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Abstract
In laser-driven inertial confinement fusion (ICF) facilities, nonuniform laser irradiation can cause significant challenges, such as hydrodynamics instability and laser plasma instability, which hinder the success of fusion. This article presents a new idea for improving the uniformity of far-field laser irradiation through a method of single-beam polarization smoothing. The method involves modulating full Poincaré beams using stress-engineered optics made from fused silica. We designed a stress birefringence system and conducted opto-mechanical modeling and analysis on it. The article elaborates on the mechanism and principles of generating large-aperture full Poincaré beams by stress birefringence, as well as the mechanism of polarization smoothing by full Poincaré beams. Near-field polarization measurements were conducted to verify these mechanisms, and the effectiveness of this method in improving the uniformity of laser irradiation in the target area was evaluated through far-field optical tests.
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1. Introduction
Laser-driven inertial confinement fusion is crucial for the future energy security of humanity. Today, the high-power laser driven fusion ignition has made significant progress, for example, the National Ignition Facility (NIF) at Lawrence-Livermore National Laboratory (LLNL) has achieved a fusion energy release greater than the input laser energy in the target area in 2022 [1]. In fact, as the power of laser increases, the hydrodynamics instability [2] and laser plasma instability [3] in the target area have become major problems hindering the success of fusion since 2012 [4], and researchers have started to focus more on improving laser irradiation performance to address these issues.
In a huge and intricate laser system like the ICF device, the beams inevitably exhibit energy non-uniformity and wavefront distortion due to uneven gains and disturbances, thus making the focal spots in the target area non-uniform [5], which can trigger laser plasma instabilities such as filamentation, stimulated Raman and Brillouin scattering, etc. [6,7], thus leading to implosion failure. To solve this problem, researchers have developed various beam smoothing techniques, including phase smoothing [8,9], temporal smoothing [10,11] and polarization smoothing (PS) [12]. Among them, phase smoothing primarily controls the shape of the focal spot and reduces low spatial frequency non-uniformity, and temporal smoothing primarily reduces high spatial frequency non-uniformity of the focal spot, while polarization smoothing can further reduce the focal spot contrast to 0.707 times, and due to its real-time characteristics, it can make up for the shortcoming that temporal smoothing requires integration time, thus significantly suppressing backscattering [13]. To maximize the irradiation uniformity, all the three smoothing techniques are employed in ICF devices [14].
Polarization smoothing enhances the uniformity of laser irradiation by the incoherent superposition of different polarization components, in order to generate two spatially separated orthogonal polarization components form the linearly polarized laser input, optics used for polarization smoothing are typically made of materials with birefringence ability. For instance, Boehly et al. proposed a scheme based on a wedge-shaped KDP crystal to separate the two polarization components in the far-field by translation [15], and Huang et al. tilted a KDP slab in the convergence beam to separate the two polarization components by interference of converging polarized light [16]. However, the large-aperture KDP crystals produce TSRS (transient simulated Raman scattering) in high-power ultraviolet laser beam, leading to crystal damage [17,18]. Although the Raman scattering gain coefficient of DKDP crystal (Deuterated KDP) is reduced by half compared to KDP, it is still challenging to adapt to higher power ultraviolet lasers in the future [19]. A simple scheme to avoid TSRS crystal damage is the use of sub-aperture beams. By adding half-wave plates to two 1ω laser beams out of a 2 × 2 beam group, when the focal spots of the four sub-aperture beams overlap in the target area, polarization smoothing can be achieved [14], but due to the fact that the beams do not always overlap during transport in the target area, the smoothing performance of this scheme is not as good as that of the single beamlet schemes. Other solutions such as liquid crystal polarization control plate are also difficult to apply in high-power ultraviolet beams due to their low resistance to laser damage [20]. Therefore, exploring new solutions is still required.
Fused silica is widely used in ultraviolet laser. Compared to the Raman scattering gain coefficient of KDP crystal (0.3 cm/GW [21]), the Raman scattering gain coefficient of fused silica is only 0.019 cm/GW [22]. Therefore, fused silica may effectively avoid the problem of stimulated Raman scattering damage. However, fused silica does not have birefringence ability itself. Bonod et al. etched a metasurface on the fused silica matrix to give it birefringence ability [23]. However, machining metasurfaces in such large-sized ICF optics may be a manufacturing challenge. Inspired by Spilman's work on modulating polarization vortex beams by stress birefringence [24], we attempt to use a stress-engineered optical element made of fused silica to modulate the polarization state of laser beam by boundary load control. Beckley’s work shows that the beam modulated by the stress-engineered optics is a full Poincaré beam [25]. In our previous work [26], we successfully reproduced the research results of generating full Poincaré beams by stress birefringence and obtained more conclusions on the modulation law.
