Effect of Proximity of Features on the Damage Threshold During Submicron Additive Manufacturing Via Two-Photon Polymerization

The absence of well-characterized scalable nanomanufacturing processes that are capable of fabricating macroscale parts with nanoscale features is a key bottleneck in transitioning several promising nano-enabled devices from the research laboratory to real-world adoption. The two-photon absorption-based laser direct write process [1] is a viable candidate for scalable nanomanufacturing as it enables fabrication of millimeter scale parts with submicron building blocks. This is achieved by a combination of the ability to: (i) cure a submicron volume around the focal spot in the resist via two-photon polymerization (TPP) [2–4], (ii) scan the focal spot via high-speed galvanometers [2,4,5], and (iii) stitch together several microscale-printed sections. As such, two-photon polymerization is a promising technique for additive manufacturing of complex three-dimensional (3D) structures with submicron features. Unfortunately, this process is currently of limited practical import for additive manufacturing of functional parts due to the inability to predictively fabricate high-quality parts that satisfy the desired tolerances. Currently, high part quality is achieved via a series of expensive and time-consuming empirical trial-and-error parameter selection procedures. Herein, we demonstrate that such empirical guide-rules cannot be successfully generalized over a broad operating region. Specifically, we demonstrate the presence of laser damage due to proximity effects at writing conditions that are otherwise considered “safe” for operation.

During direct laser writing, submicron volume pixel (“voxel”) features are printed in the interior of the resist material via localized polymerization that is initiated by two-photon absorption of the incident light [6–8]. A schematic of the process is shown in Fig. 1. Two-photon absorption is fundamentally distinct from the more commonly observed single-photon absorption. Although two-photon absorption may be used to initiate the same molecular transitions in the photoinitiator as those initiated by a single-photon absorption process, it is a weak and nonlinear effect [9]. Two-photon absorption occurs when two photons of twice the wavelength are near-simultaneously absorbed for the initiation of the molecular transition [10]. As the probability of simultaneous absorption of two photons is low, high light intensity of the order of TW/cm² is required to obtain any appreciable two-photon polymerization [8,11]. Therefore, it is possible to spatially restrict the two-photon absorption process to an interior subdiffraction volume by focusing a femtosecond pulsed laser to a diffraction-limited spot. By carefully selecting the laser wavelength and the resist absorptivity, one can ensure that the material polymerizes exclusively via two-photon absorption without any single-photon polymerization. For example, the commonly used ultraviolet (UV)-curable resists for single-photon polymerization are optically transparent at near-infrared (IR) wavelengths while being strongly absorbing at half the wavelength. Thus, two-photon polymerization can be performed by selecting a near-IR femtosecond laser and UV-curable resists; this combination has been extensively used in the past to fabricate submicron features via two-photon polymerization [2,11–14].

Laser damage occurs during TPP when the incident intensity is higher than that required for polymerization of the resist [15]. Laser damage results in the formation of bubbles in the resist that hinder further printing and weaken the printed structure. Thus, during printing one must ensure that the laser power and writing speed are properly selected to lie below the damage threshold. This is commonly achieved by running a series of empirical tests, wherein a set of small calibration structures are fabricated at increasing laser powers and speeds. Although such empirical tests can provide valuable estimates for the threshold, they are ineffective in accurately predicting the damage threshold over a wide operating range. For example, we have observed that the damage threshold varies over a wide range with changes in the feature density, i.e., the damage threshold is dependent on the proximity of the features. This proximity effect cannot be explained if laser damage in TPP is attributed to nondeterministic stochastic sources such as randomly distributed dust particles or contaminants in the resist. Here, we demonstrate that this proximity effect behavior arises primarily due to the changes in the absorption spectrum of the resist due to the curing process. The absorptivity of the polymerized resin at the incident near-IR wavelength increases to a non-negligible value postcuring. We have captured the effect of this additional absorption mode in terms of the proximity effect for laser damage.

Proximity effects reduce the laser damage threshold and make a feature-dense part susceptible to defects. We have recorded these defects arising out of proximity effects by capturing 3D X-ray computed tomography (CT) images of internal voids/bubbles that were generated via boiling of the resist during printing of the part (Sec. 2). We have quantified this proximity effect by tracking the effect of feature spacing on the damage threshold power at different writing speeds (Sec. 3). Finally, we have developed an analytical model to predict the proximity effect arising out of the additional single-photon absorption by the cured resin (Sec. 4). The analytical model and the empirical data may be used to identify the safe operating parameters during TPP to predictively fabricate high-quality 3D parts free of bubble defects.

