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Abstract
In this study, we demonstrate the elongated Type I modifications in fused silica with an axial length > 50 µm. Such extended longitudinal dimensions were obtained by deep focusing radiation of a femtosecond laser inside fused silica at a depth of 2 mm. The transition from the Type II modification (nanogratings) to the Type I modification (refraction index change) was observed with increasing focusing depth at the constant pulse energy. The refractive index changes of ∼ 1.5×10−3 for a single pass and 2.4×10−3 for multiple passes were demonstrated. The radial dimensions of the deep-focused modifications were confined to 0.5-1.5 µm size. By overlapping the modifications in radial and axial directions, 1D phase grating in the depth range from 2 to 5 mm was recorded, allowing to split of the beam with a diffraction efficiency of > 96%. We demonstrate that the aberration-based recording with a Gaussian beam in fused silica is a simple tool for fabricating complex phase diffractive optical elements.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Modification of transparent materials with femtosecond laser direct writing has a lot of interest in fields of data processing, waveguides or diffractive optical elements development, spectroscopy, wavelength division multiplexing [1–5]. Due to the nonlinear absorption processes, the deposited optical energy can trigger localized structural modifications in transparent materials, induce a refractive index change (Type I modification) [1] or form birefringence (Type II modification). The latter effects are related to nano-grating formation inside fused silica [6,7], the most common modification in the moderate laser fluence range. By combining the laser-translated track with the induced refractive index modification, the volume phase optical elements such as volume Bragg gratings (VBG) and other diffractive optical elements (DOE) can be fabricated. The DOEs have broad application interests in laser resonators, attenuators or holography [8–11]. Therefore, they can be used in laser systems to decrease the size or improve the laser beam quality [12].
The VBG are mainly recorded by exposing the interference pattern from UV laser [13] in UV-sensitive phosphate or photo-thermo-refractive glasses [5,14]. The other techniques are direct femtosecond laser writing with a cylindrical lens through a phase mask [15], implementation of the optical filaments [16,17] or Gauss-Bessel beams [18]. The maximum demonstrated diffraction efficiency for recorded VBG in such a way is up to 80% for phase mask and filament writing [15,17]. The Gauss-Bessel beams were also used for VBG fabrication and showed more promising results than the Gaussian beam recording [18], with a declared diffraction efficiency of over 90%. VBG can also be recorded in non-photosensitive glasses [19,20] and sapphire [21] by inducing refractive index change using an ultrashort pulse laser.
Almost all demonstrated DOEs were recorded near the surface at depths of up to 1.2 mm [22–24]. However, there are no reports on DOEs, recorded at higher depths. Only a few insights showing the influence of the aberrations at the deep focusing are reported in [23,25]. Usually, spherical aberrations limit the focusing depth. That increases the voxel size and lowers the recording resolution. A few investigations were carried out to explore the effect of deep focusing [23,26,27]. The increase in damage threshold and the refractive index was observed under these conditions. For objectives with a high numerical aperture, the focusing depth without any compensation is usually restricted to ∼ 100-500 µm due to the aberration effects [23] When short modifications for waveguide recording are required, aberration compensation can be used to suppress the modification to a small axial length [24]. However, for some applications, non-compensated aberrations could help to achieve the required intensity field distribution inside the material. Consequently, due to the deep focusing, the radial size of the modification can go down to the diffraction-limited size that allows the recording of high-density grating structures. The depth to radial size ratio can exceed the 1:30 value, which is very attractive for recording multi-layer gratings with minimal losses. In the recent study, the conical phase front angle was used to induce elongated modifications in fused silica [28]. It was demonstrated that the maximum achievable modification axial length was ∼ 170 µm at 2.5 mm focusing depth and 8 µJ pulse energy, but when the pulse energy was reduced the modification axial length dropped <40 µm for 1µJ energy.