In terms of improving the uniformity of focal spot, according to the theory of statistical properties of laser speckle [27], for polarization smoothing, it can change the polarization state of laser beam, causing differences in spatial distribution of the intensity or phase of two orthogonal polarization components (e.g. x-linearly and y-linearly) of the modulated beam, thus form two non-correlated speckle patterns with orthogonal polarizations in the far field with the help of continuous phase plate (CPP [9]), since these two polarizations do not interfere, the intensity patterns add incoherently, the rms nonuniformity can be reduced by a factor of $\sqrt 2 $. Polarization smoothing techniques follow this theoretical limit, regardless of the type of vector beam. For example, Ren et al. conducted a comparative study on the polarization smoothing performance of three types of beams (beam modulated by birefringent wedge, beam modulated by orthogonal polarization control plate, radially polarized beam), and they found that there was not much difference in the performance of these beams in improving the uniformity of focal spot [28]. As a beam with complex polarization distribution, full Poincaré beam clearly has the potential to be applied to polarization smoothing. In our previous simulation work [29], we evaluated the performance of full Poincaré beam in improving focal spot uniformity and found that the reduction of rms nonuniformity can approach the theoretical limit of factor $\sqrt 2 $. Besides, due to the interesting polarization distribution of full Poincaré beam, it may provide additional effects for LPI suppression. There are studies show that polarization control with near-field spatial distribution can have better performance [30,31].
Different from our preliminary studies [26,29], we go further in the work described in this article, design stress-engineered polarization smoothing optics for the large aperture rectangular optical path of ICF devices, and conduct detailed opto-mechanical modeling for this component. The mechanisms of generating large-aperture full Poincaré beams by stress birefringence and polarization smoothing by full Poincaré beams are explained, and the reliability of theoretical model and the polarization smoothing performance of this design are tested by using low-power visible light in offline experiments.
2. Configuration
In high-power ultraviolet laser beams, fused silica elements are widely used, such as shielding films, isolation windows, and focusing lenses in ICF terminal optical components. Since the additional component thickness will contribute to the B-integral (Additional phase distortion caused by nonlinear effects during the high-power laser beam propagation in medium [32]), we may consider integrating the polarization smoothing function into one of the existing components in the future. To ensure the safety of the window when applying large pressure loads (maximum pressure 2.8 MPa), in this paper, we chose a thick slab-shaped window as the basic element of stress-engineered optics. The window measures 430mm × 430mm × 40 mm. The physical parameters of fused silica we used are listed in Table 1.
Table 1. Parameters of fused silica
According to A. K. Spilman's research [24], when centrally symmetric pressure loads are applied to the side of a circular window, for example, by placing a thermal compression copper ring with three notches around the window’s side, a space-variant wave plate is generated. This wave plate can produce annular regions with a polarization vortex under linearly polarized illumination. Therefore, as shown in Fig. 1(a), considering the rectangular shaped laser beams of ICF and mechanical conditions of linearly elastic, we designed a structure consisting of a square fused silica window and four trapezoidal copper blocks. These components are spliced together into a regular 12-sided shape. When pressure loads are applied to the top edge of the copper blocks, the stress distribution in the rectangular window will approach that of a circular window with four symmetric loads. This integrated whole-face contact design can avoid stress concentration on the edges, which can occur with the block loading style mentioned in Ref. [12]. Consequently, it helps prevent significant unwanted phase distortion.
Fig. 1. (a) mounting structure diagram of the stress-engineered fused silica optical components. (b) partial enlarged view of the pressure sensor combined with the tightening screw. (c) actual physical setup in the experiment.
The loads are generated by tightening screws in the threaded holes on the four sides of the support frame. At this point, the cylindrical pressure sensors facing the screws are pressed, and they measure the value of the forces while transmitting them, which is shown in Fig. 1(b). We can read the measured values through accompanying digital display signal processors, and adjust the loads using a wrench. During experiments, we can control the loads with an error of less than 5% using this method. A hard steel block is placed between the sensors and the top edge of each trapezoidal copper block, so that the point loads can be converted into surface loads and transmitted more evenly to the copper blocks. To avoid stress concentration caused by manufacturing defects that may lead to damage to the brittle fused silica, we put 2 mm thick silicone pads on each surface where fused silica contacts with metal materials. The front frame, back frame and positioning blocks are used to maintain the structural stability of the window, and they will not be subjected to significant preload.
We conducted optical tests on the above stress-engineered polarization modulation component, the optical path constructed in the experiments is shown in Fig. 2. Due to the excessive volume of the prototype component, the experiments we conducted are offline low-power optical tests (continuous laser with a power of 1-10W), which can already reflect the polarization modulation effect caused by stress birefringence. We chose visible light for the experiments instead of ultraviolet light, considering its better economy and operability. In addition to the above reason, we believe that if visible light can be modulated into a full Poincaré beam in this experiment, then in future practical applications, the value of loads required to modulate ultraviolet light into a beam with the same polarization distribution will be smaller, making it easier for mechanical system to achieve effective polarization modulation. The reasons are as follows: simulation result shows that the radial 1st-order full Poincaré beam has the best polarization smoothing performance (see Fig. 10), at this point, the phase retardance at the inner tangent circle of the aperture must be π (see section 3.3). The phase retardance is inversely proportional to the wavelength of light (see Eq. (8), ignoring the small change in refractive index with wavelength), and is directly proportional to the value of pressure loads (see section 3.3). Therefore, when the wavelength of light becomes shorter, the value of the pressure loads required to achieve the same phase retardance will become smaller. In the optical path, a 532 nm laser served as the light source. After being polarized by a polarizing beam splitter (PBS), the laser beam became x-linearly polarized light. Then a concave lens (focal length -75 mm) was used to expand the beam, and a large aperture lens (focal length 11 m) collimated the beam to make it incident to the stress-engineered optics parallelly. The combination of the two becomes a Galileo beam expander. Considering the signal-to-noise ratio limitation of the polarimeter and the size of the fused silica window, the aperture diameter of the parallel beam was approximately 400 mm. In near-field polarization measurement, we used a polarimeter (Thorlabs PAX1000VIS) capable of outputting Stokes parameters of light. We installed it on a 2D motion table to scan and measure the near-field polarization distribution of the large-aperture beam. We measured the polarization state of the inner points of aperture at intervals of 2 cm. To approach the real application state of polarization smoothing, we passed the laser beam through a CPP for phase smoothing. The peak-valley value of the modulated phase profile is about -4π∼4π in radians. Finally, we focused the laser beam using a lens with the same model as the front lens. At the focal spot, there was a CCD to receive the focal spot image with a spatial resolution of 3µm.
