1. Introduction

Nonlinear crystals support frequency conversion and parametric amplification in laser systems. Efficient interactions require phase matching between the optical waves, which can be obtained by controlling the optical index via angular or temperature tuning [1,2]. Transverse variations in phase-matching conditions can lead to nonhomogeneous output waves having spatial variations in their fluence distribution, or in the case of broadband sources, spatial variations of their temporal or spectral properties. These variations have a detrimental impact on the output energy and beam quality for all waves, as well as on the temporal recompression of broadband waves in optical parametric chirped-pulse–amplification (OPCPA) systems. The variations in phase-matching conditions often originate from nonideal properties of the interacting waves [3,4]. They can also arise from local changes in the crystal axis angle and chemical composition, which can be caused by variations in environmental conditions and growth-solution composition, including pollutants, during the growing process, and mechanical stress [5–9].

Local phase-matching variations are a concern for all nonlinear crystals, but they can be particularly problematic for the large-aperture (∼40 × 40-cm²) potassium dihydrogen phosphate (KDP) and partially deuterated KDP (DKDP) crystals supporting the doubling and tripling of ignition-class laser systems [6,10]. These crystals are also a key component for high-intensity OPCPA systems: optimizing the critical phase-matching angle and the noncollinear angle between the signal and pump beams allows for broadband parametric amplification of a signal in the near infrared by the second harmonic of an Nd-doped laser [11–13]. Because of the large achievable aperture, laser systems reaching peak powers higher than 10 PW have been envisioned [14]. Broadband operation requires relatively tight control of the interaction angles, which depend on the deuteration level [15,16]. KDP and DKDP are negative uniaxial crystals in which the ordinary index does not depend on the propagation direction while the extraordinary index depends on the polar angle θ between the propagation direction and the z axis of the crystal [2]. In large-scale boules, z-axis wander of the order of 100 µrad has been observed [17]. Spatial variations of the local deuteration level as high as 1% have been observed in DKDP [6,18,19]. The refractive indices, and therefore the phase-matching conditions, depend on the deuteration level [20]. These variations can therefore impact the parametric amplification efficiency and the ability to maintain the broadband phase-matching conditions over the full crystal aperture. Deuteration-level variations in DKDP are also a concern for broadband second-harmonic generation (SHG) [21] and noncritical fourth-harmonic generation [22].

A broad range of techniques exists to characterize the quality and spatial uniformity of nonlinear crystals: for example, x-ray topography [23], direct observation of defects and inclusions [24], compositional analysis [25], Raman spectroscopy [18], spatially resolved characterization of wavefront and depolarization losses [26], and tomography [25]. These techniques provide important information on crystal quality but they do not directly relate to the performance of the crystal for nonlinear interactions, e.g., phase-matching conditions. The transverse variations in birefringence between the ordinary and extraordinary axes at one wavelength have been linked to spatial variations in phase matching [5,17,27,28]. The spatial homogeneity of birefringence, spectral acceptance, and peak wavelength for SHG have been mapped out by spectrally resolving the nonlinear signal resulting from SHG of a femtosecond laser [29]. For the specific case of DKDP, the deuteration level is related to the spectrally resolved transmission [15,30–32] and the phase-matching properties for parametric amplification of monochromatic signals [33]. These measurements can, in principle, be spatially resolved to assess the spatial uniformity of the phase-matching properties, although they might not be sensitive enough in practice. Directly measuring the spatial phase-matching variations would be a valuable tool for improving crystal growth, defining and selecting crystals for a specific application, and optimizing the nonlinear interaction configuration for a specific crystal.

We demonstrate the high-resolution, high-precision mapping of SHG phase-matching conditions. The SHG in the detuned nonlinear crystal under test maps out the transverse variations in phase mismatch. The high precision resulting from angular stabilization allows one to determine variations in SHG of the order of 1%, resulting in a sensitivity of the order of 0.02 rad in the phase mismatch. The technique is applied to several DKDP crystals for which variations in bulk crystal properties, typically corresponding to 10-µrad internal angle variations and 0.01% deuteration-level variations, and surface features are identified. The principle and implementation of the measurement technique are described in Secs. 2 and 3, respectively. The experimental results are presented in Sec. 4.

