1. Introduction

Temporal modes of light [1] are a burgeoning new subspace for both classical and quantum information encoding [2–7]. The temporal mode (TM) bases are sets of complex functions that span the space of the longitudinal shapes of electromagnetic wave packets. They are discrete and in principle infinite dimensional, and truly orthogonal. These orthogonal modes have overlapping spectra at the same carrier frequency, fully overlap in time (unlike time-bin encoded pulses), and share the same polarization and transverse spatial mode properties, making them convenient for long-distance communication and interfacing with optical memories [8–11]. Co-propagating TMs maintain relative coherence across the entire mode space, and experience the same reversible unitary dispersion during propagation in typical media. In order to fully exploit TMs as a resource, we need highly TM-selective, high-fidelity devices that can generate, sort, store, measure, and generally manipulate them, even at the single-photon level [12–22].

Many optically-pumped processes, such as atomic-ensemble Raman memories [8, 23–26] and nonlinear optical frequency conversion [27] via three/four-wave mixing [2, 28–30], are already known to be TM sensitive when using pulsed pumps [31–33], and show TM discriminatory behavior at low coupling strengths [12, 34–36]. Increasing the pump intensities causes the TM discrimination to peak and regress to smaller values, thus ultimately limiting the maximum selectivity of straight-forward implementations of such processes [36]. This loss in selectivity is understood to be a result of time-ordering effects between interacting temporal wave-packets of the participating fields (be they electromagnetic, polarization-phononic, acoustic, etc.) convecting through each other at varied group velocities [36, 37]. This selectivity limit, however, can be asymtotically overcome [38] by using interferometric techniques (“TM interferometry”) in cascaded setups to enhance TM separability up to 100%, even at large coupling strengths [37–42]. This new capability opens new avenues for information processing, including the complete set of operations needed for linear-optical quantum information processing [5].

Quantum frequency conversion through χ⁽²⁾- or χ⁽³⁾- nonlinear media is gaining in usefulness as an all-optical means of implementing a fully-programmeable “quantum pulse gate,” whose role is to multiplex and demultiplex TMs [12]. The ability of nonlinear frequency conversion to unitarily reshape, and preserve quantum information stored in the TM basis is well known [27, 43,44]. The TM-selectivity of frequency conversion [36] suffers from the above-mentioned time-ordering-induced upper limit, necessitating an eventual exploration of TM interferometry. Experimental exploration of TM discrimination is still in its nascent stages [14–16,20]. While the theoretically predicted selectivity limit still stands to be broken, other more specific predictions of said theory have yet to be validated. In this paper, we redress this issue both experimentally and theoretically, as well as provide a simple parameter-set guideline to aid in designing similar processes in other systems and spectral regions.

In the following, we first model pulsed-pump mediated, second-order-nonlinear-optical frequency conversion between a signal band and an idler band as a set of coupled-mode equations, and define a TM-selectivity figure of merit to characterize the process. We then showcase high TM selectivity in a parameter regime where the pump and signal pulses propagate with the same group velocity, but the idler group velocity is significantly different. We illustrate specific predictions of the model for various input conditions, and propose a means of adequately approximating the aforementioned parameter regime by choosing band-carrier frequencies around the ones phase matched for second-harmonic generation (SHG) in typical off-the-shelf nonlinear waveguides. We then demostrate TM-selective frequency conversion in a 5 µm wide, 5 mm long, MgO:PPLN waveguide periodically poled for SHG from 816.6 nm to 408.3 nm at 24.25°C. We employ a Kerr-modelocked, ultrafast titanium-sapphire laser and a folded Treacy-grating pair pulse shaper with a reflective, 2D spatial light modulator, to situate and shape the pump and signal bands at 821 nm and 812.2 nm respectively, with bandwidths ~2.5 nm. We use pump pulses with energies of order 10 nJ, and pulses of temporal widths of order 500 fs, to verify all the predicted features of our model. The first two numerically computed, natural Schmidt modes of the model achieved a 4.7-to-1 contrast (85% vs. 18%) in conversion efficiencies for the two tested pump shapes in both theory and experiment. While the experiments are carried out using weak coherent-state signals, theory indicates the same conversion efficiencies and TM selectivities would be observed using heralded single-photon wave packets [34].

