Temperature diffusivity of laser micro-structured regions in sapphire is determined by a temperature wave method with a lateral resolution reduced to ~ 10 μm using a directly sputtered micro-sensor and heater. A record high reduction of the temperature diffusivity of sapphire by 12% from its (1.26±0.02)×10-5 m2/s in-bulk value inside the femtosecond laser-structured volumes is determined; in a BK7 glass (~ 4.8×10-7 m2/s), a 2% decrease of the thermal diffusivity has been observed. Origin of the reduction is consistent with disorder and scattering of phonons around the laser photo-modified micro-volumes. The stress-induced birefringence is directly measured by polariscopy together with its radial distribution, and azimuthal orientation of the polarization ellipsis near the laser structured regions in sapphire. The maximum birefringence of Δn≃1×10-3 is achieved without crack formation and corresponds to a local stress of ~ 1.3 GPa. The stress (and birefringence) decay radially with a single-exponential constant of τR = 24 μm while the azimuthal orientation of the polarization ellipsis is spiraling around the laser structured volume. Such structures are promising in waveguiding and lasing applications of optical vortices where spatial control of birefringence and optical activity are required.
© 2010 Optical Society of America
Thermal, mechanical, and optical properties of materials are strongly interrelated and by controlling structural changes on a sub-wavelength scale inside optical materials their mechanical, thermo-electrical, and optical behavior [1–4], relevant to an increasing number of applications in the fields of micro-optics, lab-on-a-chip, optofluidics, waveguiding, and lasing should be better understood. Waveguides directly recorded by laser beam inside plastics , optically active and passive glasses [6–9] and crystals [10, 11] are used for sensing, beam splitting/multiplexing, and lasing applications. Birefringent regions formed alongside the laser-recorded micro-structures can also support waveguiding [12, 13]. Recently, a high 1 W output power with good mode quality and 60% conversion efficiency from laser structures in YAG has been reported , where the emission is guided in the birefringent region between two inscribed planes inside Nd:YAG crystal.
For future micro-lasing applications thermal properties of laser crystals and glasses should be well controlled since they are critically important for the long term stability of active micro-optical devices. The thermal properties are dependent on the dopant and defect concentration  which affect formation of thermal lensing and might be detrimental for laser operation, especially, at high output power. In the case of the laser inscribed waveguides, the local redistribution of the dopants, laser-induced defects, and even changes of local mass density are all taking place and influence optical functions . A direct measurement of the thermal properties is the required approach [15,17], however, there is lack of techniques allowing direct quantitative measurement of thermal conductivity and diffusivity with a high <1% precision on a micrometer scale. Atomic force microscopy can be used to detect the relative change of thermal properties on a micrometer scale , however it is difficult to make such measurement quantitative. The most challenging issue is miniaturization of sensors and thermal contact with sample.
Here, we report on direct measurement of thermal diffusivity by a miniaturized temperature wave (TW) method applied for a high-fidelity characterization of the micro-volumes inside laser-structured sapphire. A record high, up to 12% reduction of thermal diffusivity has been observed in laser structured volumes. The regions just adjacent (~ 3 μm separation) to the ones modified, showed no change of the bulk diffusivity. The stress-induced birefringence near the femtosecond (fs)-laser-structured micro-volumes of tens-of-micrometers in cross section is measured by polariscopy and can amount to Δn ≃ 1×10-3 suitable for waveguide formation. The stress and birefringence show exponential relaxation over length of 24 μm (with the maximum stress estimated to reach 1.3 GPa). The Stokes analysis of birefringence revealed a typical 2π spiraling phase around cylindrical laser-structured volume and could be used for generation of optical vortex.