2. Concept

2.1 Dependence of SHG efficiency on phase mismatch

The phase-matching conditions for SHG are determined from the variations in SHG energy in a detuned crystal for a monochromatic pulse at wavelength λ₁, leading to an up-converted pulse at wavelength ${\lambda _2} = {{{\lambda _1}} / 2}.$ In the plane-wave fixed-field approximation, i.e., low-conversion regime, the SHG efficiency normalized to its peak value at Δk = 0 is

 

Fig. 1. (a) Efficiency of second-harmonic generation η as a function of the phase mismatch ΔkL in the fixed-field approximation. (b) Derivative of η with respect to ΔkL/2. For birefringent phase matching, the angle θ₀ corresponds to phase matching (Δk = 0), whereas the angles $\theta_{-} $ and θ₊ correspond to phase mismatch leading to half the upconverted energy obtained at θ₀.

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In the common case of birefringent phase matching, the detuning is obtained by inducing an angular offset relative to the angle θ₀ for optimal phase matching, leading to two angles $\theta_{+} $ and $\theta_{-} $ that correspond to half the maximal SHG energy, as defined on Fig. 1(a). Each of these configurations allows for the determination of δk, including its sign, from variations in η. Numerical calculations show that detuning at the half-maximum efficiency corresponds to Δk_±L/2 = ±κ with κ ≈ 1.391. The derivative of the efficiency η indicates that the sensitivity at ±κ is ${\mp} \zeta $ with ζ = 0.538 [Fig. 1(b)]. This sensitivity observed at the easily identified half-maximum value of the SHG efficiency is close to the highest sensitivity, 0.54, which is obtained for Δk_±L/2 = ±1.304. Operation at one of the half-maximum points therefore yields

Equation (4) determines the variations in phase mismatch from the normalized variations in SHG energy using the known crystal thickness L and the calculated values of ζ. A wedged crystal has a spatially varying length, which must be taken into account in the SHG efficiency. This leads to a small additive correction to the measured energy in phase-matched and detuned conditions (Appendix 1).

2.2 Link between wave-vector mismatch and crystal properties

In the most general case, the wave-vector mismatch depends on the crystal’s properties, i.e., a set of parameters {α_j} such as temperature and the angle of the crystal’s axis relative to a reference direction. In partially deuterated crystals, the indices depend on the deuteration level. Another effect of potential interest is the variation in internal propagation direction caused by surface defect (see Appendix 2). When the parameters that impact the wave-vector mismatch are spatially varying, the wave-vector mismatch is given by the sum of the changes δα_j weighted by the corresponding partial derivative ∂_jk, following

The corresponding variations in the parameters α_j cannot be unambiguously determined from the measured δk unless an assumption is made to reduce the number of terms in the right-hand side of Eq. (5). For example, the potential impact of temperature can be ignored if the crystal is characterized in a temperature-stable environment without significant heat load caused by absorption of the laser beams. In some situations, the interest lies in the change in wave-vector mismatch between two different operating conditions: for example, the impact of the temperature change induced by absorption of heater beam could be determined from the difference in wave-vector mismatch measured with and without the heater beam, which essentially cancels all but one term in Eq. (5).

With the commonly used birefringent phase matching, the wave-vector mismatch depends on the ordinary and extraordinary index for the waves at λ₁ and λ₂, the latter index being a function of the phase-matching angle, e.g., the polar angle between the crystal axis and the common axis defined by the wavevectors at λ₁ and λ₂ for Type-I SHG. In this case, the wave-vector mismatch is

Equation (6) can be used to directly calculate the index difference at λ₁ and λ₂ from the measured variations in wave-vector mismatch. The partial derivative of Eq. (6) relative to θ, which describes the sensitivity of the wave-vector mismatch with respect to an internal change in crystal-axis angle, is given by

The measured variation in wave-vector mismatch δk and the crystal axis angle variation δθ are therefore linked by:

For SHG at 1053 nm in beta-barium borate (BBO), one can calculate ∂_θ n_e (λ₂) ‒0.093. For SHG at 1053 nm in DKDP, ∂_θ n_e (λ₂) does not significantly depend on the deuteration level X; it is equal to –0.039 for X = 70% [20].

