1. Introduction

Femtosecond micromachining has been widely utilized to fabricate microstructures inside different materials, including silica glasses, polymers and even biological tissues [1–3]. Unique properties of highly localized energy deposition and high peak laser intensity result in intense structural and morphological alterations within only the laser-material interaction region and no collateral damage is induced to surrounding areas [4–6]. One of the structural alterations of great interest is the laser-induced refractive index (RI) change or phase change below a certain optical damage threshold, which allows for the fabrication of photonic components and devices via inscribing waveguides [7–9], and recently vision correction [10]. RI change is related to phase change by calculating the optical path difference between two homogenous and isotropic regions of different refractive indices [11],

Many researchers have focused on writing inside silica glasses and positive RI changes have been found after femtosecond laser exposure, which is generally attributed to densification from rapid quenching of melted glasses or formation of color centers [12–15]. Femtosecond micromachining of polymers has been attracted attention recently, due to its easy processability and its importance in telecommunication, clinical and biological fields [4]. In the field of clinical and biological applications, our group has developed a technique “LIRIC” (which stands for laser induced refractive index change) to alter optical qualities of ophthalmic materials by writing grating lines, phase bars and other phase wrapped structures inside hydrogel-based contact lenses, intro-ocular lenses (IOL) and even corneal globes [16–19]. The magnitude of the induced RI change depends highly upon the material properties, including material type, dopant concentration, and water content [20–23].

Apart from material properties, laser system parameters also impact the magnitude of the RI change. Qualitative data of previous work showed the RI modification increases with power and decreases with scan speed, pulse duration and wavelength [24–27]. The effect of repetition rate is more complicated since both photothermal accumulation effect and photochemical effect may contribute to the induced phase change, in principle. Based on a photothermal accumulation mechanism, repetition rates affect the induced phase change via heat accumulation effect. According to a numerical simulation of temperature evolution during pulse series, a pulse train of a higher repetition rate is able to generate a higher temperature rise, thus resulting in a higher RI change [28–30]. However, our recent study on the effect of repetition rate in the four-photon absorption regime suggests that a photochemical reaction might be the dominating mechanism governing the micromachining process [31].

Building on previous work in the four-photon absorption regime using 1035 nm femtosecond laser pulses, we performed more experiments on hydrogel-based contact lenses using femtosecond laser pulses at 405 nm where two-photon absorption process takes place and studied the effects of various laser system parameters, including power, scan speed, NA and repetition rate, on the induced phase change magnitude. We find no dependence of the writing process on the polarization states of the excitation and reading beams. Experimental results are compared with NIR femtosecond writing. We report for the first time that the effect of intermediate repetition rate (1 MHz-8.3 MHz) on femtosecond micromachining hydrogel-based polymers in the two-photon absorption regime has been illustrated, in contrast to high repetition rate (> 60 MHz) and low repetition rate (< 500 KHz). By taking advantage of blue femtosecond laser pulses with an intermediate laser repetition rate, extremely high scan speed experiments were successfully performed to demonstrate the dominant effect in inducing refractive index change inside hydrogel polymers is the photochemical effect as single pulse material changes can be resolved, which eliminates the multi-pulse photothermal accumulation effects. Derived from an overwriting factor between multiple adjacent pulses and multiphoton absorption mechanism, a general photochemical model for all orders of multiphoton absorption processes is proposed for the small signal region, and we introduce a simple saturation model that can potentially explain the parametric dependence that we see in large signal regime.

