2.1 Mechanism of parametric diffraction

Multi-beam-pumped OPCPA enables energy scaling by coherently transferring energy from multiple pump beams to a single signal via parametric amplification. This approach necessitates the precise optimization of the pump beam configuration to simultaneously satisfy noncollinear geometric constraints and the phase-matching conditions across all amplification channels.

For theoretical clarity, we utilize a two-beam-pumped system as a representative case. The dynamics of this system are modeled by extending the standard three-wave coupling equations under the slowly varying envelope approximation. The evolution for both conventional OPCPA (with idler dissipation coefficient α = 0) and QPCPA (α ≠ 0) is governed by the following:

(1a) $$\begin{align}\frac{\mathrm{d}A_{\mathrm{s}}}{\mathrm{d}z}&=-i{\kappa}_{\mathrm{s}}{A}_{\mathrm{p}1}{A}_{\mathrm{i}1}^{\ast }{e}^{-i\Delta {k}_1z}-i{\kappa}_{\mathrm{s}}{A}_{\mathrm{p}2}{A}_{\mathrm{i}2}^{\ast }{e}^{-i\Delta {k}_2z},\end{align}$$

(1b) $$\begin{align}\frac{\mathrm{d}A_{\mathrm{p}1}}{\mathrm{d}z}&=-i{\kappa}_{\mathrm{p}1}{A}_{\mathrm{s}}{A}_{\mathrm{i}1}{e}^{i\Delta {k}_1z},\end{align}$$

(1c) $$\begin{align}\frac{\mathrm{d}A_{\mathrm{p}2}}{\mathrm{d}z}&=-i{\kappa}_{\mathrm{p}2}{A}_{\mathrm{s}}{A}_{\mathrm{i}2}{e}^{i\Delta {k}_2z},\end{align}$$

(1d) $$\begin{align}\frac{\mathrm{d}A_{\mathrm{i}1}}{\mathrm{d}z}&=-i{\kappa}_{\mathrm{i}1}{A}_{\mathrm{p}1}{A}_{\mathrm{s}}^{\ast }{e}^{-i\Delta {k}_1z}-\frac{\alpha }{2}{A}_{\mathrm{i}1},\end{align}$$

(1e) $$\begin{align}\frac{\mathrm{d}A_{\mathrm{i}2}}{\mathrm{d}z}&=-i{\kappa}_{\mathrm{i}2}{A}_{\mathrm{p}2}{A}_{\mathrm{s}}^{\ast }{e}^{-i\Delta {k}_2z}-\frac{\alpha }{2}{A}_{\mathrm{i}2},\end{align}$$

where A _s, A _p1,2 and A _i1,2 represent the complex amplitudes of the signal (S₀), two pump beams (P₁ and P₂) and their corresponding idlers (I₁ and I₂), respectively. The nonlinear coupling coefficient is defined as κ = ωd _eff/nc, where d _eff is the effective nonlinear coefficient, n is the refractive index, ω is the angular frequency and c is the speed of light in vacuum. The phase mismatch terms for the two parametric amplification channels are Δk ₁ = k _p1 – k _s – k _i1 and Δk ₂ = k _p2 – k _s – k _i2, respectively.

To account for first-order parametric diffraction (P₁ – I₂ → S₁₂ and P₂ – I₁ → S₂₁), we introduce two additional coupled-wave equations:

(2a) $$\begin{align}\frac{\mathrm{d}A_{\mathrm{s}12}}{\mathrm{d}z}&=-i{\kappa}_{\mathrm{s}}{A}_{\mathrm{p}1}{A}_{\mathrm{i}2}^{\ast }{e}^{-i\Delta {k}_{12}z}, \end{align}$$

(2b) $$\begin{align}\frac{\mathrm{d}A_{\mathrm{s}21}}{\mathrm{d}z}&=-i{\kappa}_{\mathrm{s}}{A}_{\mathrm{p}2}{A}_{\mathrm{i}1}^{\ast }{e}^{-i\Delta {k}_{21}z},\end{align}$$

where A _s12 and A _s21 represent the complex amplitudes of the first-order diffracted signal. The phase mismatch terms of cross-coupling processes are defined as Δk ₁₂ = k _p1 – k _s12 – k _i2 and Δk ₂₁ = k _p2 – k _s21 – k _i1. In this analysis, the theoretical model is restricted to a two-beam-pumped geometry, and only first-order parametric diffraction is considered. This dual simplification isolates the primary cross-coupling channel and allows us to clearly identify and quantify the key physical relationships. Including higher-order processes would introduce additional idler absorption for multiple diffracted beams, complicating the direct attribution of suppression to the idler dissipation coefficient. As such, the first-order model provides the most transparent physical picture for the mechanism of interest. The experimentally observed higher-order diffraction is addressed separately via quantitative measurements.

