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We demonstrate the feasibility of passive compensation of the thermal lens effect in fused silica optics, placing suitable optical materials with negative dn/dT in the beam path of a high power near IR fiber laser. Following a brief overview of the involved mechanisms, photo-thermal absorption measurements with a Hartmann-Shack sensor are described, from which coefficients for surface/coating and bulk absorption in various materials are determined. Based on comprehensive knowledge of the 2D wavefront deformations resulting from absorption, passive compensation of thermally induced aberrations in complex optical systems is possible, as illustrated for an F-Theta objective. By means of caustic measurements during high-power operation we are able to demonstrate a 60% reduction of the focal shift in F-Theta lenses through passive compensation.
© 2014 Optical Society of America
1. Introduction
Owing to the progress in the field of near IR high power solid state lasers during the last decade, almost diffraction limited multi-kilowatt systems are nowadays commercially available and already widely employed in various important industrial production processes [1,2]. However, the opportunities of such high power lasers, as e.g. tighter focusing in material processing, can be fully exploited only in combination with high-quality beam line optics and focusing objectives. In particular, these optical elements should introduce only an insignificant amount of thermally induced wavefront aberrations and focus shifts to the laser beam. At present, temperature-induced relative focal shifts of 1-2% per kW of laser power are not uncommon, even for high end fused silica objectives, e.g. F-theta multiplets with dichroitic AR coatings employed in scanning applications [3]. This issue is already of substantial concern for multimode beams, and the problem exacerbates with high quality beams and their reduced depth of focus for a given NA. Thus, any substantial improvement in reduction or compensation of thermal focal shifts and aberrations is of major importance for a large amount of industrial high power laser applications.
In order to overcome or at least minimize the severe problem of thermal lensing, three basically different approaches are viable. First of all, the absorptance of the employed optical glasses, both in the bulk material and at the surfaces, has to be kept as low as possible. This is of particular relevance for anti-reflection coatings, which strongly contribute to the overall absorption losses and thus the heating of a component [3].
Second, an active compensation of thermally induced lenses is possible, using e.g. adaptive mirrors based on piezo-electric actuators. Adaptive optics represent by far the most flexible solution for the thermal problem. Both single element mirrors, to be employed merely for defocus compensation, and multiple actuator systems for correction of higher order aberrations have been demonstrated [4–6]. However, apart from stability issues during long-term high power operation, adaptive systems introduce a considerable amount of complexity to a laser system, which is not desirable in many industrial applications. A related approach uses actively cooled mirrors instead of refractive elements for the focusing objective of a high power laser [7]. As only the thermal expansion and no refractive index variation contributes, the thermal shift can be substantially reduced. However, the optical setup is more complex, and a successful application of this method to F-theta or zoom objectives has, to our best knowledge, not been published.
As a third option, the integration of passive compensation schemes for thermally induced lenses can be considered, making use of optical materials with a negative temperature coefficient of the refractive index dn/dT. Several attempts in this direction have been published so far, the majority of contributions dealing with compensation of optical distortions in laser rods and crystals [6,8,9]. Here the effects are very large and strongly pump-power dependent and passive compensation can therefore be seriously concerned only in systems working in a fixed stationary operation mode [8]. Regarding extra-cavity beam line optics, a passive compensation of transmissive optics is proposed and numerically investigated in a paper of Scaggs and Haas [10], employing negative dn/dT materials as CaF2. However, the main focus in this approach is the athermalisation with respect to the ambient temperature, neglecting any local temperature variation. Piehler et al. [11] demonstrate a theoretical approach to the compensation of thermal lensing as a result of a transient temperature distribution.
In this paper we investigate the potential of a negative dn/dT glass (N-PK51, Schott) for passive thermal compensation of near IR fused silica optics. Following a brief section on fundamentals of thermal lensing and compensation, we present an experimental setup for spatially resolved recording of thermal wavefront distortions, making use of a Hartmann-Shack sensor previously employed for photothermal absorption measurements on deep UV optics [12,13]. Owing to the linear relationship, quantitative absorptance data are derived from the induced wavefront deformations, accomplishing also a separation of bulk and surface (or coating) absorptance of an optical element. Such data represent the prerequisite for the successful thermal compensation, as demonstrated by the achieved focal shift reduction of an F-theta objective for high power fiber lasers.
