1. Introduction

Potassium dihydrogen phosphate (KDP) and its deuterated analog (DKDP) are unique materials for large aperture laser systems owing to their low absorptivity in the UV to NIR range, sufficient nonlinearity, and ability to grow to large crystal sizes [1]. As a result, this material is exclusively used for frequency conversion (second, third and fourth harmonic generators for ICF class laser systems), beam control (plasma electrode Pockels cells) and polarization control (polarization smoothing) [2–4]. As the large aperture components are exposed to high irradiance and high fluence laser pulses, spontaneously generated Raman light traveling orthogonal to the laser beam within the optics can experience gain via transverse stimulated Raman scattering (transverse SRS or TSRS) [5,6]. Depending on the duration of the laser pulse, this gain will continue while SRS photons are traveling inside the optics and undergoing reflections at the optical faces and their edges. Although energy loss of the laser beam due to TSRS is a problem, the potential damage to the edges of the optic and the adjacent components (holders, etc.) due to the amplified TSRS light is of greater concern.

Similar issues can arise from transverse stimulated Brillouin scattering (transverse SBS or TSBS), which can be suppressed by the addition of modest bandwidths (15 to 30 GHz) to the main laser beam [7,8]. This solution is not applicable in the case of TSRS in KDP or DKDP as the bandwidth to suppress it is far too large to support efficient frequency conversion, and the operating conditions in most solid-state lasers. For example, Raman linewidths in KDP and DKDP are on the order of 600 GHz while Brillouin linewidths are 100 ~200 MHz. Thus, over 45 THz of spectral broadening on the main laser beam would be required to suppress TSRS, an extreme spectral width. Severe damage from TSBS at the third harmonic frequency was observed in a large aperture (80 cm diameter) optical glass beam splitter in NOVA laser experiments [9]. However, spectral broadening of the laser pulse had not yet been implemented at the time of these experiments. Large TSBS gain is also present in KDP and DKDP [10], but today modest bandwidths and crystal edge beveling are sufficient to suppress it in all current ICF laser designs, in harmonic converter cells, plasma electrode Pockels cells, amplifier glass, beam splitters, and final optics.

Similar to TSBS, optical damage can occur from TSRS, particularly in KDP and DKDP, which can be a more dangerous nonlinear process as it is more difficult to control. There is anecdotal evidence that such damage was observed as a darkening in certain regions of the aluminum structure supporting the 27 cm x 27 cm KDP crystals used in the 74-cm diameter crystal arrays during prototype high energy frequency tripling experiments (>8 kJ at third-harmonic) on the NOVA laser [11]. At the time, “edge-beveling” of the crystals and spectral broadening were not implemented, so TSBS would have been suppressed only by short-pulse transiency [9] while TSRS would not have been suppressed.

Among the materials used in large aperture laser systems, KDP and DKDP crystals are most vulnerable to TSRS effects due to their relatively large Raman cross section values associated with the totally symmetric mode of the PO₄ group. The problem so far has been addressed using various methods (as mentioned) including beveling the edges of the crystal plates to minimize the reflection (and thus amplification) of the SRS back into the optic, and by replacing KDP by the much more expensive DKDP [6]. The latter choice stems from the splitting of the totally symmetric breathing mode upon deuteration into two modes that have lower Raman cross sections [12–14]. But even for DKDP, fourth harmonic generation from the fundamental of an ICF class laser can be limited by TSRS, especially when considering that intrinsic and impurity defects known to exist in KDP/DKDP crystals, could resonantly enhance the Raman cross section [15–17]. In addition, the transverse Raman scattering cross section depends on the crystal cut, i.e., orientation of the crystallographic axes with respect to the light propagation direction, and polarization directions of the fundamental and harmonic fields. For these reasons, a more detailed knowledge of the Raman cross sections in KDP and DKDP is desirable to support estimation of material limits for future ICF laser designs and operational conditions.

SRS is a well understood process extensively studied following the advent of the lasers in the early 60’s with very important present applications such as for frequency generation and signal amplification in optical fibers for telecommunications purposes [18]. Furthermore, the peak SRS gain coefficient can be calculated once the spontaneous Raman cross section and spectral profile are known [19,20]. However, the measurement of the absolute value of the spontaneous Raman cross section has long been a challenge [21,22]. As a result, the values of Raman cross sections have been directly measured for only a limited set of materials and often these values can vary significantly from different reports.

