Thermal lensing effects and nonlinear refractive indices of fluoride crystals induced by high-power ultrafast lasers

Abstract

Thermo-optical and nonlinear property characterization of refractive optical components is essential for endoscopic instrumentation that utilizes high-power, high-repetition-rate ultrafast lasers. For example, ytterbium-doped fiber lasers are well suited for ultrafast laser microsurgery applications; however, the thermo-optical responses of many common lens substrates are not well understood at 1035 nm wavelength. Using a z-scan technique, we first measured the nonlinear refractive indices of CaF₂, MgF₂, and BaF₂ at 1035 nm and found values that match well with those from the literature at 1064 nm. To elucidate effects of thermal lensing, we performed z-scans at multiple laser repetition rates and multiple average powers. The results showed negligible thermal effects up to an average power of 1 W and at 10 W material-specific thermal lensing significantly altered z-scan measurements. Using a 2D temperature model, we could determine the source of the observed thermal lensing effects. Linear absorption was determined as the main source of heating in these crystals. On the other hand, inclusion of nonlinear absorption as an additional heat source in the simulations showed that thermal lensing in borosilicate glass was strongly influenced by nonlinear absorption. This method can potentially provide a sensitive method to measure small nonlinear absorption coefficients of transparent optical materials. These results can guide design of miniaturized optical systems for ultrafast laser surgery and deep-tissue imaging probes.

1. INTRODUCTION

Characterization of the thermo-optical and nonlinear properties of optical glasses and crystals is important for applications utilizing high-power, high-repetition-rate ultrafast lasers delivered through miniaturized optics. Recent developments in miniaturized ultrafast laser surgery probes for clinical use [1-4] indicate that kilohertz (kHz) to megahertz (MHz) laser repetition rates are necessary to enable rapid tissue removal. High-peak-power ultrashort laser pulses can generate strong optical nonlinearities within focusing optics, and delivery of these pulses at high repetition rates can result in thermal lensing caused by cumulative heating effects. These two effects together can lead to significant beam distortion at the focal plane and/or damage to the optical system, highlighting the need for miniaturized optical systems capable of withstanding high peak and high average powers.

Miniaturized optical systems typically require high-refractive index materials to enable tight beam focusing in small form factors, further enhancing unwanted nonlinear effects. For example, in our recently developed ultrafast laser surgery probe, we custom-fabricated a 4 mm miniaturized objective made of a highly refractive lens substrate, zinc sulfide (ZnS) [4]. We determined that multiphoton absorption in ZnS ultimately limited the laser power that could be delivered through the probe. Specifically, we found that three-photon absorption in ZnS caused deviation from linear transmission of 803 nm, 1.5 ps pulses beyond a peak intensity of 1.5 GW/cm², gradually decreasing the maximum laser power delivered to the tissue surface.

Various fluoride crystals, such as CaF₂, MgF₂, and BaF₂, are potential alternative lens materials for miniaturized optical systems. Their small nonlinear absorption coefficients, high damage thresholds, low chromatic dispersion, and efficient transmission at near-infrared wavelengths make them ideal for use in miniaturized optics [5,6]. Further, fluoride crystals are harder than many optical glasses, enabling easier manufacturing when fabricating lenses using diamond point turning. Although these crystals exhibit less nonlinear absorption when compared to ZnS, refractive index changes caused by 1) the optical Kerr effect and 2) linear and nonlinear-absorption-induced heating may cause beam distortion when using high-power, high-repetition-rate ultrafast lasers. While the nonlinear properties of CaF₂, MgF₂, and BaF₂ have been studied at 800 nm and 1064 nm wavelengths, there is little data available for their thermo-optical properties. Specifically, there is a need to characterize these properties at 1035 nm wavelength, as ytterbium-doped (Yb-doped) fiber lasers are quickly becoming the laser of choice for various biomedical applications, including ultrafast laser microsurgery and optogenetics. Therefore, we present here experimental and simulated results of the thermo-optical and nonlinear responses of these materials when exposed to high-peak and high-average-power ultrashort pulses at 1035 nm.

