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Abstract
The angular dependance of the transverse Raman scattering in potassium dihydrogen phosphate (KDP) and its deuterated analogue (DKDP) for the entire range of crystal configurations suitable for laser beam polarization control has been investigated via experimental and modeling tools. This work was made possible by simultaneously rotating a spherical sample and the pump polarization to effectively measure the angular dependance of the transverse Raman signal in 360°. This novel method, which is applicable for the investigation of the Raman scattering in optically anisotropic materials, demonstrates that the spontaneous Raman scattering signal exhibits strong angular dependence that is modulated by depolarization and polarization rotation effects generated as the Raman signal traverses the material due to its birefringence. The results show that the total signal generated by the pump beam is the sum of the signals generated by the two components that have polarization parallel and orthogonal to the optic axis. The peak signal intensity, which is of importance for high-power laser applications, depends on the orientation of the optic axis and can vary by a factor of about 2. The excellent agreement between experimental data and modeling results validates the associated models and enables one to consider optimal crystal cut designs for specific applications.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Potassium dihydrogen phosphate (KDP) and its deuterated analog (DKDP) have been extensively investigated for their technologically important ferroelectric and optical properties. Their wide use in laser systems as electro-optic and nonlinear optical materials and their crystal growth properties that make it possible to rapidly grow into large size crystals motivate a continued interest to further understand their properties and limitations. The latter property has made them unique materials for large-aperture, high-peak-power laser systems such as the Omega Laser Facility (Rochester, NY, USA), the National Ignition Facility (NIF) (Livermore, CA, USA), and the Laser Mégajoule (LMJ) (France) [1–3]. These multibeam laser systems, designed for inertial confinement fusion (ICF) research, utilize KDP and DKDP for frequency conversion (second and third harmonic), polarization control, beam smoothing, and electro-optic pulse generation. The interaction of the laser pulse with the forming plasma on a fusion-class target is affected by a number of processes governed by laser-beam parameters including the polarization state of the laser light irradiating the target [4–7]. Initial concepts for polarization control included the development of wedged wave plates and wave-plate arrays that generate two orthogonally polarized speckle patterns that do not interfere coherently, thus improving focused illumination uniformity on the target [8,9]. Such polarization control optics have been implemented in certain laser systems, including the Omega Laser Facility, and are optimally positioned in the laser system following frequency conversion to the third harmonic. Of particular interest to this team are emerging polarization-smoothing designs that generate a continuum of polarization states with spatial patterns that are statistically uncorrelated.
Polarization control optics can be based on various materials or technologies such as polarizing gratings [8,10], birefringent crystals [8,11], liquid crystal devices [8], and even electron-beam–deposited films [12], but the construction and implementation of these optics have proven to be difficult. KDP and DKDP are particularly suitable for such applications due to their ability to grow in large size and their inherent birefringence, but their performance in large-aperture (>300-mm) systems is hindered by the generation of transverse stimulated Raman scattering (TSRS) [13,14] due to the presence of a strong symmetric A1 Raman-scattering mode. This process transfers energy to parasitic transverse beams and thereby limits the maximum power output during laser operation in order to avoid damage of the optic and the surrounding hardware [13,15–18]. As a result, current-generation large-aperture laser systems lack the ability to control the polarization at the operational wavelength (typically at 351 nm), which in turn adversely affects the coupling efficiency and the hydrodynamic stability of the imploding target.
The intensity of the TSRS signal is governed by the propagation length (optic size) L; the laser intensity Ipump; and the Raman-gain coefficient g, where the latter is directly proportional to the spontaneous Raman scattering cross section, dσ/dΩ; the molecular density M; the angular frequency ωs of the Stokes Raman-scattering component; the refractive index n of the medium; and the Raman FWHM line width Δν [19–21]:
where $g = ({{{8\pi cM} / {\hbar \omega_\textrm{s}^3{n^2}\Delta \bar{\nu }}}} )\cdot ({{{\textrm{d}\sigma } / {\textrm{d}\Omega }}} ).$ The Raman-scattering cross section of a given mode is related to the mode’s Raman polarizability tensor R, which can be determined by the experimentally measurable quantity σ, the spontaneous Raman scattering, based on the fact that where the unit electric polarization vectors of the pump and scattered light are ep and es, respectively. The form of the tensor for the dominant A1 mode of KDP and DKDP is, according to group theory [22] and recently confirmed experimentally [23], symmetric:While it has been challenging to accurately determine the tensor element values for this mode [16], the key issues were recently addressed, and the Raman tensors of the A1 mode of both KDP and 70% DKDP were ascertained with high accuracy [23]. Given a well-defined Raman polarizability tensor, the strength of the transverse Raman scattering (TRS) can be modeled for rays with varying polarization states propagating in all directions, where each ray interacts with a different crystal configuration. The TRS can, in principle, also be modeled via analytical methods. However, depending on the orientation of the optic axis, both the laser beam and Raman signal can undergo polarization rotation that generates erroneous signal components [23]. A direct analytical solution cannot capture these phenomena. Other forms of modeling, such as ray tracing using proper treatment of the polarization rotation effect, are required.
