (C). The modulator (W) is excited with an RF source of frequency $f_r(t)$, and the laser beam passes through the center of the wafer that is coated with ITO. The camera detects the intensity modulated laser beam. (b) Equipment used for tracking the resonant frequency of the modulator is shown. The setup includes a signal generator driving the AOM using its first channel with center frequency $f_c$ amplitude modulated at $f_r(t) + 4~\text {Hz}$, and driving the LN modulator via a directional coupler using its second channel sending RF power at frequency $f_r(t)$ to the modulator. The oscilloscope detects both the input RF power to the modulator and reflected RF power from the modulator via a directional coupler. The oscilloscope and signal generator are interfaced to a computer running a MATLAB script that adjusts $f_r(t)$ with a negative feedback loop based on the phase difference between reflected and input waveforms. (c) Time-averaged intensity profile of the laser beam detected by the camera per pixel is shown when 160 mW of RF power at $f_r(t)$ is applied to the modulator and the second polarizer P2 is removed. (d) Time-averaged intensity profile of the laser beam detected by the camera per pixel is shown when 160 mW of RF power at $f_r(t)$ is applied to the modulator and the second polarizer P2 is present. (e) The peak-to-peak variation at 4 Hz of the laser beam is shown per pixel when 160 mW of RF power at $f_r(t)$ is applied to the modulator and the second polarizer P2 is present. (f) The phase of intensity modulation at 4 Hz of the laser beam is shown per pixel when 160 mW of RF power at $f_r(t)$ is applied to the modulator and the second polarizer P2 is present. (g) Estimated strain amplitude $S'_{yz}(x',y') = \frac {\int _{0}^L S'_{yz}(x',y',z')dz'}{L}$ per pixel is shown when 160 mW of RF power at $f_r(t)$ is applied to the modulator and the second polarizer P2 is present. (h) Estimated (Experiment) and theoretical (Theory) root mean square $S'_{yz}$ averaged over the modulator region (diameter of 1 cm) is shown for varying levels of RF excitation power.

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Essentially, the spatial $S'_{yz}$ strain profile in the wafer is mapped to the spatial intensity profile of the optical beam at the output of polarizer P2, as shown in Eq. (3). The use of a standard CMOS image sensor allows megapixel spatial resolution for reconstruction, but the sensor has an insufficient frame rate to capture $f_r$ directly. This is the reason behind using heterodyne detection; $S'_{yz}$ per pixel is extracted from the 4 Hz beat tone. The optical characterization setup is shown in Fig. 2(a).

As an addition to our previous optical characterization setup, we also include a resonant frequency tracking setup to excite the modulator at its time-varying resonant frequency. Driving the device while being impedance matched at sufficiently high RF power levels leads to heating of the wafer, causing the resonant frequency to red shift. For sufficiently high $Q$, this can cause the modulation efficiency to drop significantly. To overcome this issue, we use a resonant frequency tracking setup, where the drive frequency for the modulator $f_r(t)$ and the modulation frequency of the AOM $f_r(t) + 4~\text {Hz}$ are adjusted simultaneously as a function of time $t$ to keep a constant beat tone at 4 Hz. We observe settling to some sort of steady-state resonant frequency after tens of seconds (thermal equilibrium), where the variation in $f_r(t)$ with respect to its mean is less than 1 in 10,000 (see Supplement 1 section 5 for more details). We perform characterization only after achieving this approximate steady-state in the resonant frequency.

We perform optical measurements at four different RF power levels, and extract the pixel-wise as well as rms $S'_{yz}$ for each measurement. For each measurement, 320 frames are captured with a frame rate of 32 Hz, the exposure time for each frame is 280 $\mu s$, with 12 bit precision for the pixels. The volume rms $S'_{yz}$ is calculated by using the capture without and with a polarizer placed after the modulator (P2 in Fig. 2(a)). The peak-to-peak variation in the beat tone at 4 Hz when P2 is present (computed with a fast Fourier transform), and the average intensity value with and without P2 are used per pixel to calculate $S'_{yz}$. For the capture where P2 is present, the intensity as a function of time for each pixel is as shown in Eq. (3). For the capture without P2 present, the intensity captured by each pixel is $I_{0}(x',y')/2$. In calculating $S'_{yz}$ we use Eq. (2) and Eq. (3), and make the following assumptions: for small $\phi _D(x',y')$, $J_0(\phi _D(x',y')) \approx 1$ and $J_1(\phi _D(x',y')) \approx \frac {\phi _D(x',y')}{2}$. We also use the fact that the depth of modulation for the laser beam is 48% before being incident on the LN modulator. Volume rms $S'_{yz}$ is computed using these pixel-wise values.

