1. Introduction

Laser facilities generating high-intensity, ultrashort laser pulses at the petawatt level support the exploration of new regimes of laser–matter interaction [1–3]. These facilities rely on chirped-pulse amplification [4] in either broadband laser materials such as titanium sapphire or nonlinear crystals. In optical parametric chirped-pulse amplification (OPCPA), energy from the pump is transferred to the temporally stretched signal via three-wave nonlinear mixing [5–7]. A broadband signal can be amplified efficiently provided that the phase mismatch is small over its bandwidth. The noncollinear interaction of the pump and signal provides an additional degree of freedom to increase the phase-matched bandwidth [8,9]. Because of damage-threshold limitations, high-energy amplification requires large-aperture nonlinear crystals. KDP (potassium dihydrogen phosphate) and its partially deuterated isomorph DKDP are by far the most suitable for good-quality nonlinear crystals at aperture larger than 10 cm and relatively low cost. Growth techniques have been optimized to support not only frequency doubling and tripling but also polarization switching in high-energy ignition lasers such as the National Ignition Facility [10,11]. These crystals enable high-energy broadband parametric amplifiers pumped by frequency-converted, high-energy nanosecond Nd:glass lasers, opening the way to the generation of optical pulses with energy of hundreds of joules and bandwidth supporting sub-20-fs pulses [6,12–15]. The ordinary and extraordinary indices of refraction of a DKDP crystal depend on its deuteration level. Therefore, the deuteration level can strongly impact phase-matching conditions, e.g., for broadband second-harmonic generation [16], noncritical fourth-harmonic generation [17], and OPCPA [18–21]. Developing broadband DKDP OPCPA systems requires phase-matching models that support simulations of nonlinear wave mixing and experimental optimization. However, there are relatively few published refractive-index models, and their utility is decreased by the relative difficulty to control and determine the deuteration level of a crystal.

In this article, we present the concept and application of a novel two-wavelength phase-matching technique that precisely determines the deuteration level of a DKDP crystal consistent with known index models. The determined deuteration level and model are the much-needed combination required for performance modeling and experimental optimization of an optical parametric amplifier (OPA). By experimentally determining the deuteration level of a crystal consistent with a specific index model, the described technique allows for more accurate performance simulations as well as better identification of optimal phase-matching conditions. Section 2 generally discusses the optimization of broadband OPCPA in DKDP and the available index models. Section 3 describes the two-wavelength phase-matching technique. Section 4 presents experimental results obtained on four crystals for which the deuteration level consistent with three published index models is determined. An analysis of this technique using partial derivatives of the propagation phase and an additional comparison of these three index models in the context of previously published results for broadband second-harmonic generation are presented in two appendices. The consistency of the determined deuteration levels over multiple campaigns in different experimental configurations and the determination of confidence intervals from the experimental data demonstrate that the technique is a valuable tool for the development and operation of DKDP OPCPA systems.

2. Optimization of broadband optical parametric amplification in partially deuterated KDP

2.1 Noncollinear optical parametric amplification in DKDP

Efficient optical parametric amplification requires phase matching between the pump, signal, and idler waves at wavelengths  λ_p, λ_s, and λ_i, respectively. Energy conservation imposes ω_p = ω_s + ω_i, where these variables are the optical frequencies corresponding to the mentioned wavelengths. For a specific wave vector, a birefringent nonlinear crystal has two polarization eigenmodes that correspond to an ordinary index n_o and extraordinary index n_e. Phase matching is achieved by choosing the polarization of the interacting waves and angularly tuning the crystal to control the optical index of the extraordinary waves. In the Type-I broadband DKDP OPA’s, the signal and idler are polarized along the ordinary axis, for which the optical index is wavelength-dependent, while the pump is polarized along the extraordinary axis, for which the optical index depends on the wavelength and the angle θ between the wave vector and the crystal axis.

Broadband parametric amplification results from optimization of the crystal’s deuteration level x and the noncollinear angle α between the pump and signal beam. The deuteration level can in principle be controlled between 0% (KDP) and ∼100% (fully deuterated KDP) during crystal growth by adjusting the liquid-solution deuterium content. The noncollinear angle between pump and signal beams is chosen when setting up the OPA in the laboratory. When α is not equal to 0, there are two distinct interaction geometries for which the pump’s spatial walk-off either leads to better pump-signal spatial overlap [“walk-off compensating geometry,” Fig. 1(a)] or worse pump-signal spatial overlap [“non-walk-off compensating geometry,” Fig. 1(b)]. These two geometries have identical phase-matching properties for OPA operation, and either one of them could have been used for determining the deuteration level. However, parasitic second-harmonic generation (SHG) of the signal and idler, which can disrupt OPA operation, can be phase matched in the walk-off-compensating geometry for wavelengths and angles typically used for broadband OPCPA systems [22]. For the experiments reported in this article, parasitic SHG is not a concern because of the signal’s low intensity. However, the optical layout around the OPCPA crystal is designed for operation in the non-walk-off–compensating geometry, and the experimental results were therefore obtained in that geometry. The noncollinear geometry yields an idler beam at a wavelength-dependent angle Ω relative to the signal beam [9]. In these conditions, the wave vector mismatch along the signal wave vector is