The aim of the present work is to provide the fundamental knowledge that enables development and validation of models to quantitatively evaluate the TSRS risk, and its directional dependence in geometries relevant to polarization control. This in turn will enable optimization of the design (such as the crystal cut orientation) of KDP or DKDP polarization control optics and guide the design of future laser systems. Our approach is twofold. First, we performed a detailed experimental study using spherical samples that facilitated measurement of the TRS signal in 360° for all crystal configurations and polarization states that are relevant to polarization control applications. We subsequently employed a ray-trace modeling approach that considers beam splitting and phase difference accumulation between the ordinary and extraordinary components of each ray to reproduce the experimental results and understand the origin of measurement artifacts. The validation of this modeling approach provides the means to develop accurate simulation tools for the TSRS effect in ICF-class laser systems.
2. Experimental setup
The experiments were conducted using the vertical arm of the setup detailed in Ref. 24. Laboratory coordinates are defined using the lower-case, italicized letters x, y, and z, while crystallographic axes are defined with upper-case letters X, Y, and Z (see Fig. 1). In brief, the pump laser beam (LaserQuantum 532-nm, 100-mW GEM) propagates along the vertical laboratory y axis and is incident on a spherical KDP or (70%) DKDP sample, 32 and 30 mm in diameter, respectively. The vertical beam is also parallel to the slit of the spectrometer (HORIBA iHR320, equipped with a liquid nitrogen–cooled detector). The direction of the Raman signal collection is always along the laboratory x axis in the (horizontal) laboratory x − z plane. The azimuthal angle ϕ is found in the x–z plane, and ϕ = 0° is defined to lie along the z axis. The polar angle θ defines the position of the optic axis (OA) with respect to the vertical laboratory y axis (beam propagation direction), and the projection of the OA onto the x–z plane is along the laboratory z axis for ϕ = 0°. Independent of the orientation of the OA, the crystalline X axis lies in the x–z plane and is positioned along the laboratory x axis for ϕ = 0°.
Fig. 1. Depiction of the experimental configuration to measure the angular dependence of the transverse Raman scattering in configurations suitable for polarization control. (a) The basic configuration defines the azimuthal x–z plane, the angle ϕ = 0° along the laboratory z axis, and the principle crystalline axes aligned along the laboratory axes. The pump beam propagates along the y axis, while the Raman signal is collected along the x axis. (b) The projection of the OA at angle θ onto the x–z plane and the pump polarization are fixed at an angle α and rotate simultaneously to produce the effect of the detector revolving about the sample.
To measure the TRS signal in 360° about the sample would, in principle, require the detector to revolve about the crystal sphere. To address this problem, we developed an alternative method where the angle α between the OA (or its projection onto the azimuthal plane) and the pump polarization remained fixed, while both the sample and the laser polarization were simultaneously rotated through ϕ = 0° to 360°. Raman scattering was measured at fixed intervals. Experiments were conducted with α = 45° (where, for any angle θ ≠ 0°, the pump light will experience the greatest birefringence), as well as α = 0° and 90°, where the pump polarization is aligned in the plane containing the OA or perpendicular to it, respectively. The angle θ of the OA was varied to explore crystal configurations of interest. Data were acquired with the analyzer set to detect the Raman signal polarized either parallel or perpendicular to the laser propagation direction. The latter analyzer position (along the laboratory z axis and in the horizontal plane) is of particular interest because Eqs. (2) and (3) imply that the Raman scattering is primarily polarized in the same plane in which the pump polarization exists.