This work employs a theoretical analysis based on published Sellmeier equations for yttrium calcium oxyborate (YCOB) crystals, selected for their suitability in QPCPA implementation via rare-earth doping to facilitate idler absorption^[ Reference Ma, Wang, Yuan, Xie, Xiong, Tu, Tu, Shi, Zheng and Qian ^{23 ]}. We consider a noncollinear Type-I phase-matching geometry with two pump beams. Figure 1(a) illustrates angular relations within the dielectric frame (X, Y, Z) for a 515 nm pump and an 800 nm signal. Broadband phase-matching is achieved at θ _p = 24.55° relative to the optic axis, with a noncollinear angle α _p0 = 2.92° between the pump and signal. The signal propagates in the XZ plane (θ _s = 21.63°, φ _s = 0°). Figure 1(b) displays the phase mismatch Δk as a function of pump incidence angles. Multiple pump beams can simultaneously satisfy phase-matching conditions with a single signal, forming a conical geometry of permissible pump directions (θ _p, φ _p) around the signal axis^[ Reference Trophème, Boulanger and Mennerat ^{29 ]}.

Figure 1

Phase-matching and parametric diffraction in two-beam-pumped OPCPA. (a) Angular relations in the dielectric frame (X, Y, Z). (b) Phase mismatch Δk versus pump incidence angles, showing the range of allowed pump directions. (c) Parametric gain (red curve) and normalized phase mismatch Δk/2Γ (blue curve) as functions of noncollinear angle deviation. (d) Wave-mixing diagram in the z–y plane of the laboratory frame (x, y, z) illustrating parametric-diffraction processes P₂ – I₁ → S₂₁ and P₁ – I₂ → S₁₂. (e) Phase-matching tolerance analysis in the x–y plane of the laboratory frame (x, y, z): red and blue regions denote the angular tolerance ranges for the signal (S₀) and first idler (I₁), respectively. Parametric diffraction occurs when the second pump beam (P₂) lies within the overlap of these two tolerance cones.

Figure 1 Long description

Panel a. A three-dimensional Cartesian coordinate system with axes X, Y, and Z. A green vector k sub p and a red vector k sub s originate from the center. Angles theta sub p, phi sub p, and theta sub s define their orientation.

Panel b. A heatmap showing phase mismatch Delta k. The x-axis is theta sub p from 18 to 24 degrees and the y-axis is phi sub p from minus 10 to 10 degrees. A circular ring of high intensity (red/yellow) surrounds a central point labeled S sub 0.

Panel c. A dual-axis line graph. The x-axis is alpha sub p minus alpha sub p 0 in milliradians. The left y-axis is Parametric gain in red, showing a Gaussian-like peak centered at 0. The right y-axis is Delta k all over 2 Gamma in blue, showing a linear decrease. A horizontal arrow indicates the width Delta alpha sub p.

Panel d. A wave-mixing diagram in the z-y plane. Vectors originate from the origin. Green vectors P sub 1 and P sub 2 are separated by angle theta sub p 1 p 2. Blue vectors I sub 1 and I sub 2 point upward. Red vectors S sub 0, S sub 2 1, and S sub 1 2 point near the z-axis. Text below lists processes P sub 1 minus I sub 2 yields S sub 1 2 and P sub 2 minus I sub 1 yields S sub 2 1.

Panel e. A phase-matching tolerance analysis in the x-y plane. A red dot S sub 0 is at the origin and a blue dot I sub 1 is on the x-axis. Two overlapping curved tolerance bands (red and blue) intersect at green dots P sub 1 and P sub 2. A shaded gray cone indicates where parametric diffraction happened.

In practice, the angular tolerance Δα _p is defined as the permissible deviation of the pump wave vector from the optimal noncollinear angle α _p0. We quantify this tolerance by Taylor-expanding the phase mismatch Δk around α _p0:

(3) $$\begin{align}\Delta k\left({\alpha}_{\mathrm{p}}\right)=\Delta k\left({\alpha}_{\mathrm{p}0}\right)+\left({\alpha}_{\mathrm{p}}-{\alpha}_{\mathrm{p}0}\right)\frac{\mathrm{\partial \Delta }k}{\partial {\alpha}_{\mathrm{p}}}+\cdots .\end{align}$$

In the small-signal regime (where pump depletion is negligible), the signal parametric gain follows G ≈ exp(2gz)/4, where the gain coefficient g is as follows:

(4) $$\begin{align}g=\sqrt{\Gamma^2-{\left(\Delta k/2\right)}^2}.\end{align}$$

Here, Γ is nonlinear drive strength (Γ² = 2ω _s ω _i d _eff ² I _p /n _s n _i n _p ϵ ₀ c ³)^[ Reference Bromage, Rothhardt, Hädrich, Dorrer, Jocher, Demmler, Limpert, Tünnermann and Zuegel ^{30 ]} and ϵ ₀ is the permittivity of vacuum. Phase matching Δk dictates two distinct regimes. Here, |Δk| < 2Γ yields exponential amplification, while |Δk| > 2Γ results in oscillatory behavior without net gain. This transition at |Δk| = 2Γ defines the angular tolerance Δα _p as follows:

(5) $$\begin{align}\Delta {\alpha}_{\mathrm{p}}=\frac{4\Gamma}{\frac{\partial \varDelta k}{\partial {\alpha}_{\mathrm{p}}}}.\end{align}$$

Figure 1(c) presents the calculated parametric gain profile as a function of noncollinear angle deviation (α _p – α _p0) at a pump intensity of I _p = 10 GW/cm², a crystal length of z = 3 mm and d _eff = 1.39 pm/V. The gain spectrum exhibits peak gain at Δα _p = 0 (perfect phase matching) and an angular tolerance of 2.6 mrad defined by |Δk| = 2Γ.