2. Theory and modelling

2.1. Equations of motion and selectivity

The model for frequency conversion (FC) of temporal wave-packet modes in a χ⁽²⁾-nonlinear waveguide may be expressed as a pair of coupled-mode equations involving the electric-field envelopes, treated as quantum field operators [36]. We designate the letters p, s, and r to denote electromagnetic fields within the three participating frequency bands, namely, the pump, signal, and idler bands, respectively. The band-central, or carrier frequencies have to be constrained by energy conservation (ω_r = ω_s + ω_p), and the nonlinear-optical waveguide used for frequency conversion is assumed to be periodically poled to ensure proper phase matching (β_r − β_s − β_p − 2π/Λ = 0, where β_j are wavenumbers and Λ is the poling period) for the pulses’ carrier frequencies. If the pulses in question are sufficiently narrowband, we can ignore second- and higher-order dispersion for all three bands, and write the coupled-mode equations as

Here, γ is a composite parameter denoting the strength of the interaction, and is linearly dependent on the medium nonlinearity (χ⁽²⁾), and the square root of the pump-pulse energy, as well as the overlap integral of the transverse spatial-mode functions of the waveguide modes at the band-carrier frequencies. A_p (t) is the square-normalized, complex pump-pulse envelope shape function. The parameters ${\beta }_j^{\prime }\equiv {\partial }_{\omega }\beta (\omega ){|}_{{\omega }_j}$ are the corresponding group-slownesses (inverse group velocities) for the bands j ∈ {r, s, p}, and the operators ${\widehat{A}}_j$ (z, t) denote field-annihilation operators for square-normalized, slowly-varying, complex amplitudes of the temporal modes [1, 5] within the medium at location z along the propagation direction.

We require no additional noise operators in Eqs. (1a) and (1b) since quantum FC is in principle a noiseless, unitary process [45]. In the single-photon case, these operators can be replaced by quantum wavefunction amplitudes [1, 34]. For weak coherent pulses (as in the case of our experiment), they can be replaced by complex functions denoting the classical pulse-envelope shapes, because the same equations hold. The strong classical pump pulse A_p (t) is assumed to be undepleted in amplitude during FC. The medium is of length L, and z = 0 defines its input face. The nonlinearity of the medium is assumed uniform throughout its length.

For given values for all of the parameters defined above, and a known pump-pulse shape A_p (t), FC in a finite medium can be treated as a scattering process that relates the input temporal modes ${\widehat{A}}_j$ (0, t′) to the output modes ${\widehat{A}}_j$ (L, t) via a set of Green function (GF) relations:

Each output field j ∈ {r, s} is linearly dependent on both input fields k ∈ {r, s}, where t′ denotes an input time and t denotes an output time. This formalism is convenient for analysis of temporal-mode selectivity [36, 38], as the separability of the four Green function subkernels can be quantified via their singular-value decomposition [43, 46, 47]:

The functions ψ_n(t′), ϕ_n(t′) are the input “Schmidt modes” and Ψ_n(t), Φ_n(t) are the corresponding output Schmidt modes for the r and s bands respectively. The Schmidt modes define the sets of “natural temporal modes” for the FC problem given a particular pump shape and medium characteristics. For a given integer index n, the quartet of modes are related to each other in a beamsplitter-like transformation through the Schmidt coefficients (τ_n, ρ_n), which obey the unitarity constraint, |τ_n|² + |ρ_n|² = 1. In simple terms, if the input state in the s-band were to be a single photon (or a weak, coherent pulse) in the temporal mode ϕ_n(t′), and the r-band input were to be vacuum, then the probability (efficiency) of frequency conversion from s-band to r-band would be |ρ_n|². The converted component exits in the r-band in mode Ψ_n(t), and the unconverted component (occurring with probability/efficiency |τ_n|²) exits in the s-band in the mode Φ_n(t). Note that each of the four sets of Schmidt modes forms a complete basis set for its band and input/output context (i.e., {ψ_n(t′)} is a basis set for arbitrary r-input modes, {Φ_n(t)} for s-output modes, and so on). The effect of FC on an arbitrary input temporal mode can be easily computed by expressing said input state in the natural input Schmidt-mode basis of the device and employing the Schmidt-coefficient beamsplitter relations. The remarkable aspect of the beamplitter-like transformation is that, when working with the natural Schmidt modes defined by the process, only the modes of the same Schmidt index n interact. The temporal Schmidt modes of different n are transparent to one another. (The challenge in the experiment is to shape the TMs of the signal pulses being used to match the natural Schmidt modes of the process, which are in turn defined by the pump shape and properties of the medium.)