2. Experimental details
Femtosecond laser pulses 150 fs/800 nm were focused inside samples of sapphire and glass using an objective lens of numerical aperture NA = 1.4 . The irradiation spots were closely packed with typical separations in lateral (x and y) and axial (z) directions: 2×2×1.5 μm. Due to high refractive index of sapphire, n = 1.7 at 800 nm wavelength used for laser structuring, the actual axial period between planes was comparable with the lateral period between void-structures. Also, the regions recorded at larger depth (~ 100 μm) had axially elongated shape due to spherical aberrations . The modified regions were cylindrical with 35 μm in diameter and 90 μm in height to maximize the modified volume within a 126 μm-thick sapphire slab (Shinkoshya Inc.) and BK7 107-μm-thick glass for comparison. Pulse energy used to form void-structures (void with an amorphous shell) in sapphire was 80–120 nJ/pulse at the focus. At these conditions the void of approximately 300 nm in diameter with the amorphous 300-nmshell were formed ; in BK7 glass, pulse energies were 60–90 nJ.
The TW method used for high sensitivity measurements of thermal properties of laser host materials [15,17] was adopted for the thermal diffusivity measurements. Typical surface roughness of the thermo-sensors is <100 nm (min-max) after polishing with a 0.1-μm diamond powder. However, such sensor arrays were not suitable for the high precision direct measurements using glass and sapphire with optical grade surfaces due to imperfect thermal contacts at the sensor-sample interface. We sputtered the micro-sensors and heaters for reliable implementation of the TW method which rely on the measurement of the phase delay, Δθ, using lock-in technique [15, 17]:
where χ is the thermal diffusivity, f is the repetition rate of the heat wave generation, d is the thickness of sample, and θ0 is the phase accounting for actual experimental conditions and background noise (it has no significance for the experimental determination of χ judged from the slope according to eqn. 1).
The sputtered contacts are chemically stable, have a low electric resistivity and good thermal conductivity: 315 W/mK for Au and 237 W/mK for Al. The thickness of the sputtered Au-Al sensor layer was controlled by monitoring the electric resistance. Typically, sensor of 350-440 Ω was used which does not add up noise to measurements as compared with a thermal noise, , where k is the Boltzmann constant, R is the electrical resistance, and Δf is the bandwidth in herz over which the noise is measured. The sensor was wired via a cold junction with a copper lead. The sensor hot-junctions were simultaneously sputtered on the fabricated, non-fabricated, and adjacent to the laser-structures regions. Precision of the placement of the hot-junctions was approximately 1 μm. Series of up to 10 independent measurements of thermal diffusivity were carried out by chemically removing and resputtering the hot-junction contacts and heater.
A heat source for generation of the temperature wave was a 300-nm-thick layer of Au with an area size of 0.4×1.5 mm2 and is larger than the sensor Au-Al junction. It has an electric resistance of 30 Ω. The larger area assures the one-dimensional heat flow. The heater was wired with a copper lead and the modulated thermal power was chosen to satisfy a good signal-to-noise ratio required for detection of thermal diffusivity changes on the order of < 1%. The temperature jumps were on the order of a few degrees as confirmed by IR thermography and no any changes of structural defects in the laser structured volume occurred.
Polarimetric Stokes analysis was used to determine the birefringence and its spatial distribution around cylindrical micro-structured volumes in sapphire using Abrio IM (Cambridge Research & Instrumentation, Inc.) microscope at 546 nm wavelength . Setup is based on an Olympus BX41 microscope equipped with a UPlanFI 40× objective lens of numerical aperture NA = 1.3.
3. Results and discussion
The direct laser writing by tightly focused femtosecond laser pulses [23, 24] was used to fabricate regions of amorphised sapphire inside crystalline sample at the exactly same conditions as in the previous amorphisation studies [21, 25]. All the samples used in this work were irradiated by tightly focused fs-laser pulses at energy-per-pulse larger than the void formation when the irradiated region has a void surrounded by amorphised shell [19, 26]. Such structural modifications were the focus of this study. Irradiation spots were closely packed to maximize the modified volume.
For this study a micro-scale thermoelectric sensor of Au-Al hot-junctions with a 10×10 μm2 footprint was directly sputtered over the laser-structured regions with a heater sputtered on the opposite side as shown in Fig. 1. This made a perfect surface thermal contact otherwise not possible to achieve with a previous sensor arrangement  on high-quality optical surfaces. Next, we analyze the interplay of the thermal, mechanical (stress), and optical (birefringence and optical activity) properties of the laser structured sapphire.