Building upon the angular tolerance analysis, we now examine parametric diffraction in a two-beam-pumped OPCPA system. The laboratory frame (x, y, z) is defined with the z-axis aligned along the signal wave vector. The first-order diffraction processes involve pump–idler cross-coupling (P₁ – I₂ → S₁₂ and P₂ – I₁ → S₂₁) in the laboratory frame, as shown in Figure 1(d). Here, θ _p1p2 represents the angular separation between the two pump beams. Figure 1(e) illustrates the phase-matching geometry in the x–y plane of this frame. With S₀ and P₁ fixed, the idler I₁ is determined by the phase matching. Parametric diffraction occurs when P₂ falls within the overlapping angular tolerance regions (Δα _p) centered on S₀ and I₁, corresponding to the red and blue regions in Figure 1(e). This spatial overlap leads to undesired cross-coupling processes.

2.2 Numerical simulation parameters and results

We simulate a two-beam-pumped OPCPA system amplifying an 800 nm chirped signal pulse (2 ps duration, 150 nm full width at half maximum (FWHM) bandwidth) with two 515 nm pump beams (3 ps duration). To quantify parametric diffraction, we fix the position of P₁ and vary the angle of P₂ to maintain phase matching. Two efficiency metrics are defined: the pump-to-signal conversion efficiency η _s, and the parasitic pump-to-diffraction efficiency η _diff, which accounts for the first-order parasitic signals S₁₂ and S₂₁ as a direct energy loss mechanism. The simulations are performed with a single pump intensity I _p= 20 GW/cm² (Γ = 1384 m^–1) and a pump-to-signal intensity ratio I _p/I _s = 10³. In the following two-beam-pumping configuration, the pump intensity refers to that of a single pump beam, and the two pump beams are assumed to have identical intensities.

As illustrated in Figure 2(a), η _diff reaches 20.5% at θ _p1p2 = 1 mrad, but decreases to 5.6% at θ _p1p2 = 15 mrad, demonstrating geometric control over parametric diffraction. For each configuration, the optimal crystal length is determined by maximizing the signal conversion efficiency η _s. A smaller pump angular separation increases the probability of cross-coupling, leading to accelerated pump depletion and a commensurate reduction in η _s.

Figure 2

Numerical analysis of pump-to-diffraction conversion efficiency (η _diff) in a two-beam-pumped OPCPA system. (a) Calculated η _diff as a function of interaction length for different pump beam angular separations. (b) Calculated η _diff versus incident pump intensity.

Figure 2 Long description

The image consists of two panels, labeled a and b.

Panel a: A line graph plotting eta sub diff on the y-axis from 0 to 0.3 against Crystal length in millimeters on the x-axis from 0 to 3. Five curves represent different pump beam angular separations theta sub p 1 p 2.

- For 1 mrad (dark red), the efficiency remains near zero until 1 mm, then rises sharply to a peak of approximately 0.21 at 2.2 mm.

- For 8 mrad (red), the curve follows a similar path but peaks lower at 0.18.

- For 10 mrad (orange), the peak is approximately 0.15.

- For 15 mrad (light blue), the peak is significantly lower at 0.06.

- For 20 mrad (dark blue), the curve is nearly flat, reaching a maximum of only 0.01.

Panel b: A line graph plotting eta sub diff on the y-axis from 0 to 0.3 against Pump intensity in G W per cm squared on the x-axis from 0 to 30.

- The 1 mrad curve (dark red with circles) is a horizontal line constant at 0.21.

- The 8 mrad curve (red with squares) increases rapidly from 0.04 at 1 G W per cm squared and plateaus near 0.18.

- The 10 mrad curve (orange with circles) shows a steady logarithmic-like increase, reaching 0.16 at 30 G W per cm squared.

- The 15 mrad curve (light blue with squares) shows a linear increase from 0 to 0.08.

- The 20 mrad curve (dark blue with circles) shows a very slight linear increase, remaining below 0.02.

Parametric diffraction also exhibits a characteristic pump-intensity scaling behavior, as shown in Figure 2(b). At large angular separations (e.g., θ _p1p2 = 15 mrad), the diffraction efficiency η _diff scales nearly linearly with pump intensity, rising from less than 0.1% at I _p = 1 GW/cm² to 8.3% at I _p = 30 GW/cm². This dependence reflects the Γ² ∝ I _p scaling of nonlinear coupling strength.

In contrast, at small angular separations (e.g., θ _p1p2 = 1 mrad), η _diff saturates near 20%. This saturation originates from the coherent superposition of parametric interactions within the two-beam-pumping architecture. The signal complex amplitude A _s can be expressed as A _s = A _s1 + A _s2, where A _s1 and A _s2 represent the complex amplitudes arising from interactions with P₁ and P₂, respectively. Under ideal conditions of small angular separation (perfect phase matching) and weak signal injection, the intrinsic amplification and the parametric-diffraction channels contribute equally, yielding A _s1 = A _s2 = A _s12 = A _s21. Consequently, the signal conversion efficiency scales as η _s ∝ |A _s|² = |A _s1 + A _s2|², while the diffraction efficiency scales as η _diff ∝ |A _s12|² + |A _s21|². The total efficiency saturates at the quantum limit ω _s/ω _p = 64.4%, with η _diff reaching a theoretical maximum of 21.5% – one third of this limit.