Fig. 2. Diagram of the experimental optical path. The focal length of the large-aperture lens is scaled in the drawing for the coordination of the picture.
3. Theory and numerical simulation
3.1 Mechanical model
To clarify the stress distribution in the aperture of fused silica window when pressure loads are applied to its edges, we conducted mechanical modeling and analysis. In mechanical analysis, we simplified the structure by ignoring the frames, as the loads caused by auxiliary support and positioning are very small compared to the applied pressure loads. As shown in Fig. 3(a), the outer ends of the six pressure sensors located on the side and top edges of the window are subjected to equal pressure loads, and the outer ends of the two pressure sensors located at the bottom of the window are subjected to displacement constraints that restrict longitudinal movement. In this case of multiple forces and multi-body coupling, the mechanical calculations will become very complex, making it impossible to obtain an analytical solution. Therefore, we used a numerical solution by employing finite element method (FEM [33]). This approach decomposes the entire area into smaller sub-areas, each of which can be treated as a simple part with an easily determined stress state using the following calculation model described in this section. Finally, the overall solution can be obtained through integration. The convergence and stability of this method have been rigorously proven and will not be repeated here. We used the Ansys solver to perform specific model segmentation and calculations.
Fig. 3. (a) mechanical model of the stress-engineered optics. (b) principal stress distribution in the aperture. The arrow length is proportional to the amount of stress.
Each part in the assembly is relatively large, particularly the fused silica window, which is considered crucial for mechanical analysis. It measures 430mm × 430mm × 40 mm. Compared with this scale, the deformation is small enough and falls within the elastic range. Therefore, the mechanical analysis is conducted based on elastic mechanics. As described in the geometric equation below, when an object undergoes deformation, the strain tensor ɛ at given point can be defined as a function of the displacement vector U at that specific location.
With the object’s strain, stress distribution occurs. According to the generalized Hooke's law, the relationship between stress and strain can be described by the physical equation:
In a stable state, every point within the object should satisfy the force balance. This can be described by the equilibrium equation:
These three types of equations relate force, deformation, and stress in the case of elastic mechanics. They provide us with a solution to calculate stress distribution based on the actual boundary conditions described in Fig. 3(a): the force boundary condition ${\sigma _{ij}}{n_j} = {F_i}$ and the displacement boundary condition ${U_i} = {\bar{U}_i}$.
By using the mechanical model and method described above, we calculated the normal stress and shear stress distribution in the aperture plane for stress birefringence analysis. As shown in Fig. 3(b), under the action of designed loads, the orientation of principal stress rotates with the azimuth angle, and the value of principal stress gradually increases from the center of the aperture to the surrounding area. The spatial variation of stress state inevitably leads to the spatial variation of stress birefringence, which means that when a laser beam passes through the stress-engineered window, it will experience different polarization modulation effects in different regions of the aperture. This provides a possible means for generating vector polarization beams with relatively complex polarization distributions. Additionally, when the loads F are 12kN (the required value of loads for achieving radial 1st-order full Poincaré beam modulation mentioned in section 3.3), the maximum value of stress in the window is 2.76 MPa, which is much lower than the damage strength of fused silica.
As shown in Fig. 4, we determined the pressure load distribution on the edges of the window when the loads F are 12kN. It exhibits a symmetrical smooth curve distribution with a high middle and low end. In fact, during the modeling and simulation process, we attempted various structures and loading methods. In many cases, when a similar pressure distribution is generated on the window edges, the transmitted beam can be modulated into a full Poincaré beam, but changes in the load will inevitably cause changes in the stress field in mechanics, which in turn will change the local rotation speed of optical axis and the local change speed of phase retardance through elastic-optic effect (see section 3.2), thus causing changes in the spatial distribution of polarization state of the modulated beam. Too uneven polarization changes (such as an excessively imbalanced energy ratio between x-linearly and y-linearly polarization components) will reduce the performance of improving focal spot uniformity through incoherent superposition of orthogonal polarization components [27]. Further consideration of reasonable load optimization methods is required in our next work.
3.2 Stress birefringence
According to the theory of elasto-optical effect [34], when an isotropic material is subjected to external loads or other excitation sources, the generated stress field will change its local optical parameters. This can lead to optical anisotropy similar to that observed in uniaxial crystals. The relationship between the stress tensor and the waveplate parameters is linked by the inverse dielectric tensor B, whose change can be described as Eq. (4) [35].
