1. Introduction

Far infrared (FIR) and terahertz (THz) generation is necessary in many different applications, ranging from sensing [1–3] to communication [4,5] to spectroscopy [6]. Furthermore, control over emission angle is desirable for free-space communication and faster scanning [7,8]. While many applications can be performed with pulses, narrowband continuous wave (CW) light tends to require simpler equipment [9] and can be more easily used for real-time imaging at room temperature [10]. However, efficiently producing tunable, narrowband CW light in the FIR/THz regimes remains difficult due to high material absorption and rapidly fluctuating dispersion.

Difference frequency generation (DFG) uses the second-order optical nonlinearity to mix two optical modes that are close in frequency and produce light at the difference frequency, which can be much longer in wavelength. While DFG can produce a wide range of wavelengths, phonon resonances in nonlinear materials introduce high absorption at long wavelengths, often limiting the DFG efficiency due to short propagation lengths. Various methods to mitigate this effect have been proposed and can be separated into two categories: collinear and noncollinear.

In the first category, collinear DFG allows the FIR/THz DFG signal to travel with both input signals along the nonlinear waveguide. The DFG field increases linearly with propagation length under perfect phase matching, resulting in quadratic growth in output DFG power. However, device length is limited in practice by material absorption since DFG light experiences exponential loss as it interacts with the nonlinear material. To reduce loss while still allowing quadratic power scaling, an adjacent waveguide with lower absorption can be used to guide most of the DFG light [11]. While this scheme is advantageous for wavelengths with reasonable loss in the nonlinear material, it becomes prohibitively inefficient with higher losses.

In comparison, noncollinear DFG schemes encourage surface emission of FIR/THz light to reduce interaction with lossy materials and thus prevent absorption. One of the most common noncollinear techniques is Cherenkov phase matching, which depends on material refractive index. The Cherenkov condition can be summarized by $\phi _C = \arccos \left (\frac {n_{\rm opt}}{n_{\rm DFG}}\right )$, where $\phi _C$ is the emission angle of the DFG light from the sample, $n_{\rm opt}$ is the refractive index at optical pump wavelengths, and $n_{\rm DFG}$ is the refractive index at the longer DFG wavelength [12]. While this method works effectively in the low THz regime ($<8$ THz) in lithium niobate (LN) due to minimal dispersion, it becomes more difficult in regions where the refractive index is highly dispersive. Furthermore, emission angle $\phi _C$ is set by material dispersion and cannot be continuously tuned for steering applications.

Another method of surface emission interfaces metal antennas with integrated LN waveguides in order to couple out THz light generated from optical rectification of pulsed light [13]. While antenna design parameters provide significant control over the emitted THz waveform, efficiency is limited since DFG occurs over a short distance, around 200 $\mu$m at most per antenna. Though this approach works for generating pulses, the short length scales would severely limit efficiency for continuous wave FIR/THz light.

An alternate approach uses periodic poling to phase match DFG light to radiation modes. Instead of traveling along the waveguide as a guided mode, the DFG light immediately radiates out normal to the waveguide. Analytical derivations of this behavior [14], along with demonstrations of its performance using periodically poled lithium niobate (PPLN) [15,16], have been implemented. However, these approaches were limited by low-confinement, bulk PPLN waveguides [16,17]. Demonstrations were also focused on efficiency for normal emission and did not explore the potential for beam steering.

In recent years, thin film lithium niobate (TFLN) platforms have become commercially available, resulting in more efficient frequency conversion due to higher confinement in integrated waveguides [18,19]. Here, we focus on the potential for using periodically poled TFLN waveguides as opposed to bulk or diffused waveguides to enhance efficiency and compare performance with more recent developments in long wavelength generation via integrated photonic devices. To our knowledge, integrated TFLN waveguides have not been explored for CW surface emission via periodic poling. In this work, we demonstrate a design for a surface-emitting, periodically poled TFLN waveguide that emits tunable, narrowband FIR CW light at room temperature. We also investigate the beam steering capability of an array of TFLN waveguides, through wavelength and relative phase tuning.