2. Theory
2.1. Thermal lensing
The absorption of an intense circular laser beam (radius rb) propagating in z-direction through a cylindrical optical sample (length L, radius r0, refractive index n0) leads to a stationary temperature change δT(r,z) (cf. Figure 1), giving rise to a wavefront distortion δw. This can be captured, for example, by monitoring the wavefront of a well collimated test beam travelling parallel to the heating beam.
Considering a thermal expansion ΔL and changes of the refractive index with temperature ΔnT and induced stress Δnσ, the distortion is given by [12]
with dn/dT denoting the temperature variation of the refractive index, εzz the strain component in z-direction, σ the temperature induced stresses, B the photo-elastic tensor and the subscripts r, φ signifying radial and azimuthal test beam polarizations, respectively.As only the difference (δw(r)-δw(r0)) is of optical relevance, we take δw(r0) = 0 as understood in the following. If the sample dimensions as well as the material and laser parameters are known, Eq. (1) can be evaluated by numerical solution of the heat equation and the conditions of elastic equilibrium [12]. However, in a first order δw(r) may be approximated by its parabolic fraction, leading to an optical power 1/f:
After z-integration and averaging over the polarization states Eqs. (1) and (2) yield in the plane stress (rb >> L) or plane strain (rb << L) approximation, respectively [14,15]:with the thermo-optical parameter K:In Eq. (4) α is the coefficient of linear expansion, E Young’s modulus, ν the Poisson ratio and B||, B⊥ the stress optic coefficient for polarization parallel or perpendicular to the applied stress. From the parabolic part of the temperature profile near the axisone obtains:In Eqs. (5) and (6) P and λ are the average power of the heating laser and the heat conductivity, respectively. µ is an effective absorption coefficient, containing bulk and surface contributions i.e., µ = µbulk + 2β/L, with β the single surface absorption. δT0 denotes the temperature in the beam center and cs a coefficient, depending on the beam shape (e.g. cs = 1 for a flat top and cs = 2 for a Gaussian beam). Equation (6) holds in 1st order only if the curvature radii of the element surfaces can be neglected. A more complete discussion including this issue as well as the influence of aberrations shall be treated in a forthcoming paper.2.2. Passive compensation
In order to compensate the thermal lens generated in an optical element I with a second plane element II placed closely in front of or behind I, Eq. (6) yields for the thickness of element II (parameters indexed by I and II, respectively):
Due to the fact that in isotropic materials ΔL/L and Δnσ are always positive, the compensation of a dn/dT > 0 element requires KII < 0, i.e. at least dn/dT < -n03αE(B|| + (1 + 2ν) B⊥)/4. Table 1 shows the limiting values of the parameter K/λ for fused silica and a few materials with negative dn/dT.
Table 1. dn/dT values and thermal parameter K/λ @1060nm for fused silica and some dn/dT < 0 materials.
Despite the simple expressions given in Eq. (4), in general K depends on rb/L as well as on laser and material parameters in a complicated manner and can only be determined by a rigorous numerical thermo-elastic treatment.
Figure 2 shows the result of such a calculation for the optical power of a Schott N-PK51 sample with diameter 35 mm as a function of sample length for different diameters of a Gaussian heating laser beam. The negative dn/dT contribution accumulates along the beam path (cf. Equation (1)), whereas the competing effect of thermal expansion (i.e. bulging of the sample surfaces) saturates with increasing sample length. Thus, in good qualitative agreement with the experimental findings reported in the next sections, it turns out that a certain beam diameter dependent sample length is needed to self-compensate the material’s own expansion and stress-optical effects.
Fig. 2 Numerically calculated first order optical power of a plane N-PK51 substrate (diameter 35 mm) as a function of sample length for three diameters of a Gaussian laser beam. Beam power is 100 W and the used absorption coefficients for bulk and surface were µ = 0.09 1/m and β = 15 ppm, respectively. Stress-free mechanical boundary conditions and a constant heat transfer coefficient of κ = 10 W·m−2K−1 were applied for the calculations.
3. Photo-thermal absorptance measurement
In order to determine bulk and surface contributions to the total absorption separately, a photo-thermal method is applied in a crossed beam setup, which is described in detail elsewhere [13].