As an isotropic material, fused silica’s refractive index surface is a sphere with a radius of ${n_0}$ when it is not subjected to stress. However, when the inverse dielectric tensor changes, the surface becomes an ellipsoid known as the refractive index ellipsoid. Its equation can be represented by a formula with the elements of the inverse dielectric tensor as coefficients:
When the laser beam is incident along the z-axis, the cross-section of refractive index ellipsoid in the x-y plane determines the local birefringence characteristics, which is referred to as the refractive index ellipse. The orientation of optical axis $\theta $ and the values of the major and minor axes ${n_1}$, ${n_2}$ of the ellipse can be derived mathematically:
When a laser beam with a wavelength of $\lambda $ passes through a birefringent medium with a thickness of d, there will be phase retardance between the polarization components along the fast and slow axes of the medium:
Due to the different stress states in different regions of the aperture, there are different birefringence parameters everywhere. By using the stress distribution obtained in section 3.1 for stress birefringence analysis, we can determine the distribution of optical axis orientation and phase retardance, which are shown in Fig. 5. Compared with Fig. 3(b), it can be observed that the orientation of the optical axis is the same as the orientation of principal stress, which rotates in the aperture. Meanwhile, the phase retardance increases with the increase in principal stress value, showing a gradually increasing distribution from the center to the surrounding area in the aperture. This spatial distribution combination of these optical parameters is the potential foundation for stress-engineered optics to modulate full Poincaré beams.
Fig. 5. Simulation results of birefringence parameters distribution of stress-engineered optics with pressure loads of 24kN per point. (a) orientation of the optical axis, (b) phase retardance, a contour map is used to better demonstrate the distribution pattern.
3.3 Generation of full Poincaré beams
To calculate the change of polarization state when the laser beam passes through a birefringent medium, the Jones matrix method is used. As the birefringence parameters (optical axis orientation and phase retardance) of the stress-engineered window across the aperture plane are spatially varying and continuous (as shown in Fig. 5), in each local small region, we can treat it as an ordinary wave plate which has its own determined optical axis orientation and phase retardance. The local Jones matrix can be written as Eq. (9) [36]. This matrix is different in each part of the stress-engineered window.
By using Euler's formula, we can determine the amplitude and phase of the two polarization components along the x-axis and y-axis. We can then draw the local polarization state by using parametric equation of the endpoint trajectory of optical vibration vector. We used birefringence parameters distribution obtained in section 3.2 and made incident light an x-linearly polarized eighth-order super Gaussian beam for numerical simulation, and drew the near-field polarization distribution of emitted light, which is shown in Fig. 6(a). In order to make the image less crowded, we selected some points in the aperture at equal intervals to display the polarization state, so not all polarization states can be displayed in the picture. It can be observed that with modulation of spatially changing birefringence parameters, the polarization state of transmitted beam exhibits diverse distribution, and this distribution exhibits obvious cyclicity in both azimuthal and radial directions. We consider the inscribed circle area of the aperture, and call the number of cycles of polarization rotation changes along azimuth as azimuthal order, and the number of polarization rotation changes along radial direction as radial order. These orders also represent the number of times that the polarization state covers the entire Poincaré sphere in the manner described in [37] along azimuthal and radial directions, respectively.
Fig. 6. (a) simulation result of near-field polarization distribution of the emitted light. Red, green, and blue represent right-handed, left-handed, and linearly polarized light, respectively. (b) polarization mapping of stress modulated full Poincaré beam on the Poincaré sphere. Concentric rings of different colors represent the polarization states contained in the ring area they cover, and the color of each area is the same as the color of its mapped pattern on the Poincaré sphere.
Furthermore, we can use the Stokes parameters to describe the polarization distribution of modulated beam:
The Stokes parameters satisfy the equation: $S_0^2 = S_1^2 + S_2^2 + S_3^2$. This equation represents a sphere in Cartesian coordinates, known as the Poincaré sphere. The points on the sphere have a one-to-one mapping relationship with the polarization states of light, making it an intuitive way to describe the polarization distribution pattern and diversity. Taking the radial 1st-order full Poincaré beam part of the simulation result of the emitted beam as an example, as shown in Fig. 6(b), the polarization states of different annular regions are mapped as an 8-shaped pattern on the Poincaré sphere, and the mapping pattern becomes larger as the annular radius increases. From the center to the edge, the mapping pattern expands from the point representing the polarization state of the incident light to the equator. The polarization states of complex beams modulated by the stress-engineered optics cover the entire Poincaré sphere with this distribution rule, which conforms to the definition of full Poincaré beams.
For a stress-engineered element subjected to a central symmetrical load, the number and value of the loads are key parameters, and for a wave plate with an optical axis orientation that rotates with azimuth and a phase retardance that increases with radius (as shown in Fig. 5), the rotational speed of the optical axis and the increase rate of phase retardance are key parameters. For the stress-engineered optics, these birefringence parameters have the following relationship with mechanical parameters [26]: the number of cycles of the optical axis rotation is 2 less than the number of loads, and the increase rate of phase retardance is proportional to the value of loads.