In a partially deuterated KDP crystal, the indices of refraction n_o and n_e also depend on X, leading to

The measured variation in wave-vector mismatch δk and the deuteration level variation δX are therefore linked by

Surface variations at the input face of the crystal lead to a change of the propagation angle within the crystal via refraction, which then leads to a wave-vector mismatch following Eq. (8). As shown in Appendix 2, the change in internal propagation angle for the beam at wavelength λ₁ and the resulting change in wave-vector mismatch for SHG that result from a local surface variation can be related to the variations in surface height h(x,y) across the crystal aperture. The height function h can, for example, be derived from the phase φ measured with an interferometer operating at the wavelength λ_Test. The impact on internal propagation angle and wave-vector mismatch depends on the variations of the spatial derivatives ∂_xh and ∂_xφ along the phase-matching direction x. The induced local variation in phase mismatch is

High-frequency variations are expected to be the most detrimental. Because only refraction at the input face contributes to a change in the internal propagation angle, the contributions of surface variations can be differentiated from the contributions of crystal properties in the bulk by performing two determinations of δk for two orientations of the crystal under test differing by which surface is the input face. The wave-vector variations from bulk crystal properties are expected to be identical (after taking into account the physical rotation and symmetry that differentiate the two crystal orientations), whereas the wave-vector variations caused by surface quality are specific to each face.

2.3 Dynamic range

The largest range of δk that can unambiguously be determined corresponds to $0 < |{\delta k} |{L / {2 < \pi }},$ i.e., $0 < |{\delta k} |< {{2\pi } / L},$ for which the relative SHG efficiency spans the full range from 0 to 1 and the normalized energy variation S spans the full range from –1 to 1. Although the dependence between efficiency and phase mismatch is not linear within this range, the phase mismatch can be reconstructed using the known analytical dependence given by Eq. (1) if needed. Considering the measured experimental variations in SHG energy, which correspond to |S| < 0.2, a simple linear relation between S and δk has been used for all the results presented in Sec. 4.

2.4 Data interpretation

The presented mapping technique provides spatial resolution in the transverse plane (x,y), but not in the longitudinal propagation direction (z) because the SHG energy is obtained after propagation through the entire crystal thickness. In a relatively thin crystal, the longitudinal variations are most likely smaller compared to the transverse variations. However, longitudinal variations are expected for the thick DKDP crystals (∼50 mm) characterized in Sec. 4.2. In particular, the variations due to internal crystal stress, environmental changes, and changes of the growth-solution properties during growth are likely to depend on the position relative to the crystal axis and distance to the seed crystal, but the SHG mapping is performed by propagation at an angle suitable for SHG phase matching. Various numerical investigations show that the measured phase mismatch does not depend at first order on the linear longitudinal variations in phase mismatch, leading to a value corresponding to its longitudinal average. Therefore, in this article, we interpret the wave-vector mismatch determined from the variation in SHG energy at a given transverse position as an average over the full crystal thickness. The characterization and impact of longitudinal variations of crystal properties, with emphasis on the SHG mapping technique and the performance of OPCPA systems, will be documented in future works.

3. Experimental implementation

3.1 Experimental setup

The experimental demonstration has been performed using the setup shown in Fig. 2. A fiber front end based on a monochromatic seed laser, a Mach‒Zehnder modulator driven by an arbitrary waveform generator, and a fiber amplifier provides 350-ps seed pulses at λ₁ = 1053 nm to a diode-pumped Nd:YLF regenerative amplifier. This amplifier delivers pulses with energy of the order of 1 mJ at 5 Hz. The beam is incident on an uncoated input wedge that sends a sample beam to an energy meter. The transmitted beam is routed to the crystal under test by a piezo-actuated mirror, which is part of the beam-stabilization system. The transverse position of the crystal is set relative to the beam using two computer-controlled translation stages with a 50-mm range (MTS50-Z8, Thorlabs). A computer-controlled rotation stage (CONEX-AG-PR100P, Newport) allows for angular tuning.

 

Fig. 2. Setup for spatially resolved measurement of the SHG energy, indicating the forward-propagating beam at λ₁ (red line), the backward-propagating beam at λ₁ (orange line), and the beam at λ₂ (green line)

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The nonideal translation stages introduce angular variations that can disrupt the phase-matching conditions. To compensate these angular variations, a commercial stabilization system (MRC Systems) stabilizes the angle of the incident beam relative to the crystal. This system is composed of the previously mentioned piezo-actuated mirror, a four-quadrant photodetector, and a control unit. The four-quadrant photodetector generates two analog signals, respectively, proportional to the horizontal and vertical displacement of the beam that is incident on it relative to its reference centered position. These signals are processed by the control unit that drives the horizontal and vertical actuator of the piezo-actuated mirror to maintain the beam incident on the four-quadrant photodetector at its reference centered position. Early experiments used the beam reflected by the crystal’s input face for angular stabilization, but this implementation is detrimentally impacted by variations in the surface flatness and local surface reflectivity, which can be low in the presence of an antireflection coating. The reflection from a reference fused-silica wedge located before the crystal on the same rotation stage was used for improved performance and stability. This wedge does not introduce significant position-dependent angular disruption on the reflected beam, therefore allowing the latter to be used as a reference for beam stabilization over the full scanned aperture. After reflection off this reference wedge, the beam is again incident on the piezo-actuated mirror before being reflected by the input wedge and focused on the four-quadrant photodetector. Both the incident and reflected beam are incident on the piezo-actuated mirror to provide adequate stabilization (the configuration where only the input beam is incident on the piezo-actuated mirror is ineffective for stabilization relative to the input face).