2. Analytical model

In a waveguided DFG process, two overlapping optical modes interact in a nonlinear waveguide, generating a nonlinear polarization that acts as a source for the desired DFG mode. The frequencies of the optical modes, $f_1$ and $f_2$, are chosen such that their difference is the desired output frequency, $f_3 = f_1-f_2$. For x-cut LN waveguides with TE-like optical modes, the electric field can be written as $E_i = A_i e^{i(k_iy-\omega _i t)} \hat {z}$, where $A_i$ is the field amplitude, $k_i = 2 \pi n_{i} / \lambda _i$ is momentum dependent on effective mode index $n_{i}$ at wavelength $\lambda _i$, and $\omega _i = 2\pi f_i$ is angular frequency. The fields travel along the crystal’s $\hat {y}$ axis and are directed along its $\hat {z}$ axis. As a result, only the $\chi ^{(2)}_{zzz} = 2d_{33}$ term contributes to the nonlinear polarization. Nonlinear polarization can thus be written as:

The factor of 2 in Eq. (1) accounts for both permutations of $E_1$ and $E_2$. $P_{\rm NL}$ acts as a source for $E_3$ via the wave equation, resulting in the vector differential equation:

For long waveguides, output power begins to oscillate rather than growing due to phase mismatch, $k_1 - k_2 - k_3 \neq 0$. Quasi-phase matching through ferroelectric periodic poling is often used to counteract this effect. By periodically flipping the material domain and thus the sign of the nonlinear coefficient, phase mismatch can be mitigated, and the output power will continue to grow with longer propagation length. This periodic flip can be described by a Fourier series [20], replacing the constant $d_{33}$ by:

Depending on the choice of poling period $a$, the DFG process can be phase matched such that the DFG light travels as a collinear guided mode or emits via a noncollinear radiated mode. In the collinear case, all $k_n$ are directed along the waveguide ($\hat {y}$), and $a$ is chosen such that $k_1 - k_2 - k_3 - \frac {2\pi }{a} = 0$ (Fig. 1(a)). The resulting nonlinear polarization $P_{NL}$ gives rise to a guided wave at $f_3$ (Fig. 1(b)). The power efficiency $\Gamma$ scales quadratically with length but is also exponentially affected by loss (Fig. 1(c)).

 

Fig. 1. DFG with periodic poling for collinear (a-c) and noncollinear (d-f) approaches. a) In collinear phase matching, poling period $a$ is set so the generated DFG momentum $k_3$ copropagates with the pumps ($k_1$ and $k_2$). b) As a result, the nonlinear polarization $P_{\rm NL}$ varies slowly in space, exciting a guided mode. d) In noncollinear phase matching, $k_3$ is perpendicular to the pump light. e) $P_{\rm NL}$ varies rapidly in space, exciting a surface-emitting free-space mode. Accordingly, generation efficiencies $\Gamma$ of the two schemes exhibit very different length dependence. The collinear scheme is more efficient when loss $\alpha$ is small (c), while the noncollinear scheme is more advantageous at high loss (f).

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In comparison, our surface-emitting scheme selects the poling period to ensure DFG light immediately leaves the waveguide. In this case, $k_{3,y}$ must vanish such that $k_3$ is directed perpendicular to the waveguide. Thus, we choose $a$ such that $k_1 - k_2 - \frac {2\pi }{a} = 0$ (Fig. 1(d)). In doing so, the phase matched process only occurs for modes in which $k_{3,y} = 0$. Since $|k_3|$ is constrained by the DFG wavelength and material index to be nonzero, and the electric field $E_3$ is polarized in-plane ($\hat {z}$), the DFG light must radiate normal to the surface of the waveguide (Fig. 1(e)). Power efficiency scales linearly with length but does not experience exponential loss with longer propagation lengths since the DFG field quickly radiates away from the lossy material (Fig. 1(f)).

Note that this method of generating surface-emitting DFG is due purely to alternating the direction of the nonlinear coefficient; the linear permittivity is unaffected by periodic poling. As a result, the bandwidth of the radiated DFG is not limited by resonances, unlike typical grating structures that couple guided light to radiation modes by spatially modulating refractive index. If a particular application requires higher efficiency, introducing resonant structures like ring resonators or gratings might improve DFG efficiency at the desired wavelength. However, careful design would be required to ensure constructive interference in the radiated DFG light, and bandwidth would be reduced due to the narrow linewidth of the resonant structure. Regardless, using periodic poling to produce surface-emitting DFG does not require resonant structures to operate, enabling larger bandwidths at reasonable efficiencies.

