Abstract
This paper establishes a photothermal damage model for bubble impurities affecting laser optical field modulation based on Mie scattering theory and incorporates the effects of optical field modulation. This model elucidates the evolution mechanism of synergistic damage in fused silica, with simulation results validated through experimental verification. A novel characterization of optical breakdown due to bubble impurities is proposed, occurring on a millisecond timescale through the dynamic evolution of combustion waves. The model delineates the influence of bubble size and spacing on optical field distribution, temperature, stress distribution, and their evolutionary behaviors. The modulation of the optical field due to double bubble impurities creates a localized “hot spot,” resulting in a differential transverse contraction stress at the edges of the bubble impurities, thereby reducing the damage threshold of fused silica. The spacing of 1.1 λ represents the enhancement node for optical field modulation by double bubble impurities. Furthermore, localized oscillations in the optical field arise when the spacing between the double bubbles exceeds 1.1 λ, attributed to changes in the refractive index at the bubble defects and resonance oscillations generated by optical field modulation. This study not only enhances our understanding of the optical field modulation processes occurring at 1064 nm in the presence of bubble defects but also establishes a theoretical foundation for detecting internal defects at this wavelength without inducing surface damage.
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1. Introduction
In recent years, with the rapid development of optical systems such as space telescopes, high-energy lasers, inertial confinement fusion, laser gyroscopes, and lithography lenses, large solid-state lasers have been widely used. For instance, facilities like the Laser Megajoule (LMJ) in France, the National Ignition Facility (NIF) in the USA, and the Shenguang facility in China all incorporate fused silica optical components with defect-free structures [1,2]. However, fused silica optical elements inevitably introduce internal defective structures such as metal impurities, tiny cracks, and tiny bubbles due to mechanical processing and grinding and polishing processes during the manufacturing process [3–6]. The refractive index at the defects changes abruptly, causing the incident laser deposition energy at the defects to undergo complex interference and scattering and redistributing the laser energy to form a local optical field gradient [7,8]. Due to the uneven energy distribution at the defect sites, local optical field enhancement, large temperature gradients, and stress gradients are induced, accelerating the damage of optical components, directly impacting imaging quality and stability, and lowering the laser damage threshold [9,10]. Chunlai Zhang explored the modulation effect of cracks, scratches and bubble defects on the incident optical field of optical elements using simulation, and investigated the relationship between the width-to-depth ratio of scratches and the modulation of the optical field [11]. Ailing Tian used finite element analysis to simulate the effect of scattered optical generated on the longitudinal optic field after defect modulation on the optical intensity distribution [12]. Hui Ye theoretically and experimentally investigated the effect of nanoscale polished particles of CeO2 remaining on the surface of fused silica on the modulation of the optical field, and the optical field enhancement can increase the temperature of fused silica by 34.20%, which is easy to form a high density of low-flux damage precursors, enhance the photothermal response of fused silica, and reduce the damage threshold of fused silica [13]. Shuo Hu investigated the damage behavior induced by gas bubbles on the optical field modulation within high temperature fused silica by using the two-phase flow theory. It was obtained that the bubble impurity on the optical field modulation is a factor affecting the damage of fused silica precursor induced by high-energy laser [14]. Jin Huang investigated the effect of defect structures such as bubbles, hydroxyls, and metallic impurities on fused silica damage under nanosecond UV pulses, characterized the weak absorption of 355 nm UV nanosecond laser, and analyzed the effect of defect structures on the damage [15]. Jian Cheng investigated the relationship between optical intensity enhancement caused by cracks on the surface of KHPO4 crystals and the wavelength, geometry and location of the cracks. Transverse cracks and conical cracks on the front and back surfaces tend to trigger avalanche ionization and reduce the laser damage resistance of crystal elements. Brittle fracture and a series of surface cracks in the crystal material are important causes of induced laser damage [16]. Chao Tan investigated that scratches on the surface of a transparent optical element cause the superposition of reflected and incident light inside the element, resulting in a widening of the aberration region of the reflected light, a serious disturbance of the relative light intensity distribution, and a tendency to cause localized focusing of the locus to induce new damage [17]. Yubo Liu established a model of local optical field modulation by double bubble impurities, and systematically investigated the modulation process of local optical field by double bubble impurities in terms of size, spacing, and angle of rotation, and observed the influence of bubble impurities on the dynamic characteristics of fused silica during the process of fusible damage, which proved that the existence of bubbles in the fused silica is an important reason for the aggravation of fusible damage in the fused silica [18].
In this paper, based on the Mie scattering theory, heat conduction equation, Beer-Lambert Law, taking into account the effect of optical field modulation, Mathieson's rule, the bubble impurities on the laser optical field modulation induced fused silica synergistic damage model was established, and a combination of theoretical, simulation and experimental methods was used to study the size, aggregation properties, and the distribution density of bubble impurities on the dynamic behavior of localized optical field modulation, and to explore the effects of different The dynamic behavior of the bubble impurities on the local optical field modulation is investigated, and the effects of different bubble impurity sizes and embedding depths on the photothermal damage performance are investigated, and the dynamic characteristics of the optical breakdown of the bubble impurities to generate combustion waves and plasma are obtained [19,20].
This model clearly elucidates the mechanism by which bubble impurities act as precursors to laser damage, reducing the damage threshold of transparent optical elements such as fused silica. The research presented in this paper aims to provide theoretical guidance and reference for the optical manufacturing industry and laser technology applications. By reducing or avoiding the introduction of bubble impurities during the production of fused silica, the damage resistance of transparent optical elements can be enhanced, thereby improving the output performance and service life of high-power laser systems.