The energy of the frequency-converted beam at λ₂ = 526.5 nm, which is typically of the order of 100 µJ in phase-matched conditions, is measured after a dichroic mirror and a bandpass dielectric filter. The rms energy variation at λ₁ is 0.5%. At λ₂, the rms energy variation is 0.9% for a phase-matched crystal (angle $\theta_{-} $) and 1.1% for a crystal detuned at half efficiency ($\theta_{-} $ and θ₊), identically with and without beam stabilization. Using Eq. (4), the latter variation is consistent with a sensitivity of 0.02 rad in terms of phase mismatch δkL. The spatial resolution was estimated by placing an opaque mask with a sharp edge at the input surface of a nonlinear crystal and measuring the SHG energy as a function of the position in the direction orthogonal to the sharp edge. The resulting transverse variation in SHG energy corresponds to a Gaussian impulse response having a full-width at half maximum equal to 450 µm, which is consistent with the estimated diameter of the input beam (∼1 mm).

Figure 3 demonstrates the performance of the beam-stabilization system using various measurements made on a 3.3-mm BBO crystal. In Fig. 3(a), the energy is measured as a function of the rotation-stage angle in different stabilization conditions. Without beam stabilization, the energy curve corresponds to the efficiency expected from the linear variation in phase mismatch with angle. When the beam is stabilized for the angle θ₀ corresponding to phase matching, the SHG energy remains at its maximal value when the crystal is rotated, demonstrating that the beam-stabilization system keeps the internal angle between the beam at λ₁ and the crystal axis constant. Likewise, the SHG energy remains at half the maximal energy when the beam stabilization is set for the angles θ₊ or $\theta_{-} $. Similar stabilization results have been obtained with a DKDP crystal, with the notable difference that the SHG energy curve is significantly narrower for a ∼50-mm DKDP crystal than for the 3.3-mm BBO, therefore making the DKDP crystals even more sensitive to angular instabilities of the translation stages.

 

Fig. 3. (a) Normalized SHG energy as a function of the rotation-stage angle without (dashed blue line) and with (continuous lines) angular stabilization for the 3.3-mm BBO crystal. The energy has been normalized to its peak value and the rotation-stage angle for that value is used as a reference. The beam-crystal angle has been stabilized to θ₀ for optimal phase matching (maximal energy, red line), $\theta_{-} $ (half energy, purple line) or θ₊ (half energy, black line). (b) and (c) Normalized SHG energy variation S for the 3.3-mm BBO crystal without and with beam stabilization, respectively.

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The beam stabilization operates correctly over a range of rotation-stage angles equal to 0.15°, which is adequate for stabilization of the pitch and yaw angles of the translation stages (specified by the manufacturer as 0.05° and 0.06°, respectively). The necessity and performance of the beam stabilization are demonstrated by two scans performed on this BBO crystal. In the absence of beam stabilization, the normalized SHG energy variation S [as defined in Eq. (3)] has significant measurement artifacts caused by the angular wobbling of the translation stages, in particular large variations for small changes in the horizontal position x [Fig. 3(b)]. These variations are well compensated by the beam-stabilization system, resulting in lower-frequency variations [Fig. 3(c)], which are consistent over multiple scans performed with different angular detuning and crystal orientations [Sec. 4.1].