Different assumptions are made in solving the wave equation (Eq. (3)) depending on whether the output is a guided mode or radiated light. In the collinear case, guided modes can be treated like plane waves with an additional term that accounts for effective spot area size where all three modes overlap in the nonlinear material [11]. In the noncollinear case, the output waves radiate in a cylindrical pattern out from the waveguide, so plane wave assumptions break down. The analytical model for noncollinear phase matching in a bulk waveguide has been derived using vector Green functions [14], but our integrated waveguide model includes a few distinctions.

The first is the inclusion of mode overlap. In a bulk waveguide, the difference between optical mode sizes is negligible. However, modes in integrated TFLN waveguides are more sensitive to changes in wavelength. Furthermore, while most optical energy is concentrated in the nonlinear waveguide, some portion extends to the cladding and doesn’t contribute to DFG; this is especially true for thin, narrow waveguides. When defining overlap for noncollinear DFG, we only account for overlap between the two optical modes in the nonlinear waveguide as the DFG field immediately radiates out of the waveguide. Overlap $\eta$ can thus be written as [17]:

The second difference in our model is simulated collection efficiency. In comparing our simulation values with the expected efficiency from analytical models, simulated efficiency is affected by the definition of a measurement flux plane (Fig. 2(b)), through which we measure power outflow. In this way, we include the realistic effects of limited numerical aperture in collection optics, Fresnel coefficients that depend on dispersion, and losses towards the substrate. Note that this factor also changes with waveguide parameters since the energy distribution in the solid angle of the radiated light depends on waveguide dimensions [14].

 

Fig. 2. a) Diagram of a TFLN device that emits $f_3 = f_1 - f_2$ via DFG; black arrows indicate the direction of the $d_{33}$ component after poling with period $a$. b) Output DFG light radiates as a cylindrical wave from the waveguide; the top flux plane is used to define collected output power. TE00 mode profiles for c) $\lambda _1$ = 1.4 $\mu$m and d) $\lambda _2$ = 1.63 $\mu$m. e) Waveguide dispersion indicates the TE00 mode is guided at both input frequencies.

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With these changes, we compare our simulations to the analytical model:

Besides length scaling, noncollinear efficiency differs from collinear efficiency in a few other critical ways. Noncollinear efficiency scales cubically with frequency, as opposed to quadratically in collinear efficiency. As a result, noncollinear efficiency tends to improve at higher frequencies, and operation bandwidth is asymmetric. Furthermore, in the collinear method, effective spot area is important since higher intensity results in higher conversion efficiency [11]. In comparison, noncollinear efficiency only depends on normalized optical mode overlap; decreasing mode area does not necessarily improve efficiency.

Using integrated TFLN waveguides for noncollinear DFG still has advantages. Tightly confined optical modes ensure most electric field energy is present in the nonlinear material for DFG. Thinner devices also reduce absorption, as DFG light travels through the nonlinear material for a shorter length before radiating. Aside from potential improvements in efficiency, device size is also significantly reduced since TFLN waveguides exhibit better mode confinement.

3. Simulation parameters

Our structure (Fig. 2(a)) is comprised of a shallow-etched TFLN waveguide on top of SiO$_2$ substrate; the top cladding is air. To demonstrate the capability of this technique, we simulated emission at $\lambda _3$ = 10 $\mu$m ($f_3$ = 30 THz), allowing us to fully capture collection efficiency through 3D simulations with reasonable simulation size and mesh. However, 2D simulations demonstrate that this technique works with even longer wavelengths into the THz regime. For waveguide parameters, we used total LN thickness $t$ = 800 nm, etch depth $h$ = 320 nm, and top width $w$ = 2 $\mu$m with a sidewall angle of 60$^{\circ }$. LN waveguides with these dimensions have already been fabricated and poled, proving their experimental feasibility [21]. Poling period $a$ is set such that the DFG light emits perpendicular to the top surface unless otherwise stated.

In this structure, input wavelengths $\lambda _1$ = 1.4 $\mu$m ($f_1$ = 214 THz) and $\lambda _2$ = 1.63 $\mu$m ($f_2$ = 184 THz) have fundamental TE00 modes with effective refractive indices of $n_1 = 2.0229$ and $n_2 = 1.9843$ respectively (Fig. 2(c) and 2d). Using crystal coordinates for x-cut LN, TE modes exhibit an electric field pointing in the $\hat {z}$ direction, while the light propagates along the waveguide in the $\hat {y}$ direction. Waveguide dispersion shows that both of these TE00 modes are guided as they lie below the light line (Fig. 2(e)).