The method utilizes cuboid samples of dimension 25x25x45 mm3 polished on 4 sides with the heating beam propagating in z- and the probe beam propagating in y-direction of the cuboid reference frame. A collimated Yb-fiber laser (IPG, 500W @1070 nm, M2 < 1.2) was employed for sample heating at a beam diameter of 1.2 mm. The probe beam, a laser driven plasma source (Energetiq) coupled into an optical fiber, was collimated to a diameter of Ø30 mm and spectrally narrowed by an interference filter with 10 nm FWHM. Photo-thermal signals were captured using a 3x telescope for probe beam reduction and a highly sensitive Hartmann-Shack sensor equipped with a 12bit CCD detector. Data processing and wavefront reconstruction were performed as described in [13]. Wavefronts are acquired immediately before (reference) and directly after laser heating of the sample for a period of 10 seconds, averaging over 4 camera frames in both cases.
The sensitivity of the setup was checked by comparing the photothermal signal from an uncoated fused silica cuboid with an absorption coefficient of µ ≈2·10−3 m−1 [16] irradiated with 256 W to a zero measurement without heating. In terms of the wavefront RMS value we achieved a signal-to-noise ratio of 5.6 corresponding to a minimum detectable µ/P (S/N = 1) of 0.09 m−1/W.
Figure 3 shows on the left typical wavefront reconstructions of AR coated cuboid samples from fused silica and N-FK5, as well as an uncoated N-PK51 sample, obtained by a 12th degree Legendre-fit of the Hartmann-Shack data. The distributions exhibit a more or less pronounced central valley, caused by volume absorption, and a strong bending to the end face due to surface absorption. As expected, for fused silica with a positive thermo-optical parameter K, surface and bulk act in the same direction with a rather small bulk contribution. Qualitatively the same behavior is found with N-PK51, except for the more pronounced bulk part and the sign reversal due to the negative value of K. In contrast, the measured wavefront distribution of N-FK5 is more complex. Within the volume, stresses and lateral surface bulging are strong enough to compensate for the negative dn/dT. Closer to the surface, however, the tension of the front surfaces reduces both lateral bulging and compressive stresses, accomplishing an even negative K at the surface.
Fig. 3 Comparison between measurements (left) and simulations (right) of cuboid samples of different optical materials investigated in the crossed-beam setup. The x and y axes represent the spatial arrangement of the cuboid sample. The wavefront is displayed on the z axis for a sample height of 25 mm and reaching 20 mm (measurement) or 23 mm (simulation) into the bulk from the entry surface.
Anticipating the results given below, the right column in Fig. 3 presents simulated wavefront distributions (including the Legendre fit) using the experimental absorption coefficients of this paper and literature values for the remaining material parameters. As can be seen, the agreement is very good even for N-FK5. The oscillations in the lateral parts are associated with Legendre interpolation artefacts.
The employed strategy to determine the required absorption coefficients is outlined in Fig. 4. In a first step, two power-normalized wavefront distributions V’ and S’ for pure bulk and pure surface absorption, respectively, are simulated with arbitrary values for the absorption coefficients (left part of the figure). In a second step, as illustrated in the right part of the figure, the weighted sum is fitted to the power-normalized experimental valley-floor line we(x = 0,z)/P via a least-square approach according to
yielding the desired coefficients β and μ as fit parameters.Fig. 4 Schematic illustration of the procedure to determine absolute surface/coating and bulk absorption coefficients (cf. text).
The results for the three investigated glass materials and one or two AR coatings are shown in Table 2. With reference to the bulk absorption, all coefficients show relative standard deviations of less than 1% and are in good agreement with vendor data from Schott and Corning. The surface/coating absorption data, on the other hand, yield values of β1 ≈2.5·10−5 and β2 ≈1.2·10−5 with standard deviations comparable to the bulk coefficients in case of fused silica. For N-FK5 and N-PK51 the surface/coating contribution represents only a very small fraction of the total absorptance, resulting in an increased standard deviation of 10% – 50%.
Table 2. Coating/surface (β) and bulk (µ) absorption coefficients obtained through curve-fitting of simulated wavefront data to measurements for two different AR coatings and an uncoated N-PK51 sample.