For phase-mismatch characterization, the crystal under test is angularly tuned to determine the SHG energy at phase matching (obtained at the phase-matching angle θ₀), followed by angular detuning so that the SHG energy is half the maximal value (obtained at either θ₊ or $\theta_{-} $). The crystal’s position is then scanned so that the input 1ω beam maps out its entire aperture according to a predefined ‘snake-scan’ pattern. For efficient data acquisition, the crystal is continuously scanned at 0.5 mm/s in the vertical direction at each user-specified horizontal position. The beam-stabilization system continuously compensates for the relatively slow variations in crystal angle that occurs during the vertical scan. At the end of the vertical scanning range, one step is performed in the horizontal direction before starting a vertical scan in the opposite direction. This scanning pattern was experimentally found to yield better beam stabilization performance than horizontal scanning at each user-specified vertical position. This is attributed to the phase mismatch that is relatively insensitive to angular changes in the vertical direction, which is perpendicular to the phase-matching direction. The energies at λ₁ and λ₂ as well as the horizontal and vertical position of the crystal reported by the translation stages, are acquired at the repetition rate of the regenerative amplifier (5 Hz). Once the entire crystal aperture has been scanned, the data are numerically resampled onto an evenly spaced coordinates system. The 5-Hz repetition rate and 0.5-mm/s scanning speed correspond to a sampling size of 0.1 mm in the vertical direction. Steps equal to 0.1 mm in the horizontal direction therefore correspond to the same sampling in both directions. In these conditions, SHG mapping over the full scanning range of the two stages (50 mm) takes approximately 14 h. As demonstrated in Fig. 4, high-resolution mapping with a sampling step of 0.2 mm or smaller is necessary to characterize the high-frequency variations in phase-matching conditions caused by surface defects (Sec. 4). Such values of the sampling step correspond to oversampling relatively to the determined resolution (0.45 mm), which is beneficial for data quality.

 

Fig. 4. Example of normalized SHG energy variation S measured on a DKDP crystal with a sampling step of (a) 0.8 mm, (b) 0.4 mm, (c) 0.2 mm, and (d) 0.1 mm.

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The variations in normalized SHG energy variations S were typically acquired at both detuning angles $\theta_{-} $ and θ₊ to check the measurement consistency. Anticorrelated SHG energy variations are expected in these two configurations, leading to identical determinations of the local phase mismatch. Additionally, the phase-matching variations were measured for different orientations of the DKDP crystals corresponding to a 180° rotation of the crystal along a vertical axis. This induces a left‒right inversion of the variations induced by propagation within the crystal. This rotation swaps the input and output face, and can therefore be used to isolate the phase-matching variations induced by surface defects from the variations induced by internal propagation.

4. Experimental results

4.1 Characterization of a BBO crystal

A BBO crystal (15 × 15-mm² aperture, 3.3-mm thickness) has been characterized. This crystal has a cut angle equal to 21.6° for noncollinear optical parametric amplification around 920 nm using a pump at 526.5 nm. This angle is close to the SHG phase-matching angle for a source at 1053 nm (θ = 22.9°). The normalized energy variation S is shown for one orientation of the crystal and the two different detunings θ₊ and $\theta_{-} $ [Figs. 5(a) and 5(b)], and after rotation by 180° along the longitudinal (propagation) axis for one detuning [Fig. 5(c)]. The input face is the same face [labeled as (A) on the figure] for these three measurements. The anticorrelation of the variations measured at θ₊ and $\theta_{-} $ and the 180° rotation observed when rotating the crystal confirm the consistency of the measured energy. The three data sets lead to similar reconstructed phase mismatch δkL after applying the appropriate proportionality constants and rotation [Figs. 5(d)–5(f)]. For this crystal, there was no change in the determined phase-mismatch variations after rotation by 180° along the vertical axis, which swaps the input and output face, indicating that the measured variations are caused by bulk variations, e.g., local changes in the crystal angle. Using Eqs. (4) and (8) with the calculated value of ∂_θ n_e (λ₂), the observed 10% variations in S are consistent with 50-µrad variations in internal angle δθ.

 

Fig. 5. [(a),(b)] normalized SHG energy variation for the detuned 3.3-mm BBO crystal at $\theta_{-} $ and θ₊. (c) Normalized SHG energy variation for the detuned BBO crystal at $\theta_{-} $ after rotation by 180° around the longitudinal axis. The lower-right pictograms in these three figures indicate the crystal orientation, looking downstream. [(d)‒(f)] determined phase mismatch δkL from (a), (b), and (c) for the reference orientation, after numerical scaling, and rotation, respectively.

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4.2 Characterization of DKDP crystals

Several DKDP crystals designed for OPCPA of a broadband signal at 920 nm by a pump at 526.5 nm have been characterized: three sub-aperture crystals (25 × 25-mm² aperture, 48-mm length) with nominal deuteration level equal to 80%, 92%, and 98%, and two full-aperture crystals (63 × 63-mm² aperture, 48-mm or 52-mm length) with nominal deuteration level equal to 70%. The cut angle is crystal dependent, but it is close to the cut angle for SHG in each crystal (∼37.8° for 70% DKDP). The crystals have a sol-gel antireflection coating (centered at 527 nm on one face and 920 nm on the other face, owing to their use for parametric amplification of a signal around 920 nm by a pumped at 527 nm) on top of a GR-650 layer acting as a moisture barrier.