2. Proposed two-dimensional photochemical model

Our model is derived from an overwriting factor and pulse energies absorbed through a multiphoton absorption process. The overwriting factor arises from the overlapped displacement of two adjacent pulses. The accumulation effects in the directions both along and perpendicular to the scanning line should be considered since phase bars rather than a single grating line were written inside materials. As shown in Fig. 1, along the direction where a grating line is formed, the horizontal overwriting factor is the ratio of the laser excited region diameter and the separation between two nearby pulses which is dependent on scan speed and repetition rate of the pulse train. Similarly, the vertical overwriting factor perpendicular to the scanning direction can be expressed as the ratio of the excited region diameter and the line spacing. We assume that the linewidth of the written feature is smaller than the diffraction-limited irradiance distribution by a factor of $\sqrt m $ due to the m^th order of multiphoton absorption. Assuming a collimated Gaussian beam focus and a diffraction-limited spot size [32], the linewidth of the written feature ω after taking into account the multiphoton absorption effect can be expressed in terms of wavelength λ, NA and $\sqrt m $,

 

Fig. 1. A sketch illustrates the overwriting factors in both horizontal and vertical dimensions.

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During the micromachining process, a small fraction of the incident pulse energy can be absorbed via multiphoton absorption process and the absorbed energy then can be converted to modify the density of molecules mainly through photochemical reactions. Raman micro-spectroscopy showed photochemical reactions might occur via chain scission that causes defragmentation of polymer molecules and forms free radicals. Free radicals may diffuse out from the interaction volume and leave behind voids that can be occupied by water molecules. The degradation of polymer molecules and reoccupation of water molecules result in a lower molecular density and thus a decrease in RI [23]. The multiphoton absorption cross section scales with laser intensity to the (m-1)^th power [34,35] and together with Beer-Lambert Law, the absorption rate in the small signal region can be expressed as $\beta \cdot {I^{m - 1}} \cdot L$. We then assume the amount of molecular density change D caused by depolymerization can be related to the absorbed single pulse energy density in the weak nonlinear absorption limit by the following equation,

In the case of a pulse train exposure, the total molecular density change can be expressed as the excited density change by one pulse multiplied by the overwriting factor due to a linear accumulation assumption of multiple pulses per spot. The induced RI change is then assumed to be proportional to the total molecular density change. After mathematical manipulation and conversion from basic physical quantities in Eqs. (1)–(8), the final equation for expressing the induced phase change as a function of various experimental parameters, including average power, numerical aperture, repetition rate, pulse duration, scan speed, wavelength and line spacing, can be given by

In our case, a laser source delivering blue femtosecond pulses at 405 nm is employed and therefore the order of the nonlinear absorption process is $m = 2$. Rewriting Eq. (9) we can obtain a simplified expression for the two-photon absorption process,

3. Experimental setup

The experimental setup of the blue light writing system is shown in Fig. 2. The light source used in the writing system is a customized KM Labs Y-Fi HP laser, which includes a noncollinear optical parametric amplifier (NOPA). The laser is able to deliver femtosecond laser pulses at four wavelengths (405 nm, 517 nm, 810 nm, and 1035 nm). When operating in the blue light mode, the laser allows for tunable repetition rate in the range of 0.5-8.3 MHz. The available repetition rates are 60/N, where N is an integer varying from 7 to 120. The nominal maximum output blue light power at 8.3 MHz is 400 mW and therefore the maximum average power in the focal volume is measured to be 250 mW (corresponding to a maximum single pulse energy of 30 nJ) due to power loss on the interfaces and transmission of the microscope objective. A half-wave plate (HWP) and a polarizing beam splitter (PBS) work together to adjust the amount of transmitted light. The power in the focal volume is determined by measuring the power after the GTI mirrors and the ratio is calibrated before every experiment. A beam expander is inserted in the beam path to adjust the input beam size so that the laser beam can either overfill or underfill the entrance pupil of the microscope objective to form different effective Numerical Aperture beams, and a cylindrical lens pair reshapes the elliptical beam into a circular beam. The NAs used are 0.45 by using a beam expander ratio of 1:5 to overfill the aperture and 0.25 by using a beam expander ratio of 1:2. The expanded beam size is measured by the knife edge technique and NA can be calculated based on the effective focal length of the microscope objective. The respective diffraction limited spot size is calculated 1.1 µm for NA 0.45 and 1.9 µm for NA 0.25. Therefore, the line spacing is chosen to be 0.5 µm so as to create continuous phase bars. A Gires-Tournois Interferometer (GTI) mirror pair (Layertec) is customized at 405 nm for pulse compression and after 8 bounces reflections on the mirror the pulse duration is measured to be less than 200 fs by a commercial autocorrelator (Newport, PSCOUT2-BLUE-PMT). Mirrors are used to steer the beam into a microscope objective (Olympus LUCPLFLN 20X/0.45) with 80% transmission in the visible band. A back reflection light monitor composed of a CCD camera and a front focusing lens is used to locate the depth of the focal plane. A commercial hydrogel-based contact lens sample is sandwiched between a microscope glass slide and a cover slip and soaked with solution to maintain hydration. The sample is then mounted on a 2D XY linear stage (Aerotech, PRO115LM) for planar scanning and the vertical (Z axis) motion is provided by a step controller (Newport, GTS30 V) attached to the objective. The 2D XY Linear stage can provide a stable scan speed up to only 500 mm/s. The sample is placed at the center of the linear stage and translated 20 mm to ensure the travel distance is long enough for maintaining a constant scan speed inside the sample.

