Numerical Investigation on the Effect of Smoothing by Spectral Dispersion on Transverse Stimulated Raman Scattering Gain in KDP Crystals

Abstract

In inertial confinement fusion (ICF) laser drivers, large-aperture high-intensity third-harmonic (3ω, central wavelength 351 nm) laser pulses passing through KDP crystals (potassium dihydrogen phosphate) can produce strong transverse stimulated Raman scattering (TSRS). TSRS not only depletes the energy of the 3ω laser beam but also damages the KDP crystal, thus significantly limiting the enhancement of ICF laser driver capabilities. Therefore, effectively suppressing TSRS in KDP crystals is a critical issue in the design and construction of ICF laser driver systems. This paper first proposes that SSD has the ability to suppress TSRS through theoretical analysis of the characteristics of SSD beams. Secondly, through numerical simulations, it presents the influence of variations in three key parameters—modulation amplitude, modulation frequency, and grating dispersion coefficient—on the TSRS gain. The results show that the Stokes gain decreases with increasing modulation amplitude and modulation frequency; specifically, the suppression capability of SSD for TSRS gradually strengthens as modulation bandwidth increases. In addition, previous reports have demonstrated that SSD can significantly suppress stimulated rotational Raman scattering (SRRS) in air, which highlights the potential value of applying SSD in large laser facilities such as ICF driver systems.

1. Introduction

To enhance the output energy of an entire inertial confinement fusion (ICF) laser driver, apart from increasing the number of laser beams, enlarging the aperture of a single beam is also a crucial technical approach. Currently, the aperture of the third-harmonic KDP (potassium dihydrogen phosphate) crystal has reached or even exceeded 40 cm, with a thickness of only ~0.9 cm. Consequently, the ratio of aperture to thickness for a KDP crystal exceeds 40:1. When a high-intensity third-harmonic (3ω, central wavelength 351 nm) laser pulse passes through such a large-aperture, thin KDP crystal, it is very susceptible to transverse stimulated Raman scattering (TSRS). Indeed, significant TSRS and associated damage to KDP crystals have been observed in experiments on ICF laser driver devices such as NIF (National Ignition Facility) and SG-III (ShenGuang-III Laser Facility) [1,2,3]. Accordingly, many studies have reported on methods to suppress TSRS. The first type of scheme is to reduce the gain length of Stokes light. For example, large-aperture KDP crystals are composed of several small-aperture KDP crystals. Several grooves are etched on the rear surface of KDP crystals along the direction perpendicular to Stokes propagation [4]; the edges of KDP crystals are subjected to “beveling” or “edge chamfering” to reduce the reflection of Stokes light back into the KDP crystals [5]. The second type of scheme suppresses TSRS by controlling the polarization relationship between the laser and Stokes light, such as splitting the laser beam into several adjacent sub-beams with mutually perpendicular polarization directions via polarization control plates [6,7]. The third type of scheme is doping KDP with high-concentration deuterium [1,5,7,8].

The aforementioned suppression schemes can, to a certain extent, reduce the Stokes intensity and mitigate the impact of TSRS on KDP crystals, but they may lead to significant unintended consequences, such as beam intensity modulation, high difficulty in engineering implementation, and high cost. Moreover, what deserves more attention is that the methods mentioned above are designed merely to suppress TSRS, and they hardly consider whether they can simultaneously suppress other harmful nonlinear effects or achieve other functions. In other words, they do not address the issue of TSRS suppression from the perspective of the overall laser system design.

M. D. Skeldon et al. reported that a fundamental-frequency (1ω, central wavelength 1053 nm) laser beam employing smoothing by spectral dispersion (SSD) can significantly suppress stimulated rotational Raman scattering (SRRS) in air [9]. If SSD can similarly suppress TSRS, it would be extremely beneficial for large laser systems such as ICF drivers. Simultaneously suppressing the two nonlinear effects of TSRS and SRRS is something that previous suppression methods neither took into account nor could accomplish. Therefore, in this work we investigate the effect of SSD on the TSRS gain in KDP crystals and carry out a quantitative analysis of its suppression capability.