For 160 mW of RF excitation at resonance for a 1 cm diameter region, the rms $S'_{yz}$ is $1.1 \times 10^{-4}$. Using Eq. (2), the required rms $S'_{yz}$ to make rms $\phi _D = 1.2$ is approximately $3.0 \times 10^{-4}$. The required RF power to reach 100% intensity modulation over a $1~\text {cm}^2$ area is calculated as $0.16~\text {W} \times \Big (\frac {3.0 \times 10^{-4}}{1.1 \times 10^{-4}}\Big )^2 \times \frac {1~\text {cm}^2}{\pi \times (0.5~\text {cm})^2} \approx 1.5~\text {W}$. We use rms $S'_{yz}$ to report the modulation efficiency, similar to how voltage rms (spatial) amplitude is used in reference to electrical power. Compared to our previous demonstration relying on Y-cut LN with a resonant frequency of 3.7 MHz, the modulation efficiency has improved by a factor of $\big (7.4~\text {W}\text { cm}^{-2} / 1.5~\text {W}\text { cm}^{-2}\big ) \times \big (7~\text {MHz}/3.7~\text {MHz}\big )^2 = 17.7$. $P_{RF}$ is proportional to $f_r^2$, and this can be seen by inspecting Eq. (5). To achieve 100% intensity modulation, the product of $L$ and rms $S'_{yz}$ should be a constant. This can be expressed as: $\frac {v'}{2f_r} \times \sqrt {\frac {\int _V S'^2_{yz}(x',y',z')dV}{\pi r^2 L}} = \text {constant}$, which implies that $\sqrt {\frac {\int _V S'^2_{yz}(x',y',z')dV}{\pi r^2 L}}$ is proportional to $f_r$. Therefore, $\int _V S'^2_{yz}(x',y',z')dV$ is proportional to $f_r$, and thus $P_{RF}$ is proportional to $f_r^2$ (assuming 100% intensity modulation). We estimate the rms $S'_{yz}$ in the electrode region as a function of input RF power using Eq. (5). The analytic and experimentally extracted rms $S'_{yz}$ are plotted in Fig. 2(h) as Theory and Experiment, respectively. We also measure the insertion loss for the modulator at 532 nm wavelength by measuring the attenuation due to having the modulator in the path of the laser beam. Using a fast photodetector, we measure the optical insertion loss to be 1.1 dB. A comparison table is provided in Table 1, comparing modulator performances in terms of insertion loss and modulation efficiency.

Table 1. Comparison table. Modulator performances are compared in terms of intensity modulation at 7 MHz. The two performance metrics used are the optical insertion loss and the RF power required to achieve 100% intensity modulation of the laser beam. To show modulation efficiency, the RF power required to achieve 100% intensity modulation is calculated by extrapolating the values reported in the references.

View Table

5. Time-of-flight demonstration

We demonstrate one application of our LN modulator: phase-shift based ToF imaging [37,38]. In this imaging modality, intensity modulated light is used to illuminate targets. The phase of the intensity modulated light is shifted due to the propagation of the light from the source, to the target, and back to the receiver (after reflecting from the target). The phase shift of the intensity modulation is related to the distance $d$ of a target in the scene through Eq. (6), where $\Phi$ is the phase shift, and $c$ the speed of light in air. We use our LN modulator to down-convert the megahertz level intensity modulation $f_r$ into a hertz level beat tone that can be detected with a standard image sensor.