We choose to arrange the Schmidt modes in the four sets, and the corresponding Schmidt coefficients, in decreasing order of conversion efficiency (CE), such that |ρ₁|² ≥ |ρ₂|² ≥ |ρ₃|² … and so on. An FC device with perfect mode discrimination would have a non-zero ρ₁, and ρ_n>1 = 0, meaning the GF subkernel is separable but not necessarily 100% efficient. A device with perfect mode selectivity would have ρ_j = δ_j,₁. To characterize mode-selectivity, we define a figure of merit called selectivity $S={\left|{\rho }_1\right|}^4/{\displaystyle {\sum }_{n=1}^{\infty }{\left|{\rho }_n\right|}^2}$ [36]. It has previously been shown that perfect selectivity cannot be achieved in simple inter-pulse interaction systems due to time-ordering corrections [37]. But selectivity asymptotically approaching 100% can be achieved in cascaded, multi-stage FC implementations [37–39, 48]. This paper does not address multi-stage FC.

2.2. Group-velocity matched regime

We can condense all of the model parameters into three dimensionless quantities in order to aid mapping settings and results from diverse FC systems to this model. These are [36]:

Through an exhaustive numerical exploration [36], we have previously determined that for good GF separability at low pump energies, as well as the best selectivity (~0.83) at higher pump energies, the best parameter regime is ξ ≫ ζ ≫ 1. By designing the system such that the group-velocity of the pump pulse is identical to that of one of the other bands (the s-band, for the definitions in Eqs. (4)), and highly different from that of the remaining (r) band, we can have ξ → ∞. We call this condition the group-velocity matched (GVM) regime, and it was first considered in detail in [12]. The magnitude of ζ is limited by the maximum fabricable length of the nonlinear waveguide, and the requirement that the pump pulse be reasonably narrowband (giving a lower bound on temporal width) to avoid higher-order dispersion.

The Green function solutions for the GVM regime are known in closed analytical form [36,44], which aids in numerical analysis, as well as physical system design. We have validated these analytical solutions using regime-agnostic wave-mixing simulations based on a numerical split-step implementation of the propagation and interaction of the various fields [36, 38, 39, 48]. Specifically, we propagate the fields and apply dispersion in the Fourier domain, and “mix” them using fourth-order Runge-Kutta in the time domain, alternating between the two for every iteration. We compute the Green functions by running the simulation for a basis set of input conditions, and computing the overlap of the resulting outputs with another spanning basis set of functions. The numerical simulations verify that small deviations from the assumed conditions (GVM regime, absense of higher-order dispersion, etc.) do not cause significant departures from the predictions of the analytical solutions. We present some of these predictions here.

In the GVM regime, the group-velocity matched signal (s-band) copropagates with the pump pulse in the medium, and the unmatched idler (r-band) pulse falls behind (or, in the case of anamolous dispersion, speeds ahead) of the other two pulses. Consequently, as long as ζ ≫ 1 in addition to the GVM condition, both the input and output s-band Schmidt modes (ϕ_j (t′), Φ_j (t)) will have the same temporal widths as that of the pump pulse, and the r-band input and output modes (ψ_j (t′), Ψ_j (t)) get temporally stretched by a factor of ζ.

In Fig. 1, we plot the first two s-band input Schmidt modes and the first two r-band output Schmidt modes for a Gaussian-shaped pump pulse, and a low coupling strength ( $\tilde{\gamma }=0.141$). The temporal stretching effect between the mode widths of the two bands by a factor of ζ = 20 is clearly illustrated. Also note in Fig. 1(c) that although the CE of the first Schmidt mode is miniscule, it is large compared to that of the second Schmidt mode. The target TM (for optimum FC), which is the first s-input Schmidt mode in Fig. 1(a), is nearly identical to the pump shape, while the second mode is temporally orthogonal to the first.

 

Fig. 1 Numerically simulated results for $\tilde{\gamma }=0.141$, ζ = 20, and ξ = ∞ (perfect group-velocity matching) with Gaussian-shaped pump pulse. (a) The first two s-band input Schmidt modes. (b) The first two r-band output Schmidt modes. (c) Conversion efficiencies of first four dominant Schmidt modes (Selectivity S = 0.071). Note that ϕ₁(t′) closely resembles the pump shape, and the temporal width of Ψ₁(t) is larger by a factor of ζ = 20.