The measured SHG energy variations for the 92% DKDP crystal are plotted in Fig. 6 in three different configurations: for a reference orientation of the crystal and detuning at θ_–, for the same orientation and detuning at θ₊, and after rotation of the crystal by 180° along a vertical axis and detuning at θ_- For the first two measurements, the input face is the same [labeled as (A) on the figure], whereas the input face is the opposite face for the third measurement [labeled as (B) on the figure]. The anticorrelation between the measurements performed at $\theta_{-} $ and θ₊ confirms the consistency of the measurement [Figs. 6(a) and 6(b)] The symmetry along the x axis of the measurements performed for two different orientations [Figs. 6(a) and 6(c)] indicates that the phase-matching variations do not significantly depend on the propagation direction in the crystal, i.e., that they are linked to internal changes in properties. For this crystal, the normalized energy S varies by 10%. Using Eqs. (4), (8), and (11), with the calculated values of ∂_θ n_e (λ₂) and ∂_X n_e (λ₂)‒∂_X n_o (λ₁), the data are consistent with internal angle variations δθ of the order of 10 µrad (attributing the characterized phase mismatch solely to this effect) and deuteration level variations δX of the order of 0.01% (attributing the characterized phase mismatch solely to that effect). As previously discussed, it is not possible to determine the relative contribution of each effect from these data alone.

 

Fig. 6. Normalized SHG energy variation S for the 25 × 25 × 48 mm³ 92% DKDP crystal (a) in the reference orientation at θ_-, (b) in the reference orientation at θ₊, and (c) after rotation by 180° around a vertical axis at θ_-.

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A set of measurements performed in the same configurations on the 98% DKDP crystal is presented in Fig. 7. For the first two measurements, the input face is the same [labeled as (A) on the figure], whereas the input face is the opposite face for the third measurement [labeled as (B) on the figure]. This crystal has larger variations in normalized energy, up to 25%. Anticorrelated variations of S are obtained when operating at $\theta_{-} $ and θ₊ [Figs. 7(a) and 7(b)].

 

Fig. 7. Normalized SHG energy variation S for the 25 × 25 × 48-mm³ 98% DKDP crystal (a) in the reference orientation at $\theta_{-} $, (b) in the reference orientation at θ₊, and (c) after rotation by 180° at $\theta_{-} $.

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The high-frequency variations, e.g., in the upper-right corner on Fig. 7(a), are not measurement artefacts. They consistently correspond to a sign change between Figs. 7(a) and 7(b), indicating that they are caused by a local change in phase matching instead of another effect such as low local transmission. This is confirmed by the measured energy variations after a 180° rotation along the vertical axis [Fig. 7(c)]. The low-frequency variations in S are symmetric along the x axis for the two crystal orientations differing by the 180° crystal rotation [Figs. 7(a) and 7(c)]. This indicates, as was the case for the 92% DKDP crystal, that they are caused by bulk variations and do not depend on the propagation direction within the crystal. The observed 25% variations in S are consistent with 20 µrad angular variations and 0.025% deuteration variations. The high-frequency variations in S observed in the upper right corner of Fig. 7(a) are, however, not observed in the upper left corner of Fig. 7(c), indicating that these variations are caused by disruptions in phase matching occurring at or close to the input surface.

Two sets of mapped energy variations are shown in Fig. 8 for the 70% 48-mm DKDP crystal. They were obtained for two different orientations of the crystal related by a 180° rotation along a vertical axis, with angular detuning at θ_-, so they correspond to different input faces [labeled as (A) and (B) on the figure]. The observed low-frequency energy variations with amplitude of the order of 10% are consistent with sub-10-µrad angular variations and 0.01% deuteration variations. The energy variations are dominated by high-frequency modulations originating from coating issues on the crystal’s faces (another example of these modulations is shown in Fig. 4). The coating inhomogeneity was confirmed by visual inspection of both faces and by characterization of the wavefront reflected by each face (see Appendix 2). The crystal was set in a commercial Fizeau interferometer so that the incident beam is reflected by the input face, thereby generating the test beam. Interference with the reference beam leads to the single-pass phase φ after scaling by a factor 2. The measurement was repeated for two different crystal orientations to characterize each face, with a sampling size of 42 µm. The wavefronts φ measured over different regions of interest with surface issues clearly show local variations in coating thickness leading to reflected phase changes of the order of 0.1 wave at 1064 nm [ Fig. 9(a)]. It is worth noting that these variations, which are attributed to imperfections of the moisture barrier and sol-gel anti-reflection coating, induce a variation in transmitted wavefront that is smaller by approximately by a factor of 2. The wavefront slope along the extraordinary direction ∂_xφ was calculated by applying a Hast filter [34] and subsequent convolution with a square 10-pixel kernel to filter the high-frequency noise. The wavefront slope has values as high as 0.1 wave/mm over these regions [Fig. 9(b)] and shows good correlation with the normalized SHG energy variation [Fig. 9(c)]. This demonstrates that spatial irregularities at the input face of a nonlinear crystal can have a measurable impact on the local phase mismatch.