3.1. Thermal diffusivity in micro-volumes
Figure 2 shows normalized thermal diffusivity collected from the regions through the laser-structured, adjacent to it, and at unaffected (host) regions of sapphire sample. Each measurement was carried out after stripping out and re-deposition of contacts and heaters (Fig. 2(a)). Since every single set of sputtered contacts and heaters has slightly different thickness we present data normalized, i.e., the thermal diffusivity of the region S is divided by the value measured at host (see, Fig. 1(b)). The absolute value of thermal diffusivity of host sapphire crystal was determined as (1.28±0.016)×10-5 m2/s at 39°C. A high up to 12% decrease of the thermal diffusivity was observed through the laser structured region repeatedly in separate measurements with overall uncertainty ~1%. Good correspondence of relative changes in thermal diffusivity were observed in repeated measurements. The heat diffusion took place along c-axis (perpendicular to the image plane in Fig. 1(a,b)). Polariscope image show characteristic Maltese cross lobes due to stress-induced birefringence extending laterally up to ~20 μm from the laser-structured regions without crack initiation (Fig. 2(b)).
The region just adjacent to the laser-structured volume has no measurable decrease of thermal diffusivity (3 in Fig. 2). This is the region where stress-induced birefringence is the largest and waveguiding can be realized. Having thermal properties not changed in that particular location is a promising feature for waveguiding and especially lasing applications. The relative changes of thermal diffusivity were reliably measured over 10 cycles (Fig. 2(a)) confirming a high fidelity of the used TW method.
The cause of the decrease in thermal diffusivity is the laser-induced amorphisation and structural modifications around the irradiation spots . The pulse energy was above a threshold of void formation and strong structural modifications occurred around the irradiated volume [19,26]. The thermal diffusivity is defined by the thermal conductivity, kc, the mass density, ρm and the specific heat cp at constant pressure as χ = kc/(ρmcp) and was directly measured in our experiments. The thermal conductivity, kc, is given by :
where v is the mean velocity of the thermal carriers phonons and electrons, ρ is the density of the carriers, and Λ = vτ is the mean free path of the carriers defined via the thermal velocity v and average time interval between scattering events, τ. When dimensions of objects become smaller than Λ, which is typically several tens of nanometers, scattering increases, and τ is reduced, causing smaller Λ and kc. Because of these effects, objects with nanoscale-volumes can efficiently localize heat as was demonstrated by polymerization . Such a large 12% modification of thermal property on the scale of several micrometers can be used for the temperature localization at required location which can be freely three-dimensionally recorded inside bulk of transparent host crystal.
The same thermal analysis has been carried out for BK7 glass and the thermal diffusivity value of (4.81±0.027)×10-7 m2/s (at 45°C) has been determined for the host matrix as expected . A reduction of thermal diffusivity up to 2% in the fs-laser structured regions is observed which is considerably less than in crystalline sapphire. Since BK7 is amorphous the decrease of thermal conductivity might be caused by structural changes in ring-like structures of SiO4 tetrahedrons which occur due to densification of glasses [8, 18, 31].
3.2. Birefringence and azimuthal phase
The stress-induced birefringence was measured by polariscopy using a phase compensation technique for the changes caused by stress-induced birefringence . The intensity of the transmitted light through an optically-crossed polarizer and analyzer with a birefringent material in between is given by :
where φ is the angle of the polarizer axis with the principal stress direction in the sample, Δn is the induced birefringence, d is the axial extension of the region, and λ0 is the wavelength of light in vacuum, and Γ/2 =πΔnd/λ0 is the retardance. Equation 3 represents the well known Maltese cross intensity distribution (illustrated in the inset of Fig. 1(b)), where the principle directions of stress coincide with the dark lines and points (φ = 0), called isoclinic, in the polarization micrograph. The brightest regions are at φ = π/4 angles. By performing Stokes analysis the birefringence, Δn, and the azimuth, Ψ, or the angular distribution of an optical slow axis can be retrieved for the each pixel of a microscope image, e.g., the Stokes parameter s3 is related to the retardance via s3 = −cosΓ .
Results of the imaging polariscopy analysis are summarized in Fig. 3 for the amplitude (a) and azimuth (b). The amplitude is the retardance presented in nanometers [nm] and the azimuth shows slow axis orientation .