2. NONLINEAR REFRACTIVE INDEX MEASUREMENTS USING THE Z-SCAN TECHNIQUE

Ultrashort pulses can induce optical nonlinearities via multiphoton absorption and refractive index changes. The nonlinear response of a material is correlated to the dependence of refractive index (n) and absorption coefficient (α) on the peak intensity (I) of the pulse propagating through the medium. Refractive index is defined as

$$n(I)=n_0+\gamma I,$$ (1)

where n₀ and γ are linear and nonlinear refractive indices, respectively. Similarly, absorption is intensity dependent:

$$\alpha (I)={\alpha }_0+\sum _{n=2}^{\infty }{\alpha }_n{I}^{n-1},$$ (2)

where α₀ and α_n are the linear and nonlinear absorption coefficients, respectively.

The well-known z-scan technique allows for determination of γ and α_n in thin, transparent materials [7]. In this technique, a sample is translated along the laser propagation axis through a focused beam, and transmission is measured across a far-field pinhole aperture. The complex electric field exiting the sample contains information on the nonlinear phase shift imparted on the beam. A sample with thickness less than the diffraction length of the focused beam can be considered as a thin lens with variable focal length. Far away from the focal plane, intensities are low, and the far-field beam profile is largely unaffected by sample nonlinearities. Intensity increases as the sample is brought near the focus, and self-focusing in positive γ materials will cause a pre-focal “valley” (transmission minimum) by increasing beam divergence and reducing transmission across the pinhole aperture. Likewise, a post-focal “peak” (transmission maximum) caused by the same self-focusing in positive γ materials tends to collimate the beam and increase transmission. For a sufficiently narrow pinhole aperture, on-axis transmission for a cubic nonlinearity and a small nonlinear phase shift is commonly fit to the well-established formula [7]

$$A(x,t,s\approx 0)=1+\frac{4x\langle \Delta {\Phi }_0(t)\rangle }{(x^2+1)(x^2+9)};\mid \Delta {\Phi }_0(t)\mid <1,$$ (3)

where A(x) is normalized transmission, s is the linear transmission of the pinhole aperture, ⟨ΔΦ₀(t)⟩ is the time-averaged on-axis nonlinear phase shift, x = (z – z₀))/z_r is sample position with respect to the focus, z_r = π w₀²/λ is diffraction length of the beam, and w₀ is the 1/e² Gaussian beam waist radius. While Eq. (3) is applicable for small nonlinear phase shifts and continuous wave laser excitation, it does not hold for large phase shifts where transmission curves tend to develop asymmetries. Further, Eq. (3) does not consider a specific pulse shape, a variable that must be accounted for [8,9]. When performing z-scans with a femtosecond fiber laser, a hyperbolic secant squared (sech²) pulse shape can be assumed and the normalized on-axis transmission through a narrow pinhole aperture may then be fit to the analytical expression presented in [10]:

$$\begin{gathered} A(x,s\approx 0)=1+\frac{2}{3}\frac{4x\Delta {\Phi }_0}{(x^2+1)(x^2+9)} \\ +\frac{8}{15}\frac{4\Delta {\Phi }_0^2(3x^2-5)}{(x^2+1{)}^2(x^2+9)(x^2+25)}, \end{gathered}$$ (4)

where ΔΦ₀ is the peak on-axis nonlinear phase shift, which is related to γ by

$$\Delta {\Phi }_0=k\gamma I_0{L}_{\text{eff}}.$$ (5)

Here k = 2π/λ is the wavenumber, I₀ is peak on-axis intensity at the beam waist, and L_eff = (1 – e^−α₀L)/α₀ is the effective sample length. As α₀ ≈ 0 for all tested materials, L_eff ≈ L, where L is the sample thickness. We follow the formalism described by Paschotta to define I₀ for a squared sech² pulse [11]:

$$I_0=0.88\frac{2E_p}{{\tau }_p}(\pi w_0^2{)}^{-1},$$ (6)

where E_p and τ_p are pulse energy and pulse width (FWHM), respectively. We account for front surface reflection losses so that I₀ represents the true peak intensity within each material. The fitted value of ΔΦ₀ obtained from Eq. (4) is only applicable for thin samples (L < z_r) and far-field transmission measurements (z_a ≫ z_r), where variations in w₀ due to diffraction and nonlinear refraction can be ignored and the distance between the pinhole and beam waist z_a is much greater than the beam diffraction length.