Subsequently, we examine the influence of idler dissipation on parametric diffraction by simulating a two-beam-pumped QPCPA system. To evaluate the QPCPA performance, the dispersion parameters of pure YCOB without rare-earth ion doping are likewise employed, with the idler dissipation coefficient α treated as a numerically adjustable parameter. This approach isolates the specific contribution of idler absorption to the amplification process. For the following analysis, an idler dissipation coefficient of α = 40 cm^–1 and a crystal length of z = 20 mm are assumed.

Figure 3(a) illustrates that, in contrast to conventional OPCPA, the diffraction efficiency η _diff in a two-beam-pumped QPCPA exhibits distinct dynamics across gain regimes: while η _diff increases rapidly during the small-signal amplification region, it stabilizes at a minimal added value upon reaching saturation. This trend indicates the effective suppression of cross-coupling through idler dissipation. Quantitatively, at θ _p1p2 = 15 mrad, η _diff drops from 5.6% in the OPCPA case (Figure 2(a)) to 0.6% in the QPCPA configuration (Figure 3(a)) under identical pumping conditions.

Figure 3

Parametric-diffraction characteristics in two-beam-pumped QPCPA systems. (a) Evolution of pump-to-diffraction conversion efficiency (η _diff) with crystal length for different pump beam separations (θ _p1p2), at fixed pump intensity I _p = 20 GW/cm² and idler dissipation α = 40 cm^–1. (b) Angular dependence of pump conversion efficiency under the same dissipation condition (α = 40 cm^–1), demonstrating critical angle thresholds for parametric-diffraction suppression. (c) Comparative analysis of pump conversion efficiencies versus pump-to-signal injection ratio (I _p/I _s) for conventional OPCPA (α = 0) and QPCPA (α = 40 cm^–1) configurations. (d) Systematic suppression of η _diff with increasing idler dissipation strength at I _p = 20 GW/cm², revealing an inverse scaling relation between η _diff and α.

Figure 3 Long description

Panel a. X-axis is Crystal length in millimeters from 0 to 20. Y-axis is eta sub diff from 0 to 0.3. Five curves show that as theta sub p 1 p 2 increases from 1 to 20 milliradians, the diffraction efficiency drops significantly. The curves for 1, 8, and 10 milliradians rise sharply and plateau after 5 millimeters, while 15 and 20 milliradians remain near zero.

Panel b. X-axis is theta sub p 1 p 2 in milliradians from 0 to 30. Y-axis is Pump conversion efficiency from 0 to 0.8. Solid lines for eta sub s show a sigmoidal increase to a plateau of 0.65 as angle increases. Dashed lines for eta sub diff show a corresponding sigmoidal decrease to zero. Higher pump intensities I sub p shift these transitions to larger angles.

Panel c. X-axis is I sub p all over I sub s on a logarithmic scale from 10 to 10 super 6. Y-axis is Pump conversion efficiency from 0 to 0.6. Solid lines represent eta sub s and dashed lines represent eta sub diff. Black lines show alpha equals 0 and red lines show alpha equals 40 per centimeter. The Q P C P A configuration with dissipation (red) maintains higher signal efficiency and lower diffraction efficiency across the entire intensity ratio range.

Panel d. X-axis is alpha in 1 all over centimeter from 0 to 40. Y-axis is eta sub diff from 0 to 0.2. Two lines with circular markers show a linear decrease in diffraction efficiency as idler dissipation strength alpha increases. The line for theta sub p 1 p 2 equals 10 milliradians starts at 0.15 and ends at 0.1, while the 15 milliradians line starts at 0.05 and approaches zero.

The diffraction efficiency remains dependent on pump angular separation θ _p1p2, revealing an important design trade-off. To effectively suppress parametric diffraction while maximizing signal conversion efficiency, a sufficient angular separation is required. The critical angular separation exhibits a scaling dependence on pump intensity: approximately θ _p1p2 = 14 mrad at I _p = 20 GW/cm², versus θ _p1p2 = 8 mrad at I _p = 5 GW/cm², as shown in Figure 3(b). This demonstrates the inherent trade-off between pump beam density (number of beams) and pump intensity. Stronger pump intensity requires larger angular separations, thereby reducing maximum beam packing density.

The injection ratio I _p /I _s significantly influences system performance, as illustrated in Figure 3(c). In the I _p = 20 GW/cm² and θ _p1p2 = 10 mrad case, while strong seed injection (I _p /I _s < 10²) facilitates the suppression of parametric diffraction, the diffraction-induced energy loss becomes relatively insensitive to the injection ratio when I _p /I _s ≥ 10².