4. Collinear wavefront measurements of complex optics
In order to be able to monitor spatially resolved wavefront deformations of complex optical systems such as F-Theta lenses, a collinear setup is employed as shown in Fig. 5. This is achieved by integrating pivoted mirrors: to induce a thermal load on the sample optics by the heating beam, the mirrors are moved into their working position. Subsequently, the mirrors are moved out of the probe beam path, and the thermally affected wavefront is measured during the cooling period [17]. Compensating elements may be included in the beam path as indicated in Fig. 5.
Fig. 5 Experimental setup for the determination of focal shifts in F-Theta lenses and other optical systems using pivoted mirrors for a coaxial arrangement of probe beam and heating beam (left). The graph on the right demonstrates the linear dependence of the wavefront deformation wRMS on the laser power in a planar sample.
As seen from Fig. 5 (right), wavefront distortions induced by laser irradiation in a planar optical sample are linearly dependent on laser power. On this basis, the RMS value of the wavefront distortion wRMS can be utilized to characterize and compare the absorption properties of different optical glasses and coatings. In conjunction with a suitable calibration (e.g. numerical simulation), absolute absorption coefficients can be determined [18].
Figure 6 shows coaxial measurements of wavefront deformations induced by cylindrical samples of fused silica and N-PK51, with positive and negative thermal dispersion, respectively. As shown in section 2.2, for optical materials with dn/dT < 0 there is a transition of the wavefront curvature from positive to negative, which is confirmed by the measurement of N-PK51: In the highly heated center of the beam, the dn/dT term is already dominant, whereas the competing effects of dn/dσ and bulging of the sample surface are still prevailing in the cooler outer area, leading to the depicted inversion of the wavefront. Thus, above a certain beam diameter dependent sample length, the negative thermal dispersion of N-PK51 becomes available for passive thermal compensation of focal shifts.
Fig. 6 Wavefront deformation in plane AR coated fused silica and N-PK51 samples (length 12 mm, diameter 25 mm, laser power 168 W, beam diameter 1.2 mm). The measured area covers approx. 10x10 mm2. The effect of the negative thermal dispersion of N-PK51 is clearly visible leading to an inversion of the wavefront.
As a consequence, the compensation of thermally affected F-Theta optics is investigated. The F-Theta lenses utilized for the measurements are manufactured by Sill Optics GmbH. They have an effective focal length of 420 mm and a working distance of 499.2 mm. The entire optical system comprises four lenses and one protective glass made of fused silica (i.e. ten AR coated surfaces) with a total thickness of 46.3 mm. Two different AR coatings are available (cf. Table 2), one with a higher (“AR1”) and one with a lower absorption coefficient (“AR2”).
In Fig. 7 the thermo-optical effect for a combination of an F-theta objective for high power NIR laser applications and N-PK51 flats of different lengths is presented, indicating an inversion of optical path difference (OPD) and wavefront curvature.
Fig. 7 Thermal wavefront distortions for a combination of an F-Theta lens (AR2) and N-PK51 elements of different lengths at P = 100 W and d = 3.8 mm. The measured area covers approx. 10x10 mm2.
For the 12 mm compensating element, a combined wPV of 13 nm was measured, which indicates virtually no change compared to the uncompensated F-Theta lens. Due to the fact that N-PK51 is close to the transition point of self-compensation for this particular beam diameter and sample length, no notable compensation of the thermal lens can be achieved. Only a flattening effect in the center occurs. This agrees with the numerical results in Fig. 2, where a distinct negative wavefront curvature is calculated for a sample length of 12 mm and a beam diameter of d = 3.8 mm, but no statement about wPV is made. The plot on the right shows that a considerable overcompensation could already be measured for an N-PK51 sample length of 25 mm.
Due to the additional thermal focal shift introduced by the F-Theta lens (AR2) and the larger beam diameter of d = 3.8 mm, the transition point of the complete system (F-Theta + compensation) shifts to a greater length of the compensating element (estimated value ~16 mm), if compared to Fig. 6.
5. Determination of focal shifts
In order to substantiate the photo-thermal results at the operational wavelength of λ = 1070 nm and diameter d = 13 mm without the need for an additional probe beam, direct measurements of beam waist displacements were performed with a setup outlined in Fig. 8.
