Integrated photonic devices, consisting of diverse waveguides (optical circuits)^1-2, microcavities^{3, 4}, modulators^5-7, and laser sources^8-10, are essential for next-generation optical communication^11,12, optical-electronic systems^{13, 14}, and quantum information sciences including quantum computation^{15, 16}, quantum signal processing^{17, 18}, and quantum sensing^{19, 20}. Photonics features, such as frequency, polarization, path, pulse shape, and orbital angular momentum (OAM), have been used for the functionality in space-division multiplexing²¹, polarization encoding^{22, 23}, spatial-temporal entanglement²⁴, OAM multiplexing and de-multiplexing^{12, 25, 26}, beam shifting and steering^{27, 28}, among others. Gathering these features matches the current trends in maximizing the link capacity for classical long-distance communications²⁹ and extending the Hilbert space of quantum systems^{30, 31}.

To develop a high-fidelity interface between photonics in various frequency domains without disturbing their quantum properties^{24, 32}, nonlinear frequency conversion, steered with quadratic (χ²) processes, should be considered. These methods could be the classical second-harmonic generation (SHG)^{33, 34}, phase-matching-free SHG³⁵, difference-frequency generation³⁶, optical parametric oscillation³⁷, and nonclassical spontaneous parametric down-conversion^{38, 39}. Meanwhile, accompanying spatial mode conversion was observed in the ferroelectric domain engineerable potassium titanyl phosphate (KTP) and lithium niobate (LN) crystals for quasi-phase matching (QPM) because of the inter-mode dispersion of the waveguide geometric^40-43. This nontrivial process is promising in processing both the classical and quantum signals, such as selectively de-multiplexing the invisible multi-spatial mode signals toward the visible region⁴⁴ or preparing the high-dimension quantum states³². For inter-mode conversion, broadband lasers, or light sources with similar wavelengths^{42, 45} were applied, which rendered the prepared up-converted or down-converted spatial mode difficult to be discriminated because of their contiguous wavelengths^{41, 45}. This problem was determined using a tunable source as the fundamental wave (FW) in addition to the adjustment of the working temperature for phase matching⁴¹. However, the second harmonic (SH) wavelength of the spatial mode was un-controllable with obvious shifting. By adjusting position of the FW inside a periodically poled LN (PPLN) waveguide to excite suitable FW modes^{40, 47}, evolution in the spatial mode of the SH wave was observed, which changed from TM₀₁, TM₁₀, to TM₀₀ within a short temperature range from 23 °C to 26 °C⁴⁸. However, because the intensity of the generated space-mode was extremely weak under poor incident fundamental wave intensity (−13.5 dBm at 800 nm in ref.⁴⁰ and 7 dBm at 1066 nm in ref.⁴⁶), detail adjustment to the incident light for rigorous mode-matching was required to selectively inspire the specific spatial mode for detection. Although spatial mode of the fundamental light can be easily prepared with a bulky spatial mode modulator⁴⁴, these measures inevitably prohibit development of the spatial mode steering device for on-chip integration with other fundamental elements, such as the mircolasers⁴⁹ and fail to meet the rapid development in chip-based integrated optics.

In addition to commonly up/down converting the spatial mode between the FW and SH^{41, 47}, the parametric process, dominated by three-wave mixing, is appealing, which provides an additional degree of freedom in selecting the mixing wavelength to satisfy the inter-mode QPM condition and making the target wavelength controllable without spectral drifting or broadening^43-45. Furthermore, during a sum–frequency generation (SFG) process, the incident infrared signals could be selectively mode converted at the visible region, which can be easily detected and has ultra-low signal noise⁵⁰.

In this paper, we demonstrate a high-intensity spatial-mode steering scheme during the SFG processes with a PPLN waveguide by separately manipulating the waveguide temperature and the wavelength of the mixing wave, which takes place under sufficient high incident light intensities (2~4 order of magnitudes higher than that in the previous reports^{40, 46}) without carefully adjusting incident lights or using the spatial light modulator^{40, 44, 48}. Mode conversions at the SFG wavelength, among TM₀₁, TM₁₀, and TM₀₀ modes, are realized at each characterized broad temperature ranges (ΔT ≥ 8 °C) and wavelength ranges (Δλ ≥ 1 nm). The resulting mode intensities, un-reported in relevant literatures^40-48, are within −7 – 8 dBm, which are sufficiently high for on-chip processing of both classical and quantum signals.