Parametric diffraction can be directly suppressed by increasing the idler dissipation coefficient α. As shown in Figure 3(d), the steady-state diffraction efficiency η _diff is inversely correlated with the dissipation coefficient α. Compared to conventional OPCPA (α = 0), at θ _p1p2 = 15 mrad, η _diff is reduced by 55.4% for α = 20 cm^–1 and by 89.3% for α = 40 cm^–1 in the QPCPA case. These findings confirm that enhanced idler dissipation effectively suppresses parasitic diffraction.

It is worth noting that the nonlinear phase shift of the signal induced by inter-channel crosstalk in the two-beam-pumped QPCPA configuration has been analyzed in our previous work^[ Reference Zhu, Zhang, Wang, Ma, Yuan, Zhang, Zhu and Qian ^{22 ]}. In the present experiment, the angular separation between the two pump beams is small (7–15 mrad), leading to a negligible phase mismatch. Consequently, the associated nonlinear phase shift is minimal, and its impact on the signal’s spectral phase (e.g., group delay dispersion) is expected to be insignificant under the conditions studied here.

3.1 Experimental setup

The experimental setup for the two-beam-pumped OPCPA/QPCPA system is shown in Figure 4. A single home-built ytterbium-doped yttrium aluminum garnet (Yb:YAG) laser (3 ps, 4 mJ, 100 Hz) provides both the primary pump and the seed via supercontinuum generation (SCG), ensuring passive synchronization and eliminating active timing control^[ Reference Indra, Batysta, Hříbek, Novák, Hubka, Green, Antipenkov, Boge, Naylon, Bakule and Rus ^{31 ,} Reference Špaček, Indra, Batysta, Hříbek, Green, Novák, Antipenkov, Bakule and Rus ^{32 ]}. The pump laser chain consists of a mode-locked fiber oscillator, a regenerative amplifier and a double-pass amplifier.

Figure 4

Experimental characterization of two-beam-pumped OPCPA/QPCPA. (a) Experimental setup. λ/2, half-wave plate; TFP, thin-film polarizer; SHG, second-harmonic generation; AP, iris aperture; SCG, supercontinuum generation; AC lens, achromatic lens. The diagnostic path includes CCD cameras for spatial profile monitoring at key positions (CCD-1 for pump characterization, CCD-2 for amplified signal analysis). (b) Measured beam profiles: the first column shows input pump beam interference patterns at CCD-1, confirming spatiotemporal overlap; the second and third columns display output signal profiles from CCD-2 for single-beam (P₁ or P₂ only) and two-beam (P₁+P₂) pumping configurations, respectively. (c) Spectral characteristics: input signal spectrum (orange curve, OPCPA-1), single-pump outputs (green and blue curves) and two-beam-pumped output (red curve). (d) Energy conversion characteristics plotting output signal energy versus total pump energy.

Figure 4 Long description

The image consists of four panels labeled a through d.

Panel a: A schematic of the experimental setup. It begins at the top left with a Y b colon Y A G pumping source at 1030 nanometers. The beam path splits into green pump lines and a red signal line. The pump path includes lambda forward slash 2 waveplates, T F P, and S H G crystals, splitting into two pump beams P 1 and P 2 that pass through telescopes. The signal path includes an A P, lens, Y A G crystal for S C G, a filter, and an A C lens. These paths converge at the O P C P A forward slash Q P C P A crystal. Diagnostic C C D 1 and C C D 2 cameras and an energy meter monitor the output.

Panel b: A grid of six beam profile heatmaps. Under C C D 1, two images show the pump interference patterns labeled Before crystal and Crystal location. Under C C D 2, four images show signal profiles for P 1, P 2, the combined P 1 plus P 2, and the Far field.

Panel c: A line graph of Intensity in arbitrary units versus Wavelength in nanometers from 600 to 1000. An orange curve for O P C P A 1 shows low intensity. Green P 1 and blue P 2 curves show moderate intensity peaks around 800 nanometers. A red P 1 plus P 2 curve shows the highest intensity peak.

Panel d: A line graph of output signal energy E sub s in microjoules versus total pump energy E sub p in microjoules. Three lines for P 1 (green), P 2 (blue), and P 1 plus P 2 (red) show a linear increase. The P 1 and P 2 lines reach approximately 230 microjoules, while the combined P 1 plus P 2 line reaches approximately 150 microjoules at the highest pump energy.

To prepare the seed, a small energy fraction (∼14 μJ) from a ps pump source is utilized for SCG. This setup employs a beam conditioning system comprising thin-film polarizers (TFPs), half-wave plates (λ/2) and an adjustable iris (AP) to enable precise control over the pulse energy and numerical aperture. Subsequently, a 300 mm plano-convex lens loosely focuses the beam into a 130 mm undoped YAG crystal. By optimizing the incident energy (considering both SCG threshold and crystal damage limits), focal position and beam parameters, we obtained a stable SCG spectrum spanning 540–950 nm with good spectral uniformity. The SCG seed is filtered via a short-pass dielectric filter with a 950 nm cutoff (Thorlabs FESH950) and collimated by a 150 mm achromatic lens (AC lens, f = 150 mm). Spectral characterization was performed using an Ocean Optics USB2000 spectrometer, and pulse energy was measured with a J-10MB-HE detector and a LabMax-TOP meter.