Some reports indicate that the diffraction efficiency of thick volume gratings can be related to Talbot planes located at a specified distance from the volume element [29]. This was the case when the multi-layered elements were recorded with a non-even transition between layers. In our case, the overlapping between consecutive layers was chosen to obtain an even refractive index change along the whole element size.
In this report, we investigate the effect of the aberration on the properties of the deeply-focused modifications formed with the femtosecond laser direct writing technique (FLDW). We suggest distinguishing the prospects of the spherical aberrations in the deep focusing case. The paraxial approximation focusing method [26,30] was applied to describe the intensity distribution at a focal plane of the focused beam through the air-glass interface.
The aim of this investigation was to find a way to control the refractive index change for the recording of multilevel complex DOEs. The refractive index changes at a focusing depth from 2 to 5 mm were investigated. At relatively high focusing depths, the modification axial length stopped elongating and started to decrease. Due to the pronounced dependence of the modification axial length on the pulse energy and focusing depth, it was possible to predict the exact overlap between consecutive layers and achieve a constant refractive index change over the whole grating thickness. The motivation of this work was to simplify the recording method of the DOEs and investigate challenges in the fabrication of complex diffractive elements that were difficult to fabricate by other techniques.
2. Experiment setup and characterization
The femtosecond laser operating at 515 nm wavelength from Yb:KGW was used (Pharos, Light Conversion) and delivered the beam to the sample surface. The shortest laser pulse duration ∼ 260 fs and two different repetition rates of 200 kHz and 500 kHz were applied. The deep modifications were recorded in the 6.3 mm thick ultraviolet-grade fused silica plates (JGS1, Eksma Optics), sliced to the 4${\times} $20${\times} $6.3 mm3 samples. During the experimental work, various combinations of the Type I modifications were recorded. Single modifications were recorded to investigate their morphology and axial length. For the refractive index change and diffraction efficiency measurements, the multi-layered structures composed from the arrays of single modifications were recorded, called simply multi-layered modifications. Each modification layer partly overlapped (10-80%) with the consecutive layer, written at a different dept to build a thick, continuous structure. We call this the layers overlap.
The samples with recorded single modifications were sliced perpendicularly to the modification formation direction, as shown in Fig. 1(b) and then mechanically polished to optical quality. For a detailed analysis of the modification morphology using a scanning electron microscope (SEM), the samples were immersed in a 5% wt. aqueous solution of HF for 1 min and coated with a thin gold layer.
Fig. 1. a) Focusing geometry in the interface between air and dielectric medium when high NA objective is used. F0 is the geometrical focus in air, and fd is the paraxial focus position (focusing depth). LA is the focus depth due to the longitudinal spherical aberrations; b) Principle of volume grating formation by recording multi-layer modifications. Depth from 0.2 to 5.7 mm; pulse energy from 100 nJ to 1 µJ. Vertical modification length changes with the writing depth due to aberrations; c) Simulated intensity distribution of the Gaussian beam focused to the 700 µm depth with 0.5 NA objective.
The laser beam was reduced by a 0.5${\times} $ telescope up to 2 mm diameter to fit the entrance aperture of the NA = 0.5 focusing objective (Mplan NIR 100${\times} $, Mitutoyo). While the objective’s aperture was completely filled, the spot size at the focus had a diameter $d = 1.22\lambda /NA \cong 1.25\; \mu m$. Significant aberrations were achieved when the laser radiation was focused inside fused silica. The aberration effect is demonstrated in Fig. 1(a). The laser beam was focused to the depth range from 0.2 mm to 5.7 mm with various pulse energies. Three different scanning speeds, 0.5, 1 and 5 mm/s (at 500 kHz), were applied to ensure the required pulse density of 1000, 500 and 100 pulses/µm, correspondingly (Fig. 1(b)). The polarization was controlled by the half-wave plate or quarter-wave plate to get linear or circular polarization, correspondently.