2. Modeling of local modulation of laser optical field by bubble defects in fused silica optical elements
2.1 Establishment of the theoretical model
Laser propagation in transparent optical materials such as fused silica occurs in the form of electromagnetic waves. The bubble impurities within fused silica have a small radius, close to the wavelength of light, making them prone to cause scattering effects distinct from those caused by air clusters. To explain the scattering effect of bubble impurities on the optical field, an in-depth theoretical study was conducted using Mie scattering theory, focusing on the symmetric distribution of scattered light angles [21]. Figure 1. shows a schematic diagram of Mie scattering of a localized optical field, depicting the light path of a laser light field through a modulated by bubble impurities. The incident laser is a plane wave of unit amplitude, Ein = exp(ikz)·ex and wavelength λ = 1064 nm, incident perpendicular to the fused silica surface along the negative z-axis, and the electric vector Ei is polarized along the direction of the solution domain [22,23].
Fig. 1. Schematic diagram of Mie scattering modulation of the laser optical field by bubble impurities.
According to Mie theory, the plane wave incident field can be expanded in terms of the vector spherical harmonic function as:
Outside the bubble impurity, the scattered field can also be expanded in terms of the vector spherical harmonic function as:
Equation, the vector spherical harmonic function is expanded as:
Equation, x is the relative distance from the field point to the center of the sphere, outside the sphere x = kr, inside the sphere x = mkr, m is the relative refractive index of the sphere, θ is the scattering angle, and φ is the azimuthal angle. In the first, second and third type of vector spherical harmonic functions Zn(x) is the spherical Bessel function jn(x), the spherical Royman function yn(x), and the first type of spherical Hankel function h1n, respectively. πn and τn are the angular coefficients [24]:
The scattering coefficient expansion is obtained as:
Equation, an and bn are the scattering coefficients and p1n is the concluding Legendre function. The scattered field expansion coefficients can be obtained by using the boundary conditions as [25,26]:
Equation, µ denotes the magnetic permeability of the bubble impurity relative to the fused silica, due to the bubble impurity to the optical field enhancement induced by the fused silica occurs optical breakdown or thermal damage caused by the damage mechanism related to the optical field modulation factor. In order to investigate the relationship between bubble defects on the enhancement of optical field factor, the model of fused silica with bubble defects on optical field modulation-induced photothermal damage was established. An optical model of fused silica with bubble impurities and the optical intensity enhancement factor (LIEF) were developed to investigate the relationship between the effect of bubble impurities and the modulation factor and heat loss. The physical nature of the photothermal damage induced in fused silica by the accumulation of optical field gradients over time was explained by the growth law of the optical intensity factor (LIEF).
I0 and E0 are the values of the optical/electrical field intensity of fused silica, and Imax and Emax are the maximum values of the optical/electrical field intensity produced by the modulation of the optical field by the bubble impurities. The research process fused silica to be incompressible at high temperatures. The optical field enhancement factor modulated by bubble impurities was solved based on the incompressible Navier-Stokes equation to further analyze the characteristic behavior of bubble impurities at damage criticality [27,28]:
Equation, T is the temperature, t is the laser irradiation time, ρ is the material density, κ is the thermal conductivity, CP is the isobaric heat capacity, and Q is the laser energy injected from the interface between the bubble impurity and the fused silica. The laser energy absorbed by the bubble impurity inside the fused silica was expressed as Q(t) [29]:
Equation, α and a are the thermal absorption coefficient and radius of the bubble impurity, respectively. The decisive relationship between the modulation effect of the millisecond laser optical field of the bubble impurities and the enhancement effect of photothermal damage induced by the laser optical field on fused silica is analyzed.
2.2 For geometric model construction
The bubble-type impurity was used as the research object in the simulation, and a rectangular area of 10 µm × 40?µm was selected as the computational domain of optical field scattering, and the material was fused silica. Based on the existence of a perfectly matched layer (PML) which can make the optical wave impedance match the adjacent medium wave impedance exactly, the incident electromagnetic wave will pass through the PML layer with full reflection [30,31]. The effect of the magnitude of the angle of incidence and the shape of the wavefront on the absorptivity can be neglected, and the incident laser optical field can be absorbed without reflection, which in turn simulates the state of the laser optical field transmission in the infinite space within a very small computational region. Shorter UV wavelengths, such as 355 nm, effectively penetrate and damage the deeper regions of the fused silica, particularly near defects like bubbles. This results in increased scattering and localized energy concentration around these bubbles, leading to a more pronounced enhancement of the localized optical field. The femtosecond laser efficiently heats and ionizes electrons within the fused silica, leading to enhanced electronic resonance and an increased local electric field around the defects. This also results in more pronounced nonlinear changes in the refractive index, amplifying the optical field effect near the defects, thereby lowering the material’s damage threshold and causing a more significant breakdown at defect locations, such as bubbles. Additionally, UV lasers are commonly employed for the precise modification of materials. The relatively low absorption of fused silica at 1064 nm minimizes energy deposition per photon. This characteristic enables the laser to penetrate deeply into the fused silica without substantial energy loss prior to interacting with the defect structure. The risk of inadvertently altering or damaging the material during the study of bubble defects is diminished. Conversely, UV lasers tend to promote surface or shallow interactions due to their higher absorption rates, which limits the investigation of deeper defects. Consequently, the wavelength of 1064 nm is well-suited for probing deeper regions within the fused silica, where subsurface defects, including bubbles, reside. Nonlinear effects, such as multiphoton absorption and self-focusing, are less pronounced at 1064 nm than in the UV band. This characteristic facilitates the study of direct interactions between the optical field and bubble defects without interference from these nonlinear effects or distortions, making it more suitable for precise optical field modulation studies. The 1064 nm wavelength enables a more controlled and predictable investigation of light scattering, diffraction, and modulation around bubbles. UV lasers are more effective at inducing laser breakdown and examining shallow defect interactions, whereas 1064 nm lasers offer improved control over the energy delivered to the material when precision is necessary, without causing structural damage. Numerous industrial and high-power laser systems, particularly in telecommunications and machining, utilize infrared lasers with a wavelength of 1064 nm. Deeper penetration into the material facilitates the detection of internal defects without inducing surface damage or excessive heating.