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Figure 2 shows the same data as in Fig. 1, but for a higher coupling strength $(\tilde{\gamma }=0.707)$, and two different pump-pulse shapes: Gaussian (Fig. 2(a,b,c)), and first-order Hermite Gaussian (Fig. 2(d,e,f)). Note that the CE of the second Schmidt mode is no longer negligible.

 

Fig. 2 Numerically simulated, dominant Schmidt modes and conversion efficiencies for $\tilde{\gamma }=0.707$, ζ = 20, and ξ = ∞ (perfect group-velocity matching) with Gaussian (a, b, c) and first-order Hermite-Gaussian (d, e, f) pump pulses. Due to the lack of any complex phase structure in the pumps, the Schmidt modes end up being real valued (up to overall phase). Note that the first (n = 1) s-band input Schmidt modes (a, d) resemble the group-velocity matched, pump-pulse shapes up to temporal skewing, whereas the first r-band output Schmidt modes (b, e) get stretched relative to pump width by factor ζ. Also note the independence of the dominant Schmidt coefficients (c, f) from pump-pulse shape. Here, selectivity S = 0.77.

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In the GVM regime, even at large pump-pulse energies (coupling strengths), the complex shape of the pump-pulse envelope fully determines the shape of the dominant s-band Schmidt modes (Fig. 2(a,d)). The dominant r-band Schmidt modes, however, are influenced very little by the pump shape, and instead reflect the variation of the medium nonlinearity along the waveguide propagation direction (Fig. 2(b,e)), which in our case is considered to be uniform. At low pump-pulse energies, or small $\tilde{\gamma }$, the first (n = 1) s-band input and output Schmidt modes are nearly identical to the pump pulse shape (and the first r-band modes look like flat, square pulses). But at higher $\tilde{\gamma }$, the first Schmidt mode changes into a temporally skewed version of the pump pulse (see Fig. 3). The Schmidt coefficients (and therefore, the Schmidt-mode CE) are independent of the pump-pulse shape. Figure 3(c) depicts the rapid rise of the CEs of the second and higher Schmidt modes with increasing $\tilde{\gamma }$, which results in a cap on the maximum achievable selectivity.

 

Fig. 3 Numerically simulated, first (n = 1) s-band input Schmidt modes (ϕ₁(t′)) for ζ = 20, ξ = ∞, and various $\tilde{\gamma }$ with (a) Gaussian pump pulses, and (b) first-order Hermite-Gaussian pump pulses. These plots demonstrate the $\tilde{\gamma }$-dependent temporal skewing effect. (c) The conversion efficiencies for the first four dominant Schmidt modes. Plot (c) is identical for both pump-pulse shapes. The selectivities are given in the legend.

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Curiously, even though the s-band first/dominant Schmidt mode does not exactly resemble the original pump shape (especially at large $\tilde{\gamma }$), mutually orthogonal pump pulses seem to have mutually orthogonal corresponding s-band first Schmidt modes at any $\tilde{\gamma }$. We have only numerically confirmed this result up to discretization error, and if analytically true, could prove useful for applications such as TM tomography.

In our experiment, we validate these precise predictions by following two approaches. Firstly, for given pump-pulse shape and energy, we maximize the signal CE by attempting to match the s-input Schmidt modes predicted by theory. We numerically precompute the first Schmidt-mode shape for both Gaussian and Hermite-Gaussian pump pulses for a continuous range of $\tilde{\gamma }$. Then the problem of matching the Schmidt mode at a given pump power reduces to CE maximization through a scan of the single parameter $\tilde{\gamma }$.

A second means of verifying the model is to keep the signal input pulse shapes static, but delay them with respect to the pump pulse and chart the CE. For Gaussian and first-order Hermite-Gaussian pump and signal shapes, four surface plots of CE for various input inter-pump-signal delays (denoted by τ_d) and pump energies have been numerically generated and plotted in Fig. 4. The s-band Schmidt mode distortions show up as temporal shifts and lobe-peak asymmetries at higher $\tilde{\gamma }$.

 

Fig. 4 Numerically simulated, 3D surface plots of conversion efficiencies (CE) vs. input pump-signal time delay by pump width (τ_d/τ_p) for various $\tilde{\gamma }(\propto \sqrt{P_{\text{pump}}})$ for (a) Gaussian pump and signal, (b) Gaussian pump and first-order Hermite-Gaussian signal, (c) first-order Hermite-Gaussian pump and Gaussian signal, and (d) first-order Hermite-Gaussian pump and signal pulse shapes. Note the temporal skewness at higher $\tilde{\gamma }$, reflected in shift of CE maxima with respect to mesh grid, as well as asymmetry in lobe peak heights.