 

Fig. 2. (a) Experimental configuration of the blue femtosecond laser writing system. HWP, half-wave plate; PBS, polarizing beam splitter; GTI mirrors, Gires-Tournois Interferometer mirrors. (b) Transmission spectra of Contaflex samples and J + J contact lenses along with femtosecond laser wavelengths used in the blue writing and NIR writing processes.

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Two types of samples used are J + J contact lenses (Acuvue2, Johnson & Johnson) with a base curvature of 8.7 mm and plano hydrogels Contaflex GM Advance 58 (Contamac Inc.). J + J contact lenses are made of a soft hydrophilic material known as “etafilcon A”, a copolymer of 2-hydroxyethyl methacrylate and methacrylic acid cross-linked with 1, 1, 1-trimethylol propane trimethacrylate [36] and ethylene glycol dimethacrylate. Contaflex samples are formed by “Acofilcon A”, a synonym for “2-Butenedioic acid (2Z)-, di-2-propenyl ester, polymer with 2,3-Dihydroxypropyl 2-methyl-2-propenoate, 1-Ethenyl-2-pyrrolidinone, 2-Hydroxyethyl 2-methyl-2-propenoate and Methyl 2-methyl-2-propenoate” [37]. The transmissivities of the two samples at 405 nm are greater than 95% with linear absorption cutoff wavelengths both within the UV band around 280 nm [36]. We also measured the transmission spectra of the two samples using an Ocean Optics spectrometer (HR4000), as shown in Fig. 2(b). The samples are almost transparent in the visible waveband while having strong absorption below 330 nm.

4. Quantitative results showing effects of laser system parameters

4.1 The effect of power on the induced phase change

Power scaling experiments were performed to show the dependence of micromachining phase bars on power. Phase bars were fabricated inside both samples to obtain quantitative data and created by forming 60 scanning lines separated by half a micron to ensure that the phase carpets were continuous. As shown in Fig. 3, phase bars were written in Contaflex samples at powers from 40 mW to 75 mW at an increment of 5 mW. The scan speed was held at 200 mm/s and the NA was chosen to be 0.45. A rectangular loop scanning method was adopted to create two 30 µm x 20 mm phase bars separated by 50 µm at the same time. Different features can be found between low intensity exposure and high intensity exposure. Smooth and homogeneous phase bars exist in the small power regime. Once the power exceeds 60 mW optical damage can be observed with a very sharp threshold. In the initial phase of damage, the damage appears as part of the phase bars become distorted, while other parts remain unaffected. The damaged area is triangle shaped, showing that the damage starts at the edge of the phase bar and develops towards the center. This triangular part seems to undergo significant physical material changes, sometimes with rough surfaces. As the power continues increasing to 75 mW, no homogeneous phase bars can be seen any longer and the whole phase bars are covered with rugged surfaces and melting-like traces in this gross damage stage. Damage may result from photodegradation of polymers that causes the chemical chain cleavage as a consequence of a photochemical or photothermal decomposition. Wochnowski et al. demonstrated that in the case of UV-laser-assisted modification of PMMA, main chain scission occurs at relatively high intensities [38]. Intense laser irradiation can cause multiphoton ionization followed by avalanche ionization, which facilitates the free electron density growth to exceed the electron critical density threshold and results in optical breakdown. Energy transferred from free electrons to polymer structures via resonant electron-molecule scattering or the creation of reactive oxygen species starts the total cleavage of side chain via a photochemical reaction of either Norrish Type I or Norrish Type II [38,39].