The objective of this work is to provide an estimation of the Raman cross section of the dominant peaks of KDP and DKDP (at various deuteration levels) at the second (2ω), third (3ω) and fourth harmonic (4ω) frequencies of high-power, large-aperture ICF class laser systems. An accurate measurement of the spontaneous Raman scattering cross section will enable the evaluation of the upper limit of the material tolerance to SRS after the operational parameters (laser wavelength, temporal pulse shape and duration, and spatial distribution of the laser intensity) and material specifications (size of the plate, crystal cut and deuteration level) are taken into consideration.

2. Experimental design

Our experimental approach is based on using the Raman scattering cross section of water as a reference material to derive the Raman cross-sections of KDP and DKDP. The dominant feature in the Raman spectrum of liquid water arises from the fundamental O–H stretch mode of the water molecule centered at about 3400 cm⁻¹. The Raman scattering cross section of this broad line is fairly large and has been documented in various reports [23–27]. Most of these absolute measurements have been performed using the Raman scattering from another well-characterized material (such as benzene or nitrogen) as a reference and reported the cross section at specific excitation wavelengths or over a range of wavelengths. More recently, Faris et al. [28] complemented results of previous reports with a detailed set of measurements of the Raman cross section of water as a function of wavelength. As a result, water is now well characterized in the 200 nm to 550 nm spectral range and can be used as a reference material.

The selection of water as a reference in our study was based on a set of additional criteria. Specifically, we preferred a material that does not exhibit resonant Raman scattering enhancement due to absorption at the wavelengths of interest as this may represent a complication in the measurements including increased experimental errors. In addition, we preferred a material with a similar index of refraction to that of KDP/DKDP for minimizing the associated signal corrections and possible errors that are discussed next in more detail. Also, water is a material easy to work with in a laboratory environment that has been primarily designed for the study of solid-state optical materials.

There were two KDP and two DKDP samples used in this work. All samples were provided by Cleveland Crystals, Inc.. One of the KDP samples was grown using the rapid growth method and the other using conventional (slow) growth [1]. The difference between the two, conventional growth DKPD samples was their deuteration level which was at 70% and 99%, respectively. The samples were prepared in 2.5 cm X 2.5 cm X 2.5 cm cubes cut along the crystallographic axes, i.e., one side coincides with the optical axis of the material, denoted as Z or c. High-purity water was placed in a 2.5 cm X 2.5 cm X 2.5 cm fused-silica cuvette with wall thickness of 1 mm. The experimental system used to perform the measurements is depicted in Fig. 1 . Three compact lasers (Intelite, Inc., Minden, NV) operating at 266 nm, 355 nm, and 532 nm with average output power of about 1 mW, 5 mW, and 50 mW, respectively, were used as the excitation sources to record the Raman scattering spectra of the samples. The output laser beams had a diameter of about 2 mm and were linearly polarized. The polarization orientation of each beam was controlled using half-wave plates (WP) at the output of each laser. The laser beams were combined using dichroic mirrors and aligned to co-propagate. The polarization orientation of the combined beams was determined by a calcite broad band polarizer (P1) located before a 10-cm nominal focal length fused silica lens (L). This lens was used to focus the laser beams near the middle of the 2.5-cm thick cubic samples (S) and placed on a translation stage for optimal positioning of the focal point at each wavelength. The propagation of the excitation beam(s) was along the optical axis of the material (denoted as Z or c in Fig. 1). The average laser power transmitted through the sample during signal acquisition was measured using a power meter (PM). The generated Raman scattering signal was collected using two lenses, both having a nominal focal length of 20 cm. The first lens (L1) was at a fixed position with respect to the sample while the position of the second lens (L2) was varied for optimal focusing of the signal into the entrance slit of the spectrograph at each excitation wavelength. In this arrangement, a 4 mm section of the elongated focal region of the laser beam within the bulk of the sample was imaged along the vertical dimension of the entrance slit to the spectrograph with a magnification factor of about 1. The width of the slit was kept at 100 µm, which is larger than the size of the focused laser beam in the interrogation volume. A polarizer (P2) was positioned just before the entrance slit of the spectrograph in order to select only one

 

Fig. 1 Schematic diagram depicting the experimental system used to perform the Raman scattering measurements in the Z(XX)Y geometry. The incident k_P and scattered k_S light wave vectors are orthogonal to each other.