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2.3. Physical system design and parameter selection

In order to take advantage of theoretical predictions, one must choose the physical system parameters to match the regimes of applicability of said theory. In the context of TM-selective frequency conversion, the GVM or near-GVM condition (ξ ≫ ζ ≫ 1) must constrain the selection of waveguide material, length, band central frequencies, and TM bandwidths. At the time of this writing, only two other groups [14–16, 20] have addressed experimental frequency conversion in a TM-selective context.

The Silberhorn group [14, 15, 20] use homebuilt, periodically-poled lithium niobate (PPLN) waveguides with a poling period of about 4.4 µm. Such a period gives the optimum phase-matching for sum frequency generation (SFG) from bands centered near 1550 nm and 860 nm, into the band around 550 nm. They engineer their waveguide dispersion to achieve perfect GVM (ξ → ∞) at these wavelengths, and compensate for fabrication errors by tuning the waveguide temperature in the 150 − 200°C range. They can thus afford to use longer waveguides (~ 17–27 mm) and short pulse lengths (~ 200 fs) and obtain large ζ values without having to worry about signal-pump inter-pulse walkoff within the medium during propagation.

Although exact GVM is optimum for TM-selectivity, one can deviate from perfect signal-pump GVM and still retain most of its advantages, as long as ξ ≫ ζ ≫ 1 is satistied. The Kumar/Kanter group [16] hit upon an interesting solution that allows for minor deviation from GVM. They used a 52 mm long, custom PPLN waveguide designed for second-harmonic generation (SHG) from 1544 nm into 772 nm at 73.4°C, and situated their pump (1556.6 nm) and signal (1532.1 nm) bands symmetrically on either side of the SHG wavelength, yielding ξ ≈ 215. Their pump/signal sources and pulse shapers restricted their temporal widths to around 5 ps, implying ζ ≈ 3. The theory predicts [36, 48] that their selectivity would improve significantly with larger pump/signal bandwidths, as we have used here.

We also employ an SHG waveguide (816.6 nm to 408.3 nm at 24.25°C) for FC by situating the signal and pump bands on either side of the red SHG pump band. Typical SHG acceptance bandwidths are very narrow. As long as the FC-pump band is sufficiently detuned from the SHG-pump wavelength, so as to avoid pump-only spurious blue-light generation, the relative spectral flatness of normal dispersion ensures near-GVM conditions (see Fig. 5(a)). One must choose a temporal width that is small enough to ensure ζ ≫ 1 (for large idler-pump walkoff), but wide enough to ensure that the pump-signal inter-pulse walkoff within the medium remains a small fraction of the total pulse widths (ξ ≫ ζ). SHG waveguides, when pumped at the sum frequency can generate degenerate photon-pairs via spontaneous parametric down conversion (SPDC). The joint-spectral amplitude of the pairs are tightly anticorrelated in frequency (Fig. 5(b)), reflecting the narrowness of the SHG red-pump acceptance band [49]. But the individual photons of the pair would be wideband, allowing for sum frequency generation from two highly detuned frequency bands on either side of the SHG-pump wavelength. Here, SFG is really a band-restricted inverse of SPDC. This behavior makes off-the-shelf waveguides suitable for TM-selective FC experiments.

 

Fig. 5 (a) The wavenumber (β) and the group velocity (v_g = dω/dβ) vs. wavelength (λ) for a typical 5 µm wide, periodically-poled, MgO:LN waveguide. Also shown are the r-, s-, and p− bands that we utilize for SFG. (b) Numerically computed, peak normalized joint-spectral amplitude of the degenerate, Type-0 SPDC photon pairs that would be generated in 5 mm of such a waveguide when pumped with 0.1 nm wide blue light in the r-band. Also shown are the signal (s) and pump (p) bands for the SFG process, which are situated symmetrically on either side of the red second-harmonic generation pump wavelength at 816.6 nm. Due to the frequency anti-correlatedness of the SPDC joint-spectral amplitude, both s- and p-bands need to contain non-zero optical energies for SFG to occur into the r-band.

