## Abstract

Partially deuterated potassium dihydrogen phosphate (DKDP) allows for the optical parametric chirped-pulse amplification of high-energy broadband optical pulses to generate ultrashort, high-peak-power pulses. Modeling and experimental optimization of the noncollinear interaction geometry require the combination of a model for the deuteration-dependent optical indices and knowledge of the deuteration level of the DKDP crystal being used. We study and demonstrate a novel experimental technique that determines the deuteration level of a DKDP crystal consistent with a specific index model, based on the phase-matching angles measured at two wavelengths. Simulations and experiments show that the simple two-wavelength technique allows for consistent results with three different index models, and in particular, for the characterization of four crystals with deuteration levels ranging from 70% to 98%.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 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].

#### 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.

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 *x*_{1} and
*x*_{2}:

*j*stands for either o (ordinary axis) or e (extraordinary axis, in which case the index is also a function of the phase-matching angle

*θ*). As noted in Ref. [16], this is plausible and consistent with experimental data but it is neither fundamental nor confirmed by experimental index measurements with sufficient precision. Such interpolation has been used in various OPCPA investigations [18–20], and is the one followed in this article. Another published approach is to consider partially deuterated KDP as a mixture of pure KDP (

*x*= 0%) and fully deuterated KDP (

*x*= 100%), and apply the Lorentz–Lorenz formula taking into account their mass density [28]. We have found that this approach yields similar results as the susceptibility average, which is attributed to the fact that KDP and DKDP have essentially the same mass density. The optical index and its derivative with respect to wavelength are continuous functions of the wavelength when using the Sellmeier equation far from resonances, but models using more than two reference levels [27,28] yield small unphysical discontinuities of the derivatives of the index relative to the deuteration level at the reference levels.

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
Δ*k* = 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
*θ*_{1} 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:

- • The three models are generally in agreement over that range, although differences are clearly visible.
- • The Lozhkarev and Kirby models lead to similar phase-matching angles, which is expected at relatively high deuteration levels because the indices in the Lozhkarev model are calculated via interpolation of the Zernike model at
*x*= 0 and Kirby model at*x*= 96%. - • A given phase-matching angle corresponds to a deuteration level in the Fujioka model that is a few percent lower than in the Kirby and Lozhkarev models.
- • Considering the average variation in phase-matching angle over the considered range of deuteration level, the sensitivity is 0.0485°/%, e.g., an observed change of the phase-matching angle equal to 0.0485° corresponds to a change in deuteration level equal to 1%.

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 *θ*_{2} is defined
by

The deuteration level can be unambiguously determined from the
difference
Δ*θ* = *θ*_{2}–*θ*_{1}
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:

- • The three models approximately determine the same deuteration level (∼90%) for one specific value of
*θ*_{2}–*θ*_{1}. - • The Kirby and Lozhkarev models yield deuteration levels that are closer to one another than the Fujioka model over that range. For levels lower than ∼90%, the deuteration level predicted by the Fujioka model is higher than the level predicted by the other two models, while the opposite is true at levels higher than 90%.
- • For these two wavelengths, which are separated by 100 nm, the angular sensitivity relative to the deuteration level is 0.0035°/%. Such precision can be achieved with commercial off-the-shelf precision rotation stage (as an example, the stage used for the experimental demonstration reported in Sec. 4 has a minimal incremental motion and repeatability of the order of 0.1 mdeg, which corresponds to a 0.02% precision on the deuteration level, taking into account the above-mentioned sensitivity and scaling between internal and external angle). Higher sensitivity can be obtained by increasing the wavelength difference between the two monochromatic signals.

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.

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.

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/cm^{2} 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/cm^{2}).
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.

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)].

#### 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%.

## 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

The phase mismatch is equal to 0 for two frequencies
*ω*_{1} and
*ω*_{2} at respective angles
*θ*_{1} and
*θ*_{2}. At
*ω*_{1}, this leads to

The phase mismatch at *ω*_{2} is
developed at first order using the finite differences
*θ*_{2} = *θ*_{1} + (*θ*_{2}−*θ*_{1})
and
*ω*_{2} = *ω*_{1} + (*ω*_{2}–*ω*_{1})
as

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

For the small noncollinear angles that are typical of OPA’s, one
can approximate
sin(*α*) = *α*,
cos(*α*) = 1–*α*^{2}/2,
cos(Ω) = 1– Ω^{2}/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

Final rearrangement leads to:

Equation (16)
expresses the change in OPA phase-matching angle
*θ*_{2}–*θ*_{1}
at two different frequencies *ω*_{1} and
*ω*_{2} 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

Figure 10(a) displays the
quantities *A* and
B*α*^{2}/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
*λ*_{2}–*λ*_{1} = 100
nm, as can be seen by comparing Figs. 10(b) and the Fujioka model curves on
Figs. 3(a) and 3(b).

## 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.

## 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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