Assuming the absence of free charges and conduction charges at the interface of the medium, and considering no other impurities affecting the optical field distribution, the mesh subdivision diagram of the bubble is depicted in Fig. 2(b). To achieve the necessary higher precision in calculations while minimizing computation time and conserving computational resources, various mesh sizes were tested and compared. To balance mesh resolution with computational resources, we introduce an adaptive mesh strategy that refines the mesh in critical areas, such as defects and near-field regions, while employing coarser mesh elements in other areas to optimize this trade-off. To prevent significant skewing or deformation during the computational solution process, the mesh is composed of well-shaped cells that conform to the problem geometry. A perfectly matched layer (PML) is implemented to absorb outgoing waves and mitigate numerical reflections at the domain boundaries. To ensure the effective functioning of the PML, the mesh size is kept smaller than the wavelength to adequately resolve the oscillatory processes of the electromagnetic field. After thorough evaluation, a free triangular mesh with a maximum size of 9.2 × 10−3?µm and a minimum size of 2.7 × 10−4?µm was selected. This mesh size corresponds to 1/12 of the incident light wavelength (λ/12). The incident laser is vertically directed from above the fused silica optical element containing bubble impurities. Figure 2(a). presents a schematic model of the effect of bubble impurities on the modulation of the laser optical field. The simulation parameters utilized in this study are detailed in Tables 1 and 2. Pulse width is a crucial parameter in laser-induced breakdown, significantly influencing the interaction between laser energy and fused silica. Shorter pulse widths, such as femtoseconds or picoseconds, yield higher peak power over a brief duration, resulting in intense ionization and plasma generation without excessive heat dispersion. This characteristic is advantageous for minimizing thermal damage and enabling precise localized modifications of fused silica. However, short pulses may decrease the damage threshold of transparent optical materials, including fused silica. In comparison to longer pulses (nanoseconds or microseconds), which allow more time for thermal diffusion, shorter pulses are more penetrating and less likely to induce damage in the surface layer of fused silica. Given that 1064 nm wavelength do not cause substantial energy loss prior to interacting with the defect structure, this laser wavelength offers improved control over energy delivery to the fused silica and enhances the study of light scattering, diffraction, and modulation around bubble impurities. Therefore, utilizing a laser in the 1064 nm wavelength range facilitates the exploration of optical field modulation occurring at deeper levels within the fused silica while minimizing the risk of optical breakdown.
Fig. 2. (a) Schematic diagram of the effect of bubble impurities on optical field modulation; (b) Schematic diagram of the mesh subdivision of bubble impurities.
3. Simulation study of localized optical field modulation by bubble impurities
3.1 Modulation study of the optical field by single bubble impurities
Figure 3(a) illustrates the scattering and interference diagram of the optical field modulation by a single bubble impurity. The embedding depths of the bubble impurities in the fused silica optical element are 0.5?µm, 1.0?µm, and 1.5?µm, with laser input powers of 1 W, 5 W, and 10 W, respectively. The radius of the bubble-type impurities was varied, taking values from 1 λ to 4 λ in steps of 0.1 λ. The simulation results obtained are plotted in Fig. 3(b).
Fig. 3. (a) Schematic diagram of laser optical field modulation by single bubble impurities; (b) Relationship between bubble impurity size and embedded depth on optical field modulation.
As shown in Fig. 3(b), the modulation trend of the optical field by bubble impurities at different embedded depths remains consistent with the increase in radius size. The modulation effect of the laser optical field mainly occurs in the upper half of the bubble impurities, continuously modulating the optical field within the range of reflection angles from 0° to 90° relative to the normal. When the radius of the bubble impurity exceeds 1.5 λ, the modulation effect on the optical field diminishes, and the increase process exhibits gentle oscillations. No refraction occurs when the laser incident angle is 0° upon passing through the bubble impurity. Due to the refractive index of the bubble being lower than that of fused silica, as the incident angle increases from 0° to 90°, total internal reflection occurs at the localized interface between the fused silica and the bubble. The oscillation of the optical field, as depicted in Fig. 3(b), arises from the refraction occurring when light rays pass obliquely through the bubble region, transitioning from one medium to another. Light interacting with the surface of the bubble experiences multiple reflections, refractions, and total internal reflections, during which the interplay of incident, reflected, and refracted light generates a resonant oscillation effect. This phenomenon ultimately results in mutual interference between the optical fields and phase changes in the light waves, thereby affecting both the intensity and propagation direction of the optical field within the localized region. Additionally, the oscillations are exacerbated by the scattering of the optical field as it traverses the bubble defects, resulting in the optical field capturing and releasing energy multiple times within a localized region, which leads to oscillatory fluctuations in the optical field.
The modulation of the local optical field by bubble impurities creates an optical intensity gradient, which induces precursor damage in fused silica due to differences in laser energy absorption gradients. Based on Mie scattering theory, a model was established to analyze the enhancement damage in fused silica induced by the modulation of the laser optical field by bubble impurities, examining its effect on the enhancement of optical intensity [32]. The study did not vary the size of the single bubble impurity, focusing instead on the impact of optical field modulation, with the distribution of the optical field depicted in Fig. 4. The incident and refracted optical fields combine through the modulation effect of the bubble to form a stable, non-uniform optical intensity distribution [33]. The presence of bubble impurities increases the contact area between media with different refractive indices (bubble impurity and fused silica), thereby generating a diffraction effect at the bubble edges. The symmetrical field intensity distribution inside the bubble is due to stationary wave effects caused by reflected and refracted light. The most significant optical field attenuation occurs in the region behind the bubble impurity, which is detrimental to the transmission of laser energy.
Fig. 4. The relationship between optical field modulation and single bubble impurities of different radii.
