Laser-Inscribed Stress-Induced Birefringence of Sapphire

Birefringence of 3×10−3 is demonstrated inside cross-sectional regions of 100 μm, inscribed by axially stretched Bessel-beam-like fs-laser pulses along the c-axis inside sapphire. A high birefringence and retardance of λ/4 at mid-visible spectral range (green) can be achieved using stretched beams with axial extension of 30–40 μm. Chosen conditions of laser-writing ensure that there are no formations of self-organized nano-gratings. This method can be adopted for creation of polarization optical elements and fabrication of spatially varying birefringent patterns for optical vortex generation.


Introduction
Three-dimensional (3D) structuring of materials with a high refractive index at sub-wavelength resolution has promise to advance the field of photonic crystals (PhC) and the integration of PhC into photonic chips [1][2][3][4][5][6][7][8][9][10][11]. Femtosecond laser micro/nano-fabrication as the no contact method can directly pattern sub-wavelength ripples [12,13]). A 3D nonlinear PhC has been successfully fabricated inside lithium niobate using femtosecond laser [14]. However, this is still a challenging task [15] to deliver a close to diffraction-limited focusing at arbitrary depths required for the 3D patterning when using Gaussian-like laser pulses [16]. Compensation of spherical aberrations can be successfully achieved for laser-writing in high refractive index materials and large depths [17]. In this study, we enhance (instead of compensating [17]) the spherical aberration by tailoring axial light intensity to be stretched along the propagation direction and to form a Bessel-beam-like axial intensity profile. Similar techniques are used for 3D patterning inside high refractive index materials [5,6,8,18]. We use stretched pulses to control 3D structuring and dielectric permittivity change over tens-of-micrometers along the entire typical length of fs-laser pulses of 100-300 fs.
Permittivity ε = n 2 changes between the laser-inscribed region ε 1 (refractive index n 1 ) with width t 1 separated with a host material of permittivity ε 2 and width t 2 , creates an artificial uniaxial form-birefringent structure with ε e − ε o > 0, where e, o denotes extraordinary and ordinary beams polarized and ⊥ to the optical axis OA, respectively. Controlling the width of t 1 , its period t 1 + t 2 and depth d (along the light propagation) are essential parameters to an engineer's optical elements for polarization control. Stress-induced birefringence is a well-used phenomenon to create artificial birefringent materials and used as a method for material inspection and characterization [19,20]. Generally, the inverse permittivity tensor is ∆(1/ε ij ) = P ijkl ∂ k u l , where P ijkl is the fourth-rank photoelasticity tensor, ∂ k u l is the gradient of the displacement from equilibrium with u l being the linear displacement from equilibrium and ∂ k denotes differentiation with respect to the Cartesian coordinates. Birefringence induced by stress and expanding beyond laser-structured regions is promising for polarization micro-optics, especially when the laser modified regions are not used for optical functions due to considerable light scattering losses.
In any type of material, the modification of adjacent regions with resolution smaller than the wavelength can be used to change the effective refractive index which for extraordinary n e (along the optical axis) and ordinary n o indices depend on the volume fraction f = t/Λ [21]: where n 1 and n 2 are the indices of the host and laser-inscribed regions, respectively. If modification of a silica host n 1 = 1.40 becomes n 1 = 1.45, the form birefringence would only reach maximum of ∆n = n e − n o = −8.8 × 10 −4 when f = 0.5, i.e., t = Λ/2. Hence, for a considerable phase delay required for λ/4 or λ/2 polarization optics, a tens-of-µm-long axial modification d would be required for engineering the retardance ∆n × d. Moreover, a well-controlled laser inscription method developed here is required to reach the optimal conditions of f = 0.5 for the shortest modification d at the chosen wavelength of operation (the strongest birefringence ∆n). Such flexibility is currently not available for fabrication of micro-optical elements.
Here, we show that single nanolines array with controllable separation inscribed in crystalline sapphire (along c-axis) can reach retardance of λ/4 (π/2 in phase) for visible wavelengths using a simple approach to generate femtosecond pulses with an axially extended Bessel-like intensity distribution using a spatial light modulator (SLM).

Experimental
C-cut sapphire samples of 0.5 mm thickness were used for laser inscription. Sapphire has one of the highest Young moduluses of Y = 400 GPa and can withstand high pressures strongly localized inside the crystal, as our earlier studies as shown in our earlier studies [22]. The stress-induced birefringence inside the laser-structured region patterned with single pulse irradiation reached ∆n ≈ 1 × 10 −3 and the pressure was estimated to reach 1.3 GPa. At ∼2 GPa, micro-cracks were developed when Gaussian fs-laser pulses were used in c-cut sapphire [22].
In this study, a fs-laser beam at second harmonic λ = 515 nm wavelength and pulse duration of t p = 280 fs (Pharos, Light Conversion) was reflected from a phase mask designed on a spatial light modulator (SLM) and directed onto a tight focusing objective lens for laser structuring. Typical pulse energy used for stress-induced birefringent gratings was E p = 574 nJ (at focus) to inscribe single modification lines inside sapphire at laser repetition rate of 10 kHz at a beam scanning speed of 0.1 mm/s (if not specified otherwise). The focal diameter was ∼465 nm which can be calculated approximately using the formula d = 1.22λ/N A. The phase mask pattern on the SLM was selected to create close-to-linear intensity distribution along the propagation direction and was characterized by a stretch factor f st . For the f st = 0 the spherical aberration was compensated at the position of the Gaussian beam, while the largest value of f st = 12 was at the maximum stretch to obtain a linear intensity distribution over the entire pulse length ct p /n, here t p is the pulse duration. The inscribed structure was approximately 10-15 µm below the sample surface. A laser-structured sapphire sample was cleaved to expose the structured regions and was etched in 20% HF for 60 min before SEM observation.