First, we discuss modifications observed outside the cylindrical fs-laser structured regions (see, Fig. 2(b)). The birefringence with maximum of Δn ≃ 1×10-3 is determined before onset of micro-crack formation for the laser-structured volume used in our experiments. The typical distance over which the birefringence (and stress) relaxed single-exponentially was τR =24 μm (c). The slow axis orientation, the azimuth , has radially and azimuthally varying distribution shown by dial-markers in more details in Fig. 4. The azimuthal phase retardation over one full turn around the cylindrical laser structured region of 90 μm length and 30 μm diameter was ~ 2π as revealed by two gradient lobes with a change over π each. Hence, a light waveguided around such laser structured volume in the region of high birefringence would acquire an orbital angular momentum. By recording cylindrical structure inside sapphire the local orientation of optical axis and birefringence can be controlled.
The polariscopic analysis inside the laser structured volume is only qualitative for birefringence since a strong light scattering, as observed in earlier Raman studies , causes depolarization of light (boxed regions in Fig. 3(c)). Stress-induced birefringence can be measured with few micron resolution using polariscopy . The two dark regions inside the laser structured volume marked by arrows in Fig. 3 represent topological defects where the orientation of polarization ellipsis is undefined or has a singularity  (also, marked in Fig. 4). The location of the phase defects is related to the stress distribution, which is defined by the crystalline orientation of the sample, its thickness, geometry of the recorded pattern, and most probably to the shape of voids and amorphous regions (see Media 1 online file for details). Such singularities are common in different fields of physics and optics [36–40], e.g., pipe-dislocations in GaN crystals  and disclinations in liquid crystal films . The disclinations of the director in liquid crystals equivalent to the orientation shown in Fig. 4(b) are defined by the strength parameter s=+1/2. The azimuth of optical axis around each of two singularity points (Fig. 4(b)) is increasing anti-clockwise and in a full turn acquires a cumulative π phase change.
The stress-induced birefringence can be used to generate optical vortex  as demonstrated for the macroscopic stressed glass waveplate. The optical vortex is characterized by the azimuthal phase which is changing as exp(ilψ), where ψ is the local azimuthal angle and l is an integer number, the topological charge. The laser-structured volumes create birefringence and a smooth continuous change of azimuthal orientation of the optical axis, hence, such structure would generate optical vortex from a micro-volume in contrast to a large optical element . Alternatively, the micro-optical elements with space-varying birefringence can be recorded in an optically isotropic micro-volume of sapphire (or other material) and the micro-workpiece can then be released by wet-etching  for laser trapping applications. Once a micro-particle has birefringence and refractive index is larger than that of surrounding, it can be laser-trapped and spined [44–46], thus, performing pump, sorting, or other mechanical functions in optofluidic applications.
Since the charge, l, of an optical vortex is related to a particular 3D geometry of a spatially-variant birefringence and readout beam (a focusing angle) [47, 48], one would expect that by choosing other shapes of laser-structured micro-volumes one could obtain singularities of different topological charges. At the singularity location a minimum of light intensity is obtained and it is well localized. This has number of potential applications in physics and optics [36–40, 49].
We have chosen to record structure by laser structuring along c-axis, the direction for which sapphire has no birefringence, and to explore stress-induced birefringence and its spatial and azimuthal distribution. Crystal parameters such as photoelastic constants (discussed in next section) related to the Stokes parameters and the bulk modulus govern a birefringence change, its azimuthal distribution, handedness, and optical activity. It would be necessary to explore next, how by changing particular micro-pattern recorded inside bulk of crystal or glass, one could tailor the required changes of birefringence, handedness, optical activity using direct femtosecond laser writing. Such patterns could be used to form vortex beams which carry an orbital angular momentum due to stress-induced spatially variant birefringence around or inside regions specially tailored by laser fabrication. It might have applications in optofluidics, design of micro-optical elements, and laser trapping [50, 51].