 

Fig. 3. DIC images showing 8 sets of phase bars written in Contaflex samples at powers from 40 mW to 75 mW. Smooth and uniform phase bars can be seen at a power below 60 mW while optical damage occurs at higher irradiation doses.

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The magnitudes of phase change induced inside Contaflex samples at different powers were measured under a Mach-Zehnder Interferometer (MZI). The MZI is the same as described in Ref. [16] and the reference wavelength used in this study was 543 nm. The phase change was extracted and unwrapped via the Goldstein’s branch cut unwrapping algorithm and Fourier-transform method of fringe pattern analysis [40,41]. Interferogram and phase map corresponding to the phase bars written at 55 mW are shown in Fig. 4. In the laser untreated area, the fringes are continuous and straight while a sharp discontinuity can be seen at the boundary between the phase bar and the pristine regime, indicating a phase change inside the laser-modified area. The sign of the phase change was found negative by introducing wedge and observing the direction of fringe shift [16]. The quantitative data suggest that the magnitude of induced phase change increases as the power goes up.

 

Fig. 4. Interferogram and phase map showing one wave of phase change induced by the phase bars inside Contaflex samples written at 55 mW, 200 mm/s, NA 0.45 and 8.3 MHz.

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4.2 The effect of repetition rate on the induced phase change

Using the same writing procedures at 8.3 MHz we were able to write phase bars at two other repetition rates, 1 MHz and 5 MHz. Power functions were used to show the dependence of phase change on average power and the exponents of the fitted power functions are of interest since the order of multiphoton absorption can be estimated by the exponents according to Eq. (10). Results obtained from both Contaflex samples and J + J contact lenses are shown in Fig. 5(a) and (b) respectively. Comparison made between the blue writing and the NIR writing reveals different nonlinear processes. Exponential factors of the fitted power functions are close to 2 rather than 4 in the case of NIR writing [31,42], indicating blue femtosecond micromachining at 405 nm is governed by a two-photon absorption process because of the presence of ultraviolet blockers in the lens materials absorbing around 280 nm. The order of the power slightly decreases to 1.5 as the repetition rate increases to 8.3 MHz, which may imply a decline in the order of multiphoton absorption due to an incubation effect at a greater number of laser pulses per spot [25].

 

Fig. 5. Plots of the induced phase change as a function of average power (a, b) and as a function of single pulse energy (c, d) below the damage threshold at three different repetition rates. The experimental data at 1 MHz, 5 MHz and 8.3 MHz are fitted by power functions to illustrate the power dependence and the effect of repetition rate at NA 0.45.

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The effect of repetition rate on the induced phase change can be summarized from Fig. 5 and found to be consistent with the NIR writing results. Below the damage thresholds of all three repetition rates, the same amount of phase change can be induced at a much smaller power with a lower repetition rate; however, the dynamic range for measurable phase shifts is shorter in the case of a lower-repetition-rate writing because the optical damage threshold occurs at a much smaller power and thus the maximum achievable phase change just below the damage threshold is smaller as well. According to the study of Gandara-Montano et al., one wave of phase change is enough to fabricate any phase wrapped patterns like Fresnel lenses by creating a series of zones with phase wrapping between 0 and 1 wave [17]. As seen in Fig. 5, one wave of phase change can be obtained from both materials with 8.3 MHz using a relatively low power at ∼ 50 mW. Therefore, we are able to infer that 8.3 MHz can be regarded as the optimum repetition rate for creating phase bars in both J + J contact lenses and Contaflex samples as lower repetition rates inflict optical damage before inducing one wave of phase change while higher powers are required to induce one wave of phase change with higher repetition rates.