Among χ² optical materials, LN has received considerable attention in integrated optics because of its sufficiently high second nonlinear coefficient (d₃₃ = 27 pm/V⁵¹), engineerable ferroelectric domain for QPM, strong Pockels effect for electro-optical modulation⁵, and moderate third-order nonlinearity (1.6 × 10^–21 m²/V²) for acousto-optical modulation⁵² or Raman processes⁵³. In a few-mode PPLN waveguide with a poling period of Λ, nonlinear conversion among the signal (λ_S), pump (λ_P), and up-converted (λ_F) waves is defined by the QPM condition, which is determined by the effective refractive indices n_S^ml, n_P^uv, and n_F^jk of the waveguide. Here, jk, nv, and ml define the numbers of the modal points of the signal (S), pump (P), and SFG (F) lights along the horizontal and vertical directions. The effective index of the spatial modes can be linearly changed by the outer temperature (~2.989 × 10^–4/°C) because of the thermo-optical coefficient of the LN wafer (Fig. 1(c)) and enhanced by the waveguide geometric (Fig. 1(a)). Figure 1(b) lists the first six spatial modes of λ_F = 598.47 nm at 25 °C (see the enumerated mode profiles for λ_S = 1552.6 nm and λ_P = 973.85 nm in Tables S2–S4 in the Supplementary information). We consider Type-0 QPM to use the maximum nonlinear coefficient d₃₃ of the PPLN waveguide, in which the dominant TM mode with the electric field parallel to the waveguide height can be excited. With the calculated effective indices, the temperature-dependent inter-mode QPM equation can be described as follows:

1$\Delta (T)=\frac{n_{\text{F}}^{jk}(T)}{{\lambda }_{\text{F}}}?\frac{n_{\text{P}}^{uv}(T)}{{\lambda }_{\text{P}}}?\frac{n_{\text{S}}^{ml}(T)}{{\lambda }_{\text{S}}}?\frac{1}{\Lambda }\mathrm{.}$

Since the signal and pump lights are with high-intensity and approximately undepleted inside the waveguide, the conversion efficiency (CE) for spatial mode λ_F^jk at a given temperature is (see detailed derivation in the Supplementary information) as follows:

2${\eta }_{\text{theo}}^{jk}(T)=\frac{8{\pi }^2{L}^2{d}_{\text{eff}}^2}{n_{\text{S}}^{ml}{n}_{\text{P}}^{uv}{n}_{\text{F}}^{jk}{?}_{0}c{\lambda }_{\text{F}}^2}\Phi (x\mathrm{,}y\mathrm{,}T)\sin {\text{c}}^2\left(\frac{\Delta kL}{2}\right)\mathrm{,}$

where d_eff is the effective nonlinear coefficient of the LN crystal, ε₀ is the permittivity, c the speed of light, and Δk = 2πΔ(T), and Φ(x, y, T) indicates the overlap integral among modes λ_S^ml, λ_P^uv, and λ_F^jk at temperature T (see Supplementary information Eq. S6).

According to theoretical results (Fig. 1(d–f)), coupling among modes TM₀₁, TM₁₀, and TM₀₀ of the pump and signal and SFG lights dominates the spatial mode conversion process during frequency up-conversion. That is, at 25 °C, dominant processes λ_S⁰⁰ + λ_P⁰⁰ → λ_F⁰⁰, λ_S⁰⁰ + λ_P⁰¹ → λ_F⁰¹ and λ_S⁰⁰ + λ_P¹⁰ → λ_F¹⁰ contribute for the potential TM₀₀, TM₀₁, and TM₁₀ modes with conversion efficiencies of 0.158%/W, 0.333%/W, and 0.143%/W, respectively. Thus, the resulting SFG mode under 25 °C is at TM₀₁, which automatically takes place under high incident pump and signal light intensities without specifically modulating spatial modes of the incident lights as the conventional manners. With the theoretical process, we could detail the spatial mode to be produced at a given temperature, and characteristic broad steerable temperature ranges for modes TM₀₁, TM₁₀, and TM₀₀ are predicted (Fig. 1(g)).