Results and Discussion
By rational choice of repetition rate and scanning speed, it was possible to create a single line without movement of the sample (or beam). By scanning the beams linearly modified regions, lengths of tens-of-micrometers were recorded. For the most axially stretched intensity distribution, single lines were recorded without usual formation of nanogratings [4,23]. First, we present structural characterisation of inscribed modified lines at a wider range of parameters and subsequently present results of optical characterisation of patterns which can deliver a λ/4 waveplate performance.

Direct Write of Nano-Planes
Inscription of long axial modifications in a crystalline sapphire were made by focusing onto a c-plane sample. Typical results of laser inscription are summarized in Figure 1. For same stretch factors f st = 9, different pulse frequency was used to study the evolutionary process of the femtosecond laser irradiated area (Figure 1b,c). The property of the irradiated (modified) area has been demonstrated in our previous work, which present the phase transition of the modified area from crystalline to amorphous in sapphire after irradiated by femtosecond laser [12]. 20% HF solution was employed due to the very high contrast of etching between crystalline sapphire and laser amorphized regions (∼10 4 etching selectivity) such as those seen previously when using silica and boro-silicate glasses [24][25][26][27][28]. Etching up to 0.8 mm into the depth of sapphire sample (along z-axis; inset in Figure 1a) was observed when continuous inscriptions of strongly modified regions were formed at large frequency. For smaller pulse number and the same f st , the individual laser damaged nano-regions, which were not initially interconnected into a single line, were observed connected after wet etching. With an increasing number of pulses (larger repetition rate or slow scanning), those single damage regions formed a line, which was further revealed after etching.
The longest inscription of modifications of up to 60 µm along the propagation direction (on y-axis) were inscribed with ∼850 nJ/pulse energy at f st = 9, f = 10 kHz repetition rate and scanning speed of v s = 0.5 mm/s along z-axis (Figure 1a). At these conditions approximately n = d/v s / f ≈ 46 pulses were accumulated over the diameter of the focal spot (∼465 nm). Due to the large pulse energy, gratings with period of ∼300 nm were formed in the modified region as revealed after wet etching (inset in Figure 1c). The incident and scattered/reflected light from the strongly excited region interfere and form nanograting patterns as was described in ref. [29]. Near-sub-wavelength structures with a period of λ/n = 515 nm/1.45 = 355 nm were revealed in the strongly absorbing regions where the pulse-induced permittivity became metal-like.
As shown in Figure 2, an array of uniform non-period nanolines with 1 µm interspace was inscribed by stretched fs-laser pulse (energy 574 nJ, scanning speed 0.1 mm/s). The width and length of the single nanoline after etching in HF solution were 50 µm and ∼170 nm, respectively. The aspect ratio of the lines can reach 250, which pave the way to inscribe a strong and tunable birefringence and retardance in sapphire. The use of an even distribution of pulse energy along the propagation axis can be applied in generation of high pressure and temperature phases of materials due to better energy delivery via resonant absorption [30]. The proposed phase control using SLM can, in principle, be adopted for experiments exploring a temporal evolution of fs-laser pulse-induced micro-explosions using femtosecond X-ray pulses of a free electron laser (FEL) for probing, while coaxially propagating Bessel-like beams [31] can be used for optically triggered micro-explosions. The current study confirms formation of an amorphous phase of sapphire which is typical in conditions of high pressure [32].