The extraordinary index at an angle θ can be calculated as a function of the crystal’s ordinary index n_o and extraordinary index n_e following the equation:

The wavelength-dependent angle between signal and idler, which has a slight dependence on deuteration level via the optical indices, can be calculated as

For the specific application of optical parametric chirped-pulse amplification with a monochromatic pump pulse obtained by second-harmonic generation of the output of an Nd:glass laser, λ_p = 526.5 nm, which yields a one-to-one relation between the wavelength in the chirped signal and the wavelength of the generated idler. For a given deuteration level, optimization of the noncollinear angle α and crystal angle θ allows for broadband amplification of a signal in the near infrared [18–21].

 

Fig. 1. Wave vectors and angles definition for a noncollinear DKDP OPA in (a) the walk-off–compensating geometry, and (b) the non-walk-off–compensating geometry.

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2.2 Index models for DKDP

Optimizing parametric amplification requires a model for the ordinary and extraordinary index of refraction of the crystal as a function of deuteration level and wavelength. With a pump at 526.5 nm, the idler’s wavelength ranges from 1471 nm to 1088 nm when the signal’s wavelength ranges from 820 nm to 1020 nm. A model typically consists of a set of Sellmeier equations calculating the wavelength-dependent ordinary and extraordinary index for at least two reference deuteration levels. Interpolation or extrapolation are required to calculate the indices at other deuteration levels that are either inside or outside the range of reference levels. Table 1 shows a non-exhaustive list of published wavelength-dependent index models and their main characteristics [18,23–29]. Various studies of the temperature-dependence have also been published [25,30,31]. After consideration of the different models and numerical investigation, this study was restricted to three models: the Kirby model [26], the Fujioka model [28,29], and the combination of the Zernike index model for KDP with the Kirby index model for DKDP at x = 96%, which was first investigated by Webb in the context of broadband SHG [16]. Because that model was later used for OPCPA studies in Ref. [18], we refer to it as the Lozhkarev model in this article. As noted in Ref. [16], one limitation of the Kirby model is that it was determined using sources at wavelength lower than 1064 nm; therefore, the indices at the idler wavelengths for OPCPA are extrapolated. The Lozhkarev model relies on Sellmeier models for KDP and DKDP (x = 96%) obtained by different authors in different experimental conditions (in particular, temperature). Its use was justified by its ability to model experimental OPCPA data obtained with two crystals (x = 89%±0.5% and x = 84%±2%) and experimental SHG data at low deuteration levels when no other model was available. The Fujioka model relies on five reference levels and a wavelength range that encompasses the OPCPA signal and idler spectral ranges. There are other published models that are not considered here. The Zernike model [23] only relates to KDP, and therefore cannot be used without another model pertaining to DKDP. The Barnes model [24] is a one-pole Sellmeier model based on a narrow wavelength range that does not include the signal and idler wavelengths of interest in an OPCPA system; therefore, its accuracy for phase-matching prediction is questionable. The wavelength range used for establishing the Ghosh model [25] is unpublished, and that model has previously led to predictions that were inconsistent with experimental data [18]. The Zhu model [27] requires extrapolation for deuteration levels higher than 80%, which are preferred for OPCPA at 920 nm, and led to nonphysical results such as deuteration levels larger than 100% when applied to experimental data in our study.

Table 1. Index models for KDP and DKDP sorted by publication year.

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All models require the interpolation or extrapolation of indices determined at reference deuteration levels. The most commonly reported approach is to calculate the susceptibility at a deuteration level x as a weighted average of the susceptibility at two reference levels x₁ and x₂:

The relevance of a model depends on the accuracy of the reference levels and more generally on the experimental conditions that have been used to establish it. The deuteration level of a crystal is different from that of the liquid solution from which it is grown, leading to some uncertainty. For example, the Fujioka model relies on reference deuteration levels measured with a standard deviation of 0.4%, while experimental data used to validate the Lozhkarev model were obtained on crystals having a deuteration precision equal to either ±0.5% or ±2%. While there are techniques to determine the stoichiometric deuteration level [28,32–34], they might not be available to a typical end user and do not support direct application with index models. Furthermore, precise knowledge of the deuteration level of a crystal is not sufficient for an end user because of uncertainties in the reference levels of the available index models. Spatial deuteration variations in DKDP are also common [35,36], leading to additional uncertainties, e.g., 0.3% standard deviation for the crystals used to establish the Fujioka model. In most instances, the end user has no knowledge of the particular growth conditions and location of the crystal within the boules, and no diagnostic to directly measure the deuteration level. Moreover, the optical index of materials depends on the temperature, hence a model calculating the Sellmeier coefficients at one specific temperature might have limited accuracy at another temperature. It is therefore unclear whether the accuracy and precision on the deuteration level of a grown crystal are sufficient for direct application of an analytical model to the phase matching of an OPA, in particular, to find an optimum interaction geometry for broadband operation.