Materials exhibiting negligible nonlinear absorption α_n require only a single z-scan to determine γ, as changes in transmission through the pinhole aperture are attributed solely to refractive nonlinearities. For materials exhibiting large α_n, multiphoton absorption will contribute to the characteristic z-scan trace. In this case, to separate refractive and absorptive nonlinearities, there is a need for an additional z-scan in an “open-aperture” configuration. The pinhole aperture is removed during open-aperture z-scans, eliminating sensitivity to nonlinear refraction while maintaining sensitivity to nonlinear absorption. By normalizing the original closed-aperture z-scan data to data obtained from an open-aperture z-scan, we can isolate effects of nonlinear refraction and determine γ.

Our experimental setup was similar to the apparatus described in [7]. We used a Yb-doped fiber laser (40 W, λ = 1035 ± 5 nm, pulse width 300 fs – 10 ps, repetition rate 10 kHz–50 MHz, Monaco, Coherent Inc.) tuned to 10 kHz during initial z-scan studies. Laser pulse width τ_p = 300 fs was measured using an autocorrelator (pulseCheck, APE GmbH) [Fig. 1(a)]. We used a half-wave plate/polarizing beam cube splitter combination for power attenuation and a spatial filter, consisting of two planoconvex lenses and a 25 μm pinhole aperture, to obtain a clean TEM₀₀ beam profile at the back aperture of a f = 175 mm focal length lens. The 1/e² focal plane beam waist radius w₀ = 37.8 ± 1.1 μm [Figs. 1(c) and 1(d)] was measured using the knife-edge technique. As L ≤ 3 mm for all samples tested, the measured diffraction length z_r = 4.3 ± 0.1 mm in air meets the thin lens approximation. Samples were manually translated through the focus using a sliding stage, and a pinhole aperture (d = 600 ± 10 μm, P600D, Thorlabs) placed at a distance >40z_r from the focal plane transmitted the central portion of the beam towards an energy/power meter (PD10-pJ-C, Ophir Photonics). Transmission was recorded in 1 mm intervals and reduced to 0.5 mm increments near the focus. Transmission curves were fit to Eq. (4) to extract ΔΦ₀, and γ was determined by substituting ΔΦ₀ into Eq. (5).

Fig. 1.

Experimental setup characterization. (a) Sech² fit to autocorrelator data to determine pulse duration τ_p. (b) Spectrum of the femtosecond laser source. (c) Data from single knife-edge scan was fit to the edge spread function to determine beam radius. (d) Measured beam radii as a function of position along the optical axis. Data was fit to Gaussian beam equation to determine beam radius at the focal plane w₀. Error bars represent 95% confidence intervals for knife-edge data fit to the edge spread function.

To validate our experimental setup, we used borosilicate glass (N-BK7, L ≤ 3 mm, WG10530, Thorlabs), which has well-established nonlinear optical properties. Open-aperture z-scans on borosilicate glass revealed negligible nonlinear absorption (<2% transmission drop at the highest tested peak intensity); normalization of closed-aperture z-scans was not needed to determine γ. We determined γ = 2.35 ± 0.25 × 10⁻⁷cm²/GW, agreeing to within 3% of mean values reported by Flom et al. at 1030 nm [12] and within 6% of mean values reported by Shimada et al. at 1064 nm [13].

We measured the nonlinear optical properties of three fluoride crystals, CaF₂ (L ≤ 3 mm, WG50530, Thorlabs), MgF₂ (L ≤ 3 mm, WG60530, Thorlabs), and BaF₂ (L ≤ 3 mm, 87-700, Edmund Optics). Representative z-scan transmission curves are presented in Fig. 2. Table 1 tallies averaged nonlinear refractive indices along with pulse energies and corresponding peak intensities used during experimentation. Averaged γ values represent the average of six individual γ measurements obtained during z-scans, performed at three values of I₀ on two separate days. Uncertainty in individual γ measurements accounts for uncertainties in I₀, L_eff, and ΔΦ₀, while uncertainty of averaged γ values represents the root-mean-square error of contributing data points. Uncertainties in the fits to extract ΔΦ₀ were low, and the primary source of uncertainty in individual γ measurements was related to uncertainty in I₀. Z-scans performed on MgF₂ required higher peak intensities to obtain an appreciable nonlinear phase shift because of its relatively low nonlinear refractive index. All materials exhibited a positive refractive nonlinearity as indicated by the valley-peak shape of transmission curves, which is common in many optical glasses and crystals [14,15]. Similar to borosilicate glass, open-aperture z-scans performed at the highest peak intensity for the three fluoride crystals showed negligible variations in transmission (<2% transmission drop), indicating that nonlinear absorption was below our detection limit and open-aperture z-scan normalization was not necessary. Transmission curves fit to Eq. (4) very well (R² > 0.9 for all fits). Valley-peak separations Δz_pv of raw transmission data were consistently within 10% of Δz_pv = 1.7z_r as expected from Eq. (4) for small nonlinear phase shifts. There was a slight shift of the crossing point [i.e., where A(x) = 1] towards the positive z axis for high-peak-intensity z-scans as expected for increasing ΔΦ₀ [7,16]. While γ values at 1035 nm were not found in the literature, our results matched well with measurements reported by Adair et al. at 1064 nm [15].