3. Results
A complete set of data was acquired by varying three parameters: (1) the angular position of the OA with respect to the vertical pump beam (θ) between 0° and 90°, (2) the angular alignment of the pump-laser polarization relative to the vertical plane containing the OA (α), and (3) the transmission axis of the signal analyzer (parallel and orthogonal with respect to the beam propagation direction). KDP and DKDP are uniaxial crystals, but their crystallographic X(Y) axis can be determined by inspecting the Raman spectra [24]. In a supplementary experiment, it was found that the initial alignment of the crystallographic X axis with respect to the laboratory x axis (rotated by 0°, 22.5°, and 45° around the y axis) did not affect results. Figure 2 shows the intensity of the 915-cm−1 Raman mode integrated over 860 to 960 cm−1 produced using three different pump-polarization alignments: (1) in the plane containing the OA, α = 0°, (2) at α = 45°, and (3) perpendicular to the OA, α = 90°. This initial alignment of the pump polarization with respect to the crystal sphere positioning was held constant as both the sphere and the pump polarization were simultaneously rotated 360° in 2° increments. Data sets were acquired for 11 optic axis orientations with θ varying between 0° and 90° and the Raman scattering passing through an analyzer aligned either perpendicular or parallel to the pump beam. The data are normalized to the signal corresponding to the orientation that provides maximum spontaneous Raman scattering intensity, as discussed in detail later.
Fig. 2. Data acquired with an analyzer-aligned [(a)–(c)] parallel and [(d)–(f)] perpendicular to the pump laser. [(g)–(i)] Ray-trace modeling reproduced experimental data, including polarization rotation artifacts. The pump polarization was set at α = 0°, 45°, and 90° with respect to the vertical plane containing the crystalline OA, whose position was denoted with the polar angle θ, which was varied between 0° and 90°.
An analytical model considering both the laboratory and the crystal reference frames can be used to model the expected spontaneous Raman-scattering intensity IA1:
The ray-trace model was employed using the experimental conditions involving a 32-mm-diam sphere, an ∼0.5°incident half-angle and a collection half-angle of 5.7° to reproduce the experimental data [Figs. 2(g)–2(i) and 2(d)–2(f), respectively]. The modeling results verified that polarization rotation effects reproduce the conspicuous features observed at ϕ ∼ 90°, 270° for data acquired with α = 45° and 90°. Modeling also confirmed that the polarization rotation artifacts are reduced and essentially disappear as the collection half-angle converges to zero. Finally, the modeling shows unambiguously that while there should be no Raman scattering for certain configurations, there are several specific conditions for which the same maximum signal should be detected, as described below.
A comparison of data acquired with a pump beam parallel to the OA θ = 0° [purple traces in Figs. 2(d)–2(f)] shows that the signal peaks are lower and the minima are higher than expected based on modeling. Ray-trace modeling accounts for basic polarization rotation effects, which are typically associated with a signal “missing” in one analyzer configuration and observed in the corresponding orthogonal configuration [23] as shown in Figs. 2(b) and 2(e) or 2(c) and 2(f). An inspection of the spectra recorded for each measurement point confirms that superfluous A1-mode peaks exist where there should be no signal, even though their magnitude decreases as the OA position θ increases. This extraneous signal is likely another experimental artifact that we have not yet resolved. For θ >30°, the Raman spectra reveal that signal recorded in the spectral range of the A1 mode arises from partially overlapping neighboring modes.
According to theory, if the pump beam propagates along the OA (θ = 0°) and orthogonal to the crystallographic X − Y plane (lab z–x plane), the A1 mode should be excited to the same (maximum) degree, regardless of the pump beam’s polarization direction in the azimuthal plane. The magnitude of the detected scattering signal will depend on the relative direction of the pump polarization to the collection direction. Figure 3(a) shows the Raman signal acquired with the pump polarization set to α = 45° and the analyzer transmission along the z axis. The resulting trace produces a sinusoidal wave profile as a function of azimuthal angle ϕ, which in the given laboratory reference frame definition, produces the maximum signal when the pump polarization is orthogonal (along the z axis) to the direction of signal detection (x axis) at angles ϕ = 135°, 315°. Additional data shown in Fig. 3(b) demonstrate the reproducibility of this effect with measurements where the pump polarization alignment α was incremented in 45° steps.
Fig. 3. (a) The Raman signal is acquired with the optic axis parallel to the laser propagation direction (θ = 0°) and the pump polarization initially aligned at α = 45° from the z axis. (b) Data acquired with α shifted in 45° increments. The collected Raman signal is polarized-perpendicular to the laser propagation direction.