3. Formalism of two-wavelength phase-matching technique

In principle, it is possible to use broadband signal amplification to determine the phase-matching properties of the OPA, but this approach has several practical limitations. First, it requires a collimated coherent broadband seed, which might be readily available in a laser facility developing an OPCPA system but might not be a standard tool in laboratories developing nonlinear crystals. The determination of the wavelength-resolved gain on a chirped signal pulse can be biased by the pump temporal-intensity variations, wavelength-dependent absorption of the signal and idler, and variations in the spectral sensitivity of diagnostics such as grating-based spectrometers. This section develops the analysis of phase matching at two monochromatic signal wavelengths λ_s,1 and λ_s,2. This analysis, which is based on finite differences, is linked to calculated partial derivatives of the propagation phase for the signal, idler, and pump in Appendix 1.

3.1 Phase-matching angle in DKDP

The phase-matching angle at which Δk = 0 cm⁻¹ for interaction of a monochromatic signal at a specific wavelength λ_s,1 amplified by the pump at a given noncollinear angle can be obtained from calculation of the phase mismatch. Figure 2(a) displays an example of such calculation for λ_s,1 = 920 nm and α = 1°. This identifies the deuteration-dependent phase-matching angle θ₁ for which

Because the phase-matching conditions depend on n_o and n_e, different index models lead to different calculated phase-matching angles for a given deuteration level. Figures 2(b) and 2(c) display the calculated phase-matching angle as a function of deuteration level for the amplification of a 920-nm signal at α = 1° and α = 0°. It can be noted that:

 

Fig. 2. (a) Wave vector mismatch Δk as a function of deuteration level x and phase-matching angle θ for amplification of a 920-nm signal by a 526.5-nm pump at α = 1°, calculated with the Fujioka model. [(b),(c)] Phase-matching angle θ as a function of the deuteration level for amplification of a 920-nm signal by a 526.5-nm pump at α = 1° and α = 0°, calculated with the Fujioka, Kirby, and Lohkarev models.

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Determining the deuteration level from the measured phase-matching angle is possible, owing to the one-to-one relation between these two quantities. However, this requires the precise knowledge of the internal angle between the pump wave vector and the crystal axis, e.g., with a 0.05° degree precision to determine the deuteration level with precision equal to 1%. Such knowledge might not be readily available for finished nonlinear crystals used for OPA’s (for example, a 0.5° accuracy on the phase-matching angle is reported in [18]). The precise angular tuning of the crystal required to experimentally phase match an OPA gives a more-practical approach to determining the deuteration level from the relative change in phase-matching angle at different wavelengths, instead of the absolute phase-matching angle at one wavelength.

3.2 Difference in phase-matching angles at two wavelengths in DKDP

A more-practical approach to determining the deuteration level is to use the difference in phase-matching angles measured at two different wavelengths. That angle difference can be determined accurately in the common situation where the OPA crystal is mounted on a precise rotation stage (as is typically required for phase-matching optimization). This section presents simulations of the phase-matching angles’ difference at two signal wavelengths λ_s,1 and λ_s,2. Appendix A shows that the finite difference is related to various derivatives of the propagation phase for the signal, idler, and pump in the crystal, and confirms that the deuteration level can be unambiguously determined in a noncollinear configuration. This generally opens the way to using more than two wavelengths to increase the measurement precision.

At the second wavelength λ_s,2, the phase-matching angle θ₂ is defined by

The deuteration level can be unambiguously determined from the difference Δθ = θ₂–θ₁ provided that the noncollinear angle is known. Figure 3 shows the relation between the phase-matching-angle difference, calculated at λ_s,1 = 870 nm and λ_s,2 = 970 nm, and the deuteration levels x_Fujioka, x_Kirby, and x_Lozhkarev determined by the Fujioka, Kirby, and Lozhkarev models. This relation depends on the noncollinear angle and the index model, i.e., the three index models determine different deuteration levels from identical experimental data. Figure 4 shows the offset between models using the Kirby model as the reference, i.e., x_Fujioka–x_Kirby and x_Lozhkarev–x_Kirby. Figures 3 and 4 demonstrate that:

 

Fig. 3. Difference in phase-matching angles Δθ = θ₂–θ₁ at λ_s,1 = 870 nm and λ_s,2 = 970 nm for (a) α = 0° and (b) α = 1°.

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Fig. 4. Difference x_Fujioka–x_Kirby (blue curve) and x_Lozhkarev–x_Kirby (yellow curve) between deuteration levels determined using the Fujioka and Lozhkarev models relative to the level determined using the Kirby model for (a) α = 0° and (b) α = 1°.