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3. Experiment

3.1. Apparatus

We derived both our ~ 500 fs long, strong pump and weak signal pulses by reshaping the spectrum of a homebuilt, Kerr-lens mode-locked, ultrafast (80 fs) pulsed Ti:Sapph laser. We tuned its cavity to cause it to lase at 821 nm with a FWHM bandwidth of around 12 nm, at a pulse rate of 76 MHz. This beam was spatially expanded to a transverse width of ~ 10 mm and sent into a folded, 4 f -configured Treacy-grating-pair [50] pulse shaper, which uses a reflective spatial-light modulator (SLM) in its Fourier plane (Fig. 6). The pulse shaper utilized a 1800 lines/mm holographic grating in near-Littrow mode, and a cylindrical lens of focal length 250 mm to focus the wavelets onto the SLM. The lens is cylindrical in order to spread the beam intensity vertically so as to avoid damaging the SLM. This gave us a horizontal spot size of ~ 30 µm for a given wavelength. We used a custom-made biprism to change the height of the forward and reflected light to keep the paths symmetric, whilst sacrificing exact normal incidence on the SLM pixels, which are designed for normal incidence.

 

Fig. 6 Experimental setup. The holographic grating was used in near-Littrow mode for both incoming and outgoing beams. The m = 1 order reflection from two separated vertical blazed gratings rendered on the SLM was recombined on the holographic grating. Both the input and output couplers of the 5 µm wide, 5 mm long MgO:PPLN waveguide were single-element aspheric lenses of focal length f = 11 mm. DM stands for dichroic mirror. Some frequency filters are not shown.

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For the SLM, we used a Meadowlark 8-bit, 2D, phase-only liquid-crystal spatial light modulator of 1920 × 1152 pixel resolution and array size of 17.6 mm × 10.7 mm. The pixels were squares of size 9.2 µm and the fill factor was 95.7%. The spatial dispersion of the shaper at the SLM was 0.011 nm/pixel, although, the actual shaper resolution is limited by the spot size. In order to modulate both amplitude and phase, we used Silberberg group’s [51] first-order approach, where we form a vertical blazed grating pattern on the SLM and pick off its m = 1 reflection as the output. Different phase ramps may be applied to different wavelengths (at different horizontal positions) to affect the amount of power in the m = 1 reflection, and the phases can be manipulated by vertically shifting the blazed grating upwards/downwards. We used a vertical period of 44 pixels in the pump band, and 50 pixels in the signal band, as shown in Fig. 7.

 

Fig. 7 (a) Typical 8-bit grayscale phase mask applied to the SLM to generate a Gaussian shaped signal pulse (left band) and a first-order Hermite-Gaussian pump pulse (right band). Here, the horizontal coordinate maps to different wavelengths. The phase-contrast of the vertical gratings determines amplitude. Vertically shifting the gratings can affect phase (note relative shift between the two grating patterns generating the two frequency lobes of the pump). The curved pattern is for chirp compensation (measured using a commercial FROG/GRENOUILLE 8-50-USB), and the linear spectral phase on the signal shifts it in time relative to the pump. (b) Spectra of original Ti:Sapph laser, the signal, and the pump (first-order Hermite Gaussian, for example) generated by the SLM phase mask in (a). The three different spectra were captured under different conditions and hence, the relative heights are not to scale.

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For the bandwidths we chose to work with (2 − 2.5 nm), the pump band had sufficient power to allow for a frequency-resolved-optical-grating (FROG) measurement. A commercial GRENOUILLE 8-50-USB by Swamp Optics was employed for this. This allowed us to characterize and compensate for the frequency chirp suffered through traversal of the beam through multiple optical elements by applying quadratic and quartic phase corrections on the SLM phase mask (Fig. 7(a)). The resulting reduction in the pulse’s temporal width was independently validated on a homebuilt autocorrelator. Similar chirp compensation was optimized for the signal band by maximizing the CE at a low pump power in the nonlinear waveguide. Some linear spectral phases were added to one or both bands to overlap the pulses in time. The bandwidths for both bands were small enough to overcome any adverse effects from transverse-spatial chirp or pulse-front tilt induced in the beam by the shaper.