 

Fig. 8. Normalized SHG energy variation S for the 63 × 63 × 48 mm³ 70% DKDP crystal (a) in the reference orientation at θ_- and (b) after rotation by 180° at θ_-.

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Fig. 9. (a) Wavefront, (b) wavefront slope along the x direction, and (c) normalized energy variation in four regions of interest of the 70% 48-mm DKDP crystal. The first three columns correspond to the crystal orientation of Fig. 8(a), and the fourth column corresponds to the crystal orientation of Fig. 8(b). In each case, the wavefront induced by propagation from a reference plane to the face used as the input face for the SHG measurement is shown.

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5. Conclusions

We have demonstrated a novel approach to the characterization of transverse variations in phase-matching conditions in nonlinear crystals. SHG in the detuned crystal under test unambiguously converts the local phase mismatch onto energy at the upconverted frequency. Transverse scanning of the crystal combined with beam stabilization maps out phase-matching variations over an aperture only limited by the scanning range of the translation stages. The characterization of partially deuterated KDP crystals with submillimeter resolution over a 50 × 50-mm² aperture has revealed the impact of spatially nonuniform crystal properties and high-frequency surface variations due to coating imperfections. For these crystals the observed variations in phase mismatch are consistent with axis variations of the order of tens of µrad and deuteration variations well below 0.1%, relative to their mean values.

The experimental implementation described in this article can be directly applied to characterize and compare crystals grown in different conditions and/or with different chemical compositions. It can also be used to characterize the impact of temperature variations on the phase-matching conditions, for example caused by absorption of a high-average-power laser beam. A high-repetition-rate mode-locked laser could be used as a simpler source at the fundamental frequency. Spatial imaging of the upconverted beam resulting from detuned SHG of a larger-size beam at the fundamental frequency is an alternative acquisition approach that could increase the measurement rate for large-aperture crystals. Spatially resolving the longitudinal variations in crystal properties using several input beams crossing at a large angle would be of interest for thick crystals.

Appendix 1: Impact of crystal wedge on SHG energy

Nonlinear crystals are often cut with nonparallel faces to avoid multiple reflections from the crystal’s faces that could lead to parasitic phase-matched interactions. A wedged crystal, however, introduces a spatially dependent thickness; therefore, Eq. (1) cannot be directly applied. Rewriting the efficiency to take into account thickness variations yields

(19)$$\eta = {\left( {\frac{L}{{{L_0}}}} \right)^2}{\left[ {\frac{{\sin ({{{\Delta kL} / 2}} )}}{{{{\Delta kL} / 2}}}} \right]^{{\kern 1pt} 2}}.$$

The derivative of η with respect to Δk (at L = L₀) is unchanged. Its derivative with respect to L is

(20)$$\frac{{\partial \eta }}{{\partial L}} = \frac{2}{{{L_0}}}\frac{{\sin ({\Delta k{L_0}} )}}{{\Delta k{L_0}}}. $$

At η = 1 (Δk = 0), the derivative is simply

(21)$$\frac{{\partial \eta }}{{\partial L}} = \frac{2}{{{L_0}}},$$

which is consistent with the fact that a longer phase-matched crystal yields an increase in SHG signal. At η = 1/2, the derivative can be expressed as

(22)$$\frac{{\partial \eta }}{{\partial L}} = {{\sqrt {2 - {\kappa ^2}} } / {{L_0}}}.$$

For the crystal detuned at the half-efficiency point, an increase in crystal length also leads to an increase in SHG signal. Noting that Eq. (22) pertains to the peak-normalized efficiency defined by Eq. (1), the variation of the normalized energy S is

(23)$$\frac{{\partial S}}{{\partial L}} = {{2\sqrt {2 - {\kappa ^2}} } / {{L_0}}}.$$

Considering that $\sqrt {2 - {\kappa ^2}} \simeq 0.255,$ the relative impact of the wedge on S is therefore approximately 4 × smaller than the impact on the energy for the phase-matched crystal.