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The difference between phase-matching angles at λ_s,1 and λ_s,2 can be directly determined from the relatively small angular tuning that is performed to phase-match the parametric process. The parametric gain can be measured simultaneously at λ_s,1 and λ_s,2 as a function of the tuning angle if two combined monochromatic sources are used as the OPA seed and the corresponding gain is measured independently at these two wavelengths. Measuring the gain for monochromatic signals simplifies the data acquisition, while making the technique relatively insensitive to the pump pulse-shape variations and wavelength-dependent absorption. Additionally, spatially resolved data acquisition can be implemented with cameras after wavelength demultiplexing to spatially resolve the deuteration variations.

3.3 Relevance of determined deuteration levels

The deuteration level determined from phase matching depends on the index model because different models have their own uncertainties and have been determined in different conditions, e.g., temperature. This means that the determined level is not exactly equal to the stoichiometric level, and in particular, is model-dependent. This applies to any technique that determines the deuteration level from phase-matching conditions, e.g., from parametric gain. Determining a deuteration level that is consistent with a specific index model is nevertheless highly relevant to OPA optimization and simulation. Figures 5 and 6 demonstrate this point by displaying the small-signal parametric gain calculated with the three index models at different deuteration levels that are adjusted to be consistent across models, following the results shown in Fig. 4. For each sub-figure, the gain calculated using elliptic integrals is plotted as a function of the signal wavelength and the difference δθ = θ–θ_max, where the model-dependent θ_max corresponds to the maximal average gain in the wavelength range [870 nm, 970 nm]. For Fig. 5, the deuteration level was chosen to be 90% for the Fujioka and Lozhkarev models and 90.5% for the Kirby model, and α = 0.8° was used for calculations of the parametric gain as a function of wavelength and crystal angle. For Fig. 6, the deuteration level was set to 70% for the Kirby model, while it was set to 73.1% and 67.3% for the Fujioka and Lozhkarev models, respectively, with operation at α = 0.4°. These two figures demonstrate the similarity of the phase-matching properties calculated with the three index models and their respective deuteration levels. In particular, the gain calculated with the three models at a phase-matching angle that maximizes the average gain over a 100-nm wavelength range centered at 920 nm, i.e., at δθ = 0 on each sub-figure, is essentially the same over the optimization range [Figs. 5(d) and 6(d)]. This shows that using the deuteration level determined for an OPA crystal using a specific index model allows for consistent simulations of the OPA performance. Additionally, the optimal interaction geometry, i.e., noncollinear angle, can be determined from the determined deuteration level.

 

Fig. 5. Small-signal gain of a 48-mm DKDP crystal at α = 0.8° with index model from (a) Fujioka, (b) Kirby, and (c) Lozhkarev. A deuteration level equal to 90.5% is used for the Kirby model, while equivalent deuteration levels are used for the Fujioka (90%) and Lozhkzrev (90%) models. (d) The spectrally resolved gain at the θ angle that maximizes the gain averaged between 870 nm and 970 nm (highlighted range).

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Fig. 6. Small-signal gain of a 48-mm DKDP crystal at α = 0.4° with index model from (a) Fujioka, (b) Kirby, and (c) Lozhkarev. A deuteration level equal to 70% is used for the Kirby model, while equivalent deuteration levels are used for the Fujioka (73.1%) and Lozhazrev (67.7%) models. (d) The spectrally resolved gain at the θ angle that maximizes the gain averaged between 870 nm and 970 nm (highlighted range).

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The two-wavelength technique relies on the combination of index definitions and experimental data. Temperature changes modify the experimental data; therefore, the deuteration level determined using a specific index definition depends on the temperature. The lack of temperature-dependent DKDP index definitions as a function of deuteration level suitable for OPCPA phase-matching makes it difficult to quantify this dependence. However, the predicted performance (for example conditions for broadband gain) is consistent with the combination of index model and two-wavelength data; therefore, it is predicted at the temperature at which the data is measured.

4. Experimental results

4.1 Experimental setup

The two-wavelength phase-matching technique has been experimentally demonstrated at the last OPCPA stage of the MTW-OPAL Laser System, an optical parametric amplifier line (OPAL) pumped by the Multi-Terawatt (MTW) laser at the Laboratory for Laser Energetics [37]. That system is composed of a sequence of OPA stages pumped either in the picosecond regime or in the nanosecond regime. The five low-energy stages are based on BBO pumped by several Nd:YLF lasers operating at 5 Hz. The final high-energy stage relies on a partially deuterated KDP crystal pumped by a high-energy laser system operating at a low repetition rate (typically one shot every 20 minutes). This laser system generates the pump pulse using a fiber front end, a sequence of Nd:YLF and Nd:glass amplifiers, and frequency conversion from 1053 nm to 526.5 nm in KDP. Precise temporal shaping by a Mach–Zehnder modulator driven by an arbitrary waveform generator in the front end allows for compensation of square-pulse distortion, leading to an approximately flat-in-time 1.5-ns pump pulse. In regular operation as a laser facility, a broadband coherent seed around 920 nm is obtained by spectral filtering of a supercontinuum generated by a short-pulse at 1053 nm in a YAG plate [38]. For crystal characterization, commercial fiber-coupled laser diodes are combined before being launched in free space as a single collimated beam with size of the order of a few millimeters (Fig. 7). This signal is combined with the pump pulse at the crystal under test.