To verify theoretical prediction, we prepared a few-mode PPLN waveguide array consisting of seven waveguides with a length of 30 mm (Fig. 2(d)), which was diffusion-boned on a lithium tantalate (LT) wafer with the assistance of a SiO₂ layer (see Fig. S1 for the fabrication flowchart). The third waveguide, with well-protected cutting edge and lowest loss (0.29 dB/cm), was used throughout the experiment process (Fig. 2(e)). The ridge waveguide is characterized with a width of 11.2 μm and height of 10.9 μm to support at least 47, 11, and 50 eigenmodes at the pump, signal, and SFG wavelengths, respectively, at which three typical SFG modes (TM₀₀, TM₀₁, and TM₁₀), meeting the temperature controlled inter-mode QPM conditions are distinguished on a white broad (Fig. 2(b)). As with the designed poling period, Λ was measured to be 10.2 μm using the microscopy image (Fig. 2(f)). We coupled signal light at the c-band from a tunable Erbium fiber-based master oscillator power-amplifier and pump light at 973.85 nm from a single-mode fiber-coupled diode laser into the ridge waveguide (Fig. S3), which was mounted inside a thermoelectric cooler (Fig. 2(a)). The pump and signal lights have maximum output powers of 25 dBm and 27.8 dBm respectively to implement an automatic spatial mode steering process, without specially preparing the signal or pump modes in advance. Moreover, to avoid other potential spatial modes at the SFG wavelength caused by the broad bandwidth of the pump and signal waves^{45, 47}, their bandwidths are narrow and with values of 0.15 and 0.017 nm, respectively. Therefore, pure TM₀₁, TM₁₀, and TM₀₀ modes are prepared stably at temperature ranges from 25 °C to 32 °C, 35 °C to 43 °C, and 54 °C to 70 °C, respectively (Fig. 2(b)). By tuning the signal wavelength from 1546 to 1556 nm, evolution in the spatial mode recurs and changes from TM₀₀, TM₀₁, to TM₁₀ (Fig. 2(c)). Both temperature- and wavelength-dependent steering methods are appealing in integrated optics through preparing target spatial modes with accessible wavelengths in the quantum or classical optical networks and overcoming the growing complexity of quantum information protocols.

On fixing the incident power of the pump light at 25 dBm, we measured the power curves of the SFG process at various temperatures from 25 °C to 70 °C (Fig. 3(a)) at which the signal wavelength was 1552.6 nm. The spatial mode did not change during each power scaling process at a given TEC temperature. Under an incident signal power of 27.8 dBm, Fig. 3(b) depicts the maximum SFG power versus the working temperature. The regions for pure spatial mode are denoted with colors. Figure 3(c) displays the evolution of SFG efficiency (denoting by η_exp= 100·P_SFG/P_P/P_S), which is consistent with the calculation result by denoting the maximum CE among F₀₁, F₁₀, and F₀₀ (Fig. 1(g)) as the theoretical efficiency η_cal = max(η_F10, η_F01, η_F10). Controlling the waveguide temperature from 25 °C to 70 °C, the spatial mode begins with TM₀₁ and subsequently changes from TM₀₁ to TM₁₀, and TM₁₀ to TM₀₀ (Fig. 3(e)). Figure 3(d) depicts the width of temperature windows for TM₀₁, TM₁₀, and TM₀₀ modes, with values of 8 °C, 9 °C, and 17 °C, respectively, which are sufficiently broad for a stable pure spatial mode generation. At each temperature window, the maximum SFG intensities are −3.82, −2.08, and 7.82 dBm, with corresponding conversion efficiencies of 0.21%/W, 0.29%/W, and 3.16%/W. The SFG spectrum is verified to be stabilizing at 598.47 ± 0.1 nm when changing the waveguide temperature (Fig. 3(f)).