2. Fundamental Theory of TSRS

When the 3ω laser pulse passes through the third-harmonic KDP crystal, stimulated Raman scattering (SRS) occurs due to lattice vibrations within the crystal. At the initial stage of scattering, Stokes light scatters into the entire solid angle (4π) with random polarization directions. However, as the pump laser continues to inject, the Stokes light will propagate in the direction of maximum gain to achieve the greatest amplification—namely, along the direction perpendicular to the incident laser beam (as illustrated in Figure 1)—and the Stokes component with polarization parallel to the incident laser’s polarization has the strongest gain. Therefore, when the 3ω laser pulse passes through the KDP crystal, the Stokes light is mainly amplified in the direction perpendicular to the incident laser’s polarization; this transverse scattering is known as TSRS. In general, the side faces of the KDP crystal reflect a portion of the Stokes light, which further exacerbates the generation and growth of TSRS and increases the risk of damage to the KDP crystal. Inside the KDP crystal, the 3ω laser intensity increases from the entrance face to the exit face. Therefore, the TSRS effect is strongest and the risk of damage highest at the rear surface of the KDP crystal, making this surface the focus of our study.

The governing Raman equations under slowly varying envelop approximation are as follows [10,11,12]:

$$\left({\nabla }_{\perp }^2+2\mathrm{i}{k}_{\mathrm{L}}\frac{\partial }{\partial z}\right){\mathit{E}}_{\mathrm{L}}=2k_3{k}_{\mathrm{L}}\mathit{Q}{\mathit{E}}_{\mathrm{S}},$$

(1)

$$\left({\nabla }_{\perp }^2+2\mathrm{i}{k}_{\mathrm{S}}\frac{\partial }{\partial z}\right){\mathit{E}}_{\mathrm{S}}=2k_2{k}_{\mathrm{S}}{\mathit{Q}}^{\ast }{\mathit{E}}_{\mathrm{L}},$$

(2)

$$\frac{\partial }{\partial t}{\mathit{Q}}^{\mathrm{*}}=-\Gamma {\mathit{Q}}^{\mathrm{*}}+\mathrm{i}{k}_1{\mathit{E}}_{\mathrm{L}}^{\mathrm{*}}{\mathit{E}}_{\mathrm{S}}$$

(3)

where ${\mathit{E}}_{\mathrm{L}}$ and ${\mathit{E}}_{\mathrm{S}}$ are the laser and the Stokes complex amplitudes, and their respective wave numbers are $k_{\mathrm{L}}$ and $k_{\mathrm{S}}$; Q represents the medium polarization, k₁, k₂, and k₃ are the coupling coefficients, and $\Gamma$ is the Raman bandwidth. Regard the rear surface of the KDP crystal (the research object) as a thin slice, Stokes light is generated and amplified in the transverse direction, with no need to consider diffraction effects. Each incremental step that the Stokes light propagates upward (or downward) will encounter a fresh portion of pump laser (3ω light) arriving at the next moment, which couples with the Stokes light and amplifies it. Therefore, the TSRS process can be regarded as a steady-state process in which the 3ω pump intensity is effectively constant. Ignoring diffraction effects, the TSRS process can be described by the following coupled-wave Equations (4) and (5):

$$\frac{\partial }{\partial x}{\mathit{E}}_{\mathrm{S}}=-\mathrm{i}{k}_2{\mathit{Q}}^{\ast }{\mathit{E}}_{\mathrm{L}},$$

(4)

$$\frac{\partial }{\partial t}{\mathit{Q}}^{\ast }=-\Gamma {\mathit{Q}}^{\ast }+\mathrm{i}{k}_1{\mathit{E}}_{\mathrm{L}}^{\ast }{\mathit{E}}_{\mathrm{S}}.$$

(5)