As shown in Fig. 4(a), the interaction area between the laser optical field and small-sized bubble impurities is relatively small, resulting in a minor modulation gradient of the local optical field due to scattered and refraction effects. In contrast, as depicted in Fig. 4(d), when the laser optical field passes through larger bubble impurities, modulation occurs on both sides of the bubble impurity, while the region directly beneath remains unaffected. The modulation of the optical field by bubble impurities was attributed to the interference, scattering, and total internal reflection of the incident and reflected optical fields through the bubble impurities. The modulated optical field, coherent with the original optical field, creates a periodic pattern of alternating light and dark regions, forming a local optical field gradient. The extent of modulation by bubble impurities varies with their size, initially increasing and then decreasing as the size of the bubble impurities increases. The local optical field gradient induced by bubble impurities after laser modulation results in localized temperature and stress gradients over time. These gradients reduce the damage resistance threshold of fused silica.
To further investigate the influence of laser input energy on local optical field modulation under the condition of fixed bubble embedment depth, finite element analysis was employed. The embedding depth of bubble impurities was set at 1?µm. Laser power levels of 1 W, 2 W, 3 W, and 4 W were applied, and the relationship between bubble impurity size and laser power evolution was illustrated in Fig. 5.
As shown in Fig. 5, with the increase of laser power, the modulation of the optical field by bubble impurities of various sizes initially enhances and then levels off, exhibiting periodic oscillations at the threshold of optical field modulation. This phenomenon occurs because the incident angle for total internal reflection increases continuously as the area of the bubble impurities grows, thereby enhancing total internal reflection within a certain range. Under the condition of a fixed embedding depth, changing the laser input power affects the modulation. When the size of the bubble impurities is extremely small, the modulation effect of the optical field is weak due to diffraction and the limited scattering area. As the size of the bubble impurities gradually increases, the surface area increases, leading to stronger total internal reflection modulation effects in the local region, manifested as an increase in the electric field amplitude. When the size of the bubble impurities reaches 1.1 λ, the modulation of the optical field increases steeply. For bubble impurities larger than 1.1 λ, the modulation of the optical field tends to level off. This is because the increased radius of the bubble impurities leads to a larger, yet stable, contact area with the incident laser, increasing the unit optical field transmission while reducing the coupling modulation effects of interference and scattering. This indicates that 1.1 λ is the critical size for local optical field modulation by bubble impurities. When the size of the bubble impurities exceeds 2 λ, the modulation effect reaches a threshold, resulting in periodic oscillations. The oscillations of the optical field, as illustrated in Fig. 5, can be attributed to mechanisms involving local changes in the refractive index and resonance oscillations. Changes in the refractive index within the vicinity of the bubble defect result in a phase shift of the light wavefront, which in turn causes fluctuations in both the intensity and propagation direction of the light waves during their interference. The bubble defect functions as a resonance cavity or scattering center for the light, effectively trapping and releasing light over a short timescale, which results in oscillatory fluctuations of the light within the confined region. As the radius of the bubble impurities increases to the micron scale, the modulation threshold observed aligns with the findings of Chunlai Zhang [11]. The results of the study of the modulation of the optical field by bubble impurities embedded in fused silica at a fixed depth and varying the input power of the laser are plotted in Fig. 6.
Fig. 6. Relationship between the electric field modulus and laser power modulation by single bubble impurities of different radii.
When the input laser power is relatively low, the modulation effect of bubble impurities of different radii on the internal optical field of fused silica is similar. However, as the input laser power increases, larger bubble impurities exhibit a more pronounced modulation effect on the optical field, with the modulation value approximately corresponding to a radius size of 1.1 λ. The embedding depth has a minimal effect on the optical field modulation within fused silica due to the material's high transmittance and the high coherence of the laser. In areas without bubble impurities, no optical field interference or scattering occurs, resulting in negligible energy loss. Considering the distribution trends in Fig. 4 and Fig. 3(b), the modulation trend of bubble impurities on the local optical field within fused silica remains consistent when varying the embedding depth under a certain input laser power. In transparent optical components with only bubble impurities as defects, the embedding depth of the bubble impurities has a negligible impact on the optical field, whereas the input laser power and the size of the bubble impurities significantly influence the optical field modulation effect.
3.2 Coupled synergistic study of optical field modulation by double bubble impurities
As shown in Fig. 7(a), the schematic diagram illustrates the modulation of the laser optical field by dual bubble defects. It can be observed that there was a significant enhancement of the local optical field modulation on both sides of the bubble impurities. To further investigate the modulation effect of dual bubble impurities on the laser field, a two-dimensional cross-sectional line was constructed along the central axis of the dual bubble impurities (from the interface between fused silica and air at y = 0?µm to y = -20?µm). The results of the laser optical field modulation are presented in Fig. 7(b).
Fig. 7. (a) Schematic diagram of laser optical field modulation by double bubble impurities; (b) Modulation distribution of the optical field along the perpendicular line through the double bubble impurities.
The spacing of the double bubble impurities is double the laser wavelength, the modulation of the optical field inside the fused silica will reach the maximum effect, after which the modulation of the bubble group begins to decline rapidly as the spacing of the two bubbles in the combination continues to increase and is accompanied by a periodic oscillatory tendency in the process of decreasing modulation of the optical field. Using the fused silica-air interface as a reference plane, we observe the modulation of the optical field inside the fused silica under the effect of double bubble coupling. Analyzing the position of the maximum value of the optical field generated by the modulation, it is not difficult to see that, with the increasing spacing of the two bubbles in the coupling effect, the position of the maximum value of the optical field modulation shows a tendency of periodic oscillation, and the position of the maximum optical field modulation is gradually moving away from the reference surface toward the interior of the fused silica, and the modulation of the optical field by the coupling of the combination of bubbles in the process is gradually weakened.
The interference effect of dual bubble impurities is more pronounced along the perpendicular bisector of the dual bubbles. To investigate the enhanced interference and the more complex optical field gradient relationship along this line, the modulation of the optical field intensity was simulated using the finite element method. The simulation data were plotted as shown in Fig. 8.
Fig. 8. Modulation distribution of the optical field along the two-dimensional cross-section of double bubble impurities.