3.3. Determination of stress
Let us determine the stress responsible for birefringence of Δn≃10-3. The refractive index of a crystal is related to stress, σ, through the stress-optic (piezo-optic) tensor qij or to strain through the strain-optic (elasto-optic or photoelastic) tensor pij; in the Voigt nomenclature elements of tensors are interrelated pij = qijcij with cij being the stiffness tensor . Strain (ε) is related to stress (σ) through the compliance tensor, sij, and vice versa. The Hook’s law defines a stressto-strain relation via stiffness σi = cijεj .
The stress-induced birefringence Δn = n∥ - n⊥ ≃ qσ where n0 is the average refractive index, n∥ and n⊥ are the refractive indices for light with its electric vector parallel and perpendicular to the direction of stress, respectively. The pij [54, 55] and cij  values of sapphire were used for calculation of the stress-induced birefringence. In the employed imaging geometry along c-axis (Fig. 1(b)) the stress component, σ3, can be estimated assuming transversely isotropic conditions with symmetry axis along e3 as σ3 = c13ε1+c13ε2+c33ε3 ≃(2c13+c33)ε. We used sapphire constants c13 = 117.2 GPa, c33 = 501.8 GPa , p13 = 0.005, p33 = 0.23  for evaluation of effective coefficient q33. It is noteworthy that when the recorded pattern inside sapphire exert a complex 3D stress pattern this proposed estimation can only be used for a qualitative measure since the model is valid for the patterns which are 2D and when their cross sections are considerably larger than the wavelength of light. One would find that a σ3 = 2Δn/(n30q33) ≃ 125 MPa stress is required for the Δn = 1×10-4 birefringence when an average refractive is n0 = 1.7. This is one of the highest photoelastic sensitivities among crystals which is caused by the high p33/p13 ratio as first reported in ref. : an experimentally validated value of Δn = 1×10-4 birefringence was obtained for the 1 kbar (0.1 GPa) hydrostatic pressure.
The experimentally measured birefringence of 10-3 corresponds to the stress of ~ 1.3 GPa for the employed geometry of laser structuring along c-axis. Since sapphire has the Young modulus of 400 GPa, pressures up to 1 GPa can be created by fs-laser structuring without crack formation  and can be used for waveguiding.
It has been established that the thermal diffusivity of fs-laser structured micro-volumes in sapphire is reduced by 12% as compared with the host crystalline matrix. Such large change of thermal property can be localized with several micrometers precision inside the bulk of crystalline host. Due to high crack propagation resistance  (mainly determined by a high value of Young modulus ~ 400 GPa) the crack-free regions of birefringence up to Δn ≃ 1×10-3 can be formed adjacent to the laser-structured volume and spanning tens of microns. This opens a possibility to form 3D waveguides inside sapphire, ruby (Cr-doped sapphire ), and other laser crystals which have comparable mechanical properties and can be used for formation of active lasing waveguiding structures. Thermal diffusivity in the stress-birefringent regions was found unchanged.
Polariscopy imaging is used to quantify the stress-induced birefringence around laser-structured volumes. The peak birefringence of Δn≃10-3 at the edge of laser structured volume is decaying exponentially with spatial constant τr = 24 μm. It is found that azimuth of slow axis is spinning around the laser structured region. This opens possibility to engineer and tailor optical activity and record functional polarization-sensitive micro-devices, e.g., for generating optical vortex beams.
The quantitative measurement of birefringence is used to estimate the stress. The cylindrical volume composed of void-structures exerted a 2D stress pattern. This simplified calculations and the following correspondence has been established: Δn ≃ 10-3 is equivalent to 1.3 GPa stress.
Presented combined assessment of optical, mechanical, and thermal properties inside and around laser structured regions in sapphire should help for miniaturization of optically active and passive functional micro-devices, polarization micro-optics, and can be applied to other crystalline materials and glasses.
Support via a Discovery ARC Grant DP0988054, Grant-in-Aid from the Ministry of Education, Science, Sports, and Culture of Japan No. 19360322, and a grant from Japan Science and Technology Agency (Development of System and Technology for Advanced Measurement and Analysis) are gratefully acknowledged. Authors would like to thank Etienne Brasselet for useful discussions, and Tokyo Instruments, Inc. for agreement to publish data obtained by Abrio IM microscope.
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