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polarization component of the Raman scattering signal. The spectrograph (Acton Research, SpectraPro 750) was equipped with a 2400 g/mm grating blazed at 240 nm (which was used to analyze the Raman scattering spectra under 266 nm and 355 nm excitations) and a 1200 g/mm grating blazed at 700 nm (which was used to analyze the Raman scattering spectra under 532 nm excitation). The spectra were recorded using a liquid nitrogen cooled CCD, 1340 X 400 pixels array detector (Princeton Instruments). The spectral resolution of the system was evaluated by measuring the spectral width of the laser lines which were found to be about 3 cm⁻¹ at full width at half maximum for all three excitation wavelengths.

As TSRS in large aperture optical components builds up in a direction perpendicular to the incoming beam, the Raman scattering cross section to be measured is in the corresponding 90° geometry shown in Fig. 1 and can be written as ${\left(\frac{d\sigma }{d\Omega }\right)}_{90°}$. This represents the differential scattering cross section integrated over a selected portion of the entire Raman spectral profile, per molecule. In our measurements, we integrated the Raman scattering of the water line centered at about 3400 cm⁻¹ from 2900 cm⁻¹ to 3800 cm⁻¹. For the case of KDP and DKDP crystals, we integrated the signal intensity over a window of 100 cm⁻¹ centered at the peak of the Raman line of each sample. Specifically, the signal integration for KDP was performed from 860 cm⁻¹ to 960 cm⁻¹ with the peak centered at 914 cm⁻¹, for the 70% deuterated DKDP from 825 cm⁻¹ to 925 cm⁻¹ with the peak is at 888 cm⁻¹, and for the 99% deuterated DKDP from 820 cm⁻¹ to 920 cm⁻¹ with the peak centered at 879 cm⁻¹. We should note that the Raman spectrum of the A₁ vibration in DKDP is split into two peaks, of which we selected the major peak in each case for determination of the differential cross sections listed in Table 1 .

Table 1. The measured Raman scattering cross sections integrated over a 100 cm⁻¹ spectral range centered at the main peaks of KDP/DKDP samples

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The overall spectral response of the instrument was calibrated for each of the spectrograph settings, i.e., including one of two gratings depending on excitation wavelength and grating angle/center wavelengths for either KDP/DKDP or water, as follows. For the two settings used to acquire the spectra under 266 nm excitation in the UV range, a Deuterium calibration lamp was used and the results were compared with the spectral response provided by the manufacturers of the various components involved (spectrograph, grating, CCD). There was up to a 6% difference between these two methods and we chose to use the calibration results from the Deuterium lamp for our analysis. For the four settings used to acquire the spectra under 355 nm and 532 nm excitation, two tungsten calibration lamps were used to characterize the instrument’s spectral response. There was less than 1% difference in the results for the settings covering the Raman spectra of KDP/DKDP and water under 355 nm excitation and up to 5% difference for the settings used to acquire the Raman spectra under 532 nm excitation. In both cases, we utilized the results of the tungsten calibration lamp that was acquired and calibrated within less than two months from the execution of the experiment.

There are several other wavelength-dependent corrections that must be performed when measuring the Raman cross-section using another material as reference. These include attenuation of the excitation and Raman scattering signal that is due to i) transmission losses incurred by the pump beam and Raman scattering signal, ii) the variation in Raman scattering collection solid angle (Ω) that is due to the difference in index of refraction of the materials, and iii) the polarization state of the detected signal. A general form describing the correction parameters that must be included to obtain the Raman scattering cross section of a material A using a material B as reference can be written as:

where

In our case, A represents our sample and B represents water. It must be noted that the values of the Raman cross-section of water is provided for the total scattering signal integrated over the entire line-width. As we are measuring the polarized Raman signal components of KDP/DKDP, we need to multiply our measured signal for water by 1.17 to account for the depolarized signal that was not recorded. This is included in the last correction factor in the above equation where we can use ρ_B = 0.17 and ρ_A = 0. Furthermore, the molecular densities at room temperature are: M_KDP = 1.0322x10²² molecules/cm³, M_DKDP = 1.0318x10²² molecules/cm³ (as provided by the manufacturer, Cleveland Crystals, Inc.) and M_WATER = 3.335x10²² molecules/cm³.