As the distance between two bubble impurities increases, their modulation effect on the optical field coupling diminishes, resulting in periodic oscillatory movement of the bubbles wake. Through a simulation study focused on dual-bubble impurities with a radius of 0.3 λ, findings indicate that the peak optical field modulation arises when the bubbles are spaced 1 λ apart. This configuration, situated 5.43?µm from the air-fused silica interface, generates an optical field modulation value of 5969.88 V/m. This value contrasts with the 3747.85 V/m modulation observed under identical incident optical power (1 W) but devoid of bubble impurities, thereby presenting an optical field enhancement factor (LIEF) of 1.6.
To further investigate the reasons behind the enhancement of photothermal damage in fused silica induced by laser optical field modulation due to varying bubble spacing, the study was conducted with dual bubble radii set at 0.25 λ, 0.30 λ, 0.50 λ, 1.00 λ, 1.50 λ, and 2.00 λ. The research explores the spacing between two bubbles ranging from [0.1?µm to 10?µm] with a step size of 0.1?µm, under an incident laser power of 1 W. The evolution of optical field modulation as a function of the spacing between dual-bubble impurities was illustrated in Fig. 9.
In Fig. 9, the horizontal axis represents the spacing between dual-bubble impurities, while the vertical axis denotes the optical field values within fused silica. The study results indicate that when two bubbles were positioned parallel to the fused silica-air interface, the modulated optical fields combine. The coupling effect of the two bubbles significantly enhances the optical field intensity within the fused silica. An optical field maximum was observed when the distance between the dual-bubble impurities reaches approximately 1 λ. When the dual-bubble impurities have a spacing of 1 λ and a radius of 0.25 λ, the peak modulation of the optical field is 5432.00 V/m. This modulation is equivalent to that of a single bubble impurity with a radius of 0.5 λ, resulting in the optical field enhancement factor (LIEF) of 1.44. For dual-bubble impurities with a radius of 0.3 λ spaced 1 λ apart, the modulation effect is comparable to that of a single bubble impurity with a radius of 0.8 λ, with an LIEF of 1.62. Dual-bubble impurities with a radius of 1 λ exhibit a modulation effect that surpasses that of a single bubble impurity with a radius of 4 λ, achieving an LIEF of 2.14.
Upon modulation by dual-bubble impurities, the total internal reflection, originally incident light, reflected light, and refracted light coherently superimpose, forming alternating bright and dark interference fringes in the areas on both sides and behind the dual-bubble impurities. The first interference fringe exhibits the strongest optical field intensity, while the intensity of each subsequent coherent optical field gradually decreases outward, creating a local gradient within the fused silica. The spacing between the dual-bubble impurities affects the probability of interference between the incident and scattered light. When the spacing between the dual-bubble impurities is 1 λ, the maximum optical field modulation occurs at the midpoint (0, 8?µm). The distribution of scattered and interfered optical fields due to varying dual-bubble impurity spacing is illustrated in Fig. 10.
Fig. 10. Interference distribution relationship of optical field modulated by double bubble impurities.
After optical field modulation, the free electron density within the fused silica increases, affecting its optical response. Certain sites within the fused silica begin to exhibit metallic-like behavior, with increased kinetic energy of electrons as the bandgap rises. Thermal expansion causes changes in the refractive index, leading to phase shifts. Energy becomes localized and absorption is enhanced at the peaks of the standing electric field, thereby influencing the field strength distribution within the fused silica. The coupling modulation effect on the local optical field decreases when the spacing between dual-bubble impurities exceeds 1.1 λ. When the spacing is greater than 6 λ, the dual-bubble impurities no longer enhance each other's optical fields, and the modulation effect of dual-bubble impurities resembles that of two independent single-bubble impurities. This indicates that the coupling effect between the two bubbles occurs only within a spacing of less than 6 λ.
Laser irradiation causes a temperature increase in fused silica, leading to the dissolution of internal air, which is unable to escape. This results in the rapid diffusion and aggregation of air within the fused silica, forming vacuum bubbles. Since the vacuum regions cannot conduct heat, the average temperature inside the bubble impurities is considered to be 300 K. There is a temperature difference of approximately 3000 K between the vacuum bubbles and the fused silica in the remelting state, creating a significant temperature gradient at the boundary of the bubble impurities. As the duration of laser energy deposition increases, the lattice of the fused silica exhibits gradual thermal expansion, resulting in thermal stress accumulation at the boundaries of the bubble impurities. The temperature distribution resulting from the scattering and interference of the optical field by dual-bubble impurities is illustrated in Fig. 11.
Fig. 11. Isothermal distribution relationship under optical field modulation for double bubble impurities.
In the isothermal distribution map of fused silica, it is observed that the heat accumulation resulting from laser optical field modulation forms local “hot spots” behind the vertical axis of the bubble impurities. These hot spots create a significant temperature gradient, acting as precursors to damage. When fused silica reaches its melting point, localized remelting occurs. Upon cessation of laser irradiation, the fused silica undergoes cooling and annealing, and the areas that experienced localized remelting and cooling exhibit a reorganization of atomic arrangement at the micro-nano scale, slightly altering the physical properties of the fused silica. This is characterized by more pronounced nonlinear temperature absorption and larger temperature gradient variations, leading to multiple local hot spots.The increase in local hot spot density elevates the probability of damage in fused silica, thereby lowering the damage threshold. The temperature changes induced by bubble impurity modulation affect the structure of fused silica, causing significant variations in the refractive index in the hot spot regions. This results in stronger local modulation effects on laser optical fields, including scattering, interference, and refraction, ultimately reducing the output quality of the laser beam.