Engineering of Birefringence
Optical characterization of laser-inscribed gratings are shown in Figure 3. Gratings with Λ = 10 µm period were inscribed with duty cycle of 0.5, i.e., 10 µm were inscribed with a separation of ∆x ranging from 170 nm to 500 nm between two single nanolines. The footprint of the gratings were 100 × 100 µm 2 , an acceptable size for many applications in the field of micro-optical elements. '  Single lines without the formation of self-organized nano-gratings were inscribed by stretching the incoming fs-laser pulses. The stretch factors of 10 and 11 corresponded to a single line (a plane under scanning) inscription for 30 µm and 40 µm, respectively. Inspection of the laser-inscribed regions with scanning electron microscopy (SEM) revealed the width of the structurally modified lines corresponding to ∼170 nm ( Figure 2). When those modifications were written with ∆x = 200 nm separation, cracks formed during the laser-writing ( Figure 3); however, for larger separations the gratings were stable. The writing depth was approximately 10 µm below the surface and extending into the sample. Strong stress-induced birefringence was observed inside the gratings in the regions without laser damage as well as between the gratings as revealed by cross-polarized imaging.
To determine the sign of refractive index change ∆n = n e − n o , a λ-waveplate at 530 nm was inserted at 45 • in respect to the orientation of polarizer and analyzer. In this setting, Michael-Levy color charts can be used to determine color changes corresponding to +|∆n| and −|∆n| (Figure 3c). For the λ-plate of 530 nm wavelength oriented vertically, the blue color indicates the stress-induced regions. Since the blue color on the Michael-Levy chart corresponds to the higher absolute birefringence and the orange to the lower, the change ∆n has to be negative n e > n o where n o is refractive index of the ordinary beam (perpendicular to the optical axis OA). The slow axis of the form-birefringent pattern (grating) is along the vertical direction (Figure 3c) and the refractive index is ∆n = n − n ⊥ = n e − n o > 0. Hence, the form-birefringent structure acts as a negative uniaxial crystal.
To determine birefringence ∆n(λ) at several wavelengths, a recently developed method was used for measuring [33]. A single wavelength measurement of birefringence (as for example in the popular Abrio tool) leaves an ambiguity of the true ∆n value due to a possible 2π folding of the phase and is avoided by carrying out measurements at several wavelengths in our method [33]. The measurement of ∆n was made at five different wavelengths while using narrow 10-nm-bandpass filters spanning the visible spectral range of 475-650 nm. A linear fit through the origin point in the retardance (∆n × d [nm]) vs. wavenumber (1/λ) plot was obtained with a single slope which defines the ∆n averaged over the tested spectral range [33]. Figure 4 shows the experimentally determined retardance for the grating patterns. Regions of interest (ROIs) were set to average retardance on the grating and in its vicinity. The error bars are the max and min retardance measured in the ROIs. The highest ∆n × d was observed in between gratings and was scaling with separation ∆x between nano-planes. For the largest retardance of 112 nm and stretch parameter of 11 (d = 40 µm), ∆n = 2.8 × 10 −3 . Only slightly smaller birefringence was determined for the stretch factor of 10 and 30-µm-long inscribed gratings (Figure 4). . Retardance |∆n|d measured at several wavelengths from 475 nm to 650 nm with 10-nm-bandpass filters. Sample was fs-laser-inscribed at pulse energy E p = 574 nJ; sample is shown in Figure 3. Rectangular regions of interest (ROIs) show locations from where an average retardance was measured. Two lines of gratings with different stretch factors of 11 (the length of inscribed line d = 40 µm) and 10 (d = 30 µm) were analyzed using liquid crystal compensator [33]. Note logarithmic ordinate was used to reveal single exponential decay of retardance with ∆x.
Even the larger ∆n values were observed inside gratings in the 10-µm-wide openings reaching 22% retardance at the longest wavelength of 650 nm, selected for the measurement of birefringence (Figure 5a). For shorter wavelengths in visible range, a λ/4 waveplate condition was achieved by direct write of nano-inscribed modifications without changing the axial position of the modified region during inscription. Single-pixel cross-sections (Figure 5b) of the retardance maps show difference in the gradient of ∆n between laser-inscribed regions. By placing a few regions of nano-planes at different depths from the surface, it should be possible to fabricate λ/2 waveplate retarders. Stress-induced regions between larger extended laser-structured patterns should allow reduction of light scattering observed from laser-inscribed areas which are used for fabrication of optical elements [4,34].
Optical resolution of large area birefringence mapping ( Figure 5) have provided only a couple of points measured per 10 µm regions inside the grating. It illustrates a 0.1λ modulation of retardance. Measurements at approximately twice a higher resolution ( Figure 6) confirmed the modulation amplitude of retardance shown in Figure 5. Retardance at 625 nm wavelength ( Figure 6) is shown in relative units of waves since the reference retardance required to make calibration was not possible to measure from the stress-free region on the same image (as area "Ref." in Figure 3). The cross sections of the measured retardance map clearly shows that stress-induced phase delay between inscribed 20-µm period and 0.5 duty cycle grating were clearly resolved between the laser-inscribed regions with axial extension tens-of-µm.

Conclusions and Outlook
We demonstrate λ/4 phase retardance at visible wavelengths in sapphire recorded by direct write of nano-inscribed modifications tens-of-micrometers long. This modality of laser structuring opens a flexibility in stress-induced optical element fabrication and eliminates light scattering since the regions of tailored birefringence are outside of laser-structured regions. Patterns of tailored birefringence can be produced at different depths along the light propagation direction or even in different micro-plates for the final optical element. One particular field of application can be spin-orbital couplers where spatially variant birefringence can be inscribed with complex 3D topology similar to the polymerized 3D couplers [35].