As the rotation of optical axis exhibits cyclicity with a period of 360°, the polarization modulation effect also exhibits cyclicity along the azimuthal direction with the number of cycles equal to the number of cycles of the optical axis rotation. That is, the azimuthal order of full Poincaré beam is also 2 less than the number of loads, so for the rectangular window subjected to quadruple symmetric loads described in this article, the azimuthal order of modulated beam is 2. Meanwhile, as the phase retardance exhibits cyclicity with a period of $2\pi $, the polarization modulation effect also exhibits cyclicity along the radial direction with the number of cycles equal to the quotient of the maximum phase retardance at the boundary of inscribed circle of the aperture divided by $2\pi $. That is, the radial order of full Poincaré beam is also proportional to the value of loads. Due to the symmetry of optical axis, when the incident light is linearly polarized, the polarization state on the concentric circles in the aperture of modulated beam is mapped as an 8-shaped pattern on the Poincaré sphere. Scanning along the radial direction, this pattern will cover the Poincaré sphere in both directions simultaneously, so only a maximum phase retardance of $\pi $ is required to allow the pattern to reach the equator of Poincaré sphere, thus completing a full coverage of the Poincaré sphere. Therefore, the radial order is actually equal to the quotient of the maximum phase retardance divided by $\pi $. For the optical-mechanical system described in this article, numerical simulation results indicate that pressure loads of 12kN per point are required for the generation of radial 1st-order full Poincaré beam. However, the value of loads can vary continuously, so the maximum value of phase retardance is not necessarily an integer multiple of $\pi $. In this case, we defined fractional radial order for full Poincaré beams based on the ratio of the maximum phase retardance in the inscribed circle of the aperture to the value of $\pi $.
3.4 Polarization smoothing mechanism of full Poincaré beams
In the analysis of section 3.3, we learned that the near-field polarization state of the emitted light modulated by stress-engineered optics exhibits diversity and special distribution. If we decompose the modulated polarization state of the beam in x and y directions, the spatial distribution of the intensity of these two polarization components will be different during the propagation of the beam (such as the complementary patterns in the far field which are shown in Fig. 7(a)). In this way, when the modulated beam passes through CPP, the x-linearly and y-linearly polarization components will be smoothed by phase smoothing in different regions of the CPP and form different focal spot patterns (as shown in Fig. 7(b)), which are then superimposed on the focal plane. Moreover, due to the orthogonality between x-linearly and y-linearly polarization components, this superposition is also a non-coherent superposition, thus improving the uniformity of the focal spot [27]. The detailed analysis is as follows.
Fig. 7. (a) focal spot images of x-linearly (left) and y-linearly (right) polarization components when only stress-engineered optics is used for polarization smoothing. (b) focal spot images of x-linearly (left) and y-linearly (right) polarization components when polarization smoothing and phase smoothing are used in combination.
As the target is usually placed at the focal point of the lens used to focus the large aperture laser beams, we performed far-field diffraction calculations on the light field modulated by stress-engineered optics, that is, the Fourier transform:
As shown in Fig. 7(a), we calculated focal spot images of x-linearly and y-linearly polarization components without CPP phase smoothing, respectively. The energy distribution of two polarization components has a unique spatial separation form on the target: the focal spot image of x-linearly polarization component is a cross-shaped pattern, while the focal spot image of y-linearly polarization component is a complementary four-point pattern with former. This is only the result when radial order of full Poincaré beam is 1. When the radial order increases, images of focal spot will be more complex. However, simulation results on the relationship between the radial order and smoothing performance of full Poincaré beams (shown as the blue curve in Fig. 10) have shown that higher-order full Poincaré beams do not lead to better polarization smoothing performance, so they will not be elaborated on further in this article. This interesting form of polarization component separation is different from the pattern translation form in Ref. [15] and the converging polarized light interference form in Ref. [16]. Then we added CPP phase modulation to light field and calculated the focal spot images of x-linearly and y-linearly polarization components. These two orthogonal polarization components form different focal spot patterns which are shown in Fig. 7(b). It can be observed that polarization smoothing separates the speckles in CPP modulated focal spot in the manner described in Fig. 7(a), especially in the intensity image of y-linearly polarization component, many small patterns with four-point features can be seen.
4. Experimental results and discussion
4.1 Near-field polarization modulation
To verify the reliability of the theoretical and predictive model for generating full Poincaré beams by stress modulation, we measured the near-field polarization distribution of the modulated beam using a polarimeter and mapped the polarization states onto a Poincaré sphere to observe the distribution pattern. Due to the strength limitation of experimental system, we conducted experiments under different conditions: no external load, pressure loads of 4kN per point and pressure loads of 8kN per point. These conditions correspond to no polarization modulation, modulation of radial 1/3 order full Poincaré beam, and modulation of radial 2/3 order full Poincaré beam, respectively. Figure 8 shows a comparison between the experimental results and the simulation results obtained by the theoretical model under the same boundary conditions.
Fig. 8. Experimental results (below) and simulation results (upper) under the same boundary conditions. (a) no external load (only the small amount of preload required to maintain the structure). At this point, the mapping pattern on the Poincaré sphere is only a dot. To illustrate it clearly, we enlarged it in the picture. (b) pressure loads of 4kN per point, (c) pressure loads of 8kN per point. The left images show the polarization map where red, green, and blue represent right-handed, left-handed, and linearly polarized light, respectively. The right images show the corresponding mapping to the Poincaré sphere, the ratio in the picture is the coverage of the Poincaré sphere.