Regarding the signal transmission losses, we need to account for the transmission losses of the pump and scattered light due to i) reflections at the sample/air interface and ii) absorption throughout the bulk material before and after the focal region, respectively. As discussed earlier, the power meter PM is used to measure the transmitted pump beam through the sample. The polarization of the pump and Raman scattering light are kept the same and orthogonal with respect to the optical axis (c-axis) of the crystals (for KDP/DKDP). In addition, the Raman scattering wavelength is very close to that of the pump beam, thus the losses experienced by the pump beam traveling through the material after the focal point are almost identical to those we need to include for the Raman scattered signal. This provides for a simple method to include a set of reflections and an unknown amount of absorption (especially at 266 nm) while it introduces a very small error that we estimate to be less than 1%. Next, to account for the variation in the collection solid angle between KDP/DKDP and water, their index of refraction at the corresponding Raman scattering wavelengths for each excitation wavelength were used. Finally, the Raman scattering cross section of water at the different frequencies was obtained using the fitting curve of the experimental data provided in the report of Faris et al. [28].

3. Experimental results

The results obtained from the experimentally measured values of the Raman scattering signal (F) and Eq. (1) above are summarized in Table 1. The measurements were performed in the Z(XX)Y geometry, where Z (parallel to the optical axis of the material), X and Y are the pump, polarization and scattered wave vectors, respectively. It must be noted that there was no measurable variation in the Raman scattering strength from different locations within each material within the resolution of our instrumentation (on the order of 1%). Additional errors arise from the calibration process of the instrumentation (which we believe should be on the order of 10% or less) and the accuracy of the values used for water [28] (which the authors of the report state that should be within 20% over the wavelength range of 215–550 nm). For reasons that will be discussed later, we also measured the corresponding values of the Raman scattering cross section in the X(ZZ)Y geometry. We found that, for all crystals and laser excitation wavelengths used in this work, the cross section value of the breathing mode in the X(ZZ)Y geometry is (61.5 ± 2)% of that measured in the Z(XX)Y geometry.

The spectra of the main Raman peaks of our samples after they were normalized to the peak intensity of conventionally grown KDP and plotted over the signal integration spectral range (100 cm⁻¹ wide) are shown in Fig. 2 . These spectra were obtained under 532 nm excitation. These peaks were fit to a Lorentzian function that yielded a coefficient of determination (R-squared) of better that 0.997 for the DKDP spectra and better that 0.999 for the KDP spectra. These fits yielded the scale parameter w, which specifies the full-width at half-maximum (FWHM), and a background shift y₀ (offset) with respective values which are summarized in Table 2 . Some broadening of the peaks (value of w) is expected due to the line-width of the laser beam and the loss of spectral resolution arising from the width of the spectrograph’s slit. However, the measurement of the Raman cross section over a relatively wide spectral range makes it effectively independent of these instrument related effects as it represents an accurate measurement over that spectra range. This value can then be used to retrieve the actual peak values and the corresponding SRS gain coefficient after using the actual value of the spectral line-width.

 

Fig. 2 The spectra of the main Raman peaks of our samples (1: rapid, 2: conventional grown KDP, 3: 70% DKDP, 4: 99% DKDP) after normalization to the peak intensity of conventionally grown KDP plotted over the signal integration spectral range (100 cm⁻¹ wide).

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Table 2. Fitting parameters using a Lorentzian function of the main Raman lines in KDP/DKDP vs. excitation wavelength

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Direct measurement of TSRS gain in large aperture optical components can be performed by i) using a high power pump laser plus a probe laser that can be tuned to the Raman wavelength to measure the gain [5], ii) by estimating the transverse gain from the stimulated threshold [6], or iii) by employing an intra-cavity SRS gain measurement technique [29–31]. Alternatively, since TSRS gain is proportional to the peak spontaneous Raman scattering cross section, it can be estimated using the integrated spontaneous Raman scattering cross section, the spontaneous Raman linewidth, and known material parameters following [19]:

where M is the molecular density in cm⁻³, ω_s is the angular frequency of the Stokes Raman scattering component, n is the refractive index, Δν is the Raman FWHM line width in cm⁻¹ (assuming a Lorentzian spectral profile) and dσ/dΩ is the corresponding Raman scattering cross section for the particular crystal cut orientation in units of cm² molecule⁻¹ sr⁻¹. Theoretically, the integrated Raman scattering cross section for a specific crystal cut, pump and, scattered wave propagation directions is proportional to the respective tensor coupling terms given by {e_s^T ⋅ R ⋅ e_p}², where e_p,s are the unit electric polarization vectors of the pump and scattered light respectively, and R is the Raman polarizability tensor [15]. These are listed for specific crystal symmetry groups by Cardona [16].