The coupling coefficients $k_1$ and $k_2$ are, respectively, as follows:

$$k_1=\sqrt{\frac{\Gamma c^{2}g}{8{\mathsf{\pi }}^{2}n\mathrm{\hslash }{\omega }_{\mathrm{S}}}},$$

(6)

$$k_2=\frac{2\mathsf{\pi }n\mathrm{\hslash }{\omega }_{\mathrm{S}}}{c}{k}_1^{\mathrm{*}}.$$

(7)

c is the speed of light in vacuum, n is the medium refractivity, g is the TSRS gain coefficient, and ${\omega }_{\mathrm{S}}$ is the angular frequency of Stokes. The results presented below were all derived using the coupled-wave Equations (4) and (5), with calculations implemented via the finite difference method using custom-written code in MATLAB (R2020b).

The TSRS fluence damage threshold for KDP crystals is $F_{\mathrm{t}\mathrm{h}}=6\sqrt{t_{\mathrm{L}}}$, where $F_{\mathrm{t}\mathrm{h}}$ is in $\mathrm{J}/{\mathrm{c}\mathrm{m}}^2$ and ${ t}_{\mathrm{L}}$ is the 3ω laser pulse width [6]. Using typical design parameters of current ICF laser drivers (beam aperture 36 × 36 cm², pulse width $t_{\mathrm{L}}=3$ ns with an 18th-order super-Gaussian intensity distribution in time and space; KDP crystal aperture 40 × 40 cm² with side-face reflectivity R = 4%; TSRS gain coefficient g = 0.345 cm/GW [6]), it can be calculated that the safe operating intensity threshold for a narrowband (unmodulated) 3ω laser pulse in the KDP crystal is $I_{\mathrm{t}\mathrm{h}0}^{\mathrm{N}\mathrm{B}}=1.30 \mathrm{G}\mathrm{W}/{\mathrm{c}\mathrm{m}}^2$, meaning that exceeding this intensity would cause the generated TSRS fluence to surpass the damage threshold $F_{\mathrm{t}\mathrm{h}}=10.4 \mathrm{J}/{\mathrm{c}\mathrm{m}}^2$, thereby damaging the KDP crystal. However, the required 3ω laser pulse intensity for ICF drivers typically ranges from 2 GW/cm² to 4 GW/cm². In fact, significant TSRS and damage to KDP crystals have already been observed in experiments at large ICF laser facilities such as NIF and SG-III. Thus, studying the growth behavior and suppression methods of TSRS in KDP crystals is of great significance for designing large laser systems like ICF drivers.

3. Results Fundamental Theory of SSD

The principle of SSD is illustrated in Figure 2. The SSD system mainly consists of an electro-optic phase modulator and a pair of diffraction gratings. The first grating is placed before the phase modulator to introduce a transverse phase delay to the initially narrowband laser beam. The phase modulator applies a time-dependent phase modulation to the narrowband laser pulse to broaden its spectral linewidth. The second grating, located after the phase modulator, compensates for the phase delay introduced by the first grating, maintaining the original pulse shape while simultaneously inducing transverse dispersion and spatial frequency cycling in the broadband laser beam.

Under the slowly varying envelope approximation, the electric field of an SSD beam can be expressed as follows [13]:

$${\mathit{E}}_{\mathrm{L}}\left(x,t\right)=E_{\mathrm{L}0}\left(x,t\right)\mathrm{e}\mathrm{x}\mathrm{p}\left[\mathrm{i}\delta \sin \left({\omega }_{\mathrm{m}}t+\alpha x\right)\right],$$

(8)

$$\alpha =2\mathsf{\pi }\frac{\mathsf{\Delta }\theta }{\mathsf{\Delta }\lambda }\frac{{\omega }_{\mathrm{m}}}{{\omega }_0}$$