Spherical aberrations appeared when the beam was focused through an air-glass interface. The paraxial focusing model (4) was applied to simulate the intensity distribution at different focusing depths (Fig. 1(c)). More pronounced axial elongation of the laser-induced modification was observed when the laser beam was focused with a high-numerical-aperture objective. The appearance of the longitudinal spherical aberrations at the deeply focused beam is presented in Fig. 1(a). The elongation of the focus depth (LA) due to the spherical aberrations can be expressed with the following formula [31]:
where NA is the numerical aperture of the focusing objective in air, n is the refractive index of the sample, fd is the focusing depth. The Eq. (1) describes only elongation due to the geometrical optics effects and helps to understand how important it is to choose the right focusing objective. The influence of the nonlinear absorption was not considered.2.1 Refractive index measurement and classification
The refractive index was measured using the interference fringes method. The setup for the refractive index changes measurement with a Michelson interferometer is shown in Fig. 2(a). The He-Ne laser beam passing through the sample with recorded multilayer modification was magnified with an objective lens (10${\times} $, NA = 0.25), split into two beams and reflected backwards, resulting in a superposition beam which intensity was subsequently imaged onto a CMOS camera. By adjusting one of the mirrors, an interference pattern with equally spaced, straight parallel lines near the beam axis was obtained (Fig. 2(b)). When the mirror angle was changed, the laser beams overlapped only partly. This means that phase change could not be observed in the part where the beams overlapped entirely. In place of the partial beam overlap, the interference between the beam passing through the modified and non-modified sample parts appeared. In this case, the phase change could be measured.
Fig. 2. a) The experimental setup for the refractive index changes measurement with a Michelson interferometer; b) Interference fringes registered with a CMOS camera.
The refractive index change in respect of nonmodified material can be evaluated as follows:
where h is the height of the multi-layered modification and $\varphi = \mathrm{\Delta }d/d$ ($\mathrm{\Delta }d$ is the fringe shift of the same line and d is the period of the interference fringe) [32].3. Results and discussion
3.1 Modification axial length
The optical microscope and SEM images of single modifications at each processing depth were taken, and the axial modification length was measured. The microscope and SEM images with the modifications recorded at different processing parameters are demonstrated in Fig. 3(a), b. While maintaining the constant focusing depth, the modification morphology differed depending on the scanning speed. When the processing speed was low, corresponding to 1000 pules/µm pulse density, the nanogratings were formed due to the high energy dose. This corresponds to the dark-looking structure with birefringent properties. At higher scanning speeds, corresponding to 100 pules/µm pulse density, the energy dose was sufficiently lower, and only refractive index modifications (Type I) were induced. The axial lengths of the Type I modification recorded with different pulse repetition rates were comparable when the processing conditions were set constant. The maximum achieved axial length of the Type I modification was ∼ 60-70 µm. Variation of the focusing depth and pulse energy combination allowed us to create a 2D map of the axial modification length in a parametric process window, distinguishing the Type I area. The analysis of the given 2D map led to an empirical formula predicting the axial modification length depending on the pulse energy and focusing depth. Consequently, for multi-layered phase gratings, the given dependence keeps the constant overlap between each layer recorded in different depths.
Fig. 3. The microscope images of the modifications, recorded at various focusing depths, different pulse energies and different scanning speeds. a) modifications recorded with 0.2, 0.4 and 2 mm/s speed at 200 kHz repetition; b) modifications recorded with 0.5, 1 and 5 mm/s speed at 500 kHz repetition rate. The insets show the SEM images of only Type I modifications.