The maximum achievable phase changes and the power damage thresholds are roughly the same for both materials at the three repetition rates in the case of 405 nm writing. However, the maximum achievable phase changes of the two materials obtained from the NIR writing differ by 40%: the largest phase change induced in J + J contact lenses was −1 wave while the largest phase shift in Contaflex sample was only −0.56 waves [31]. Additionally, the average power of the NIR femtosecond pulse needed to obtain the maximum phase change was 3.7 W in J + J contact lenses and 5 W in Contaflex samples, which are roughly 100-fold greater than the powers needed to induce the same amount of phase change at 405 nm. The remarkable difference between the blue writing and the NIR writing can be attributed to the difference in the magnitude of the nonlinear multiphoton absorption coefficient. The experimental results suggest that two-photon absorption coefficients of the two materials are comparable and much greater than four photon absorption coefficients.

The relationship between the induced phase change and the single pulse energy is plotted in Fig. 5(c) and (d) by converting the average power to single pulse energy damage threshold via Eq. (5). At a given single pulse energy, the induced phase change increases with the repetition rate, which is reasonable since more pulse energies could be deposited into the material with a higher repetition rate laser source. According to Eq. (2), the number of laser pulses per spot is ∼1.6, ∼ 6, ∼14 at 1 MHz, 5 MHz and 8.3 MHz, respectively. Our previous study comparing a wide range of repetition rates (5 MHz - 60 MHz) in the four-photon absorption regime demonstrates that the damage threshold in terms of single pulse energy decreases with the repetition rate. Same conclusion can be found here in the two-photon absorption regime: for Contaflex samples, the damage threshold is calculated to be ∼10 nJ, ∼6 nJ and ∼7 nJ at 1 MHz, 5 MHz and 8.3 MHz, respectively; ∼10 nJ, ∼7 nJ and ∼7 nJ at 1 MHz, 5 MHz and 8.3 MHz respectively for J + J contact lenses.

4.3 The effect of numerical aperture on the induced phase change

The effect of NA on the magnitude of induced phase change is studied by comparing the results obtained at NA 0.45 and NA 0.25. A high NA of 0.45 is obtained by using a 1:5 beam expander to overfill the entrance pupil of the objective while a 1:2 beam expander is used to form a low NA of 0.25. Quantitative data obtained at NA 0.45 and at NA 0.25 are shown in Fig. 5 and Fig. 6 respectively for both J + J contact lenses and Contaflex samples. Power function fitting is again used for estimating the order of multiphoton absorption process. The complications due to nonlinear effect like self-focusing and aberrations are minimized due to the usage of low peak powers (on the order of kW) and relatively low NAs (< 0.5).

 

Fig. 6. Below optical damage threshold, quantitative results showing the power dependency of phase change induced at NA 0.25 for Contaflex samples (a) and J + J contact lenses (b) via the fitting of power functions.

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In the small signal regime where the magnitudes of phase changes are smaller than half a wave, the induced phase changes obtained from both high NA and low NA can be found to be comparable at different powers and at three different repetition rates, as predicted by Eq. (10). However, dramatic differences can be seen in the large signal domain where optical damage (as shown in Fig. 3) has already taken place at the high NA while in contrast, uniform phase bars can still be observed at the low NA. It is noteworthy that the dynamic range reaching from the power for producing barely measurable phase change to the one inducing maximum phase change just below damage threshold has been extended significantly in the case of a low NA writing. By comparing the power damage thresholds at two NAs, one could find that in the case of NA 0.25, the power damage threshold of J + J contact lenses has been increased from 10 mW to 21 mW at 1 MHz, 35 mW to 60 mW at 5 MHz, and 60 mW to 90 mW at 8.3 MHz; similarly, the damage threshold of Contaflex samples also rises. Owing to the increased damage threshold, the magnitude of the maximum achievable phase change induced at NA 0.25 is therefore much higher than the one obtained at NA 0.45.