(9)

where E_L0(x, t) is the real amplitude of the laser field, δ is the phase modulation amplitude, ω_m is the modulation angular frequency, ω₀ is the laser center frequency, $\mathsf{\Delta }\theta$ is the beam dispersion angle (mrad), and $\mathsf{\Delta }\lambda$ is the laser bandwidth (nm). After passing through the second grating, the different frequency components of the broadband beam disperse laterally. The number of spatial “color cycles” across the beam aperture is given by Equation (10):

$$N_{\mathrm{C}}=D\alpha {\nu }_{\mathrm{m}}.$$

(10)

Equation (8) can be further rewritten in a simplified form, yielding expressions (11) and (12):

$${\mathit{E}}_{\mathrm{L}}\left(x,t\right)=E_{\mathrm{L}0}\left(x,t\right)\mathrm{e}\mathrm{x}\mathrm{p}\left\{\mathrm{i}\delta \sin \left[2\mathsf{\pi }\left({\nu }_{\mathrm{m}}t+\beta \frac{{\nu }_{\mathrm{m}}}{{\nu }_0}x\right)\right]\right\},$$

(11)

$$\beta =\Delta \theta /\Delta \lambda$$

(12)

where β is the grating dispersion coefficient (mrad/nm), ν₀ is the laser center frequency, and ν_m is the modulation frequency. From Equation (11), the initial phase distribution of the SSD beam can be obtained as follows:

$$\varphi =\delta \sin \left[2\mathsf{\pi }\left({\nu }_{\mathrm{m}}t+\beta \frac{{\nu }_{\mathrm{m}}}{{\nu }_0}x\right)\right].$$

(13)

It can be seen that $\varphi$ is a coupled spatiotemporal function, with the phase varying sinusoidally in both space and time. As illustrated in Figure 3, the phase of the SSD beam varies continuously along the spatial dimension and over time (in this example, the phase spans two full cycles in both the spatial and temporal domains). This continuous variation in the beam’s phase in space and time effectively reduces the beam’s spatiotemporal coherence, thereby lowering the TSRS gain. Additionally, it should be noted that, consistent with the previous research on the suppression effect of SSD on SRRS in air [9], in this research, SSD is also applied to the generation or amplification stage of 1ω laser pulses at the front end of the ICF laser driver. This is more in line with practical application scenarios and also conforms to the research objective of simultaneously suppressing SRRS in air and TSRS in KDP. Additionally, limited by conditions such as modulation devices, the laser bandwidths of SSD are generally relatively narrow. Currently, in the design of ICF laser drivers, the guiding parameters provided by the SSD technical units indicate that the maximum bandwidth of 1ω light is approximately 0.3 nm, corresponding to a modulation bandwidth Δν of about 80 GHz. Moreover, limited by the modulation voltage, the modulation amplitude δ is generally relatively small. Considering the above conditions, in the following simulation, the values are set as 0 ≤ δ ≤ 5 and 0 ≤ ν_m ≤ 15 GHz.

4. Effect of SSD on TSRS Gain and Suppression Effectiveness

4.1. Effect of SSD on TSRS Gain

As indicated by Equation (11), the phase modulation of an SSD beam is governed by three key parameters: the modulation amplitude δ, the modulation frequency ν_m, and the grating dispersion coefficient β. We performed numerical simulations to examine how the Stokes peak fluence F_P and peak intensity I_P in the KDP crystal vary with these modulation parameters, and the results are shown in Figure 4. Figure 4a,b indicates that the Stokes peak fluence F_P and peak intensity I_P both decrease as the modulation amplitude δ and modulation frequency ν_m increase. This is because the SSD laser pulse’s electric field phase is continuously varying in both time and space, so photons at adjacent times have different phases, and photons at adjacent spatial positions likewise have different phases. During the TSRS process, the stimulated phonon fields generated by those photons at different times/positions will have mismatched phases. Thus, when a later photon encounters the phonon field produced by an earlier photon, the phase mismatch leads to a lower coupling efficiency. In other words, the three-wave coupling among the pump laser, the Stokes light, and the phonon wave fails to achieve phase locking, resulting in a reduced TSRS gain. Therefore, the TSRS gain of an SSD-broadened laser pulse is lower than that of a narrowband laser pulse. Moreover, as the modulation amplitude δ and the modulation frequency ν_m increase, the degree of phase mismatch becomes larger, causing the TSRS gain to decrease further.