Between adjacent spatial modes, an obvious temperature window, covering a few Celsius degrees as the transition state, exists. This window can be attributed to the competition among the spatial modes with comparable conversion efficiencies. In the first transition state, mode rotation from TM₀₁ to TM₁₀ occurs (Fig. 4(d)), which was first observed in an SHG PPLN waveguide (within a small temperature range of ~2 °C) and explained as the coupling between TM₁₀ and TM₀₁⁴⁸. However, in the second transition state, the rotated mode profile is distorted (Fig. 4(e)). According to the theoretical CE for modes TM₀₀, TM₀₁, and TM₁₀, we could see CE for TM₀₀ gradually increases and participates in the coupling between TM₁₀ and TM₀₁ with the increased waveguide temperature (Fig. 4(c)). Since the mode overlap integral Φ(x, y, T) for TM₀₀, TM₀₁, and TM₁₀ is relatively stable (Fig. 4(b)), the resulting mixing mode patterns in transition states are originated by the temperature-dependent QPM efficiencies (denoted by sinc(ΔkL/2)) (Fig. 4(a)). We derive the mixing mode amplitude E_F(T) as follows:

3$E_{\text{F}}^i(T)={\displaystyle \sum _{j\text{k}}}{A}_{\text{F}}^{\text{jk}}{E}_{\text{F}}^{j\text{k}}(T)\mathrm{,}$

where i denotes the i^th transition state, and jk the fundamental TM modes for the SFG light. Here jk = 00, 01, and 10 is selected and A_F^jk denotes the complex amplitudes for mode jk. Denoting A_F⁰¹ = 1, we obtain A_F¹⁰ = $r_1{\text{e}}^{\text{i}{\theta }_1}$ and A_F⁰⁰ = $r_2{\text{e}}^{\text{i}{\theta }_2}$, where (r₁ = η₁₀/η₀₁, θ₁) and (r₂ = η₀₀/η₀₁, θ₂) are the relative amplitudes and phase angles of A_F⁰¹ and A_F⁰⁰ to A_F¹⁰, respectively. In the first state, as the CE of TM₀₀ is sufficiently small to be ignored, mixing mode patterns between TM₁₀ and TM₀₁ are computed and classified as the quasi-TM01, quasi-TM10, and highly coupling modes, in Fig. S4(a). The coupling modes occur at 0.7 ≤ r₁ ≤ 3 with 0.8 rad ≤ θ₁≤ 2.2 rad or −2.2 rad ≤ θ₁ ≤ −0.8 rad. Therefore, the coupling mode patterns at 32.5 °C and 33.7 °C can be reproduced by setting (r₁, θ₁) as (0.77, π/3) and (1, –π) (Fig. 4(d)). In the second transition state, the mixing mode pattern is blurred because of the growing CE of TM₀₀. We address this three-mode coupling state by initializing the mode pattern under the two-mode coupling condition before sweeping parameters between r₂ and θ₂ (Table S5). In addition to setting (r₁, θ₁) as (1.14, π/2), (r₂, θ₂) = (1.2, –π/2) is selected (Fig. 4(e)). In the third transition state, TM₀₀ dominates the mixing mode, where the mode profile becomes convergent into a quasi TM₀₀ mode with the increased waveguide temperature (Fig. 4(f)) and is insensitive to r₁, θ₁, and θ₂ (Fig. S5).