As the laser energy deposition time increases, the absorption of laser energy by fused silica causes the temperature to rise, simultaneously generating thermal stress. At the end of the pulse, thermal stress reaches its peak accumulation value, with bubble impurities forming a stress concentration point at the bottom of the laser-modulated area. Observations of the isobaric lines indicate a stress difference of up to 21 MPa on either side of the vacuum bubble. This phenomenon arises due to the thermal expansion of the lattice, which increases the distance between the lattice points of the fused silica. As the laser irradiation time extends, thermal stress and deformation at the boundary between the fused silica and bubble impurities continuously increase. The stress distribution after scattering and interference of the optical field by double bubble impurities is illustrated in Fig. 12.
Fig. 12. Stress distribution relationship under optical field modulation for double bubble impurities.
At the end of laser irradiation, the vacuum bubbles exhibit noticeable elliptical plastic deformation. The external fused silica, undergoing thermal expansion, exerts inward pressure on the vacuum bubbles. Due to the inability of the vacuum inside the bubble impurities to conduct heat, there is no accumulation of thermal stress within the bubbles, resulting in a significant stress gradient at their boundaries. The accumulated thermal stress from laser exposure, exceeding the yield strength of fused silica, causes the bubble impurities to fracture. Consequently, the damage threshold at the junction of the vacuum bubbles is significantly reduced. As depicted in Fig. 11 and Fig. 12, “local hotspots” formed by optical field modulation lead to substantial temperature and stress gradients. The lateral compressive stress exerted on the vacuum bubble impurities causes the fused silica to fracture, demonstrating that optical field modulation is a critical factor in lowering the damage threshold of fused silica.
4. Experimental study on the enhancement of fused silica damage induced by laser optical field modulation by bubble impurities
To verify the accuracy of the simulations, a laser damage experimental system was established to irradiate fused silica optical elements containing impurity defects. By modulating the experimental parameters, the results were obtained, and the experimental optical path system is shown in Fig. 13. The experimental platform mainly comprises a laser emission and focusing system, combustion wave detection, damage morphology observation, and optical performance detection of the damaged area. The laser used for laser emission is a Melar-100 Nd:YAG laser manufactured by Beamtech Optronics Co., Ltd. with an output wavelength of 1064 nm, an adjustable output energy of 10-100 J, an adjustable pulse width of 0.5-3.0 ms, and a repetition frequency of 10 Hz. The focusing system uses a focusing lens with a focal length of 500 mm. The energy meter model PE25BF-C has an energy measurement range of 60µJ - 10J and is manufactured by Ophir. A beam splitter placed at an angle of 45° throughout the laser path separates a small portion of the laser light from the original path and projects it onto an energy meter probe, where the laser energy at which damage to the fused silica occurs is calculated [34,35]. The mobile control platform is used to control the distance between the focusing lens and the fused silica and to adjust the radius of the laser spot on the irradiated fused silica. A 532 nm laser model LSR532H-OEM 800 manufactured by Ningbo Yuanming Laser Technology Co., Ltd. with an adjustable output power of 800-2000mW was used to provide background light as a combustion wave in order to enhance the observation of the combustion wave evolution process. A high-speed camera model PhantomV641-16 G, manufactured by YORK TECH, with a maximum shooting rate of 2.19× 105 fps and a minimum exposure time of 1 × 10−6 s was used to investigate the effect of impurity defects on the millisecond laser-induced combustion wave generation in fused silica. A metallurgical microscope model Leica DMI3000 M manufactured by Megphy was used to observe the damage morphology of fused silica.
The optical damage to fused silica induced by millisecond pulse laser primarily includes thermal damage and optical breakdown [36]. Energy densities of 1.73 × 103 J/cm2, 1.93 × 103 J/cm2, 3.24 × 103 J/cm2, and 3.84 × 103 J/cm2 were selected for this study. Due to the modulation effect of bubble impurities on the laser optical field, nonlinear absorption of laser energy occurs in the fused silica. Processes such as multiphoton ionization, tunnel ionization, and avalanche ionization lead to a rapid increase in electrons in the conduction band of the fused silica (Overcoming the optical gap width, -7.8 eV photoionization energy) [37]. When the free electron density exceeds the threshold density, intense absorption of laser energy is triggered, resulting in the formation of high-temperature, high-density plasma [38,39]. The morphology of photothermal damage in fused silica induced by millisecond laser irradiation of bubble impurities, after local optical field modulation, is organized and shown in Fig. 14.
The localized optical field gradients induced by bubble impurities modulating the laser optical field exhibit photothermal effects over time. These photothermal effects lead to enhanced local absorption, and the accumulation of thermal stress at the bubble impurities exceeds the yield strength of the fused silica, resulting in damage. The photothermal effects caused by optical field modulation are crucial in reducing the breakdown threshold of fused silica. The presence of circumferential and radial stress facilitates crack formation and delamination in the damaged area. As shown in Fig. 14(a), at a laser energy density of 1.73 × 103 J/cm2, there is no significant damage observed on the surface of the fused silica. The scattering and refraction caused by the modulation of the laser optical field by bubble impurities lead to nonlinear energy conduction, forming subsurface channels of varying depths and positions. Subsurface ablation microchannels are formed in the overlapping areas of bubble impurities. As illustrated in Fig. 14(b), at a laser energy density of 1.93 × 103 J/cm2, damage appears on the surface of the fused silica. The localized “hot spots” formed due to bubble impurity modulation create a temperature gradient. The bubble impurities reaching the melting point form melt pits, with ablation marks extending from the central region to the interior. This demonstrates that the accumulation of thermal stress due to bubble impurity modulation is a major factor in precursor damage caused by bubble impurities. Figure 14(c) shows that at a laser energy density of 3.24 × 103 J/cm2, multiple bubble impurity regions jointly modulate the optical field. Smaller bubble impurities induce radial cracks, and as the radius of the bubble impurities increases, the local optical field modulation strengthens nonlinear heat conduction. The radial cracks merge to form irregular large damage pits, and the microcracks from localized melting and spallation exhibit distinct ring patterns with extended laser irradiation time, forming radial and circumferential cracks on the outer surface. Figure 14(d) indicates that at a laser energy density of 3.84 × 103 J/cm2, in the laser-irradiated region without bubble impurities, the edges of the damage melt pits show ablation marks.