The polarization distribution pattern measured in the experiments is consistent with the simulation results: when there is no external load, there is no polarization modulation effect (affected by the small amount of preload required to maintain the structure, there is negligible polarization modulation here). However, as the value of external loads increases, the polarization modulation effect becomes more significant, and the emitted light shows a diverse polarization state in the near-field. The polarization states of modulated light map to an 8-shaped pattern on Poincaré sphere, and the pattern becomes larger with increase in external loads, indicating increase in diversity of polarization states. Although current external loads are insufficient to include all the polarization states in emitted laser beam, agreement between experimental results and theoretical predictions can still be used to indicate the reliability of calculation model.
4.2 Improvement of uniformity of far-field laser irradiation
The goal of beam smoothing is to improve the physical effects in the target area, and improving the laser irradiation uniformity of the focal spot is fundamental and important. In this study, optical testing experiments were conducted to examine the influence of modulation of stress-engineered optics on far-field laser irradiation performance. The experimental conditions were the same as those of near-field polarization measurement. We captured far-field focal spot images of laser beams modulated by CPP phase smoothing only, combination of phase smoothing and polarization smoothing with stress-engineered optics subjected to pressure loads of 4kN per point, combination of phase smoothing and polarization smoothing with stress-engineered optics subjected to pressure loads of 8kN per point, respectively. The images of focal spot are shown in Fig. 9.
Fig. 9. Focal spot images of laser beam modulated by: (a) phase smoothing only, (b) phase smoothing & stress-engineered optics with pressure loads of 4kN per point, (c) phase smoothing & stress-engineered optics with pressure loads of 8kN per point.
Fig. 10. Normalized focal spot contrast with polarization smoothing by full Poincaré beam and KDP birefringent wedge, respectively.
When adding polarization smoothing, the contrast of the focal spot images significantly decreases. The high energy density hot spots that originally exist in the focal spot are smoothed out. As the total energy of laser beam remains basically unchanged, the overall brightness of the image improves. To quantitatively evaluate the irradiation uniformity of far-field focal spots, we consider focal spot contrast, fractional power above intensity and power spectral density as evaluating indicators.
Focal spot contrast is often used to describe the nonuniformity of images. The incoherent superposition of two polarization components caused by polarization smoothing can reduce the focal spot contrast to 0.707 times at most. The mathematical definition of focal spot contrast is:
In the simulation results, the radial 1st-order full Poincaré beam exhibits the best smoothing performance, with the contrast decreasing to near the theoretical limit of polarization smoothing. However, higher order full Poincaré beams do not lead to better smoothing performance, and the contrast fluctuates around a convergence value. This behavior is similar to that observed in polarization smoothing method based on KDP birefringent wedge [38]. As shown by the red line in Fig. 10, we figured out the relationship between the angular shift induced by a KDP birefringent wedge and the normalized focal spot contrast based on the theoretical model in Ref. [12]. When the angular shift exceeds the minimum separation required to decorrelate a speckle pattern $\lambda /D$ (where $\lambda $ is the wavelength of laser, D is the size of the beam), further improvements in smoothing performance are not achieved. The optimal smoothing performance of a full Poincaré beam is 93% of that achieved with a birefringent wedge. However, the primary objective of this scheme is to replace crystal materials that are susceptible to TSRS damage. In the experimental results, the radial 1/3 and 2/3 order full Poincaré beams modulated by stress-engineered optics reduce focal spot contrast to 0.886 and 0.763 times, respectively, which is consistent with the simulation curve. The experimental data represents the mean and standard deviation of five parallel tests. Before each measurement, we unloaded the loads and then reloaded them. The results show that the polarization smoothing performance in five parallel experiments exhibit small and irregular changes, as indicated by the error bars for data points. We attribute this to inevitable engineering errors during loading, rather than the effect of stress cycles, as the mechanical behavior of fused silica is limited to the linear elastic range.
Fractional power above intensity (FOPAI) can describe the energy share of high-energy-density hot spots in the focal spot. These hot spots are an important source of LPI initial density perturbations. The mathematical definition of fractional power above intensity is:
Figure 11 shows the fractional power above intensity curves of the focal spots obtained under the three different experimental conditions. It can be observed that the energy share of ultra-high energy density hotspots exceeding 6 times the average energy is significantly reduced, while the energy share with about 5 times the average energy is slightly increased. Stress-engineered optics with pressure loads of 8kN per point exhibit a more significant polarization modulation effect, resulting in better suppression of high energy density hotspots than optics with pressure loads of 4kN per point.
Fig. 11. Fractional power above intensity curves of the focal spots under different experimental conditions.
Power spectral density (PSD) is also an important indicator for evaluating the uniformity of focal spot. It describes the share of power in each spatial frequency. To improve the irradiation uniformity of focal spot, we aim to suppress the speckles with higher spatial frequencies. The mathematical definition of power spectral density is:
The normalized power spectral density curves of the focal spots obtained under the three different experimental conditions are plotted in Fig. 12, and both the power spectral density in x and y orthogonal directions are calculated. It can be observed that the addition of polarization smoothing significantly suppresses the speckles smaller than 100 microns (space frequency greater than ${10^{ - 2}}\mu {m^{ - 1}}$). This indicates that the spatial distribution of focal spot energy has been optimized, and the small-scale high-frequency fluctuations are smoothed out.