Material dispersion varies widely across pump and DFG wavelengths, requiring multiple models. We used the Sellmeier equation for LN at telecom wavelengths to model dispersion for the input light [22]; loss at these wavelengths is small [23]. For LN in the FIR regime, we utilized a double Lorentz model and measured data to model birefringence and loss [24,25]. For SiO$_2$, we used data in the frequency range 0.6 - 295 THz to model dispersion across the full spectrum [26]. For air, we set real refractive index $n$ = 1 and imaginary index $\kappa$ = 0 for all wavelengths.

We used 30 pm/V as the $d_{33}$ value for our DFG simulations. While there is evidence that $d_{33}$ can be much larger around 181-195 pm/V in the THz regime [11,27], absolute $\chi ^{(2)}$ measurements involving frequency mixing between vastly different wavelengths are limited. Furthermore, the $\chi ^{(2)}$ values are expected to dramatically fluctuate in the the Reststrahlen regime due to strong dispersion. For LN in the region between 1-100 THz, $\chi ^{(2)}_{zzz}$ is expected to vary between values as high as 2000 pm/V to values as low as 2 pm/V [24]. However, this dramatic fluctuation has not been rigorously verified experimentally. As such, we set $d_{33}$ to be 30 pm/V as a characteristic value. Note that any changes in this value due to dispersion will cause efficiency to scale proportionally to $d_{33}^2$. In general, additional experimental data on $\chi ^{(2)}$ values in the THz regime would be helpful to confirm nonlinear strength at particular wavelengths.

We used COMSOL Multiphysics software to simulate the surface-emitting DFG process in periodically poled TFLN waveguides. Input modes were simulated and injected into the waveguide. These optical fields were used to define the nonlinear polarization, which acted as the source for the DFG light at frequency $f_3$. We then simulated the resulting DFG radiation as a cylindrical wave originating from the waveguide (Fig. 2(b)). To determine expected efficiency, we evaluated the surface integral for power emitted through the flux plane at the top surface. The collection efficiency factor therefore includes losses due to emission towards the substrate and sides, which better reflects the expected experimental efficiency.

4. Results

In this scheme, efficiency scales linearly with power (Fig. 3). If the structure is too short, around $L < 10a$, the efficiency is lower than expected due to irregular power distribution at the ends of the waveguide. However, as length is increased and these edge effects diminish, efficiency trends towards linear scaling with length as expected by the analytical model. Reasonable waveguide lengths are much greater than this transition point, so experimental devices will exhibit a linear increase in efficiency with length. We confirmed that output power continues to linearly scale with device length up to centimeter scales via 2D simulations. By scaling the efficiency from our 3D simulations, we expect a power efficiency of 9.16 $\times$ 10$^{-6}$ W$^{-1}$ for a 1 cm device at 30 THz.

 

Fig. 3. Power efficiency $\Gamma$ scales linearly with length $L$. Deviation from linear scaling at shorter lengths is due to edge effects.

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When poling period is adjusted, radiated output light emits at an angle that deviates from fully perpendicular (Fig. 4). This emission angle can be controlled by choosing $a$ such that $k_1 - k_2 - \frac {2\pi }{a} = k_{3,y} = k_3 \sin \theta _{LN}$, where $\theta _{LN}$ is the emission angle in LN. The emission angle in air, $\theta _z$, is related by refraction and can be calculated as $n_{air} \sin \theta _z = n_3 \sin \theta _{LN}$. $\theta _z$ can be positive or negative depending on emission direction but cannot exceed 90$^{\circ }$ in either direction, limiting the range for poling periods for a particular wavelength. For our device, poling period must be within the range 3.1 $\mu$m < $a$ < 7.9 $\mu$m to allow emission at 30 THz.