For frequency conversion, we used a standard, off-the-shelf, 5 mm long, SHG chip made of MgO-doped lithium niobate from AdvR. It had been poled at a period of ~ 3.1 µm for Type-0 SHG from wavelengths around 810 − 820 nm, and consisted (as is typical) of several waveguides of varying widths. These waveguides had been etched using a diffusive ion-exchange technique, yielding an effective depth of 5 − 8 µm. We chose to work with the 5 µm wide waveguide, which at a temperature of 24.25°C phase-matched SHG from 816.6 nm to 408.3 nm. The waveguide was placed in a homebuilt oven whose temperature was controlled using a PID circuit. We confirmed the narrowness of the SHG red-pump acceptance bandwidth (< 0.5 nm) as well as the wide SFG acceptance bandwidth (> 15 nm, see Fig. 5(b)) by scanning the frequencies of single-band and dual-band red inputs generated from the pulse shaper [49] to generate blue light.

We coupled light into and out of the waveguide using f = 11 mm aspheric lenses, which after some post-pulse-shaper beam resizing gave us a red-light coupling efficiency of about 30%. The blue and red beams were separated at the output by a Thorlabs DMLP650 longpass dichroic mirror, and the pump and signal bands in the red beam were split by angle-tuning two Semrock FF01-810/10 bandpass filters. Although the SHG/SFG process for the phase-matched frequencies dominates when controlled for input powers, imperfections in the waveguide resulted in spurious SHG blue light at all “red” wavelengths. Our process used a strong pump, but a weak signal, which generated some pump-induced spurious SHG (Fig. 8) at higher pump powers, thus possibly violating the unchanging, undepleting pump approximation. The SFG and spurious-SHG blue bands were separated by an angle-tuned Semrock TBP01-400/16 bandpass filter.

 

Fig. 8 Blue light spectra generated from the waveguide in a typical run. The SFG peak requires both the pump and signal to be present at the input, whereas the pump-only SHG peak remains even without the signal, and occurs due to imperfections in poling. The latter peak can compete with the former at higher pump powers. To use very weak signals (say sub-single-photon level), very tight spectral filtering will be needed at the blue output arm.

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3.2. Measurements

The choice of central wavelengths for the pump (821 nm) and the signal (812.2 nm) bands afforded us ξ > 200. The pump-pulse width was set by the pulse shaper to be ~ 530 fs, yielding ζ ≈ 20, landing us well within the near-GVM regime. The pulse shaper allowed us a sufficient range for time shifting the pump and signal pulses independently of each other. The average signal powers were chosen around 20 – 40 μW (measured at waveguide output), which for a laser pulse rate of 76 MHz, translates to 0:26 – 0:53 pJ per pulse. The pump power coupled into the waveguide was varied from 0 to ~ 3:5 mW (46 nJ per pulse), which was sufficient for significant CE [52] without much pump depletion via spurious SHG. In order to compare theoretical predictions with experimental data, we needed to map the square-root of the pump power to $\tilde{\gamma }$ through a proportionality factor σ. We fit all the diverse data for different input pulse-shape combinations and inter-pulse delays to a single σ value of $~18/\sqrt{\mathrm{W}}$.

Before attempting to create ideal input signal shapes matched to the system’s Schmidt modes, we first studied the system’s behavior when both the pump and signal inputs were Gaussian and/or first-order Hermite Gaussian. Figure 9 shows the conversion efficiencies recorded for Gaussian- and (first-order) Hermite-Gaussian-shaped pump and signal input pulses for various pump powers and initial pump-signal time delays (τ_d). The pump powers were changed by changing the phase-contrast of the vertical gratings in the pump band on the SLM. The pump-signal time delay was scanned in steps of ~ 18.3 fs by applying a linear spectral phase ramp to the signal band on the SLM and changing its slope. The data reproduces the broad features predicted by theory in Fig. 4, namely, the temporal-shift (both extent and direction) of the peaks and troughs for the various shape combinations, as well as the numbers and relative heights of the peaks. This scan ensures that we aren’t seeing an artificial contrast in CE between pump-signal shape-matched vs. shape-mismatched cases owing to a setting dependent, systemically applied, extreme time delay between the pump and signal input pulses. The vertical error bars are all of order 10⁻³, and are not shown for the sake of clarity.

 

Fig. 9 Experimental data: 3D point plots of conversion efficiencies (CE) vs. input pump-signal time delay (τ_d) for various $\tilde{\gamma }=\sigma \sqrt{P_{\text{pump}}}$ (where $\sigma =18/\sqrt{W}$, and Ppump is average pump power) for (a) Gaussian pump and signal, (b) Gaussian pump and first-order Hermite-Gaussian signal, (c) first-order Hermite-Gaussian pump and Gaussian signal, and (d) first-order Hermite-Gaussian pump and signal pulse shapes. Note the temporal skewness at higher $\tilde{\gamma }$, as well as asymmetry in lobe peak heights, matching the theoretically predicted trends from Fig. 4. Vertical error bars are all of order 10⁻³, not shown.