Changing the signal wavelength from 1546 to 1556 nm, the SFG wavelength changes from 597.54 to 599 nm (Fig. 5(a)), where the slop rate is fit to be Δλ_P/Δλ_S = 0.14. Mode conversion occurs during red shifting the SFG wavelength (Fig. 5(c)). Theoretically, this method can be described by the inter-mode QPM condition (Eq. (1)), where the changed signal and pump wavelengths and the resulting effective indexes are considered:

4$\Delta (T\mathrm{,}{\lambda }_{\text{P}}\mathrm{,}{\lambda }_{\text{S}})=\frac{n_{\text{F}}^{jk}(T\mathrm{,}{\lambda }_{\text{F}})}{{\lambda }_{\text{F}}}?\frac{n_{\text{P}}^{uv}(T\mathrm{,}{\lambda }_{\text{P}})}{{\lambda }_{\text{P}}}?\frac{n_{\text{S}}^{ml}(T\mathrm{,}{\lambda }_{\text{S}})}{{\lambda }_{\text{S}}}?\frac{1}{\Lambda }\mathrm{,}$

here, λ_F = 1/(1/λ_P+1/λ_S). Experimentally, Fig. 6(a) illustrates the measured SFG power during turning the signal wavelength at various waveguide temperatures (see Tables S6–S9 for the evolution processes of the spatial mode at 35 °C, 45 °C, 55 °C, and 65 °C, respectively), where the wavelength range for pure TM₀₁, TM₁₀, and TM₀₀ modes is denoted. As with the temperature-dependent steering scheme, characterized broad wavelength windows to prepare the pure spatial modes by changing the signal wavelength are displayed (Fig. 6(b)). With the working temperature and incident mixing wavelengths, a 2D route map for preparing demanded spatial mode with up/down-conversion wavelength within the transparent window of the QPM χ² media can be determined.

We have comprehensively demonstrated a high-intensity spatial-mode steerable frequency up-converter in both temperature and wavelength manipulating methods. Stable TM₀₁, TM₁₀, and TM₀₀ modes at the SFG wavelength could be prepared automatically within a few-mode PPLN waveguide without prefabricating the spatial modes of the incident signal and pump lights by adjusting the incident combined lights or using a spatial light modulator. This result is appealing for on-chip integration with the microlasers such as distributed feedback laser (DFB)⁴⁹ or semiconductor optical amplifier⁵⁴. The conversion processes can be theoretically explained by the competition among the SFG spatial modes under a high-intensity three wave mixing process described with the inter-mode QPM model, where a considerable broad temperature range with a width of approximately 8 °C and a waveband width of approximately 1 nm is displayed for the prepared spatial modes. The maximum mode switching speed is theoretically limited by the frequency-switching speed of the tunable pump or signal sources.

Moreover, because of the sufficiently high pump and signal light intensities, output power for the prepared up-converted spatial modes is within −7 to 8 dBm (200 μW to 6.3 mW), which has not been recorded in other literatures due to the poor fundamental wave intensity. This method is sufficiently high for on-chip preparation of the high-dimensional quantum information carriers in extending the Hilbert space and can be used in long-distance quantum communication and hyperentangled photon generation for increasing information capacity, improving the noise resistance, and making the quantum cryptographic schemes difficult to hack despite errors. Furthermore, the study has the potential in other functional devices including de-multiplexing the multi-mode signal by frequency conversion or encoding the target wavelength with characterized mode pattern⁵⁵. Unlimited by the SFG scheme, the automatic spatial mode conversion process could also be implemented by differential frequency generation or optical parameter oscillation, which extends the desired wavelengths within transparent windows of the ferroelectric χ² media from ultra-violet to mid-infrared regions.

We are grateful for financial supports from National Key Research and Development Program of China (2021YFB3602500), Self-deployment Project of Fujian Science & Technology Innovation Laboratory for Optoelectronic Information of China (2021ZZ101), and National Natural Science Foundation of China (Grant Nos. 62275247 and 61905246).

HZ Huang: Methodology, Investigation, Visualization, Writing–original draft; HX Chen：Device preparing, Data curation, Investigation, Writing–review & editing; HG Liu：Data curation, Writing–review & editing; Z Zhang, XK Feng, and JY Chen: Data curation, Visualization; HC Wu and J Deng: Formal analysis, Visualization; WG Liang: Supervision, Writing–review & editing; WX Lin: Conceptualization, Supervision.

The authors declare no competing financial interests.

Supplementary information for this paper is available athttps://doi.org/10.29026/oes.2024.230036