Fig. 12. Normalized power spectral density curves of the focal spots under different experimental conditions. (a) x direction, (b) y direction.
After the above discussions, we can conclude from the evaluation of different evaluation indicators that stress modulated full Poincaré beam can significantly improve the uniformity of far-field laser irradiation. Unfortunately, the existing experimental system requires single point loads of 12kN to modulate the radial 1st-order full Poincaré beam to achieve optimal smoothing performance. However, the inevitable manufacturing defects in the support frame make it unable to withstand such high loads. Enhancing the strength of the support frame and optimizing the loading method may help solve this problem.
5. Conclusion
Aiming to prepare for the exploration of new polarization smoothing schemes suitable for ultra-high power ultraviolet lasers, we designed stress-engineered optics based on a fused silica window, and the opto-mechanical model, numerical simulation and experimental verification are performed in this article. We can modulate the transmitted beam into a fully Poincaré beam with diverse polarization states by applying centrally symmetric pressure loads on the edges of the window. The pressure load on each edge should be distributed symmetrically with high middle and low ends. We elucidate the polarization distribution pattern of the full Poincaré beam generated by stress birefringence using Poincaré sphere mapping. The consistency between the experimental results of near-field polarization measurement and the simulation results verifies the reliability of the theoretical model used for opto-mechanical analysis. Far-field analysis shows that the full Poincaré beam can separate the two orthogonal polarization components in a special complementary pattern, thereby achieving their incoherent superposition in the target area and improving the uniformity of laser irradiation. The results of far-field experiment combined with phase smoothing show that the stress modulated full Poincaré beam can significantly improve the uniformity of far-field laser irradiation, and we evaluate its smoothing performance in detail using three different quantitative indicators.
Considering the difficulty of mechanical control of large aperture optics, we adopted a method of fewer points loading in the design of prototype component. However, this leads to limitations in load distribution optimization, an increase in the volume of the component, and single point loads that the frame cannot withstand. To overcome the load limitation in mechanics, a reasonable loading method design should be the main direction of effort to prevent the load from being concentrated at weak positions such as the midpoint of frame. In future work, we will focus on the distribution optimization and precise control methods of boundary loads, and we will design a stress birefringence polarization smoothing component compatible with the ICF optical path under the guidance of theoretical basis presented in this article. This will allow us to further verify the safety of this scheme and the changes this interesting beam can bring to laser plasma interaction.
Funding
National Natural Science Foundation of China (52375475, 51975322); Beijing Municipal Natural Science Foundation (3212006).
Acknowledgements
The authors appreciate the contributions and efforts of engineers in Research Center of Laser Fusion, China Academy of Engineering Physics.
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.
References
1. J. Tollefson and E. Gibney, “Nuclear-fusion lab achieves ‘ignition': what does it mean?” Nature 612(7941), 597–598 (2022). [CrossRef]
2. M. Desselberger, O. Willi, M. Savage, et al., “Measurement of the Rayleigh-Taylor instability in targets driven by optically smoothed laser beams,” Phys. Rev. Lett. 65(24), 2997–3000 (1990). [CrossRef]
3. J. D. Lindl, O. Landen, J. Edwards, et al., “Review of the national ignition campaign 2009-2012,” Phys. Plasmas 21(2), 020501 (2014). [CrossRef]
4. D. Clery, “Ignition facility misses goal, ponders new course,” Science 337(6101), 1444–1445 (2012). [CrossRef]
5. J. D. Lindl, P. Amendt, Richard L. Berger, et al., “The physics basis for ignition using indirect-drive targets on the National Ignition Facility,” Phys. Plasmas 11(2), 339–491 (2004). [CrossRef]
6. R. L. Berger, C. H. Still, E. A. Williams, et al., “On the dominant and subdominant behavior of stimulated Raman and Brillouin scattering driven by nonuniform laser beams,” Phys. Plasmas 5(12), 4337–4356 (1998). [CrossRef]
7. C. H. Still, R. L. Berger, A. B. Langdon, et al., “Filamentation and forward Brillouin scatter of entire smoothed and aberrated laser beams,” Phys. Plasmas 7(5), 2023–2032 (2000). [CrossRef]
8. S. N. Dixit, I. M. Thomas, B. W. Woods, et al., “Random-phase plates for beam smoothing on the Nova laser,” Appl. Opt. 32(14), 2543–2554 (1993). [CrossRef]