Fig. 2.

Representative closed-aperture z-scan transmission curves. (a) CaF₂, (b) MgF₂, and (c) BaF₂. Solid lines represent the fit of collected data to Eq. (4), and open black circles represent normalized transmission during open-aperture z-scans at the highest tested peak intensity.

Table 1.

Average Nonlinear Index of Refraction (γ) at 1035 nm for Tested Materials ^{a, b}

Material	γ × 10−7 [cm2/GW]	Ep [μJ]	I0 [GW/cm2]
N-BK7	2.35 ± 0.25	0.5–2	63–251
CaF2	1.34 ± 0.12	1–3	127–380
MgF2	0.72 ± 0.07	2–4	255–511
BaF2	1.96 ± 0.23	1–3	126–379

^aThe table indicates the range of pulse energies used (E_p) and corresponding peak intensities (I₀).

^bPulse energies were measured after the focusing lens, and reported I₀ values were calculated at the beam focus according to Eq. (6) after accounting for reflection losses at the front surface of each sample.

3. THERMAL LENSING EFFECTS IN Z-SCAN MEASUREMENTS

Laser repetition rate is commonly kept below 1 kHz during z-scans to minimize thermal contributions to the apparent nonlinear index, as heating can produce a lensing effect in many optical materials [17-19]. The initial radial temperature profile within a material will approximately follow the intensity distribution of the illuminating beam and will eventually evolve into a parabolic profile due to the interplay of heat generation and heat diffusion [17]. A material with a nonzero thermo-optic coefficient will exhibit a temperature-dependent radial refractive index profile, leading to the formation of a thermal lens. The presence of thermal lensing during z-scan studies will lead to inaccurate estimation of the nonlinear phase shift ΔΦ₀ and, by extension, γ. If the characteristic z-scan trace is attributed solely to electronic nonlinearities (i.e., optical Kerr effect), ΔΦ₀ should increase linearly with peak intensity. If thermal lensing is present, the additional contribution ΔΦ_TL to the total accumulated phase will cause the measured (albeit incorrect) ΔΦ₀ to roughly scale linearly with the average laser power, as the thermal lens strength depends on the total amount of heat transferred to the sample [20-22]. This linear scaling is approximate, as predicted valley-peak separations Δz_pv are smaller for nonlinear lensing when compared to thermal lensing.

To better understand the effects of thermal lensing in our experimental setup, we simulated the temperature evolution in borosilicate glass, CaF₂, MgF₂, and BaF₂ using the 2D heat transfer equation:

$$\rho C_p\frac{\partial T}{dt}=\stackrel{.}{q}(r)+K\left(\frac{{\partial }^{2}T}{d{x}^2}+\frac{{\partial }^{2}T}{d{y}^2}\right),$$ (7)

where ρ is density, C_p is specific heat, K is the thermal conduction coefficient, and $\stackrel{.}{q}(r)$ is a heat source. While all tested materials exhibited low linear absorption at 1035 nm, cumulative heating may induce thermal lensing if time between successive pulses is shorter than the characteristic thermal diffusion time τ_c:

$${\tau }_c=\frac{C_p\rho w_0^2}{4K}.$$ (8)