Due to the modulation of local optical fields by bubble impurities, energy conduction differences arise, leading to temporal variations in the combustion wave generated by fused silica damage. High-speed cameras were employed to record the evolution of waveforms over time during the propagation of the combustion wave. By analyzing these recordings, we can infer the relationship between local optical field modulation by bubble impurities and the induced damage process in fused silica [40,41]. The experimental results are summarized in Fig. 15.
Fig. 15. Temporal evolution of combustion waves after optical field modulation by bubble impurities.
As illustrated in Fig. 15(a), the dynamical process of the combustion wave is induced by bubble impurities modulating the local optical field at a laser energy density of 1.73 × 103 J/cm2, resulting in damage to the fused silica. The millisecond laser ionizes a large number of free electrons and ions. The continuous energy deposition causes these free electrons and ions to reach critical density, forming plasma. The sustained laser energy deposition extends the plasma plume along the laser beam direction, gradually forming a combustion wave that propagates in the opposite direction of the laser source. As the expansion distance and volume increase, the high-temperature combustion wave loses internal energy through thermal radiation to the surrounding medium, causing the propagation speed of the combustion wave to gradually decrease. As the internal energy of the combustion wave decreases, the brightness in the image also diminishes. The experimental study shown in Fig. 15(a) observed that the waveform of the combustion wave is longitudinally compressed during its expansion away from the fused silica, confirming that the longitudinal pressure generated by the bubble impurities modulating the laser optical field leads to initial damage to the vacuum bubbles. By recording the morphological expansion of the combustion wave generated during the fracture of fused silica, we amplify the characterization of the reduction in damage threshold caused by longitudinal pressure, further validating the accuracy of the established model. The modulation of laser energy by bubble impurities results in local interference phenomena during the propagation of the combustion wave. When the laser output energy density is 1.93 × 103 J/cm2, two shadows are observed behind the hemispherical combustion wave in Fig. 15(b). The dual bubble impurities locally modulate the optical field, altering the energy deposition distribution within the fused silica. This differential energy distribution during the laser-induced combustion damage results in local variations in the waveform of the combustion wave, with lower internal energy regions appearing relatively darker. Combining this with the damage morphology shown in Fig. 14(b), the near-spherical combustion wave in Fig. 15(b) corresponds to the damage threshold of dual bubbles in fused silica. At a laser output energy density of 3.84 × 103 J/cm2, the bubble impurities within the fused silica undergo optical breakdown first. Most of the gas energy is compressed into the collapsing bubble impurities, causing the surrounding fused silica to undergo phase change explosion due to significant longitudinal compressive stress. The nonlinear absorption of vapor energy by the laser within the bubble impurities results in the release of stored energy through an expanding combustion wave during the initial stage of expansion, driven by inertia. The explosion of bubble impurities forms new irregular cavities within the fused silica, and the bubble walls exhibit Rayleigh-Taylor instability deformation. Multiple wave oscillations and interferences occur within the cavity during the expansion of the combustion wave, leading to irregular waveforms. The differences in depth and number of bubble impurities result in more complex scattering and interference after laser optical field modulation, manifesting as more complex dynamic characteristics after laser-induced combustion damage. Figure 15(c) and Fig. 15(d) show the instantaneous “lag” observed when the combustion wave passes through the vacuum region of bubble impurities during breakdown or collapse. The absence of such “lag” when the combustion wave does not pass through the vacuum region indicates that the local vacuum region caused by bubble impurities leads to differences in the local extension of the combustion wave. The local interference phenomena during the expansion of the combustion wave are magnified to characterize the optical response of bubble impurities undergoing breakdown and collapse on the millisecond scale.
Figure 16 illustrates the relationship between the transmittance of the damaged fused silica regions and wavelength, combining the Beer-Lambert Law and the least squares method. The transmittance increases with the wavelength. In non-damaged regions, the maximum transmittance reaches 93% after laser energy deposition. When bubble impurities modulate the laser optical field, the local absorption rate increases with the gradual increase in laser energy input, resulting in a decrease in the transmittance of the damaged regions.
Fig. 16. Evolution of UV transmittance with wavelength in the damaged area after optical field modulation by bubble impurities.
The modulation of the laser optical field by bubble impurities creates local optical field gradients. With prolonged laser irradiation, the resulting local thermal stress gradients increase the density of oxygen vacancy defects around the bubble impurities, which become precursors to laser damage and one of the reasons for the reduced damage threshold of fused silica. In the ultraviolet spectrum, scattering light absorption deviation, photon oscillations at the band edge, and defect-induced electronic transitions affect the refraction of the laser optical field by bubble impurities. The resonance absorption of phonons and photons increases, thereby enhancing the absorption coefficient. In the limit of weak spherical scattering of the optical field, the total optical field scattering rate can be calculated as the sum of several scattering rates according to Matthiessen ‘s [42,43]. The differences in the refractive index at the edges of bubble impurities lead to increased scattering and reflection of the optical field. As the wavelength increases, the transmittance gradually reaches a threshold. Higher transmittance after energy deposition indicates a smaller modulating effect of bubble impurities on the optical field. The modulation of the local optical field by bubble impurities induces significant thermal stress gradients, leading to internal rupturing of the bubble impurities. The resulting impact causes more severe damage to the region, manifested as decreased laser transmittance and increased absorption, thereby exacerbating the likelihood of damage to fused silica and lowering the damage threshold of transparent optical components like fused silica.
To further investigate the effects of bubble impurity modulation on the optical properties of fused silica, a study on the absorbance and transmittance of the damaged regions was conducted. The average measurement data of wavenumber and relative absorption rate in the damaged fused silica regions were obtained by performing 20 consecutive scans at room temperature using a spectrophotometer. The results, shown in Fig. 17, illustrate the evolution of wavenumber and relative absorption rate in the local regions of fused silica after modulation by bubble impurities.