The increase in damage threshold and enhancement in the maximum phase change can be attributed to a broadened interaction volume in the case of a lower NA writing. The cross sections of the micromachined phase bars written at two NAs are shown in Fig. 7. Sectioning the hydrogels was achieved by firstly freezing a sample in a mold using OCT compound, then precisely cutting the sample into slices with 60 µm thickness, and finally storing the slices with aqueous gel to maintain hydrated. At a given power of 40 mW, a confined and highly localized laser treated area is observed at NA 0.45 with an estimated longitudinal length of ∼20 µm while an elongated and enlarged interaction area can be seen at NA 0.25 and the overall length is measured to be ∼40 µm, longer than that at NA 0.45. To the first order approximation, the damage threshold is dependent on pulse energy deposition per volume u_th which scales with the average power and is inversely proportional to the interaction volume (i.e. ${u_{th}} \propto {P_{avg}}/(\upsilon \cdot VOL)$), implying a higher average power is necessary at a low NA to attain the threshold. Therefore, one can picture that damage occurs at a lower u_th if NA is increased to 0.9 and the maximum achievable phase change before damage is accordingly small while in the case of NA 0.1, damage should occur at a much higher u_th along with a higher maximum attainable phase change. Phase bar structures can be seen clearly from the cross-section images and the length of the interaction region is found to increase with the optical power, which might be a manifestation of material saturation discussed in the following section.

 

Fig. 7. DIC images showing cross sections of phase bars written inside Contaflex samples at different powers with NA 0.45 (a, b) and NA 0.25 (c, d). The length of the interaction region at 40 mW is estimated to be ∼20 µm at NA 0.45 and ∼40 µm at NA 0.25. Interaction volume generated at high NA is more confined and localized than low NA.

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The optimum repetition rate shifts as the NA changes: in the case of NA 0.25, one can see the optimum repetition rate has shifted towards a smaller value ∼1 MHz and one wave of phase change can be achieved at an extremely small power of ∼15 mW (Fig. 6).

4.4 The effect of scan speed on the induced phase change

Scan speed scaling experiments were performed in both J + J contact lenses and Contaflex samples at six different scan speeds, 10 mm/s, 50 mm/s, 100 mm/s, 150 mm/s, 200 mm/s and 250 mm/s, to demonstrate the scan speed dependence of the magnitudes of phase changes. Experimental data were collected at 8.3 MHz and three different powers 20 mW, 30 mW and 40 mW were tested. Data at each writing condition are fitted by respective power functions. As shown in Fig. 8(a) and (b), the exponents of the fitted power functions are found to be close to −0.5, showing that the phase change and the scan speed forms a sub-linear inverse relationship. However, based on the photochemical model (Eq. (10)), we were expecting that the phase change and the scan speed should make an inverse ratio since the scan speed only impacts the overlapping factor by linearly affecting the distance between two adjacent laser pulses. One possible explanation for the sub-linear inverse dependence on scan speed is the presence of a saturation factor, of which the effect becomes dominant in a large signal regime and prevents the phase change from increasing without bound.

 

Fig. 8. Power function fitting for quantitative data obtained at NA 0.45 from Contaflex samples (a) and from J + J contact lenses (b).