Since our pulse width was orders of magnitude shorter than τ_c of borosilicate glass and the three fluoride crystals, we assumed each pulse gave rise to an instantaneous radial temperature distribution ΔT(r, t = 0):

$$\Delta T(r,t=0)=\frac{{\alpha }_0{F}_0}{C_p\rho }{e}^{-\left(2\frac{r^2}{w_0^2}\right)},$$ (9)

where F₀ is the peak fluence at the focal plane. In our simulations, we applied Eq. (9) as a temperature source within the sample every t_laser, defined as the time between successive pulses for a given laser repetition rate, to model sample heating as induced by a laser pulse train. Assuming radial symmetry, we used a finite-difference approximation of Eq. (7) to estimate the radial temperature profile within each material prior to the arrival of the next pulse. We repeated this process to determine the temperature profile after a series of many pulses, ΔT(r, t_final). Once a steady-state radial temperature profile has been established and a quasi-permanent thermal lens is formed, the thermal phase shift ΔΦ_TL imparted on the beam can be found by

$$\Delta {\Phi }_{\text{TL}}(r,t_{\text{final}})=kL\frac{dn}{dT}[\Delta T(0,t_{\text{final}})-\Delta T(r,t_{\text{final}})].$$ (10)

Here dn/dT is a material-specific thermo-optic coefficient and ΔT(0, t_final) is temperature at the beam center in the middle of the sample. For our simulations, we used a spatial step size of w₀/5 across a 12.7 × 12.7 mm² plane and a temporal step size of t_laser/5. High spatial and temporal discretization ensured simulation convergence and provided an accurate estimation of the final radial temperature profile.

To determine the contribution of thermal lensing (i.e., the thermally induced phase shift ΔΦ_TL) to the measured ΔΦ₀ in our initial experiments, we performed z-scans at average powers ranging between 10 mW–10 W. To maintain a constant peak intensity, we varied the laser repetition rate between 10 kHz–10 MHz while maintaining a constant pulse energy. We also performed z-scans at 50 MHz with a slightly increased pulse duration of τ_p = 1 ps at the highest tested average power of 10 W (I₀ < 8 GW/cm²) to evaluate thermal lensing without the added contribution of nonlinearities. Pulse energy and pulse width were measured at every repetition rate to ensure constant I₀. We added an electronic shutter before the beam attenuator and a nonpolarizing beam cube splitter (BS074, Thorlabs) after the sliding stage to avoid pinhole aperture heating during high-repetition-rate/high-average-power measurements. We measured transmission across the pinhole aperture at every sample position 3 sec after opening the shutter to allow the thermal lens to fully develop and to enable accurate modeling of the expected radial temperature profile within each sample. The shutter was closed for 1 min between each measurement to allow samples to return to a baseline temperature. For brevity, we refer to high-repetition-rate/high-average-power z-scans as high-repetition-rate z-scans in the following paragraphs.

The data presented in Figs. 3(a)-3(d) show z-scan transmission curves for borosilicate glass and the three fluoride crystals at laser repetition rates/average power combinations of 10 kHz/10 mW 500 kHz/500 mW, and 1 MHz/1 W. Below 1 MHz/1 W, transmission curves were similar; measured ΔΦ₀ was largely insensitive to increasing average power (i.e., ΔΦ_TL was negligible), allowing us to conclude that nonlinear indices measured during 10 kHz z-scans (Table 1) were not skewed by cumulative heating effects. For high-repetition-rate z-scans (10 MHz/10 W), we noted multiple differences in the transmission curves [Figs. 3(e)-3(h)], all pointing to the possible formation of thermal lensing.

Fig. 3.

Closed-aperture z-scan transmission curves for varying laser repetition rates/average powers. Z-scans for (a) borosilicate glass, (b) CaF₂, (c) MgF₂, and (d) BaF₂ at low repetition rate/low average power. Z-scans for (e) borosilicate glass, (f) CaF₂, (g) MgF₂, and (h) BaF₂ at high repetition rate/high average power. Peak intensities in each material were constant for all repetition rate/average power combinations except for 50 MHz/10 W/1 ps (I₀ < 8 GW/cm²). Solid lines represent the fit of collected data to Eq. (4). Fits to Eq. (4) are omitted for 50 MHz/10 W data. Beam radii measured in front of a pinhole aperture during (i) 1 MHz/1 W and (j) 10 MHz/10 W z-scans performed on CaF₂. (f) Closed-aperture z-scan transmission curves are overlaid in (i) and (j) for reference.