Regarding pump preparation, 515 nm pulses are generated via second-harmonic generation (SHG) in a 2.5 mm Type-I barium borate (BBO) crystal (θ _p = 23.4°). This process achieves a conversion efficiency of 42.8%, yielding a pump energy of 1.2 mJ. The residual 1030 nm fundamental light is further frequency-doubled to give 0.2 mJ (515 nm) for the first OPCPA stage. This initial stage utilizes a 3 mm Type-I BBO crystal (θ _p = 24.6°) in a noncollinear configuration (α _p = 2.6°), amplifying the seed pulse energy from approximately 50 nJ to 10 μJ. In the theoretical study, we used YCOB for the OPCPA baseline and Sm:YCOB for QPCPA, where these two crystals differ essentially only in the idler absorption coefficient – constituting a clean comparison. In the experiment, however, we adopted BBO for the OPCPA reference despite these differences. The primary reason is that BBO, under a walk-off-compensated noncollinear Type-I phase-matching configuration, generates strong parasitic SHG from both the signal and idler^[ Reference Lang, Harth, Matyschok, Binhammer, Schultze and Morgner ^{33 ]}, making the parametric-diffraction pattern directly visible to the naked eye. This visualization capability was essential for unambiguously demonstrating the underlying cascaded nonlinear processes.

The pre-amplified pulse is then injected into the main amplifier; this stage employs two-beam pumping with independent control of each pump arm. Pump energy is balanced and adjusted using a half-wave plate and a TFP in each path. The first pump beam (P₁) and the signal (S₀) are precisely positioned within the crystal principal plane in a walk-off compensating geometry. The second pump beam (P₂) is injected with an initial angular separation θ _p1p2 = 15 mrad relative to P₁. Temporal overlap better than 100 fs is achieved using adjustable delay lines in both pump paths. The pump beams are collimated to 0.95 mm diameter using a telescope system, while the signal beam is slightly larger than the pump beams, acting as a soft aperture.

3.2 Characterization of the amplified signal

The performance of the two-beam-pumped OPCPA was evaluated through the spatial beam profiles, spectrum and energy measurements of the amplified signal. As shown in Figure 4(b), spatial beam profiles of both pump and signal beams were captured using a high-resolution charge-coupled device (CCD) camera (BeamOn U3). Distinct interference fringes were observed at the crystal plane for the two pump beams, confirming their precise spatial and temporal overlap. Nevertheless, the amplified signal retained a near-Gaussian profile in both the near- and far-field under both single- and two-beam pumping conditions. This indicates that the scheme avoids the spatial interference modulation effects commonly encountered in coherent beam combining, ensuring robust beam quality.

As illustrated in Figure 4(c), the pre-amplified signal exhibited a smooth broadband spectrum (700–950 nm, 110.8 nm bandwidth FWHM), and single-pump configurations (P₁ or P₂ alone) produced nearly identical output spectra. In the two-beam pumping architecture, the signal spectrum was amplified uniformly across the entire bandwidth without the introduction of perceptible spectral distortion.

Energy conversion efficiency was comparable (~28.7%) across the two configurations at low pump energies (E _p < 300 μJ), verifying proper system alignment. However, at higher pump energy (E _p = 700 μJ), the two-beam pumping configuration exhibited a 36.5% reduction in efficiency. This observation is consistent with the predicted onset of parametric diffraction, which redirects pump energy away from the primary signal. These results quantitatively match our theoretical predictions (Section 2.2) regarding nonlinear wave-mixing at high intensities. These experimental results provide definitive evidence of parametric diffraction’s role in impairing energy extraction efficiency for multi-beam-pumped OPCPA systems.

3.3 Experimental observation of cascaded cross-couplings

The cascaded cross-coupling in two-beam-pumped OPCPA is directly visualized in Figure 5(a), which shows the beam pattern on a diffusing screen. The image includes the two pump beams (P₁, P₂), the amplified signal (S₀) and their corresponding idler waves (I₁, I₂), along with multiple higher-order diffraction spots – a signature of cascaded wave-mixing.

Figure 5

Experimental observation of parametric diffraction. (a)–(c) Diffraction patterns in two-beam-pumped OPCPA under different conditions: (a) at 7.5 GW/cm² pump intensity and 15 mrad pump beam separation; (b) at 13 GW/cm² (15 mrad) showing stronger diffraction; (c) at 13 GW/cm² with tighter 7 mrad spacing, displaying more diffraction orders. (d) Calculated angular positions of first- to third-order diffraction spots. (e) QPCPA at 13 GW/cm² (15 mrad) showing effective diffraction suppression. No artificial enhancement was applied to the QPCPA image; the lack of visible spots is due to the suppression of parasitic SHG.

Figure 5 Long description

A five-panel figure labeled a through e.

Panels a, b, and c are photos of diffraction patterns in a two-beam-pumped O P C P A. Each shows three vertical columns of light spots labeled Idler (red, left), Pump (green, center), and Signal (blue, right).

* Panel a: At 7.5 G W per cm squared and 15 mrad separation, distinct spots are visible. Labels point to specific orders: I sub 21212 down to I sub 12121 for Idler; P sub 21212 down to P sub 12121 for Pump; and S sub 21212 down to S sub 12121 for Signal.