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4.5 Extremely high scan speed experiment

Owing to the unique properties of low-repetition-rate writing, extremely high scan speed experiments were performed with the highest output power of 240 mW in the focal volume, at a repetition rate of 8.3 MHz and at a NA of 0.3. Figure 9(a) shows DIC images of grating lines written inside a Contaflex sample at scan speeds of 5 mm/s, 50 mm/s, 100 mm/s and 200 mm/s. Extremely high scan speed scaling (> 1 m/s) was achieved by the employment of a rotational scanner. Since the 2D linear stage can provide highest scan speed up to only 500 mm/s, a rotational scanning stage is placed on top of the 2D XY linear stage and the sample is then mounted at the edge of the rotation scanner in order to proceed with experiments demanding high scan speeds of several meters per second. The rotational scanning stage is actually an optical chopper with 6 slots and chopping frequencies from 0 to 350 Hz, thus providing linear scan speeds up to 11.72 m/s at 350 Hz after measuring the distance between the center of the stage and the sample to be 32 mm. The scanned curves are actually separated concentric circles formed by rotating the optical chopper and moving the linear stage towards X direction simultaneously; however, the circles resemble lines on the micron scale under the microscope. Qualitative results of micromachined grating lines inside contact lenses obtained at scan speeds from 500 mm/s (15 Hz) to 11.72 m/s (350 Hz) are shown in Fig. 9(b-e). By carefully selecting the dwell time, the grating lines within each grating group were scanned only once. A higher magnification was used to image the grating lines written at 11.72 m/s shown in Fig. 9(e). Due to the employment of such a high scan speed and a low repetition rate of 8.3 MHz, the spacing between two adjacent laser pulses along the same grating line can be calculated to be 1.41 µm. Assuming a 405 nm Gaussian beam focus through a NA 0.3 microscope objective, the diffraction-limited laser spot size in the focal plane is estimated to be 0.61 µm Eq. (2), which is smaller than the displacement of two laser pulses. Therefore, there is only one pulse per spot and no thermal accumulation effect should exist during the writing process. The grating lines can still be seen clearly even at one pulse irradiation and single pulse material change can be resolved in Fig. 9(e), which allow us to conclude that the photochemical reaction is the main mechanism responsible for inducing phase changes inside hydrogel-based polymers.

 

Fig. 9. (a) DIC image of grating lines written at scan speeds of 5 mm/s, 50 mm/s, 100 mm/s and 150 mm/s. (b) DIC image of grating lines written by a rotational stage at scan speeds of 500 mm/s, 1.34 m/s and 2 m/s. (c) DIC image of grating lines written at 2.68 m/s, 3.35 m/s and 5 m/s. (d) DIC image of grating lines written at 5 m/s, 6.7 m/s and 8.37 m/s. (e) DIC image of grating lines written at 11.72 m/s on a smaller scale.

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5. Photochemical model fitting and discussion

5.1 Photochemical model fitting

Quantitative data under various writing conditions can be fitted well by Eq. (10) and the results are shown in Fig. 10. The least-squares fitting method is employed to determine the material parameter γ₂ at 5 MHz and the values of the fitted material constant γ₂ are shown in Table 1. The same material parameter γ₂ is then used to fit the experimental data at 1 MHz and 8.3 MHz. The values do not change significantly between both materials in contrast to the distinctly different material constants found in the NIR writing [31], implying a similar material response in the case of blue laser irradiation. The best fitting results yield a coefficient of determination R² greater than 96% for J + J contact lenses and greater than 93% for Contaflex for all the three repetition rates and both NAs. The R² values are not high enough due to the deviation of the photochemical model from the experimental results at high irradiation intensities.

 

Fig. 10. Phase changes induced at different conditions are fitted by the photochemical model. Quantitative data were collected from Contaflex samples at NA 0.45 (a); from J + J contact lenses at NA 0.45 (b); from Contaflex samples at NA 0.25 (c); from J + J contact lenses at NA 0.25 (d).