First, measured ΔΦ₀ was altered with increasing repetition rate/average power. The larger phase shift (i.e., related to the larger valley-peak excursion) observed in borosilicate glass and MgF₂ at 10 MHz/10 W [Figs. 3(e) and 3(g)] indicated formation of a positive thermal lens as expected for these dn/dT > 0 materials at room temperature [23,24]. For dn/dT < 0 materials, such as CaF₂ and BaF₂, we would expect a decrease in valley-peak excursions, as a negative thermal lens, if strong enough, would partially offset a positive nonlinear lens. The measured low- and high-repetition-rate transmission curves were similar for BaF₂ [Figs. 3(d) and 3(h)], indicating negligible thermal effects. However, we saw an increase in valley-peak excursion during 10 MHz/10 W z scans on CaF₂ [Fig. 3(f)], indicating an unexpected positive thermal lensing effect despite having dn/dT < 0. We further confirmed this observation by imaging the beam at the pinhole aperture plane using a beam profiler (SP928, Ophir Photonics) during 1 MHz/1 W and 10 MHz/10 W z-scans on CaF₂. As shown in Figs. 3(i) and 3(j), maximum and minimum 1/e² beam radii occurred near the axial positions corresponding to minimum and maximum transmission, respectively. Further, the relative beam size variations during 10 MHz/10 W z-scans [Fig. 3(j)] were larger as expected from the larger valley-peak excursion of the 10 MHz/10 W transmission curve [Fig. 3(f)].

Second, valley-peak separations Δz_pv were larger during 10 MHz/10 W z-scans on CaF₂ and MgF₂ as expected when thermal lensing contributes to the characteristic z-scan trace. In the case of linear-absorption-induced thermal lensing only, we would expect Δz_pv = 3.46z_r, while Δz_pv = 1.72z_r for a refractive nonlinearity [25]. In our case, we would expect the added thermal contributions to our high-repetition-rate measurements to broaden the valley-peak separation (Δz_pv > 1.72z_r) and cause a worse fit of collected data to Eq. (4). For example, during the 10 kHz/10 mW z-scan in CaF₂, Δz_pv = 1.65z_r, and an uncertainty in measured ΔΦ₀ of ±4%, while Δz_pv = 1.96z_r during the 10 MHz/10 W z-scan exhibited a ΔΦ₀ measurement uncertainty of ±9%.

Third, there was a shift of the crossing point [i.e., where A(x) = 1] towards the positive z axis during 10 MHz/10 W measurements on borosilicate glass, CaF₂, and MgF₂. This shift, along with the transmission curve asymmetry, indicated a stronger lensing effect [7,16] that was not seen during low-repetition-rate z-scans at the same I₀. Finally, we saw significant changes in transmission measurements during 50 MHz, τ_p = 1 ps z-scans that were not caused by self-focusing. Low-repetition-rate measurements revealed negligible transmission variations [A(x) ~ 1 across the entire z-scan] for I₀ < 28 GW/cm², confirming that the peak intensity, I₀ < 8 GW/cm², used during 50 MHz, 1 ps experiments was below the intensity required to generate an appreciable nonlinear phase shift in any of the tested materials.

Simulations revealed that material-specific thermal lensing effects may contribute to measurements during high-repetition-rate z-scans (see Table 2). At low-repetition-rates/average powers, for example, 10 kHz/10 mW, we predicted a temperature increase smaller than 2 × 10⁻³ K in all materials, leading to a weak lensing effect. In this case, the resulting thermal phase shift (ΔΦ_TL) was orders of magnitude smaller than our nonlinear phase shift (ΔΦ₀) measurements and below the resolution of our experimental setup. On the other hand, we predicted a temperature increase of 1.83 K was reached after 3 s at the beam center in the middle of the MgF₂ sample during 10 MHz/10 W z-scans [Figs. 4(a) and 4(b)], leading to a thermal phase shift of ΔΦ_TL = 33 mrad. This value agrees well with the observed increase in measured phase shift (ΔΦ_{10 MHs/10 W} – ΔΦ₀ = 40 ± 25 mrad) for 10 MHz/10 W experiments on MgF₂. The predicted 0.03 K temperature increase in BaF₂ produced a ΔΦ_TL = −5 mrad, agreeing with the decrease in measured phase shift (ΔΦ_{10 MHs/10 W} – ΔΦ₀ = –7 ± 18 mrad) observed during 10 MHz/10 W experiments. The weak thermal lens was a consequence of low linear absorption and rapid thermal diffusion in BaF₂.