The onset of polarization rotation artifacts (depolarization) becomes evident as soon as the OA is tilted away from the pump-laser direction (θ ≠ 0°), and its effect is most pronounced when the pump polarization is set at α = 45°. Depolarization arises as the two orthogonal components of a ray experience different indices of refraction in a birefringent crystal. While some rays undergo polarization rotation (producing a modified polarization state as one component experiences a relative phase delay), most incident rays split into ordinary and extraordinary rays with orthogonal polarizations and very small spatial displacements. In Fig. 4(a), the polar angle θ = 5° and the traces acquired for α = 0°, 90° resemble the sinusoidal wave profile measured for θ = 0° in Fig. 3. Depolarization effects remain comparatively small when the pump polarization is aligned along and rotates with either the projection of the OA onto the azimuthal plane (α = 0°) or the crystallographic X axis (90°). However, the trace for α = 45° (orange) flattens and has an average value of about half the peak values of the other two traces. This behavior approaches the flat line, as predicted by the model. The asymmetries in the α = 45° trace are assigned to misalignments in the experimental setup that are difficult to control since a slight adjustment of the sphere position, of the order of a few milliradians, can change the shape of the trace drastically. This sensitivity diminishes with increasing θ.
Fig. 4. The Raman signal acquired for OA polar angle (a) θ = 5° and (b) θ = 90° from the laser propagation direction. Pump polarization is set at α = 0°, 45°, and 90°.
Polarization rotation artifacts (small dips or peaks) grow as the polar angle increases and more Raman scattering rays travel along the OA [Fig. 4(b)]. In the extreme case for θ = 90°, two differing principal crystallographic axes X and Z (OA), with no and ne indices of refraction, respectively, are found in the azimuthal plane. The signal maxima for α = 90° (green) are greater than for α = 0° (purple) because the pump polarization is aligned with the X axis, and the strength of the A1 mode excitation is determined by the polarizability tensor element A (value is proportional to A2). For α = 0°, the signal strength is determined by the tensor element B (value is proportional to B2 = 0.64*A2, see [23]). Again, the trace observed at α = 45° has essentially half the magnitude of the summed traces for α = 0°, 90°.
Ray-trace modeling confirmed that as the collection aperture size is reduced and polarization rotation artifacts diminish, the Raman scattering detected with the analyzer perpendicular to the pump beam increases (while that of its orthogonal component decreases) and approaches the shape of the total signal (sum of the two analyzer positions). Figure 5 explored the collection cone half-angles of 0°, 1.0°, and 5.7°. Note that, even for the largest cone angle, the polarization rotation artifacts are only barely visible for an OA θ = 60° (see Fig. 5(b)). The analytical solution [Eq. (4)] also accurately reproduces the results shown in Fig. 5 as the collection angle converges to 0°.
Fig. 5. Ray-trace modeling of Raman scattering acquired for the configuration where α = 45° and the analyzer is orthogonal to the pump beam. Behavior is compared for collection cone half-angles of 0°, 1.0°, and 5.7°.
Modeling confirms that summing the vertical and horizontal Raman signals removes the signal polarization rotation artifacts. The experimental data and modeling results of the total signal are presented in Fig. 6. Specifically, the results in Figs. 6(a) and 6(d) show that for α = 0°, the maximum signal gradually decreases with increasing θ. The total scattering signal does change for α = 90° and the profiles overlap [results in Figs. 6(c) and 6(f)], irrespective of the OA position. The results for α = 45° [graphs in Figs. 6(b) and 6(e)] demonstrate that the maximum scattering signal is reduced by half, but it also never reaches zero. Again, a sinusoidal wave profile is observed for OA θ = 0° because depolarization effects are not present for the specific configuration.
Fig. 6. The sum of the parallel and perpendicular polarizations is shown as the total Raman signal for each pump polarization configuration. [(a)–(c)] Data and [(d)–(f)] modeling are presented as the polar angle θ varies between 0° and 90°. Experimental data are normalized to the single largest signal found in the data set for α = 90°.
This behavior is quantified in Fig. 7, where the maximum values (independent of the observed angle) of both the experimentally measured and theoretical curves are plotted as a function of the polar angle θ for the case of KDP. The signal for α = 90° (green curve) remains constant; the pump polarization is always aligned along the crystallographic X axis, and the ability to excite the A1 mode remains the same, regardless of the OA position. For α = 0°, the maximum signal gradually diminishes with increasing θ (blue curve) as the strength of the A1 mode changes from its maximum (θ = 0° and Raman polarizability is driven by tensor element A) to its minimum (θ = 90° and polarizability is driven by tensor element B). When the pump polarization is set for α = 45°, the signal is reduced by half as soon as the OA is tilted away from the pump beam and depolarization arises.