 

Fig. 4. DFG emission of $f_3$ = 30 THz with varying poling period. a) Efficiency $\Gamma$ v. poling period $a$; $\Gamma$ is normalized by maximum efficiency at $a$ = 4.4 $\mu$m (red circle). b) Emission angle in air $\theta _z$ v. poling period $a$. At $a$ = 4.4 $\mu$m (red circle), $\theta _z$ = 0$^{\circ }$. c) The DFG field $E_{3,z}$ emits at different $\theta _z$ for different poling periods. Emission towards the substrate dies out quickly due to higher loss in SiO$_2$ at 30 THz.

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Efficiency is highest for normal emission ($\theta _z$ = 0$^{\circ }$, $a$ = 4.4 $\mu$m) since our efficiency is defined by evaluating output power through the top surface. Normal emission also reduces loss due to Fresnel reflection coefficients. A steeper angle can result in internal reflection, reducing outgoing light. Fresnel coefficients change according to material dispersion, so critical angles for total internal reflection will depend on the desired central frequency and tuning range.

To determine bandwidth for a single device, we fixed the poling period to be 4.4 $\mu$m and swept output frequency (Fig. 5). Due to phase matching, emission angle $\theta _z$ changes as frequency is swept, but we maintained the same flux plane to determine efficiency of radiation normal to the waveguide. Material dispersion of both LN and SiO$_2$ was included, resulting in a large peak around 35 THz. At this frequency, phonon resonances in SiO$_2$ affect its refractive index such that imaginary index is larger than real index. The substrate therefore acts like a metal, eliminating substrate loss and greatly increasing efficiency. In general, higher frequencies with similar loss tend to be more efficient due to the cubic dependence of efficiency on output frequency $f_3 = c/\lambda _3$.

 

Fig. 5. Efficiency $\Gamma$ v. output frequency $f_3$ for a waveguide with $a$ = 4.4 $\mu$m. $\Gamma$ is normalized to efficiency at $f_3$ = 30 THz (red circle). The large peak around 35 THz is due to a phonon resonance in the SiO$_2$ substrate, which effectively eliminates substrate loss.

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The bandwidth around 30 THz, here defined as the full width at half of the efficiency at 30 THz, ranges from about 25 THz to 50 THz (Fig. 5). While asymmetrical, the overall bandwidth spans 25 THz, allowing wide tuning of output wavelengths for different applications. In this range, $\theta _z$ shifts from -27$^{\circ }$ at 25 THz to 64$^{\circ }$ at 50 THz. While this angle variation may limit effective bandwidth for some applications, the measured output power was still defined using the flux plane normal to the waveguide. As such, bandwidth could be larger for applications that are less sensitive to emission angle.

5. Comparison to other sources

In collinear DFG, output power scales quadratically with propagation length but is exponentially affected by absorption. Though effective loss can be reduced by guiding the DFG light in a lower loss material [11], loss still exponentially reduces output power. After a certain length, exponential absorption overpowers the quadratic growth with length, and efficiency decreases with longer devices (Fig. 6); this is especially problematic at frequencies above 5 THz where imaginary refractive index $\kappa$ spikes due to phonon resonances in LN. When $\kappa$ is large, the length at which maximum efficiency occurs is too short to efficiently generate FIR/THz light.

 

Fig. 6. Efficiency $\Gamma$ v. propagation length $L$ in collinear (solid line) and noncollinear (dashed line) schemes for $f_3$ = 30 THz, $d_{33}$ = 30 pm/V, and different losses set by imaginary refractive index $\kappa$. In practice, $\kappa$ changes due to dispersion, but here we keep $f_3$ = 30 THz and artificially scale $\kappa$ to better illustrate the effect of loss on efficiency. In the collinear case, oscillations around 5 mm for $\kappa = 0$ are caused by phase mismatch, which affects efficiency through a sinc squared term. For $\kappa \ge 0.01$, absorption begins to dominate, washing out effects of phase mismatch.

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In comparison, noncollinear efficiency continues to scale linearly with length since DFG light is immediately emitted, experiencing a constant level of absorption regardless of total device length. At low losses ($\kappa < 0.1$), noncollinear efficiency shows minimal change. Higher loss reduces efficiency by a constant value, but increasing propagation length only improves efficiency. As a result, the noncollinear scheme becomes more efficient than the collinear approach for long enough devices; this crossover length depends on the material absorption at the desired wavelength. At 30 THz with $\kappa = 0.0431$ for LN [24], the noncollinear case is more efficient after propagation lengths above 1 mm (Fig. 6), which is reasonable for integrated optics. Noncollinear DFG may be even more beneficial for generating other frequencies that are close to phonon resonances, and thus experience higher loss. In LN, $\kappa$ is above 1 for for most frequencies between 3 and 25 THz and can reach values as high as 10.1 [25]. For $\kappa > 1$, a noncollinear device is more efficient than a collinear one regardless of length (Fig. 6).