The axial modification length varied depending on the focusing depth and pulse energy. To maintain the constant distance between the modified layers or to keep the constant overlap, the empirical relation was found to estimate the axial modification length. The measured axial lengths for various processing parameters were fitted with a two-dimensional polynomial function applying the MATLAB tool “Curve Fitting”. For approximation, the following second-order polynomial function was used:
where x corresponds to the pulse energy and y is the focusing depth.3.2 Change in the axial modification length with the focusing depth
The critical power Pc for beam self-focusing could be described according to the formula Pc=$\mathrm{\pi }{({\textrm{0}\textrm{.61}} )^\textrm{2}}{\mathrm{\lambda }^\textrm{2}}/\textrm{8}{\textrm{n}_\textrm{0}}{\textrm{n}_\textrm{2}}$ [33] and it is $\textrm{0}{.707\; }$MW for fused silica. The calculated peak power used during the modification recording was P0 = 0.09-0.7 MW. Therefore, during all experimental works, P0 < Pc, or P0 ∼ Pc and the beam propagation remained linear with a possible minor contribution of nonlinearity. That means that the modification length elongation for pulse energies up to ∼ 200 nJ was mainly impacted by the spherical aberrations. For deep focusing, the laser power was dissipated. Hence, the linear propagation behavior remained even for the highest used pulse energies in the bulk of fused silica.
Figure 4(b) demonstrates the aberration-caused focus region elongation depending on the focusing depth and NA of the applied objective. This dependence includes only the aberration effect without considering the intensity drop with depth in the material. Therefore, only the focus elongation was demonstrated. For NA < 0.4, this extent can be considered small for focusing up to 4 mm. However, for the NA > 0.4, the elongation is sufficiently high, which can cause a significant laser intensity drop.
Fig. 4. a) Modification axial length dependence on the focusing depth. The inserts show SEM pictures of the modification morphology under certain conditions. The red arrow shows the maximum modification length shift at higher energy. b) The geometrical focus length elongation on the applied numerical aperture for different focusing depths, calculated according to the formula (1).
Detailed analysis of the modification morphology has shown that the axial modification length maximum could be observed precisely when the modification type changes (Fig. 4(a)), i. e. the transition between one modification type to another modification type. This was observed for the energy range where the Type I modification was formed in a wide focusing depth range.
When the beam was focused near the surface, the pulses with high intensity exceeded the threshold for nanogratings (Type II) or micro-explosions (Type III) formation. Slowly going down into the bulk of the material, the axial modification length increased and the peak intensity dropped to a lower level. At some transitional points, the peak intensity could only induce Type I or Type II modifications. This could be explained by the multiphoton excitation that restricted the modification length only to the focal region where the highest intensity was induced. While focusing deeper into the sample, the aberration-induced axial modification length increase was noted. In this case, the laser intensity was sufficient to initiate a multiphoton modification even with increasing depth of focus. At transitional focusing depth, further depth of focus elongation (LA) reduced laser intensity limiting the multiphoton modification axial length. At some level, the axial modification length shrunk until it disappeared completely.
According to the observations, the transition point corresponded to the modification with the largest axial length. With the higher pulse energy, this transition moved to the higher focusing depths, as shown in Fig. 4(a) by the red arrow. For pulse energies up to ∼ 400 nJ, this transition corresponded to the transformation from Type II modification to Type I modification. However, only Type II modifications were observed for pulse energies higher than 400 nJ. In this case, the transition also exists.
The paraxial approximation focusing model was applied to understand the evolution of the intensity distribution for the deeply focused beam [26]. Usually, the paraxial approximation shows excellent results when the numerical aperture of the focusing objective is NA ≤ 0.5. However, the vector model should be used for NA starting from NA ≥ 0.5 [34]. As the value of the refractive index modification depends on the laser intensity, it is crucial to find how much energy is dissipated at different focusing depths. The paraxial focusing model was described in [26] and was expressed in the following way:
Depending on the focusing depth and pulse energy, permanent refractive modifications or nanograting formation occurred in the focal volume. The correlation between the modelling and experimental data was analyzed.
The beam was focused to 0.2-5.7 mm depth, and intensity profiles were analyzed. All profiles were normalized to the intensity at focusing depth near the sample surface to have comparable data for further analysis.