* Panel b: At 13 G W per cm squared and 15 mrad, the spots are brighter and more elongated.

* Panel c: At 13 G W per cm squared and 7 mrad, the vertical density of spots increases significantly, showing higher diffraction orders.

Panel d is a scatter plot of angular positions. The x-axis represents degrees from negative 8 to 0. The y-axis represents degrees from negative 6 to 6.

* Blue triangles and circles (Idler) are clustered around x equals negative 8.

* Green triangles and circles (Pump) are clustered around x equals negative 3.

* Red triangles and circles (Signal) are clustered along the y-axis at x equals 0.

Panel e shows a two-beam-pumped Q P C P A at 13 G W per cm squared and 15 mrad. It consists of two side-by-side frames. The left frame (Pump) shows two bright green central spots with no surrounding diffraction. The right frame (Signal) shows a faint orange-yellow background with two very dim central spots, demonstrating suppressed parasitic diffraction.

To enable visual detection, the two-beam-pumped OPCPA employs a noncollinear walk-off compensating Type-I phase-matching geometry in BBO. Here, the signal is injected along the plane defined by the pump wave vector and the optic axis, on the same side of the Poynting vector walk-off of the e-polarized pump. This configuration promotes parasitic SHG for both the signal and idler beams, a phenomenon previously observed in BBO-based OPCPA systems^[ Reference Lang, Harth, Matyschok, Binhammer, Schultze and Morgner ^{33 ]}. The amplified signal (~820 nm) appears violet (~410 nm) due to SHG, consistent with the spectral dip in Figure 4(c). Similarly, the idler (~1445 nm) generates visible SHG components that exhibit angular dispersion across the beam profile, producing a color gradient from green to red.

The underlying process begins with conventional parametric amplification:

(6) $$\begin{align}{\mathrm{P}}_1-{\mathrm{S}}_0\to {\mathrm{I}}_1,\quad {\mathrm{P}}_2-{\mathrm{S}}_0\to {\mathrm{I}}_2.\end{align}$$

Cross-coupling between idlers and pumps then generates first-order diffracted waves:

(7) $$\begin{align}{\mathrm{P}}_1-{\mathrm{I}}_2\to {\mathrm{S}}_{12},\quad {\mathrm{P}}_2-{\mathrm{I}}_1\to {\mathrm{S}}_{21}.\end{align}$$

These diffracted waves then drive the second-order parametric diffraction via the following:

(8a) $$\begin{align}{\mathrm{P}}_1-{\mathrm{S}}_{21}\to {\mathrm{I}}_{121},\quad {\mathrm{P}}_2-{\mathrm{S}}_{12}\to {\mathrm{I}}_{212},\end{align}$$

(8b) $$\begin{align}{\mathrm{S}}_{12}+{\mathrm{I}}_1\to {\mathrm{P}}_{121},\quad {\mathrm{S}}_{21}+{\mathrm{I}}_2\to {\mathrm{P}}_{212}.\end{align}$$

The cascaded nonlinear interactions further produce the third-order diffraction as described by the following:

(9a) $$\begin{align}&{\mathrm{P}}_{121}-{\mathrm{I}}_2\to {\mathrm{S}}_{1212},\quad {\mathrm{P}}_{212}-{\mathrm{I}}_1\to {\mathrm{S}}_{2121},\end{align}$$

(9b) $$\begin{align}&{\mathrm{P}}_{121}-{\mathrm{S}}_{21}\to {\mathrm{I}}_{12121},\quad {\mathrm{P}}_{212}-{\mathrm{S}}_{12}\to {\mathrm{I}}_{21212},\end{align}$$

(9c) $$\begin{align}&{\mathrm{S}}_{12}+{\mathrm{I}}_{121}\to {\mathrm{P}}_{12121},\quad {\mathrm{S}}_{21}+{\mathrm{I}}_{212}\to {\mathrm{P}}_{21212}.\end{align}$$

Figure 5(d) shows the calculated angular distribution of these diffraction orders based on phase-matching conditions. The excellent agreement between the measured spot positions in Figure 5(a) and the simulated angles in Figure 5(d) confirms that the observed pattern arises from phase-matched parametric diffraction rather than scattering light.

Furthermore, we characterize the dependence of parametric diffraction on key pump parameters. As the pump intensity increases from 7.5 to 13 GW/cm² (Figures 5(a) and 5(b)), higher-order diffraction becomes markedly more discernible, and the pump-to-diffraction conversion efficiency increases approximately twofold, from 19.8% to 35% (Figure 6(a)). This strong nonlinear scaling reflects the enhanced efficiency of parametric diffraction at elevated intensities, driven by strengthened nonlinear coupling.

Figure 6

Experimental results of parametric-diffraction efficiency η _diff. (a) Measured η _diff versus pump intensity at 15 mrad beam angular separation. (b) Experimental angular dependence of diffraction efficiency at fixed 13 GW/cm² pump intensity.

Figure 6 Long description

Two-panel line graph.