Table 2.

Summary of Thermal Lensing Simulations Based on Linear-Absorption-Induced Heating ^{a, b, c, d, e}

Material	τc [μs]	α0 [m−1]	dn/dT × 10−6 [K−1]	ΔT(0, t=3 s) [K]	ΔΦTL [mrad]	ΔΦ10 MHz/10 W – ΔΦ0 [mrad]
N-BK7	690	0.12	1.2 [23]	0.82	18	334 ± 36
CaF2	100	0.08	−11.5 [24]	0.07	−16	172 ±29
MgF2	54	4.00	1.0 [24]	1.83	33	40 ± 25
BaF2	61	0.03	−16.2 [24]	0.03	−5	−7 ± 18

^aThermo-optic coefficients (dn/dT) for CaF₂, Baf₂, and MgF₂ were taken from Feldman et al. [24] at 293 K and λ = 1.15 μm.

^bdn/dT for borosilicate glass (N-BK7) was taken from Frey et al. [23] at 295 K and λ = 1 μm.

^cTemperatures at the beam center after 3 s exposure to 1 μJ, 300 fs laser pulses at 10 MHz repetition rate ΔT(0, t = 3 s) were determined from heat transfer simulations (Fig. 4).

^dΔΦ_TL the predicted thermal phase shift from our simulations, ΔΦ₀ is the measured average nonlinear phase shift from 10 kHz z-scans, and ΔΦ_{10 WHz/10 W} is the measured phase shift during 10 MHz/10 W z-scans.

^eThe last column shows the difference between ΔΦ_{10 MHz/10 W} and ΔΦ₀, which should be similar to ΔΦ_TL if linear-absorption-induced thermal lensing is the primary mechanism contributing to the increase in measured phase shift during high-repetition-rate/high-average-power z-scans.

Fig. 4.

2D temperature simulation results. Temperature variations are based on heating due to linear absorption only. (a) Radial temperature profile in the MgF₂ sample after 3 s exposure to 1 μJ, 300 fs pulses at 10 MHz laser repetition rate (10 W average power). Temperature evolution due to linear absorption at the beam center in (b) MgF₂ and borosilicate glass and (c) CaF₂ and BaF₂.

The predicted thermal phase shift, ΔΦ_TL = 18 mrad, in borosilicate glass based on pure linear absorption was too small to account for the 334 ± 36 mrad increase in measured phase shift during high-repetition-rate z-scans. We hypothesized that additional positive thermal lensing effects due to nonlinear-absorption-induced heating might be the cause for such an increase. We modified the expression used for the radial temperature distribution induced by a single pulse to include nonlinear absorption:

$$\Delta T(r,t=0)=\frac{{\alpha }_0{F}_0}{C_p\rho }{e}^{-\left(2\frac{r^2}{w_0^2}\right)}+\frac{{\alpha }_n{I}_0^n{\tau }_p}{0.88C_p\rho }{e}^{-\left(2n\frac{r^2}{w_0^2}\right)},$$ (11)

where n is the minimum number of photons required for nonlinear absorption. For borosilicate glass, a four-photon absorption process was required to initiate nonlinear absorption, considering the borosilicate glass band-gap energy (~4 eV [26]) and photon energy of 1035 nm laser light. By replacing Eq. (9) with Eq. (11) as the time-dependent temperature source in our heat transfer simulations, we predicted a nonlinear absorption coefficient of α₄ ≈ 3 × 10⁻⁴⁵ m⁵/W³ was required to match the observed thermal lensing effects. Following the theoretical analysis of Corrêa et al., we estimated the intensities required to observe a transmission reduction due to four-photon absorption during open-aperture z-scans on borosilicate glass [27]. Using the predicted α₄, we expected a very low (~0.2%) transmission reduction for the used peak intensity of I₀ = 126 GW/cm² and an observable 10% transmission decrease at much higher peak intensities (I₀ > 565 GW/cm²). We concluded that measuring such a low nonlinear absorption coefficient was not possible in our experiments, where we needed to use peak intensities that avoided damaging the sample.