Furthermore, collinear schemes are subject to strict phase matching demands, and output power plateaus and oscillates after a certain length. In comparison, noncollinear phase matching requirements are less stringent since a slight deviation in phase matching simply matches the output light to an angled radiation mode. Efficiency will continue to increase with propagation length, until depletion of input wavelengths starts to become significant.

In comparison to Cherenkov surface emission schemes, our design is less affected by strong dispersion. In bulk devices, Cherenkov emission is not possible between 10-15 THz or 20-27 THz since LN refractive index for THz light dips below that for optical light. In waveguide devices, adjusting waveguide parameters and using other adjacent materials can tune the effective index for THz light, allowing for Cherenkov emission at otherwise impossible frequencies [24]. However, Cherenkov surface emission for continuous wave DFG has only been demonstrated in LN ion-planted slab and channel waveguides with low index contrast [12], resulting in relatively weak confinement. Furthermore, many devices that rely on Cherenkov emission require silicon prisms on top of the nonlinear material to improve emission towards the top surface since internal reflection occurs at the Cherenkov angle [12,24]. Since emission angle in our approach is determined by poling period and target wavelength, we have greater freedom in our design to maximize efficiency.

A fair comparison between our simulated efficiency and previous work on surface emission based on periodic poling is difficult since these works used different output wavelengths, nonlinear strength, and underlying assumptions. Since many contributing parameters are strongly wavelength dependent, it becomes even more difficult to separate differences in models from how well efficiency is optimized in the waveguide structure. One analytical calculation estimated efficiency around $1\times 10^{-7}$ W$^{-1}$ for a 3 cm device emitting light at 150 $\mu$m [14]. Another analytical calculation considered theoretical increase in $\chi ^{(2)}$ due to polaritonic resonances and predicted an efficiency of 2.1$\times$10$^{-4}$ W$^{-1}$ at 16 $\mu$m [17]. However, their efficiency was scaled by a factor of 260 due to calculated $\chi ^{(2)}$ values which have not been experimentally verified. Accounting for this, their device length of 10 cm, and the difference in wavelength, their efficiency would be 3.3$\times$10$^{-7}$ W$^{-1}$ for a 1 cm device at 30 THz with $d_{33}$ = 30 pm/V, which is 30 times smaller than our calculated efficiency. Experimental approaches using ridge-type periodically poled LN reported efficiencies around $4\times 10^{-10}$ W$^{-1}$ at 203 $\mu$m [15] and $2\times 10^{-9}$ W$^{-1}$ at 200 $\mu$m [16].

Our approach uses simulations rather than relying solely on analytical calculations. It thus includes many contributing effects, ranging from material dispersion, mode overlap, limited collection angle, Fresnel coefficients, and other physical effects that are difficult to include analytically. Given our starting parameters, the predicted efficiency of $9.16 \times 10^{-6}$ W$^{-1}$ for a 1 cm device emitting 10 $\mu$m light is reasonable, particularly for an integrated photonics platform.

By using the TFLN platform, our proposed design is capable of reasonable DFG efficiency in more compact devices. Better confinement in TFLN compared to bulk or diffused waveguides significantly reduces device footprint. Using thinner LN also reduces absorption of FIR/THz light since it travels a shorter distance through the LN before radiating out. Additionally, thicker LN requires phase matching in the emission direction when the thickness approaches the DFG wavelength in LN [17]. TFLN enables thinner waveguides while maintaining strong confinement of optical modes, eliminating the need for additional phase matching without compromising efficiency. While reducing mode area does not automatically improve efficiency in noncollinear DFG, finely tuning mode overlap between desired input wavelengths is possible in integrated waveguide design since thickness, width, and etch depth can all be adjusted. In comparison, techniques to adjust mode overlap in bulk and diffused waveguides are more limited. As such, the TFLN platform enables great improvement in comparison to prior surface-emitting DFG based on bulk waveguides.