Panel a. The y-axis is eta sub diff ranging from 0 to 0.6. The x-axis is I sub p 1 in G W / cm super 2 ranging from 0 to 15. A blue line with circular markers for Two-beam-pumped O P C P A shows a steady linear increase from approximately 0.08 at 2.5 G W / cm super 2 to 0.35 at 13 G W / cm super 2. A red line with circular markers for Two-beam-pumped Q P C P A shows a much flatter trend, starting at 0.04 and reaching a plateau near 0.08 beyond 10 G W / cm super 2.

Panel b. The y-axis is eta sub diff ranging from 0 to 0.6. The x-axis is theta sub p 1 p 2 in mrad ranging from 0 to 30. The blue line for O P C P A starts at a peak of 0.42 at 7 mrad and shows a downward trend with some fluctuations, dropping to 0.2 at 22 mrad. The red line for Q P C P A starts lower at 0.18 at 6 mrad, remains relatively stable until 10 mrad, then drops sharply to a plateau around 0.08 between 12 and 22 mrad, finally tapering off toward zero at 28 mrad.

Similarly, the angular separation is a critical determinant of diffraction behavior. Reducing the pump beam angular separation from 15 to 7 mrad at a constant pump intensity of 13 GW/cm² (Figures 5(b) and 5(c)) yields more prominent diffraction patterns and additional higher-order spots, accompanied by a 25.1% increase in diffraction efficiency (Figure 6(b)). This pronounced angular sensitivity stems from the phase-matching conditions that govern energy transfer between the interacting beams.

3.4 Suppression of parametric diffraction via idler dissipation

For the two-beam-pumped OPCPA scheme, a 3 mm BBO crystal is used as the nonlinear medium. To implement the QPCPA configuration, we employ a 3.9-cm-long Sm:YCOB crystal with 30% Sm³⁺ doping, providing an average idler absorption coefficient of approximately 1.5 cm^–1[ Reference Ma, Wang, Yuan, Xie, Xiong, Tu, Tu, Shi, Zheng and Qian ^{23 ]}. It is important to note that the relevant physical parameter for the suppression mechanism is the absorption coefficient (~1.5 cm^–1) rather than the Sm³⁺ doping concentration (30%) itself, as the absorption depends on the specific material context. Given the peak pump intensity of approximately 13 GW/cm² and the nonlinear refractive index of Sm:YCOB (n ₂ ≈ 3.2 × 10⁻¹⁶ cm²/W), the calculated B-integral is approximately 1 rad, well below the typical threshold for small-scale self-focusing (3 rad). Hence, no B-integral-related degradation or crystal damage was observed under our experimental conditions.

Unlike the BBO-based OPCPA, parasitic SHG in Sm:YCOB is too weak for direct visual observation of the diffracted idler and signal beams. To detect the diffracted light, we therefore used laser viewing cards to record the spatial distribution of the emitted beams. As shown in Figure 5(e), the QPCPA configuration exhibits negligible diffraction. Both the number of observable diffraction orders and their intensities are drastically reduced compared to the conventional OPCPA scheme. This suppression results from strong absorption of idler waves (1200–1600 nm) by Sm³⁺ ions, which interrupts the cascaded nonlinear processes that generate higher-order diffraction.

Quantitative analysis of the pump-to-diffraction conversion efficiency (η _diff) further reveals the performance improvement achieved by QPCPA. As shown in Figure 6(a), at a fixed angular separation of 15 mrad and pump intensities ranging from 2.6 to 13 GW/cm², the conventional OPCPA exhibits a sharp increase in η _diff, reaching 35% at 13 GW/cm². In contrast, η _diff for QPCPA remains stable near 7%. The angular dependence at I _p = 13 GW/cm² further elucidates this suppression effect (Figure 6(b)): the OPCPA shows a dramatic escalation in η _diff as the separation decreases, reaching 42.8% at a 7 mrad spacing, whereas the QPCPA mitigates this loss to only 17.2% under the same conditions. At larger pump angular separations (e.g., 22 mrad), the QPCPA achieves a 67.2% reduction in η _diff relative to the OPCPA configuration.

It should be noted that the idler dissipation coefficient in the currently available Sm:YCOB crystal is approximately 1.5 cm⁻¹, limited by the Sm doping concentration of approximately 30%. This value is significantly lower than the ideal case considered in the numerical simulations (α = 40 cm⁻¹). Nevertheless, even at this modest dissipation level, the idler absorption mechanism is expected to effectively disrupt the cascaded nonlinear processes that drive parametric diffraction, enabling observable suppression in the subsequent experimental results.

It is also worth emphasizing that pump beam interference does not degrade the main signal beam quality due to the spatial-filtering effect of phase matching. Instead, beam-quality degradation arises primarily from the parametric-diffraction beams, which are suppressed by idler dissipation. Hence, QPCPA inherently protects both the conversion efficiency and beam profile. These results demonstrate that idler dissipation not only suppresses parametric diffraction but also provides significantly greater flexibility in system design. By maintaining low parametric-diffraction losses across a broad range of pump intensities and angular separations, the QPCPA can achieve both high efficiency and a compact geometry, effectively overcoming the fundamental trade-offs inherent in parametric diffraction in conventional multi-beam OPCPA systems.