 

Fig. 7. Layout of the experimental setup and example of measured pump pulse (green line) and measured signal pulses after amplification (red and orange lines).

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The crystal is mounted on a high-precision motorized rotation stage (Newport RGV100BL-S, angular repeatability ∼0.1 mdeg) for phase-matching in the type-I configuration, for which the pump pulse is horizontally polarized while the input signal and the generated idler are vertically polarized. The crystals’ temperature is set by the laboratory environment, which is controlled at 20°C±0.1°C. The experimentally measured gains are plotted as a function of the stage angle Θ, which corresponds to the deviation from normal incidence at the input face for each crystal. For analysis, the angle difference at the two cw wavelengths is converted into an internal angle using Fresnel’s law. During setup, the signal and pump beams are intercepted by a kinematic mirror in front of the crystal and sent to an achromatic pointing diagnostic to determine the external noncollinear angle, from which the internal noncollinear α is calculated. After the crystal, the amplified signal beam is directed to a diffraction grating that angularly separates the beams at λ_s,1 and λ_s,2 before detection by two photodiodes and an oscilloscope. An aperture is placed before the diffraction grating to minimize parametric fluorescence from unseeded regions of the crystal. The voltage levels corresponding to the cw sources without amplification are precisely measured at the beginning of each campaign. The gain for a particular configuration is obtained by averaging the measured on-shot voltage over the temporal range when the pump and signal temporally overlap in the crystal. Pump intensities of the order of 1 GW/cm² typically correspond to a small-signal gain of the order of 100. The signal intensity is extremely low (for reference, the intensity of a cw flattop beam with 10-mW average power and 1-mm radius is ∼0.3 W/cm²). Therefore the OPA operates in the undepleted-pump regime, and absorption of the generated idler in the infrared does not lead to local heating that could modify the phase-matching conditions. For a specific crystal and noncollinear angle, the parametric gain is measured at λ_s,1 and λ_s,2 for about ten shots taken at different stage angles, resulting in one determination of the deuteration level. For some of the nonlinear crystals, this process was repeated for a second noncollinear angle determined via modeling to phase-match the two wavelengths λ_s,1 and λ_s,2 for the same crystal angle. The deuteration levels determined with the first and second set of data are generally in very good agreement.

4.2 Data processing and error analysis

The experimental setup allows for the simultaneous determination of the gain at λ_s,1 and λ_s,2 as a function of the stage angle Θ. Because the deuteration level is obtained from the difference in internal angles corresponding to phase matching at these two wavelengths, only relative internal angles are of interest, which can be determined from the external rotation-stage angles. For a specific crystal at a given noncollinear angle, a data set is composed of the measured gain at λ_s,1 and λ_s,2 as a function of the stage angle. The data sets corresponding to the nominal 70% crystal (α = 0.32°) and nominal 80% crystal (α = 0.49° and α = 0.61°) are displayed in Fig. 8 as examples. Each gain curve was fitted with a Gaussian function having three parameters (peak gain, width, and center Θ_C). The deuteration level was then calculated, for a specific index model, from the difference of the determined centers Θ_C,1–Θ_C,2, taking into account the noncollinear angle.

 

Fig. 8. Measured gain versus rotation-stage angle for (a) crystal #1 at α = 0.32°, (b) crystal #2 at α = 0.49°, and (c) crystal #2 at α = 0.61°. The data at each wavelength (square markers) is fitted by a Gaussian function (continuous and dashed lines).

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A confidence interval on the determined deuteration level can be calculated using the confidence intervals on the fit parameters obtained by linear regression. The width of the 100 × (1–a)% confidence interval (e.g., 95% confidence interval for a = 0.05) for the center of the Gaussian function Θ_C, is given by 2 × t(1−a/2, n−p) × SE(Θ_C). In this expression, SE(Θ_C) is the standard error of the center estimate, which is commonly returned by linear-regression software routines, and t(1–a/2, n–p) is the 100(1–a/2) percentile of the t distribution with n–p degrees of freedom, where n is the number of observations and p is the number of regression coefficients, which is equal to 3. The confidence interval on the difference in phase-matching angles at λ_s,1 and λ_s,2 depends on the correlation between errors on the two data sets [39]. When these correlations are unknown, the width of the 100 × (1–a)% confidence interval is 2 × t(1–a/2, n–p) × [SE(Θ_C,1) + SE(Θ_C,2)]. Because the deuteration level is locally linear relative to Θ_C,1–Θ_C,2 (see for example Fig. 3), the confidence interval on this angular difference translates into a confidence interval on the deuteration level at the same significance level.