9. J. Neauport, X. Ribeyre, J. Daurios, et al., “Design and optical characterization of a large continuous phase plate for laser integration line and laser megajoule facilities,” Appl. Opt. 42(13), 2377–2382 (2003). [CrossRef]
10. R. H. Lehmberg and Y. Chan, “Near-field nonuniformities in angularly multiplexed KrF fusion lasers with induced spatial incoherence,” Appl. Opt. 44(14), 2805–2817 (2005). [CrossRef]
11. S. Skupsky, R. W. Short, T. Kessler, et al., “Improved laser-beam uniformity using the angular dispersion of frequency-modulated light,” J. Appl. Phys. 66(8), 3456–3462 (1989). [CrossRef]
12. J. E. Rothenberg, “Polarization beam smoothing for inertial confinement fusion,” J. Appl. Phys. 87(8), 3654–3662 (2000). [CrossRef]
13. J. Fuchs, C. Labaune, S. Depierreux, et al., “Modification of spatial and temporal gains of stimulated Brillouin and Raman scattering by polarization smoothing,” Phys. Rev. Lett. 84(14), 3089–3092 (2000). [CrossRef]
14. M. Spaeth, K. Manes, D. Kalantar, et al., “Description of the NIF Laser,” Fusion Sci. Technol. 69(1), 25–145 (2016). [CrossRef]
15. T. R. Boehly, R. L. Mccrory, R. L. Verdon, et al., “Inertial confinement fusion experiments with OMEGA-A 30-kJ, 60-beam UV laser,” Fusion Eng. Des. 44(1-4), 35–42 (1999). [CrossRef]
16. H. Xiaoxia, J. Huaiting, W. Zhou, et al., “Experimental demonstration of polarization smoothing in a convergent beam,” Appl. Opt. 54(33), 9786–9790 (2015). [CrossRef]
17. S. G. Demos, R. N. Raman, Steven T. Yang, et al., “Measurement of the Raman scattering cross section of the breathing mode in KDP and DKDP crystals,” Opt. Express 9(21), 21050–21059 (2011). [CrossRef]
18. K. R. Manes, M. L. Spaeth, J. J. Adams, et al., “Damage mechanisms avoided or managed for NIF large optics,” Fusion Sci. Technol. 69(1), 146–249 (2016). [CrossRef]
19. H. Huang, T. Z. Kosc, T. J. Kessler, et al., “Modeling of transverse stimulated Raman scattering in KDP/DKDP in large-aperture plates suitable for polarization control,” High Power Laser Sci. Eng. 11(5), e54 (2023). [CrossRef]
20. T. Z. Kosc, A. A. Kozlov, S. Papernov, et al., “Investigation of parameters governing damage resistance of nematic liquid crystals for high-power or peak-intensity laser applications,” Sci. Rep. 9(1), 16435 (2019). [CrossRef]
21. C. Xiangxu, L. Fuquan, W. Sheng-Lai, et al., “Influence of deuteration degree on the transverse stimulated Raman scattering gain coefficient of DKDP crystal,” Acta Phys. Sin. 64(3), 034213 (2015). [CrossRef]
22. G. P. Agrawal, Nonlinear fiber optics2nd ed. (Academic, 1995).
23. N. Bonod, P. Brianceau, J. Neauport, et al., “Full-silica metamaterial wave plate for high-intensity UV lasers,” Optica 8(11), 1372–1379 (2021). [CrossRef]
24. A. K. Spilman and T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt. 46(1), 61–66 (2007). [CrossRef]
25. A. M. Beckley and T. G. Brown, “Full Poincaré beams,” Opt. Express 18(10), 10777 (2010). [CrossRef]
26. L. Bowu, T. Menjiya, C. Dongya, et al., “Modulating linearly polarized light into a full Poincare beam by stress-engineered optical element,” Proc. SPIE 11849, 1184912 (2021). [CrossRef]
27. J. W. goodman, Topics in Applied Physics (Springer, 1984).
28. R. Guangsen, S. Quan, W. Wuming, et al., “Effect of radial polarization modulation on smoothing and polarization properties of focal speckle,” High power laser and particle beams 27(12), 122008 (2015). [CrossRef]
29. L. Bowu, W. Hui, S. Xibo, et al., “Simulation analysis of polarization smoothing based on full Poincaré beam modulated by single transmission element,” Proc. SPIE 12501, 125011E (2022). [CrossRef]
30. D. H. Froula, L. Divol, R. L. Berger, et al., “Direct measurements of an increased threshold for stimulated Brillouin scattering with polarization smoothing in ignition hohlraum plasmas,” Phys. Rev. Lett. 101(11), 115002 (2008). [CrossRef]
31. T. J. B. Collins, J. A. Marozas, K. S. Anderson, et al., “A polar-drive-ignition design for the National Ignition Facility,” Phys. Plasmas 19(5), 2841 (2012). [CrossRef]
32. V. Ginzburg, M. Martyanov, D. Silin, et al., “Small-scale fluctuations of laser beam fluence at the large B-integral in ultra-high intensity lasers,” Opt. Express 31(3), 4667–4674 (2023). [CrossRef]
33. M. G. Larson and F. Bengzon, The Finite Element Method: Theory, Implementation, and Applications (Springer, 2013).
34. C. Malgrange, C. Ricolleau, and M. Schlenker, Symmetry and Physical Properties of Crystals (Springer, 2014), Chap. 19.
35. A. Feldman, “Photoelastic properties of optical materials,” Proc. SPIE 0681, 127–134 (1986). [CrossRef]
36. R. C. Jones, “A new calculus for the treatment of optical systems I. Description and discussion of the calculus,” J. Opt. Soc. Am. 31(7), 488–493 (1941). [CrossRef]
37. B. Radhakrishna, G. Kadiri, and G. Raghavan, “Polarimetric method of generating full Poincare beams within a finite extent,” J. Opt. Soc. Am. A 39(4), 662–666 (2022). [CrossRef]
38. T. R. Boehly, V. A. Smalyuk, D. D. Meyerhofer, et al., “Reduction of laser imprinting using polarization smoothing on a solid-state fusion laser,” J. Appl. Phys. 85(7), 3444–3447 (1999). [CrossRef]