In Fig. 5(a),(b), two intensity distributions along the beam propagation axis are demonstrated when the focusing depth in fused silica was 200 µm and 1500 µm. It is evident from the modelling data, that the intensity at 1500 µm depth dropped ∼ 7 times. For even deeper focusing, more energy was dissipated, and modifications disappeared. The experimental results shown in Fig. 4(a) revealed that as focusing depth increased, the length of axial modification increased to a maximum value, while intensity was sufficient to induce elongated modification and decreased further when intensity dropped. This behavior could be explained by the intensity drop presented in Fig. 5(c). During processing near the sample surface, the intensity was much higher than the modification threshold and, going deeper, maximum intensity dropped below the modification threshold. The intensity distribution along the beam propagation spread when the focusing went deeper. At the defined focusing depth, the intensity was sufficient only at a narrow intensity range, and only slight modification was formed. The pulse energy should be increased for deeper focusing to compensate the losses due to the intensity spreading.
Fig. 5. Intensity distribution, modelled according to the paraxial approximation focusing model for linearly-polarized Gaussian beam. a) focusing to the 200 µm depth; b) focusing to the 1500 µm depth; c) The first maximum of the intensity distribution dependence on the focusing depth.
3.3 Classification of modifications in fused silica
It was complicated to distinguish the modification type from the optical microscope images. The extended SEM analysis was applied to explore the exact transition between all types of modifications. As the modified and unmodified parts of fused silica are etched at different rates [36], the samples with modifications were immersed in HF for 1 min to reveal the transitions between modified and unmodified zones. The refractive index change was observed as monotonous modifications with poor contrast (Fig. 6(b), SEM image marked with red rectangle). From the extracted data, modifications were classified and represented depending on the focusing depth and applied pulse energy. A 2D map with the classified modifications was prepared (Fig. 6).
Fig. 6. Modifications morphologies for various focusing depths and pulse energies. a) The blue-marked area indicates the processing window for the Type I modification. The red-marked area indicates the Type III modifications. The unmarked area between Type I and Type III areas corresponds to the Type II modification area. The blue scale bar corresponds to the type I modifications. The red scale bar corresponds to type II and type III modifications. b) The classification map of the modifications recorded with 1 mm/s scan speed, 500 kHz repetition rate and 290 fs pulse duration.
The boundaries between modification types were recognized. Four zones were distinguished: 1) the unmodified zone for pulse energies from ∼76 to 180 nJ and focusing depth from 1.7 to 5.7 mm; 2) the Type I modification zone, which started at a smaller focusing depth starting from 0.7 mm and could be extended up to 5.7 mm depth varying pulse energy from ∼76 nJ to ∼313 nJ; 3) the Type II modification zone with nanogratings formation. It started from the sample surface and could be extended to the maximum tested depth by choosing the appropriate pulse energy from 76 nJ to 534 nJ; 4) the Type III modifications (micro explosions) were usually formed near the sample surface when the pulse energy was exceeding 180 nJ, and selecting a proper ratio between the pulse energy and focusing depth could be extended to a wide parameter range.
The Type II modifications can be divided into three subtypes: 1) single nano strips with the width of several tens of nanometers; 2) clearly distinguishable single-period nanogratings in transversal direction; 3) nanogratings with double periodicity: one periodicity in transversal direction and one periodicity in the longitudinal direction. Each of those subtypes has its own modification window. The largest processing window corresponds to the nanogratings with dual periodicity. However, nanogratings with a single well-defined period are usually preferable for selective chemical etching. The 2D classification map is presented in Fig. 6(b). In this work, we mainly focused on the Type I modifications - refraction index modification. Therefore, no deeper discussions regarding the Type II modification are provided.
3.4 Exploring the process window for the type I modifications
The refractive index modification (Type I) can be achieved in a narrow pulse energy window. However, the depth range to achieve a phase change of 2π for phase elements was quite broad. The refractive index change dependence on various processing parameters was investigated in the next step. Due to the small refractive index change, several layers of modifications should be inscribed to allow the measurements with the interference method.