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For closer comparison, we take a τ_d = 0 slice of the theoretical graphs and the measured data from Figs. 4 and 9 respectively, and plot them in Fig. 10. Note that the four possible input shape configurations follow the expected contrasts in CE. The data points for the first-order Hermite-Gaussian-shaped pumps are shifted horizontally forward relative to those for the Gaussian-shaped pumps. This is because for a given temporal-width scale, the first-order Hermite Gaussian spectrum has a slightly larger bandwidth, giving us more available power to be syphoned off from the ultrafast seed laser. Also note that for a given pump shape, the signal-shape matched points are shifted horizontally slightly backward relative to the signal-shape mismatched points, and the shift is larger at higher CE. This is because, due to energy conservation, some amount of power from the pump pulse is lost along with depletion of signal power during FC. The effect, a violation of the undepleting pump approximation, is negligible for weak signals, as demonstrated by the close match of the data with theory.

 

Fig. 10 CE (conversion efficiency) vs. $\tilde{\gamma }(\propto \sqrt{P_{\text{pump}}})$ for various pump and signal input pulse shapes at “zero” delay (defined as delay that maximizes CE at low pump power). The legend label (HG_j, HG_k) denotes j-th order Hermite-Gaussian pump pulse, and k-th order Hermite-Gaussian signal pulse. The solid lines are theory and the markers are measurements.

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In Fig. 11 we show the CE vs. $\tilde{\gamma }$ for our attempt to match the exact, first input Schmidt mode at every $\tilde{\gamma }$ via numerically computed TM shapes. The plotted points are for Gaussian, and first-order Hermite-Gaussian pump pulses. The solid lines are the theoretical prediction. The CE with matched (appropriate to pump shape) Schmidt mode inputs exceeds those of the pure Hermite-Gaussian-shaped signal inputs from Fig. 10. The measured data falls short of theory at larger pump powers. We suspect this is due to pump pulse reshaping within the waveguide due to spurious pump-only second-harmonic generation (see Fig. 8). Despite this, we achieve a CE contrast of 4.7 to 1 (85% vs. 18%) between the two pump shapes and their corresponding, first Schmidt modes. The error bars are of order 10⁻³, and are not shown.

 

Fig. 11 CE (conversion efficiency) vs. $\tilde{\gamma }(\propto \sqrt{P_{\text{pump}}})$ for j-th order Hermite-Gaussian pump pulses (HG_j) and the corresponding first input Schmidt modes (SM_j). Also shown are CE with the pump shapes swapped for j ∈ {0, 1}. Solid lines are theory, and markers are experiment (type: “exp.”). Error bars (not shown) are of order 10⁻³.

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4. Conclusion

We have quantified and demonstrated the importance of group-velocity and temporal-width parameter regimes that enhance temporal-mode selectivity in nonlinear optical frequency conversion in χ⁽²⁾-waveguides. We have confirmed Kumar/Kanter group’s [16] observation of the viability of using standard frequency-doubling waveguides around the SHG wavelengths, and shown one such case functioning at near room temperature. And we have illustrated the use of a single, large, Treacy-grating-pair pulse shaper, in conjunction with an ultrafast, titanium-sapphire laser and a reflective spatial-light modulator, to manipulate both the pump and signal fields, shapes and time delays. And we have demonstrated the accuracy of the coupled-mode model equations in predicting the experimental results.

Combined with the good mode separability achieved at 50% conversion, the good agreement between experiment and theory paves the way to the implementation of a proposed two-stage cascaded FC scheme that is predicted by the presently confirmed theory to reach near-perfect (100%) selectivity [37–39]. The two-stage scheme (temporal-mode interferometry) is a generalization of two-color Ramsey interferometry, proposed independently in [38] and [41]. This will bring us a step closer to a highly-selective quantum pulse gate (QPG) [5, 12, 13, 36]. The use of off-the-shelf waveguides and other ubiquitous resources should empower other groups without access to waveguide fabrication facilities to invest in temporal-mode selective frequency conversion research, which is sorely needed in this new, rapidly evolving field.