For a crystal with an angle α between its input and output faces, the length variation at a transverse coordinate x is x · tan(α). The normalized SHG energy variation S is therefore expected to change by 2x · tan(α)/L₀ when the crystal is phase matched, and by ${{2\sqrt {2 - \kappa _0^2} x \cdot \tan (\alpha )} / {{L_0}}}$ at one of the half-efficiency points. As shown in Fig. 10(a), these analytical derivations are in excellent agreement with split-step Runge‒Kutta simulations of SHG. They can be used to correct the impact of the spatially varying crystal length. If the wedge angle and orientation are known, the wedge effect can be subtracted from S before calculating the wave-vector mismatch. If the wedge characteristics are not known, the correction factor can be calculated from the measured normalized SHG energy for the phase-matched crystal. The impact of the wedge remains relatively small for the small incremental change in length x tan(α)relative to the average crystal length L₀ for most crystals used in practice.

 

Fig. 10. (a) Variation in SHG energy as a function of relative crystal length increase from Runge‒Kutta simulations (lines) and analytical derivations (circles) for a phase-matched crystal (blue lines/circles), i.e., ΔkL = 0 and η = 1, and a crystal detuned at half-efficiency (red lines/circles), i.e., ΔkL= 2κ and η = 1/2 (Appendix 1). (b) Definition of variables for the calculation of the impact of input surface irregularities on phase mismatch (Appendix 2), where the angles have been exaggerated for clarity.

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Appendix 2: Impact of input surface irregularities on phase mismatch

The variations in wave-vector mismatch induced by a defect of the input face of the crystal can be determined from the induced change in internal angle caused by refraction [Fig. 10(b)]. The input surface is described by the height function z = h(x,y). The variations in surface height across the aperture are assumed to be small enough that they do not introduce changes in the propagation distance throughout the crystal. The local incident angle in the phase-matching direction is assumed small so that it can be expressed as α(x,y) = −∂_xh(x,y). For SHG of a source at wavelength λ₁, a change in the incident angle α in air leads to a change in the angle β in the crystal calculated from the refraction law as n_o(λ₁)sin(β) = sin(α), where all angles are defined relative to the local surface normal. For small angles, one can approximate n_o(λ₁)β ≈ α. Following Fig. 10(b), the local change of internal propagation angle relative to the crystal axis is therefore $\alpha - \beta \approx \alpha [{1 - {1 / {{n_\textrm{o}}({{\lambda_1}} )}}} ].$ This change has the same impact on phase matching as the local variation of the crystal axis relative to the internal propagation direction considered to derive Eq. (8). Combining the expression for the local change of internal propagation angle with Eq. (8) therefore yields the local change in wave-vector mismatch:

(24)$$\delta k ={-} \frac{{2\pi }}{{{\lambda _2}}}[{1 - {1 / {{n_\textrm{o}}({{\lambda_1}} )}}} ]{\partial _\theta }{n_\textrm{e}} \cdot \delta {\partial _x}h.$$

The variations in surface height can be characterized via optical means, for example, by measuring the wavefront of a wave reflected by the input face. That wavefront is twice the wavefront φ(x,y) that is introduced by propagation from a reference plane to the surface, the latter being expressed as

(25)$$\varphi ({x,y} )= {\varphi _0} - \frac{{2\pi }}{{{\lambda _{\textrm{Test}}}}}h({x,y} ),$$

where λ_Test is the wavelength of the source used to test the surface and φ₀ is a piston term that is constant across the aperture. This allows one to rewrite Eq. (24) as a function of the spatial variations of the phase derivative along the phase-matching direction δ∂_xφ

(26)$$\delta k = \frac{{{\lambda _{\textrm{Test}}}}}{{{\lambda _2}}}[{1 - {1 / {{n_\textrm{o}}({{\lambda_1}} )}}} ]{\partial _\theta }{n_\textrm{e}} \cdot \delta {\partial _x}\;\varphi .$$

Funding

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

Acknowledgment

The authors thank A. Bolognesi, M. Barczys, T. McKean, and M. Spilatro for experimental assistance with the 1053-nm source. 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.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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