The processing windows of the Type I modification for the linear and circular polarizations were compared. Figure 7 presents the axial length dependence on the focusing depth and applied pulse energy. The Type I modification window is marked by the grey frame and was quantitatively compared depending on the processing speed by measuring the modification process window area in the arbitrary square units in the Fig. 7 frame. The Type I modification zone, either for linear or circular polarization, shrunk when the translation speed was increased (Fig. 7(c)). The faster scanning speed caused lower energy dose in the unit area due to the decreased overlap, which increased of the surface area where no modification was observed.
Fig. 7. The axial length of Type I modifications (L, indicated in the color bar) dependence on the focusing depth and pulse energy at 5 mm/s scanning speed. The focusing depth range was kept from 2 mm to 5.7 mm. a) modifications recorded with a linear polarization; b) modifications recorded with a circular polarization. The marked grey boundary line indicated the type I modification area measured in arbitrary square units; c) The Type I modification area data, extracted from dependences at different scanning speeds (in focusing depth and energy space). The pulse repetition rate was kept constant at 500 kHz.
According to the measured data, the processing window for circular polarization was slightly narrower compared to the linear polarization case. However, it can be distinguished that the axial modification length was larger for the circular polarization under the same conditions and remained almost constant for the focusing depth from 3 mm to 5.7 mm. It makes the circular polarization more attractive to record the multilayer modifications and keep the constant refractive index change.
The difference between circular and linear polarizations can be explained by the fact that during the multiphoton ionization and tunnelling ionization processes, the free carrier generation rate depends on the polarization state. Temnov et al. demonstrated that photo-ionization rates for fused silica and linear polarization were 3–4 times higher than for circular polarization at low irradiances (<15 TW/cm2) [37]. The photo-ionization rates became comparable for higher irradiances (30-35 TW/cm2) because the multi-photon ionization regime was shifted to mixed multi-photon and tunnelling ionization mechanisms. For even higher irradiances > 42 TW/cm2, the photo-ionization rate for circular polarization appeared higher, which confirms the higher refractive index change compared to linear polarization. In this experimental work, the calculated irradiances for the Type I modification, considering the spot diameter of 1.5 µm, were from 15 TW/cm2 to 61 TW/cm2.
The multilayer phase elements with maximum efficiency could be recorded only when the overlap between the layers had a constant value, and the refractive index change for each layer was equal. This target could be approached if the axial length of modifications could be predicted during the inscription procedure. The proposed solution was a 2D polynomial approximation of the axial modification length as a function of two process parameters: focusing depth and pulse energy. The data, shown in Fig. 7 (represented by Fig. 8(a)), was fitted with a second-degree polynomial function described by Eq. (3). The generated approximation is demonstrated in Fig. 8(b). The polynomial coefficients for the circular polarization case are shown in Table 1.
Fig. 8. a) Modification axial length dependence on the pulse energy and focusing depth at 5 mm/s scanning speed and b) an empirical 2D surface fit of Fig. 8(a) according to the second-degree polynomial function.
Table 1. The calculated polynomial coefficients of the second-degree polynomial function for the modification axial length dependence on the focusing depth and pulse energy for circular polarization case and 5 mm/s scanning speed.