Fig. 8 Caustic measurement setup for the determination of thermal beam waist displacements of high power laser optics. Displayed are also some beam profiles around the waist, taken at high power operation and with a 70 mm compensating element implemented.
For this purpose the high power laser beam has to be strongly attenuated without introducing additional thermal aberrations. This is achieved by using the reflected beam from high-quality fused silica wedges in combination with additional filters. As the coefficient of thermal expansion of fused silica is very low (α ≈0,5·10−6 K−1) and only the reflected beam is used for measurements, thermal expansion of the attenuating wedges can be neglected. Caustic measurements according to ISO 11146 [19] are carried out to characterize the behavior of the F-Theta lens (AR1) under thermal load with and without compensation. The beam profiles in Fig. 8 on the right are taken around the beam waist at 381 W with a compensating N-PK51 element of 70 mm in length (because of the larger beam diameter), showing a small astigmatism due to stress birefringence in N-PK51.
Figure 9 displays the power dependent focal shift as well as the resulting beam diameter in the working plane.
Fig. 9 Thermal focal shifts of an F-Theta lens (AR1) with and without compensating element (left). It is assumed that focal shifts at P = 25 W are very small and are therefore set to zero. The graph on the right shows average beam diameters at a fixed working distance of 500 mm.
The compensation element is clearly able to reduce the displacement of the average beam waist by a factor of ~2.5 at 381 W laser power. Although stress induced birefringence in N-PK51 introduces an astigmatic waist difference of ~0.3 mm at 381 W, the spot diameter at working distance increases more slowly with laser power for the compensated setup. The values were corrected to match at 25 W (i.e. nearly zero focal shift) to account for imperfections of the N-PK51 element and statistic measurement errors.
6. Conclusion
In this paper it has been demonstrated, both experimentally and by numerical means, that 1st order compensation of thermally induced wavefront distortions in complex high power fused silica optics is possible by round flats of the glass type Schott N-PK51. For layout purposes of the compensating element and as input for numerical calculations, bulk and surface absorption coefficients of fused silica, Schott N-FK5 and Schott N-PK51 have been determined by a specifically adapted photothermal method. The results for bulk absorption are in good agreement with literature values, and the concordance between measured and simulated wavefront distributions (Fig. 3), even for complicated cases like N-FK5, indicates the reliability of the fitted surface absorptions. Measurements of thermally induced wavefront distortions in round flats of N-PK51 show for various sample and beam parameters a good qualitative agreement with the numerical results presented in Fig. 2. Obviously, a minimum beam diameter dependent sample length is necessary to self-compensate the effects of thermal expansion and stress in order to achieve a negative thermal lens for compensation of fused silica. For a 3.8 mm diameter beam a self-compensating sample length of ~10 mm was measured, whereas for compensation of the F-theta objective N-PK51 flats of 16 mm are required.
In order to investigate the F-theta performance at operating conditions (1070 nm wavelength and 13 mm beam diameter), direct beam profile measurements were carried out near the beam waist, yielding a focal shift for the uncompensated F-theta lens of Δf = 5.6 mm/kW or Δf/f = 1.3%/kW corresponding to approximately two Rayleigh ranges per kilowatt laser power. These values could be reduced by a 70 mm N-PK51 plane substrate by a factor of 2.5 to 2.1 mm/kW or 0.8 zR/kW. Compared to that, the stress induced astigmatism (~0.3 mm at 381 W laser power) is considerably smaller, resulting only in a minor beam quality reduction.
Plane substrates are compact, easy to fabricate and, if placed in the collimated part of the beam, simple to replace or substitute for a different operation mode or beam diameter. Thus, in view of our results, the use of N-PK51 flats as passive compensating elements seems a promising alternative in high power fused silica optics. Field trials with industrial laser systems are planned in the near future to further examine the compensation potential, including important issues such as spherical aberration or dynamical response.
Acknowledgments
The authors like to thank Sill Optics GmbH and Heraeus Quarzglas for providing F-theta objectives and fused silica samples, as well as T. Graf (IFSW, Univ. Stuttgart) for stimulating discussions.
This work was partly conducted within the ZIM joint research project KF2411810DF2, supported by the German federal ministry of economics and technology.
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