4.3 Crystals under test

Four partially deuterated KDP (θ ∼ 37.5 °, ϕ = 45 °) have been commercially procured [40] and tested (Table 2). The nominal deuteration level x_nominal, from 70% to 98%, is used for general denomination of the crystals. The deuteration level x_vendor is calculated for three of the four crystals as the average of the deuteration levels determined by the vendor from pycnometer measurements of the solution during crystal growth (that data is not available for crystal #1, for which x_vendor is set to x_nominal = 70%). The crystals have a thickness equal to 48 mm and a small wedge to avoid phase-matched multiple reflections from the crystal’s faces. The crystals’ faces have a GR650 moisture barrier and a sol-gel antireflection coating optimized at 526.5 nm for the input face and at 920 nm for the output face. Signal sources at 850 nm and 976 nm were used to characterize crystals #2, #3, and #4. Sources at 915 nm and 976 nm were used for crystal #1 (nominal deuteration level equal to 70%) to avoid operation at a short wavelength that would not be relevant for broadband operation [see Fig. 6(d)].

Table 2. Crystals’ description and experimental parameters.

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4.4 Experimental results

Eight campaigns have been carried out, leading to determinations of the deuteration levels x_Fujioka, x_Kirby, and x_Lozhkarev for the four crystals. The difference between the deuteration level determined with the two-wavelength technique and the vendor-determined deuteration level x_vendor is plotted as a function of x_vendor on Fig. 9. The dimension of each plotted marker along the deuteration-level axis indicates the 95% confidence interval. The confidence intervals do not depend significantly on the index model. The width of the confidence interval varies from 0.0009° to 0.0098° (average = 0.004°) over the 16 fits (eight campaigns, each with two wavelengths), which translates into deuteration confidence interval with width varying from 0.04% to 2.95%, (average = 1.4%). The deuteration level determined across the eight campaigns are within 5% of the vendor-determined deuteration level for the three index models. The relative variations in deuteration level across models are consistent with what is expected from Fig. 4, in particular the Fujioka model yields lower deuteration levels than the Kirby and Lozhkarev models for x > 90%, and higher deuteration levels for x < 90%. The Fujioka model consistently predicts deuteration levels that are higher than the vendor-determined levels, but is generally closer to those values than other models over the range of deuteration levels that have been tested (70% to 98%). This observation suggests that the Fujioka model is closer to predicting optical indices from the stoichiometric deuteration level if one assumes that the latter is equal to x_vendor. This is consistent with the calculations of noncritical second-harmonic generation presented in Appendix B, which show better consistency of experimental data with the Fujioka model applied with reference levels equal to 0% and 39.5%. Our data is consistent with the observation that the combination of the Zernike model (x = 0%) and Kirby model (x = 96%), which was used by Lozhkarev and coworkers, provides a somewhat better match between deuteration level and observed phase-matching conditions than the Kirby model for crystals at 84% and 89%.

 

Fig. 9. Determined deuteration level for the four crystals over eight campaigns using the index model from (a) Fujioka, (b) Kirby, and (c) Lozhkarev.

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5. Conclusions

A novel characterization technique for partially deuterated KDP crystals has been described and experimentally demonstrated. The deuteration level consistent with a specific model of the refraction indices is directly determined using phase-matching curves at two wavelengths. Consistent experimental results obtained with four different crystals (deuteration level ranging from 70% to 98%) have been obtained, showing that this technique offers valuable information for experimental optimization and simulations of DKDP-based OPCPA systems. These results have successfully supported the optimization and characterization of a high-energy (10-J) broadband OPCPA stage based on a 70% DKDP crystal, for the MTW-OPAL Laser System [37].

While the work has been described in the framework of optical parametric amplification, it can also benefit DKDP-based second-harmonic generation, which can be made broadband by operating at a deuteration-dependent noncritical wavelength. Because this technique only requires gain measurements at two wavelengths, it could be implemented with spatially resolved photodetection to characterize the spatial uniformity of phase matching in the large-aperture DKDP crystals required for multipetawatt laser systems.

Appendix A

In this appendix, the two-wavelength characterization technique is described using partial derivatives of the propagation phase for the interacting waves in a crystal with deuteration level x (omitted from the equations for clarity). The phase mismatch per unit length for parametric amplification in a crystal as a function of the propagation phase per unit length for the signal (frequency ω), idler (frequency ω_p–ω), and pump (frequency ω_p) is

(7)$$\Delta \varphi (\omega )= {\varphi_\textrm{p}}({{\omega_\textrm{p}},\theta } )\cos (\alpha )- {\varphi_\textrm{s}}(\omega )- {\varphi_\textrm{i}}({{\omega_\textrm{p}} - \omega } )\cos [{\Omega (\omega ,\theta )} ]$$

with

(8)$${\varphi_\textrm{s}}(\omega )= {{{n_\textrm{o}}(\omega )\omega } / {c,}}$$

(9)$$\varphi {_\textrm{i}}({{\omega_\textrm{p}} - \omega } )= {{{n_o}({{\omega_\textrm{p}} - \omega } )({{\omega_\textrm{p}} - \omega } )} / {c,}}$$