The early experiments to measure the SRS in KDP crystals [5] involved a direct measurement of stimulated gain in KDP with a high irradiance narrow-band pump beam at the second harmonic and a single-frequency CW probe beam (scanned in wavelength through the Raman line profile). Transverse SRS gain coefficients were estimated from the longitudinal SRS gain measurements. This required a detailed knowledge of the Raman tensor for the A₁ vibration at 914 cm⁻¹. Relative spontaneous Raman scattering amplitudes from cube samples of KDP (cut along the crystal axes) similar to those used in this work, and a small type II second harmonic crystal, indicated a noticeable angular anisotropy in the depolarized scattering. It was postulated at the time that the A₁ vibrational mode in KDP would be better described by the general tetragonal point group (4 or $4/m$) rather than the more restricted group ($422$, 4mm, $\overline{4}2m$). This would contribute off-diagonal elements to the diagonal A₁ tensor, and perturbations with the polar vibrational modes B₂(z), E(x), and E(y) would thereby introduce an additional dependence of these Raman tensor components on the scattering angles [32].

For a more accurate description of these early spontaneous and stimulated Raman experimental data, a tensor with non-zero off-diagonal elements was adopted which would depend in amplitude on the directions of the incident and scattered wave vectors relative to the crystal axes. A first approximation that incorporated the most important features of the Raman scattering behavior of the 914 cm⁻¹ mode was provided in the following form [5]:

where

and Ψ is the polar angle between the q vector of the Raman vibration (defined as the difference between the k vectors of the pump and scattered light waves, k_p - k_s) and the Z-axis of the crystal. The two independent scattering coefficients were denoted as “a” and “b” following Cardona [16]. Furthermore, the Raman scattering cross section in the Z(XX)Y geometry is proportional to a² (which is the corresponding tensor coupling element) and proportional to b² in the X(ZZ)Y geometry. As mentioned earlier, the cross section value of the breathing mode in the X(ZZ)Y geometry was measured to be (61.5 ± 2)% of that measured in the Z(XX)Y geometry. This is in agreement with the early LLNL work that provided b²/a² of about 0.60 (Table 6-7 in Ref. [5]). The addition of this relative value of b provides for a complete description of the Raman tensor in terms of only one parameter (a) in the approximation provided by Eq. (3).

Using Eq. (2) and the data presented in Table 2, the SRS gain for the a² coefficient (corresponding to the Z(XX)Y scattering configuration) can be calculated. This value represents the upper limit for the expected SRS from each material (at optimal SRS gain configuration) for each excitation wavelength. The results are summarized in Table 3 . Virtually no difference between rapid growth and conventionally grown KDP was observed, while deuteration at 70% reduces the gain significantly in DKDP. With increasing deuteration to 99% DKDP (nearly pure DKDP), the SRS gain is again increased but still remains below that of KDP (at ~2/3 that of KDP). In order to estimate the TSRS coefficients for crystal orientations corresponding to the various type I and type II phase matching schemes commonly employed for frequency harmonic generation or polarization smoothing applications, the Raman tensor in a form similar to Eq. (3) must be transformed to the X’, Y’, Z’ coordinate system of the harmonic crystal. The results presented in this work provide sufficient information to estimate the TSRS gain coefficients for various wavelengths and polarizations of the electric fields present in common harmonic converter designs, for example see Wegner et al. [11], and the possible side-scattered fields [5,6,30,31].

Table 3. Calculated TSRS gain for a² tensor coupling element (XZZY geometry)