6. Beam steering

In addition to FIR/THz generation on an integrated platform, our approach provides control over emitted light wavelength and direction. Continuous beam steering in a 1D array of TFLN waveguides (Fig. 7(a)) with set poling period is possible through a combination of phase shifting and wavelength tuning. A variety of approaches from metal antennas [13] to integrated phased arrays [28,29] to metasurfaces [30] have been used to control FIR/THz light emission patterns. However, our approach works for continuous wave FIR/THz light and provides steering along two axes. It also directs light as it is generated, streamlining the process and reducing absorption loss.

 

Fig. 7. Integrated beam steering device for FIR/THz light with poled TFLN waveguides. a) $f_1$ and $f_2$ are combined in a 1D array of periodically poled TFLN waveguides such that the DFG light at $f_3=f_1-f_2$ is normally emitted. b) By phase shifting $f_2$ input light by incrementally larger phases in steps of $\phi$, the DFG signal can be strategically delayed in adjacent waveguides, angling the wavefront by $\theta _y$. c) For a device with fixed poling period, tuning input wavelength will shift the DFG wavelength, adjusting the phase matching condition and angling the wavefront by different $\theta _z$.

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In a phased array of TFLN waveguides, adjusting the phase of the DFG light shifts emission angle by $\theta _y$. To accomplish this, an electro-optic phase shifter adds phase $-\phi$ to one of the incoming pump wavelengths, resulting in the electric field $E_2 = A_2 e^{i(k_2y-\omega _2 t)} e^{-i\phi } \hat {z}$. The nonlinear polarization from frequency mixing is proportional to the input fields’ product as $P_{\rm NL} \sim E_1 E_2^*$, and thus becomes:

Since $P_{\rm NL}$ acts as the source for DFG emission, the DFG field has a corresponding phase shift of $\phi$ [31]. In this way, control over the phase of the output DFG signal is possible by adjusting the phase of one of the input signals.

In an array of adjacent waveguides with gradually larger phase shifts, emitted light from subsequent waveguides is slightly delayed, angling the wavefront towards one side of the array by some angle $\theta _y$ (Fig. 7(b)). The allowable range of $\theta _y$ is limited by the phase shifters’ efficiency, which for electro-optic tuning, is limited by length and voltage. For an applied voltage of 1 V and electrode separation of 3 $\mu$m, the change in refractive index is $\Delta n = 5 \times 10^{-5}$ using $r_{33} = 30$ pm/V [32]. For a 1 mm phase shifter, the resulting phase shift $\phi$ for $\lambda _2$ = 1.63 $\mu$m is 10$^{\circ }$. This corresponds to $\theta _y$ based on the phased array equation [33]:

Wavelength tuning shifts the phase matching condition, resulting in deviation from the normal emission direction by some angle $\theta _z$ (Fig. 7(c)). In this way, light can be angled along the waveguide direction. The limits on $\theta _z$ depend on the target output wavelength, poling period, waveguide structure, and dispersion. Since beam steering uses the emission angle in air, the main limitation is the application’s tolerance to wavelength changes. Based on LN dispersion and phase matching calculations, a device with normal emission at 30 THz will emit 25 THz light at $\theta _z$ around -27$^{\circ }$, while 35 THz light emits at $\theta _z$ of 19$^{\circ }$. The tuning range for $\theta _z$ can be quite broad as wavelength can be adjusted further, but practical use depends on the desired range of wavelengths, changes in efficiency as wavelength is tuned, and subsequent design considerations. While rapid fluctuations in dispersion can complicate phase matching, even slight changes in wavelength can still result in significant changes in far field radiation. Using both techniques, multi-directional beam steering for FIR/THz light is possible.

7. Conclusion

We have simulated surface emission of 10 $\mu$m light using periodically poled TFLN waveguides and demonstrated their potential for efficient generation of narrowband, tunable CW light at long wavelengths. While material dispersion still affects device performance at different frequencies, efficiency scales linearly with length and is not exponentially affected by loss. Output power can continually be improved by increasing device length until other effects like pump depletion arise. Furthermore, our device remains reasonably efficient in generating CW light rather than requiring strong pulses, facilitating applications that require a more precise wavelength or CW light. Through wavelength tuning and phase modulation, the emission angle from an array of waveguides can be continuously adjusted, enabling beam steering along two directions.