3.5 Refractive index change measurement
Further, the refractive index changes versus processing parameters were investigated. The matrix of multilayer structures was recorded by varying the pulse energy, focusing depth and overlap between consecutive layers (Fig. 9(a)). The single structure was composed of 8 layers, as shown in Fig. 9(a) on the right side enlarged picture. The single layer was square-shaped on the XY plane, and it was filled with a lines that were spaced by 1 µm distance. The separate line was scanned with a 5 mm/s translation speed. The setup depicted in Fig. 2 was used to measure phase changes on the entire multilayer structure, which represented a uniform modification. The focusing depth was varied from 2.5 mm to 5.5 mm, and the pulse energy was chosen in the range where the Type I modifications were observed: from 250 nJ to 400 nJ. The layer overlap was changed from 10% to 80%, considering the axial modification length described by the two-dimensional polynomial function (Table 1). The refractive index dependence on the focusing depth is presented in Fig. 9(b) and (c). To achieve a better phase contrast between the modified and unmodified materials, the modification with 8 layers was chosen. Effects of various pulse energies and two polarization states (linear and circular) were compared. The measured refractive index change values increased when the pulse energy was increased. For low values of layers overlap, when only small parts of the modification areas were overlapping, the refractive index change for each pulse energy was similar. The reason was that at the lowest overlap of 10% and linear polarization, the refractive index change was observed only for a very limited focusing depth range (Fig. 9(b)), while, for circular polarization, the focusing depth range was much broader, permitting a more flexible selection of the processing parameters. At this regime, the refractive index change was only ∼ 0.5${\times} $10−3. Therefore, it was too low in many cases to create a sufficient phase change in a limited focusing depth range. When the modification overlapping increased, the nominal refractive index change values increased to 1.5${\times} $10−3 for the 50% overlap. A reduction in the refractive index change was observed when the focusing depth was increased. This was related to the spherical aberrations that induced intensity outspread and the axial modification length elongated.
Fig. 9. a) The processing algorithm to record multilayer structures at different depths for refractive index investigation. To better understand, a single structure with 8 layers was enlarged on the right side of the figure; Refractive index change dependence on the focusing depth for various polarization states; b) linear polarization; c) circular polarization. The modifications were recorded with 290 fs pulse duration at 500 kHz repetition rate and 5 mm/s scanning speed.
For circular polarization, the refractive index change was observed in a broader focusing depth range. The refractive index change decrease was observed when the focusing depth was increased only for overlap below 50%. At 50% overlap, the refractive index change was almost constant while changing the focus depth in the range from 3 to 5.5 mm. For overlaps > 50%, the refractive index change started to increase with the increase of the focusing depth. The maximum refractive index change value was achieved for the 80% vertical overlap, which was ∼ 2.46${\times} $10−3. At the maximal overlap, the modification lines were recorded almost at the same depth, which could be considered as a multiple pass inscription. In this case, the biggest refractive change was observed.
The refractive index change for linear polarization and the 80% vertical overlap exhibited an opposite behavior: it was lower than for the 50% overlap. It means that multiple passes reduced the refractive index change. In both cases (linear and circular polarization), the optimal refractive index change was achieved for the 50% vertical overlap. Comparing the modifications recorded with linear and circular polarizations, it could be noted that the maximal phase change and refractive index change were observed for circular polarization. The required element thickness could be smaller using circular polarization for phase element recording. The application of circular polarization allows recording phase modifications at the smaller focusing depth range avoiding the nanograting formation and achieving higher refractive index change values.
3.6 Recording of 1D volume phase grating
The theoretical diffraction efficiency for a binary 1D square wave phase is ∼ 81% and is determined according to a simple formula [38]:
Fig. 10. a) Representation of the 1D square wave phase grating; b) The theoretical simulation of the binary phase grating diffraction efficiency angular distribution. The insert shows the picture of the measured diffraction orders distributions; c) Binary phase grating diffraction efficiency of the first order dependence on the unmodified zone width b. The inserts show microscope pictures of the recorded 1D gratings with different fill factors. The maximum efficiency was achieved when the measured fill factor corresponded to ∼ 50%.