In summary, nonlinear absorption may have increased thermal lensing effects in high-repetition-rate z-scan measurements in borosilicate glass. It is interesting to note that higher order absorption processes, whose detection was far below the measurement resolution of a conventional open-aperture z-scan, could have such a substantial effect on the thermal lens strength. The low predicted value of α₄ shows that large nonlinear absorption is not needed to induce significant cumulative thermal effects. Our combined experimental- and simulation-based approach to predict small nonlinear absorption coefficients is similar to the methods outlined by Falconieri et al., who used a 2D temperature model to calculate the two-photon absorption coefficient required to recapitulate femtosecond z-scan traces in CS₂ [28]. It was not possible to verify the accuracy of the predicted α₄, as fourth-order absorption coefficients of borosilicate glass at 1035 nm were not found in the literature. One study found α₄ ≈ 10 × 10⁻⁴⁵ m⁵/W³ at 1035 nm for lithium niobate, a crystal with a similar band-gap energy as borosilicate glass [29]. The apparent high sensitivity of thermal lensing to nonlinear absorption suggests that the methodology presented here could be used to measure multiphoton absorption coefficients at peak intensities substantially lower than those required for conventional open-aperture z-scan measurements; however, validation requires further studies.

It is also worth noting that our simulations could not correctly predict the CaF₂ data, even when considering heating due to nonlinear absorption. Since CaF₂ exhibits a dn/dT < 0, any thermal lensing effect should partially offset the positive nonlinear lens, leading to a smaller measured phase shift. However, we observed a ~50% increase in measured phase shift during high-repetition-rate z-scans, which could not be explained by any thermal lensing effect.

The results presented here can be useful when designing miniaturized ultrafast laser surgery probes. For example, in one of our projects, we are developing an ultrafast laser surgery probe to treat scarred vocal folds. The surgical probe utilizes a miniaturized, high-numerical-aperture CaF₂ objective. In the present study, we observed no appreciable valley-peak excursion, where A(x) ~ 1, during z-scans on CaF₂ at peak intensities below 30 GW/cm² and repetition rates below 1 MHz. Using 300 fs laser pulses and assuming that 10 μJ pulse energies are required for surgery, we predict that a minimum 1/e² beam radius of 250 μm on all CaF₂ surfaces will ensure I₀ < 30 GW/cm² and avoid self-focusing effects. Further, our results suggest that thermal effects can be disregarded when laser repetition rates below 1 MHz are used. However, care must be taken when using higher-repetition-rate/higher-average-power combinations.

4. CONCLUSIONS

Using a z-scan technique, we measured the nonlinear refractive indices of three fluoride crystals at 1035 nm and studied effects of thermal lensing in these materials when exposed to high-repetition-rate, high-average-power ultrashort laser pulses. Nonlinear index measurements fit well with values found in the literature at 1064 nm wavelength. We showed that thermal lensing did not contribute significantly to apparent nonlinear index measurements below a laser repetition rate of 1 MHz, allowing us to conclude that nonlinear indices measured at 10 kHz z-scans were not skewed by cumulative heating effects. While time between successive pulses at 10 kHz (100 μs) was on par with the thermal diffusion times of tested materials, indicating that cumulative heating may cause thermal lensing in our experimental setup, the small absorption coefficients of all materials led to a minimal temperature rise and a weak thermal lens. In contrast, we saw variations in transmission data during high-repetition-rate/high-average-power z-scans, which we attributed to thermal lensing. This claim was supported by simulations that predicted linear-absorption-induced thermal lensing would contribute to high-repetition-rate/high-average-power z-scan measurements in MgF₂ and BaF₂. However, when considering linear absorption only, we could not predict the strong lensing effect observed in borosilicate glass and CaF₂. Including nonlinear absorption as an additional heat source in our simulations could explain thermal lensing observed during high-repetition-rate z-scans on borosilicate glass. We found that small nonlinear absorption could significantly enhance the thermal lens strength, given the n th-order intensity dependence of heat generation. The apparent high sensitivity of thermal lensing to nonlinear absorption could be used to measure small nonlinear absorption coefficients in transparent materials. These results can be used to guide optical system designs for ultrafast laser microsurgery and deep-tissue nonlinear imaging probes.