Fig. 17. Relative absorption relationship with wavenumber in the damaged area after optical field modulation by bubble impurities.
The modulation of the optical field by dual bubble impurities causes more severe damage compared to single bubble modulation. The temperature of the irradiated fused silica region reaches the melting point, leading to the melting or breaking of covalent bonds, which in turn causes micro- and nanoscale damage. Due to differences in the average scattering angle after optical field modulation, variations in the absorption spectrum are observed. The relative rates of energy deposition and heat diffusion determine the damage threshold, with permanent structural changes occurring in the bubble impurity regions due to heat diffusion and thermal transfer. At the defect sites, the Si-O bonds experience enhanced vibration due to laser energy deposition. During laser irradiation, Si-O and Si-O-Si bonds undergo breaking and reformation. The increased thermal stress and plastic deformation lead to a higher density in the topological structure of fused silica. The transient breaking and reformation of bonds reduce the bond angles of Si-O and Si-O-Si locally, resulting in increased absorption of infrared laser energy at the edges of bubble impurities. The modulation of the optical field by bubble impurities alters the Si-O-Si structure, increasing the reflectivity of the bubble impurity surface and causing the appearance of amorphous phases in the local regions of fused silica. The band gap absorption rate of Si-O bonds increases with wavenumber. The nonlinear absorption of laser energy by oxygen vacancies after optical field modulation influences the formation of absorption bands, manifesting as oscillations in the wavenumber absorption peaks. The transfer of laser energy from excited electrons to the lattice results in Si-O stretching vibrations. Lattice vibrations in fused silica cause dual infrared absorption peaks at wavenumbers of 2300?cm?1 and 3600?cm?1, whereas regions without optical field modulation only show an infrared absorption peak at 2300?cm?1. The Si-O-Si lattice in bubble impurities shows higher infrared laser absorption band intensity after local optical field modulation, which is a contributing factor to the reduced damage threshold in these regions.
To further investigate the absorption characteristics of fused silica in different spectral bands after modulation by bubble impurities, an absorbance study was conducted on the damaged regions. The study revealed changes in the absorbance of local regions after laser optical field modulation, elucidating the relationship between precursor damage and different wavenumbers. At room temperature, 20 consecutive scans of the wavenumber and relative absorption rate in the damaged fused silica regions were performed using a spectrophotometer. The average measurement data are illustrated in Fig. 18.
Fig. 18. Evolution of UV absorption with wavelength in the damaged area after optical field modulation by bubble impurities.
As shown in Fig. 18, the relationship between wavelength and relative UV absorption in regions where the optical field is modulated by bubble impurities is depicted. The local thermal stress accumulation induced by bubble impurities modulating the optical field leads to Si-O bond stretching, O-Si bond bending, and Si-O-Si rocking vibrations. This modulation changes the optical absorption properties at local sites, resulting in enhanced UV laser absorption at 200 nm, which reduces the energy required for photon transitions. The enhanced local absorption due to bubble defects is one of the reasons for the reduced damage threshold of fused silica in the UV spectrum. Bubble impurities increase the density of defect states, raising the defect density in fused silica. The likelihood of damage increases with higher laser energy and prolonged irradiation time. The presence of a small number of free electrons in the interstices of fused silica and their interaction with intrinsic defects lower the bond energy damage threshold, making multiphoton absorption and electron avalanche ionization more likely. The strain in Si-O-Si bonds is a key factor in increased energy absorption in bubble impurity lattices, causing wavelength absorption shifts. As the wavelength increases, photons lack sufficient energy to interact with fused silica, leading to enhanced scattering by long-wave photons. With increased laser energy deposition, scattering at bubble impurity sites intensifies, and the local absorbance increases, further reducing the optical damage threshold of fused silica.
5. Conclusions
Based on Mie scattering theory, the heat conduction equation, the Beer-Lambert Law, and considering the optical field modulation effect and Matthiessen ‘s rule, this study establishes a model that describes the local optical field modulation in transparent optical components containing bubble impurities by combining theoretical, simulation, and experimental approaches. The relationship between bubble impurity size and spacing on laser optical field modulation was investigated, and the optical shadow method was used to characterize the dynamic “lag” phenomenon of bubble impurities during optical breakdown at the millisecond scale. The following key results were obtained: The embedding depth of bubble impurities shows a consistent trend in optical field modulation. The optical field modulation node for dual bubble impurities occurs at a spacing of 1.1 λ. When the spacing is greater than 6 λ, the modulation effect is similar to that of two independent single bubble impurities. The optical field modulation by dual bubble impurities induces local “hot spots,” which evolve into strong temperature and stress gradients over time. The lateral compressive stress exerted on the bubble impurities leads to the fracture of fused silica, reducing its damage threshold.
The local optical field modulation by bubble impurities increases the density of oxygen vacancy defects, affecting the local optical field refraction. This modulation results in Si-O bond stretching, O-Si bond bending, and Si-O-Si rocking vibrations. The breaking and reformation of Si-O and Si-O-Si bonds due to modulation lead to increased topological density and reduced bond angles, resulting in lattice vibrations of fused silica that produce dual infrared absorption peaks at wavenumbers 2300?cm?1 and 3600?cm?1. A small number of free electrons in the gaps of fused silica, interacting with intrinsic defects, make multiphoton absorption and electron avalanche ionization more likely, thus lowering the bond energy damage threshold.
In conclusion, the experimental results are consistent with simulation studies, validating the accuracy of the model and providing important theoretical support for understanding and predicting the damage behavior of transparent optical components like fused silica under laser energy deposition. This study elucidates the physical mechanism by which bubble impurities in transparent optical components reduce the damage threshold through local optical field modulation.
Funding
Jilin Provincial Scientific and Technological Development Program (Approval No.:20230402078GH).
Acknowledgments
Jilin Provincial Scientific and Technological Development Program.
Disclosures
The authors declare no conflicts of interest.
Data availability
All data that support the findings of this study are included within the article.
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