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Table 1. Material constants γ₂ of two materials at two NAs determined numerically by fitting the experimental data at 5 MHz with the photochemical model

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5.2 Phenomenological saturation factor and discussion

The photochemical model in form of Eq. (9) is valid only at low irradiances. In order to apply the photochemical model to the large signal region, we propose a modified photochemical model incorporating a phenomenological saturation factor that can effectively suppress the acceleration rate of the induced phase change. The phenomenological saturation factor can be attached to the material constant γ on the assumption that the saturation factor originates from material saturation due to the limitation on the number of available polymer molecules per volume that can be defragmented. From a macroscopic point of view, the phenomenological saturation factor can be exerted on the overall photochemical model and an overall saturated molecular density D_tot is embedded in analogy with saturation of an absorber or a gain medium for the case of homogeneous broadening of the absorption line [43],

The experimental data obtained from both J + J contact lenses and Contaflex samples at different writing conditions are fitted by the modified photochemical model, as shown in Fig. 11. The material constant γ₂ and the saturation phase change ϕ_s were determined by fitting the experimental data at 8.3 MHz and at 30 mW for power scaling experiments (Fig. 11(a-d) and scan speed scaling experiments (Fig. 11(e) and (f)), respectively. The values obtained from power scaling experiments are listed in Table 2. The saturation phase changes obtained from scan speed scaling experiments were calculated to be 2.86 waves for J + J contact lenses and 2.47 waves for Contaflex samples at NA 0.45, which agree well with those extracted from power scaling data. The saturation model is more reliable to reproduce the experimental data as evidenced by the increased R². The predicted saturation phase change is found to decrease with the numerical aperture, which is consistent with the experimental results. We posit that the saturation phase change is a quantity dependent on the interaction volume. In other words, the saturated phase change per volume is assumed to be invariant due to a fixed available polymer molecular density or a constant refractive index difference; however, the interaction volume expands and thus the total saturation phase change within the whole interaction volume increases as the NA decreases. Future work should attempt to identify the exact physical quantity associated with the saturation factor and to investigate the mechanisms for both the material saturation and damage. The proposed saturation model in the form of Eq. (12) might not be the only function to include the material saturation effect and other expressions for electron and hole mobilities in silicon [44] or intrinsic carrier density in non-degenerate semiconductors [45] could also be analogized and applied to introduce a saturation factor. More theoretical work needs to be done to determine the most appropriate form for describing the saturation effect.

 

Fig. 11. The modified photochemical model incorporating a saturation factor is used to fit qualitative data obtained from Contaflex samples and J + J contact lenses under different exposure conditions.

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Table 2. Material constants γ₂ and saturation phase changes ϕ_S determined by fitting the experimental results at 8.3 MHz with the modified photochemical model

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6. Conclusion

Femtosecond micromachining hydrogel-based ophthalmic materials including contact lenses, IOL and even corneal tissues has been demonstrated to be feasible and enables the possibilities of fabricating customized contact lenses and vision correction [19]. The investigation of optimum laser system parameters used in micromachining process has become more and more significant. The effects of multiple laser system parameters on the induced phase change inside hydrogel-based contact lenses have been extensively studied in this manuscript. Assembling all the quantitative results, we are able to conclude that the blue femtosecond micromachining hydrogel polymers is dominated by the two-photon absorption process. We also find the same amount of phase change can be achieved at a much smaller power with a lower-repetition-rate writing; however, too low repetition rates can cause optical damage more easily while too high repetition rates considerably increase the required power. Therefore, taking into account both eye safety issues and a significant amount (one wave) of achievable phase change, we believe an intermediate repetition rate (∼8.3 MHz) writing is favorable for femtosecond micromachining ophthalmic materials. The power damage threshold of a low NA writing can be extended significantly and accordingly the maximum achievable phase change can be enhanced greatly due to a broadened interaction volume. A photochemical model is proposed for low laser irradiation along with a modified photochemical model incorporating a phenomenological saturation factor for working at high intensities. The presence of the saturation effect helps interpret the deviation in the large signal region, the sub-linear inverse scan speed dependence of the induced phase change and the noticeable higher maximum achievable phase change at the lower NA. We count on the photochemical model to help predict the scaling behavior of the induced phase change under different laser exposure conditions and serve as a straightforward and convenient tool to increase the efficacy of femtosecond micromachining hydrogel-based polymers.