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4. Discussion

The experimentally measured transverse stimulated gain at the third harmonic reported by Barker et al. [6] in a conventionally grown, type II DKDP third harmonic crystal (80% deuterated) was 0.098 cm/GW. In this work, the gain that we calculate using the measured spontaneous Raman cross section yields a value of 0.13 cm/GW. Other authors reported a stimulated gain of 0.15 +/− 0.03 cm/GW for conventionally grown DKDP (80% deuterated) at the third harmonic, and 0.6 +/− 0.1 cm/GW for rapid growth KDP at the third harmonic [29,30]. We are in agreement with the prediction of TSRS gain in the conventionally grown DKDP, but find ourselves in disagreement with the TSRS gain measured for the rapid growth KDP at the third harmonic. Similarly, our calculation of TSRS gain for a conventionally grown KDP crystal cut for type II doubling of 1053 nm light yields a peak of 0.22 cm/GW at the second harmonic, while the estimated value was 0.15 +/− 0.03 cm/GW, inferred from the measured longitudinal Raman gain by Smith et al. [5]. This discrepancy in the stimulated gain coefficient with the early work might be attributed in part to sample quality issues including the presence of scattering centers (from previous optical damage, inclusions, and lattice defects) or resonant absorption [15–17] via pre-existing or transient defects (generated by the intense pump pulses) that can lead to gain loss or gain enhancement, respectively. We are currently performing a more detailed study on the angular dependencies of the spontaneous Raman scattering cross section [32] to resolve these discrepancies as well as better understand and describe the Raman tensor in KDP and DKDP crystals.

The generation of SRS in large aperture optical components is related to the corresponding spontaneous Raman scattering cross section as shown by Eq. (2). It must be noted that TSRS can involve gain along propagation at 90° with respect to the surface of the optic as well as over a range of angles around 90° that can support propagation via total internal reflections. In the latter case, the transverse distance traveled by the parasitic SRS will be increased. In terms of the TSRS-imposed operational limitations in high-power, large aperture laser systems, it is essential to know the spontaneous Raman scattering cross section within the range of angles that can support transverse signal amplification. The results presented in this work can help address this issue in two ways. First, the absolute value of the spontaneous Raman scattering cross section provided in this work can be used to calculate (using the Raman polarizability tensor) the spontaneous Raman scattering cross section within the range of possible orientations of interest and thereafter estimate the TSRS gain for the most favorable orientation. The second approach is to measure the spontaneous Raman cross section of samples with a particular crystal cut (suitable for the intended application) and use as a reference sample that is cut in the same orientation as used in this work. In this case, the measurement is very simple for the 90° orientation but may be more complicated for angles much different from 90°. Cylindrical samples (where the cylinder’s axis is along specific crystallographic orientation) can be used to perform this type of measurement [33], but other methods are possible.

As discussed above, deuterated KDP has lower SRS gain owing to the addition of deuterium which splits the degeneracy of the dominant KDP Raman mode at 915 cm⁻¹ and hence reduces the peak Raman gain in DKDP (see Fig. 2). Recently, our group demonstrated non-critically phase-matched fourth harmonic generation (FHG) of Nd:YLF laser using 70% deuterated DKDP at near room temperature [34]. This suggests that FHG in a large aperture laser system with sufficient output power to perform various measurements including plasma diagnostics is possible. However, this also necessitates an estimation of the TSRS expected in DKDP at these operational conditions. Substitution of the corresponding parameter values into Eq. (2) (assuming for simplicity gain at 90° propagation) yields a TSRS gain at 4ω of 0.17 cm/GW. A similar calculation was carried out using DKDP material parameters at 2ω to arrive at a TSRS gain at 2ω of 0.03 cm/GW. It must be noted that estimation of the TSRS gain at both wavelengths is required, as the input side of the crystal plate will be exposed mainly to intense 2ω laser light while the output side will be exposed to mostly intense 4ω laser light. Based on the TSRS gain derived above, we can assess limits imposed by TSRS on FHG in DKDP by calculating the Raman gain G = exp(gI_maxL) after one traversal through the transverse dimension of the crystal at the expected maximum pump intensity. For this analysis, we assume moderate 2ω laser irradiance of 1.0 GW/cm² and an output 4ω irradiance of 0.5 GW/cm², and a transverse crystal dimension of L = 35 cm. Using these values, we calculate the single pass gain to be 2.86 and 19.6 at 2ω and 4ω, respectively.

To prevent build-up of the Raman scattered light from multiple traversals through the crystal in a long laser pulse case, at a minimum, it is necessary to reduce reflectivity at all crystal edges to be less than 1/G. Thus, the edge reflectivity for a DKDP fourth harmonic crystal must be less than 35% at 2ω, and 5% at 4ω, both of which are easily achieved. However, from engineering experience on large aperture (>27 cm) harmonic converter cells at either the second, third or fourth harmonic, and because of total internal reflection possibilities, the crystal edges and corners should be chamfered and beveled in a compound manner to prevent multiple-passes of side-scattered light at all wavelengths.