Fig. 7. Total Raman signal in KDP detected for varying pump polarization configurations. Solid lines are from ray-trace modeling results.
The maximum Raman scattering signal for the dominant A1 Raman mode in KDP [produced when the pump and scattering polarizations are both aligned parallel to the crystallographic X(Y) axis] corresponds to the orientation that provides maximum spontaneous Raman scattering intensity [23]. These peak values have been previously measured for different laser wavelengths [21]; therefore, the experimental and modeling results discussed below can be used to obtain the exact value of the cross section.
The above experiments were repeated using a 70% deuterated DKDP. As expected, the behavior is practically identical to that of KDP and the DKDP signal strength is lower due to the difference in the relative values of the Raman polarizability matrix elements A and B [23]. Specifically, comparing an analogous set of experimental data acquired for DKDP and KDP, we find that the DKDP results scale by a factor of 0.75, in close agreement with previous reports [21,23].
4. Discussion
The experimental results can be directly applied for the assessment of the TSRS risk in large-aperture laser systems and help optimize crystal cut designs for different polarization control applications. Although estimation of TSRS in KDP and DKDP crystal plates for harmonic generation is not the focus of the current work, it is worth comparing the relative strength of the TRS signal generated in such crystal configurations (for frequency conversion) to that for polarization control. For example, a type-II third-harmonic–generation 70% DKDP crystal is cut in a configuration that corresponds with θ = 58°, while the polarization of the incident laser beam is aligned at 90° with respect to the plane containing the OA [2]. Using the same OA angle (θ = 58°) for a polarization control optic would still require the laser polarization to be aligned at 45° with respect to the OA. Therefore, the angular dependence of the Raman cross section for the OA angle θ = 60° and the two different laser (pump) polarizations (effectively α = 45°, 90°) used in each application is compared in Fig. 8.
Fig. 8. (a) The Raman cross section for type-II THG DKDP crystal is compared to that of a polarization rotator nearly optimized for reduction of TSRS. (b) Experimental results are superimposed onto the fixed geometry and polarization direction (α = 45°) of a polarization rotator plate. Polar plots are shown for θ = 60° and 90°.
The results obtained from a 360° rotation in the azimuthal plane can be superimposed onto the fixed geometry and polarization direction of a polarization rotator plate in the geometry currently used in fusion-class laser systems (such as the NIF and LMJ). Exemplifying data for α = 45° are shown in Fig. 8(b) using a polar plot to help visualize the dependence of the transverse spontaneous Raman signal intensity with regards to the crystal-plate orientation and laser-beam polarization. While the Raman signal is relatively small along the OA direction, it is maximal in the diagonal orthogonal direction.
The validation of the model and methodology by the experimental results presented in this work provide confidence on its application to guide crystal cut optimization needed to minimize TSRS gain, predicting maximum operational fluence, or helping to develop novel designs with complex polarization control properties in large-aperture optics. Future work will consider the design of specialized optics and include the ray paths contained by total internal reflection or retroreflected conditions that introduce longer gain paths.
5. Conclusion
The transverse spontaneous Raman scattering of the dominant Raman mode in KDP and DKDP in configurations relevant to polarization control in large-aperture, high-power laser systems was experimentally measured. The measurements effectively detected the Raman signal in 360° around the beam propagation direction for several crystal OA alignments. Both analytical and ray-tracing models captured the depolarization that dominates Raman-scattering behavior for most configurations. However, the latter model was required to predict the polarization rotation that occurs for Raman-scattered rays traveling along the OA, especially as the polar angle of the OA increases. These results enable one to consider optimal crystal cut designs for specific applications, to estimate the corresponding TSRS gain, and to consider laser designs that minimize the effect of TSRS such as controlling the laser polarization so that the direction of maximum TRS is located along an optics’ shortest geometrical path. In addition, the experimental and modeling treatment presented in this work can be used to investigate the Raman scattering properties in birefringent materials where the experimental studies are strongly affected by artifacts detailed in this work.
Funding
National Nuclear Security Administration (DE-NA0003856).
Acknowledgments
The authors acknowledge John Lambropoulos and Kyle Kafka for their support with modeling and interpretation of experimental results.
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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