According to previous investigations, the refractive index change tendencies were evaluated and applied to record the 1D phase grating with a binary phase step from 0 to $\pi $. The phase gratings were recorded by combining Type I modifications layer by layer. The grating was constructed from modified a and unmodified b zones. The modified zone was composed of separate modifications spaced by $\mathrm{\Delta }x$=1 µm distance in the horizontal direction. The modifications were overlapping, and uniform refractive index change over the area was obtained. 1D phase grating operates as a beam splitter that splits the beam into two separate beams, and the maximum efficiency is achieved when the phase difference is equal to $\pi $. Based on this assumption, the 1D gratings were optimized to set optimal separation between modified and non-modified zones as well optimal thickness.
The spacing between modified zones b was optimized because the dimension of the modified zone size a, set in the software, was slightly different from that measured with an optical microscope (Fig. 10(c), insert). For this purpose, the size b was increased in each trial by 0.5 µm by adding an additional scanning line. The modified zone width a was composed of 8 recorded tracks to get ∼ 10 µm size (a = 8$\Delta x$), so the unmodified zone width was varied in the range ${b_{min}} < 8\Delta x < {b_{max}}$. In all cases the grating period Λ was constant.
The diffraction efficiency optimization was separated into two steps: grating thickness optimization and fill factor optimization. The initial grating thickness was taken from the refractive index change investigations. The grating thickness was controlled by changing the number of the modification layers and spacing between the layers. The maximum achieved diffraction efficiency at the optimal fill factor was ∼ 96%, which is an excellent result.
It is worth mentioning that the main aim at this research stage was to get the beam splitting into two beams. In the case of splitting the beam into three beams in a line, the grating is the same geometrically; only the thickness of the grating should be different to fit the phase change of $0.64\pi $. In this work, we were limited only to 1D diffraction gratings optimization. The additional optimization to record the complex phase gratings will be done further.
4. Summary and conclusions
The extended experimental work investigating the aberration-affected Type I and Type II modifications was performed. The detailed modification classification was demonstrated depending on the pulse energy and focusing depth. It distinguished the Type I modification area and how it changed depending on the processing speed and applied polarization state. The Type I modification area increased at higher pulse density. Using a fixed 5 mm/s scan speed (100 pulse/µm density), the Type I modification area for linear polarization was ∼ 8% higher than that for circular polarization. The highest axial length for the Type I modification was in the range of 60-70 µm, which allowed to reach > 1:30 width to the length aspect ratio, and it was comparable to the results achieved with Bessel-Gauss beam processing. Because the axial length is a function of the pulse energy and focusing depth, the 2D polynomial approximation allowed us to precisely predict the axial modification length that was critical to get the required overlapping of modification layers.
The refractive index change was measured by recording the modification containing the multi-layered overlapped structure. The optimal conditions for the maximal refractive index change were obtained for the 50% axial overlap, and the maximal refractive index change was ∼ 1.5${\times} $10−3 for both polarization states. For linear polarization, the significant refractive index change dependence on the pulse energy and focusing depth was observed, while, for circular polarization, the refractive index change was ∼ 8 times less sensitive to the focusing depth. This could be related to different photo-ionization rates in fused silica for different ionization states. Based on the paraxial approximation focusing model, the intensity distributions for different focusing depths were calculated, and the dependence of the axial modification length on the focusing dept was explained.
As the refractive index change induces the phase delay, this technology allows to design and record of the volume phase elements for various applications. This work demonstrated the basic concept of the 1D thick phase grating operation, which splits the beam into two beams with equal energy. The > 96% diffraction efficiency was demonstrated for such a diffractive element.
The presented results open new possibilities to build complex phase elements inscribed inside fused silica for diverse applications by using the positive effect of the spherical aberrations and all control possibilities coming together.
Funding
Lietuvos Mokslo Taryba (09.3.3-LMT-K-712-23-0145).
Acknowledgements
The research leading to these results has received funding from European Social Fund (project No 09.3.3-LMT-K-712-23-0145) under a grant agreement with the Research Council of Lithuania (LMTLT).
Disclosures
The authors declare no conflicts of interest
Data availability
Data underlying the results presented in this paper are not publicly available but may be obtained from the authors upon reasonable request.
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