(10)$${\varphi_\textrm{p}}({{\omega_\textrm{p}},\theta } )= {{{n_\textrm{e}}({{\omega_\textrm{p}},\theta } ){\omega_\textrm{p}}} / {c,}}$$

(11)$$\sin [{\Omega ({\omega ,\theta } )} ]= \frac{{{\varphi_\textrm{p}}({{\omega_\textrm{p}},\theta } )}}{{{\varphi_\textrm{i}}({{\omega_\textrm{p}} - \omega } )}}\sin (\alpha ).$$

The phase mismatch is equal to 0 for two frequencies ω₁ and ω₂ at respective angles θ₁ and θ₂. At ω₁, this leads to

(12)$$\Delta \varphi ({{\omega_1}} )= {\varphi_\textrm{p}}({{\omega_\textrm{p}},{\theta_1}} )\cos (\alpha )- {\varphi_\textrm{s}}({{\omega_1}} )- {\varphi_\textrm{i}}({{\omega_\textrm{p}} - \omega {_1}} )\cos [{\Omega ({{\omega_1},{\theta_1}} )} ]= 0.$$

The phase mismatch at ω₂ is developed at first order using the finite differences θ₂ = θ₁ + (θ₂−θ₁) and ω₂ = ω₁ + (ω₂–ω₁) as

(13)$$\begin{aligned} \Delta \varphi ({{\omega_2}}) &= \Delta \varphi ({{\omega_1}} )+ \frac{{\partial {\varphi_\textrm{p}}}}{{\partial \theta }}\cos (\alpha )({{\theta_2} - {\theta_1}} )- \frac{{\partial {\varphi_\textrm{s}}}}{{\partial \omega }}({{\omega_2} - {\omega_1}} )\\ &+ \left[ {\frac{{\partial {\varphi_\textrm{i}}}}{{\partial \omega }}\cos (\Omega )+ {\varphi_\textrm{i}}\frac{{\partial \Omega }}{{\partial \omega }}\sin (\Omega )} \right]({{\omega_2} - {\omega_1}} )+ {\varphi_\textrm{i}}\frac{{\partial \Omega }}{{\partial \theta }}\sin (\Omega )({{\theta_2} - {\theta_1}} )= 0. \end{aligned}$$

Taking into account Eqs. (11) and (12), Eq. (13) leads to

(14)$$\frac{{{\theta_2} - {\theta_1}}}{{{\omega_2} - {\omega_1}}} = \frac{{\frac{{\partial {\varphi_\textrm{s}}}}{{\partial \omega }} - \frac{{\partial \varphi_{\textrm{i}}}}{{\partial \omega }}\cos (\Omega )- {\varphi_\textrm{p}}\frac{{\partial \Omega }}{{\partial \omega }}\sin (\alpha )}}{{\frac{{\partial {\varphi_\textrm{p}}}}{{\partial \theta }}\cos (\alpha )+ {\varphi_\textrm{p}}\frac{{\partial \Omega }}{{\partial \theta }}\sin (\alpha )}}.$$

For the small noncollinear angles that are typical of OPA’s, one can approximate sin(α) = α, cos(α) = 1–α²/2, cos(Ω) = 1– Ω²/2, and $\Omega ({\omega ,\theta } )= \frac{{{\varphi_\textrm{p}}({{\omega_\textrm{p}},\theta } )}}{{{\varphi_\textrm{i}}({{\omega_\textrm{p}} - \omega } )}}\alpha .$ The latter can be used to calculate

$$\frac{{\partial \Omega }}{{\partial \omega }} = \alpha \frac{{{\varphi_\textrm{p}}}}{{\varphi_\textrm{i}^\textrm{2}}}\frac{{\partial \varphi_{\textrm{i}}}}{{\partial \omega}}\; \textrm{and} \;\frac{{\partial \Omega }}{{\partial \theta }} = \alpha \frac{1}{{\varphi_{\textrm{i}}}}\frac{{\partial \varphi_{\textrm{p}}}}{{\partial \theta }},$$

which simplifies Eq. (14) to

(15)$$\frac{{{\theta_2} - {\theta_1}}}{{{\omega_2} - {\omega_1}}} = \frac{{\frac{{\partial {\varphi_\textrm{s}}}}{{\partial \omega }} - \frac{{\partial {\varphi_\textrm{i}}}}{{\partial \omega }}\left[ {1 + \frac{{{\alpha^2}}}{2}\frac{{\varphi_\textrm{p}^2}}{{\varphi_\textrm{i}^\textrm{2}}}} \right]}}{{\frac{{\partial {\varphi_\textrm{p}}}}{{\partial \theta }}\left( {1 - {{{\alpha^2}} / 2} + {\alpha^2}\frac{{{\varphi_\textrm{p}}}}{{{\varphi_i}}}} \right)}}.$$

Final rearrangement leads to:

(16)$$\frac{{{\theta_2} - {\theta_1}}}{{{\omega_2} - {\omega_1}}}\; = \;\frac{{\frac{{\partial {\varphi_\textrm{s}}}}{{\partial \omega }} - \frac{{\partial {\varphi_\textrm{i}}}}{{\partial \omega }}}}{{\frac{{\partial {\varphi_{\textrm{p}}}}}{{\partial \theta }}}}\; + \;\frac{{{\alpha ^2}}}{2}\,\;\frac{{\frac{{\partial {\varphi_\textrm{s}}}}{{\partial \omega }}\left( {1 - \frac{{2{\varphi_{\textrm{p}}}}}{{{\varphi_{\textrm{i}}}}}} \right) - \frac{{\partial {\varphi_\textrm{i}}}}{{\partial \omega }}\left( {1 + \frac{{\varphi_\textrm{p}^2}}{{\varphi_\textrm{i}^2}} - \frac{{2{\varphi_{\textrm{p}}}}}{{{\varphi_{\textrm{i}}}}}} \right)}}{{\frac{{\partial \varphi {_\textrm{p}}}}{{\partial \theta }}}}.$$

Equation (16) expresses the change in OPA phase-matching angle θ₂–θ₁ at two different frequencies ω₁ and ω₂ as a function of the noncollinear angle α and material-dependent parameters via the propagation phase for the signal, idler, and pump. These parameters can be calculated as a function of the central wavelength for the signal and the deuteration level to express

(17)$$\frac{{{\theta_2} - {\theta_1}}}{{{\lambda_2} - {\lambda_1}}} = A({\lambda ,x} )+ \frac{{{\alpha ^2}}}{2}B({\lambda ,x} ).$$

Figure 10(a) displays the quantities A and Bα²/2 (calculated at α = 1° for ease of comparison), calculated at λ = 920 nm with the Fujioka model. A strongly depends on the deuteration level while B is approximately constant (variations of the order of a few percent are observed for deuteration levels between 60% and 100%). This explains that changing the noncollinear index mostly adds a constant angular offset to the relation between phase-matching angle difference and deuteration level, as seen in Fig. 3. The variation in phase-matching angle calculated at α = 0° and α = 1° with Eq. (17) is in excellent agreement with the finite-difference calculations for λ₂–λ₁ = 100 nm, as can be seen by comparing Figs. 10(b) and the Fujioka model curves on Figs. 3(a) and 3(b).

 

Fig. 10. (a) Calculated A and Bα²/2, following the definitions of Eqs. (16) and (17). (b) Variation in phase-matching angle calculated a α = 0° and α = 1° with the Fujioka model.

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Appendix B

The noncritical wavelength for second-harmonic generation λ_nc in partially deuterated KDP, for which SHG of broadband light around λ_nc yields broadband light around λ_nc/2, can be calculated from derivatives of the ordinary and extraordinary indices at these two wavelengths [16]. In this appendix, calculations of λ_nc with the Fujioka, Kirby, and Lozhkarev model are compared with the experimental data reported in Refs. [16,41], and [42]. For the Fujioka model, the optical indices are calculated from interpolation of the indices given at x = 0% and x = 39.5%, while the two other models use interpolation between x = 0% and x = 96%. SHG around 1 µm uses index calculations at wavelengths that are expected to be within the validity domain of the three models.

As can be seen in Fig. 11, the Fujioka model yields noncritical wavelengths that are between the wavelengths calculated with the two other models and are generally in better agreement with the data (keeping in mind the possible uncertainties in the deuteration level of the crystals). The Fujioka model predicts noncritical SHG at 1053 nm for x = 11.2% (versus x = 7.3% for the Kirby model and x = 15.6% for the Lozhkarev model). The spectral data reported for broadband SHG of femtosecond pulses in a 12% DKDP crystal does not allow for a precise comparison of the three index models [41]. The spectral data reported for broadband SHG of spectrally incoherent pulses in a 15% DKDP crystal is consistent with a noncritical wavelength higher than 1056 nm [42], which is in better agreement with the Kirby and Fujioka models, which predict λ_nc = 1063 nm and λ_nc = 1058 nm, respectively, than with the Lozhkarev model, which predicts λ_nc = 1052 nm.

 

Fig. 11. Noncritical wavelength for SHG calculated with the Fujioka model (blue line), Kirby model (red line), and Lozhkarev model (yellow line). The black markers correspond to experimental data obtained from Fig. 5 in Ref. [16].

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Funding

National Nuclear Security Administration (DE-NA0003856); University of Rochester; New York State Energy Research and Development Authority.

Acknowledgments

The authors thank M. Guardalben for technical discussions as well as for assistance with the specification and procurement of the DKDP crystals. Technical discussions with the MTW-OPAL team, in particular C. Feng, C. Jeon, R. Roides, M. Spilatro, and B. Webb, are also acknowledged.

Disclosures

This report was prepared as an account of work sponsored by an agency of the U.S. Government. Neither the U.S. Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the U.S. Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the U.S. Government or any agency thereof. The support of DOE does not constitute an endorsement by DOE of the views expressed in this paper.

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