On the other hand, as the grating dispersion coefficient β increases, the Stokes peak fluence F_P and peak intensity I_P are found to increase (see Figure 4c). This is because a larger β reduces the spatial phase variation period across the beam cross-section (equivalently, it increases the number of equal-frequency spectral lines across the aperture). Consequently, as the Stokes light propagates transversely, the probability of it encountering in-phase 3ω pump light is higher, which leads to a higher TSRS gain. In the simulations, the laser pulse peak intensity was 2 GW/cm², and the other parameters were the same as in Section 2.

4.2. TSRS Suppression Performance

Numerical results presented in the preceding section demonstrate that SSD can markedly suppress TSRS. To quantify this capability, the damage-threshold intensity $I_{\mathrm{t}\mathrm{h}0}^{\mathrm{NB}}=1.30 \mathrm{GW}/{\mathrm{cm}}^2$ of a narrow-band 3ω pulse is taken as the reference level. Figure 5 plots the safe operating intensity threshold $I_{\mathrm{th}}^{\mathrm{SSD}}$ for SSD-broadened 3ω pulses in a KDP crystal, together with the corresponding enhancement factor $I_{\mathrm{th}}^{\mathrm{SSD}}/I_{\mathrm{th}0}^{\mathrm{NB}}$, as functions of the modulation amplitude δ at several spectral bandwidths ($\Delta \nu = 2\delta {\nu }_{\mathrm{m}}$). Throughout these simulations the grating-dispersion coefficient was fixed at β = 1.0 mrad/nm. From Figure 5, the following can be observed: for a fixed laser bandwidth, as δ increases (i.e., as the modulation frequency ν_m decreases), the safe operating intensity threshold initially rises rapidly. When δ exceeds about 1, the threshold tends to level off and approach a constant value; as the laser modulation bandwidth increases, the safe operating intensity threshold in the KDP crystal consistently increases. For example, when the fundamental laser’s modulation bandwidth $\Delta {\nu }_{1\mathsf{\omega }}$ is increased from 40 GHz to 80 GHz (corresponding to increasing the 3ω bandwidth $\Delta {\nu }_{3\mathsf{\omega }}$ from 120 GHz to 240 GHz), the damage-threshold intensity of the 3ω pulses in a KDP crystal rises from 1.91 GW/cm² to 2.41 GW/cm², which are 1.47 times and 1.85 times that of the narrowband case, respectively. Clearly, SSD provides a significant suppressive effect on TSRS. Moreover, since SSD likewise has a strong suppressive effect on SRRS in air, it holds great promise for improving the performance of large laser systems such as ICF drivers.

5. Conclusions

This study indicates that SSD has a significant suppressing effect on TSRS in KDP crystals. The main reason is that the phase of an SSD beam varies continuously in time and space, which reduces the beam’s spatiotemporal coherence and thus suppresses the TSRS gain. This suppression effect becomes more pronounced as the modulation amplitude and modulation frequency are increased (i.e., with increasing modulation bandwidth). In addition, previous reports have shown that SSD can significantly suppress SRRS in air as well. In other words, applying SSD to the initial fundamental-frequency laser pulse can simultaneously suppress SRRS in the air path and TSRS in the KDP crystal. Furthermore, SSD is known to mitigate laser–plasma instabilities (LPI), thereby improving the prospects for ignition in fusion targets. Therefore, SSD may hold great application value for large high-energy laser facilities such as ICF drivers. However, it should be noted that SSD increases the bandwidth and divergence angle of the fundamental frequency laser, which poses challenges to third-harmonic generation. Fortunately, extensive research is already underway to improve the frequency-conversion efficiency for broadband, low-coherence laser light, and many important advances have been achieved [14,15,16,17,18,19,20]. These developments may further enhance the practical value of implementing SSD in ICF laser driver systems.