The modulator is excited with 160 mW of RF power with frequency $f_r(t)$ via the resonant frequency tracking setup, and a 520 nm wavelength laser diode with output power 80 mW is intensity modulated at $f_r(t) + 4$ Hz by modulating the injection current to the laser diode. We measure a depth of modulation of 72% for the intensity modulated laser beam using a fast photodetector. The divergence of the laser beam is shaped by passing through a lens, and this beam is used to illuminate targets placed 1-2 m away from the laser. A standard CMOS camera offering 4 megapixel resolution is placed next to the laser. A camera lens with diameter 4.5 cm is attached to the camera (to resolve the targets). A polarizer - modulator - polarizer optical chain is placed in front of the camera. A pupil is placed farther from the camera, between polarizer P2 and modulator W as illustrated in Fig. 3(a) so that the reflected laser light from the targets only passes through the active region of the modulator.

 

Fig. 3. Time-of-flight imaging demonstration. (a) Schematic of the time-of-flight imaging setup. The setup includes a laser (L) with a wavelength of 520 nm that is intensity modulated at $f_r(t) + 4~\text {Hz}$, aperture (A) with a diameter of 1 cm, two polarizers (P1) and (P2) with transmission axis ${\hat {t} = (\hat {a}'_x + \hat {a}'_y)}/\sqrt {2}$, the modulator (W), camera lens (CL), and a standard CMOS camera (C). The modulator (W) is excited with an RF power of 160 mW at frequency $f_r(t)$, and the laser beam is directed to targets placed away from the laser and the camera. A beam shaping lens (F) is placed in front of the laser to adjust the illumination area of the laser. (b) The targets illuminated with the laser are shown. (c) Time-averaged intensity detected by the camera per pixel is shown. (d) Reconstructed depth map per pixel by the camera. Depth reconstruction is performed by converting the phase of the beat tone at 4 Hz to distance using Eq. (6). Pixels that receive very few photons are displayed in black. (e) Dimensions of the targets used for the imaging experiment. (f) Distance distribution of the pixels corresponding to the different targets are shown. Approximately 20,000 pixels are used for each target to construct the histogram. The mean distance estimated by averaging the pixels corresponding to the targets are 1.03 m, 1.28 m, 1.56 m, and 1.84 m, respectively. The standard deviation of distance per pixel for the targets are 29.7 cm, 27.4 cm, 27.0 cm, and 28.8 cm, respectively.

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We use white wooden targets that reflect diffusely to perform a ToF imaging experiment, and place them more than a meter away from the laser and the camera (as shown in Fig. 3). Black cardboard is placed underneath and behind the targets to limit multi-path interference (multiple light paths incident on the targets by following different paths) and to remove background light, respectively.

For the ToF measurement, 960 frames are captured with a frame rate of 16 Hz, the exposure time for each frame is 62 ms, camera gain of 10 dB is used, with 12 bit precision for the pixels. Figure 3(c) shows the brightness of the targets detected by the camera per pixel, and Fig. 3(d) shows the reconstructed color coded depth per pixel (by mapping the phase at the beat tone of 4 Hz to distance using Eq. (6) after performing a fast Fourier transform across time for each pixel). Figure 3(f) shows the depth estimate distribution for the different targets. The ranging accuracy per pixel is approximately 27 cm based on the standard deviation of distance per pixel for the same target. The ranging accuracy is limited by the illumination power, noise of the laser, RF power driving the device, the aperture of the modulator, integration time, and the resonant frequency.

We have demonstrated ToF capability for our system on diffusely reflecting targets. Moving to higher modulation frequencies (by using thinner wafers) and improving the $Q$ through carefully choosing $\beta$ and the electrode thickness would improve the ToF performance.

6. Conclusion

In summary, we demonstrated Y-Z cut lithium niobate longitudinal piezoelectric resonant photoelastic modulators. These modulators consist of a simple design and allow intensity modulation of free-space beams at megahertz frequencies with record-high efficiency. The modulator described in this work allows simultaneous high modulation frequency and efficiency, and can find use in applications requiring free-space beams to be intensity modulated with low-power at megahertz frequencies. As a potential use case, we have demonstrated 4 megapixel ToF imaging relying on a standard image sensor. This work opens up the path for reaching higher frequencies (20 MHz and beyond by using thinner wafers) and with even higher modulation efficiencies. This advance would enable low-power, high performance, and high spatial resolution (exceeding hundred megapixels) ToF imaging through integrating this modulator with a standard CMOS image sensor.