We have observed the significant role of proximity effects in reducing the laser damage threshold power by recording X-ray 3D CT images of a nonhomogenous-printed part. The phase contrast CT image of the part and a solid model of the part are shown in Fig. 2. The defects are visible in the form of internal voids in the CT images that arise due to boiling of the resist at the feature-dense cavity walls. Such internal voids are not visible via surface imaging techniques such as scanning electron microscopy. However, these internal voids adversely affect the properties of the part, such as strength and stiffness, and must be eliminated when a high part quality is desired.

We have printed the part on a commercially available Nanoscribe GT laser lithography system that implements a TPP based 3D writing technique via Z-piezo scanner and high-speed X–Y galvanometric scanners. Printing was performed with a proprietary negative tone resist (IP-DIP) that is available from Nanoscribe GmbH (Eggenstein, Germany). The parameters of the writing process are summarized in Table 1. The printed part consists of three sets of cavities within a right circular cylinder. The cylinder was 125 μm tall and had a designed diameter of 55 μm. For ease of material handling during CT imaging, the cylinder was built on top of a large 750 μm tall pillar base that has a square cross section of side 125 μm. In the top cylindrical section, each set of cavities lies on a different horizontal plane and comprises three types of cavities: flat topped, concave topped, and convex topped cavities. The cavities on the top most plane visible in Fig. 2(b) are 15 μm long and 10 μm deep. The height of the cavities is 10 μm, 10 μm, and 12.5 μm for the flat, convex, and concave tops, respectively.

For 3D printing, the part was discretized and printed layer-by-layer, wherein each Z layer was spaced by 0.6 μm. A log-pile structure was generated with a line spacing of 0.4 μm by laying lines along the X-axis and then along Y-axis in alternate layers. This in-plane spacing was reduced to 0.25 μm over an in-plane distance of 2 μm at each of the cavity walls and the external wall. The design intent of this high-density wall was to provide a higher wall strength and stiffness to prevent cavity and part collapse. Printing was performed at a linear writing speed of 20 mm/s and an average laser power of 30 mW. These parameters were selected by first writing a small test structure to ensure the absence of laser damage. The absence of laser damage was also verified during printing of the first few layers that had a homogenous in-plane spacing of 0.4 μm. However, laser damage was observed in the layers with the dense cavity walls; this damage is evident from the CT images in Fig. 2. This suggests that the laser damage arises due to the proximity effect during printing of the feature-dense cavity walls.

X-ray CT imaging of the printed part was performed on the commercially available Zeiss Xradia UltraXRM-L200 system. This system has an 8 keV X-ray source and utilizes X-ray diffractive optics to image the part. The 3D image was computationally reconstructed from several two-dimensional (2D) projection images. These 2D projections were taken by rotating the part relative to a stationary source and detector. A total of 721 projections were collected over a range of 180 deg with each projection lasting for 2 mins. To correct for system drift during image acquisition, the part was marked with a 2 μm gold microsphere, which was tracked using a semi-automated approach. Projections were registered using the drift tracking and reconstructed using the in-house tomography software livermore tomography tools (LTT). The field of view was 65 μm with a resolution of 0.3 μm. Thus, only about the top 60 μm of the part is visible in the CT image (Fig. 2(c)). In Fig. 2(c), the dotted lines indicate the location of the CT cross section planes relative to each other. The line in left panel indicates the plane through which the vertical slice (right panel) has been taken, and the line in the right panel indicates the plane through which the horizontal slice (left panel) has been taken.

Computed tomography imaging was performed in the phase contrast mode. Thus, the gray scale image represents the phase shift of the X-ray beam as it passes through the material. The phase contrast mode was selected instead of the more conventional transmission mode because of the negligible attenuation of the transmitted beam; this low attenuation is caused by the low atomic number of the components of the acrylate-based resist. Distinct features in the phase contrast CT images can be identified from the feature edges that comprise a pair of adjacent bright and dark edges. The side with the bright edge corresponds to the high density-cured material, whereas the side with the dark edge corresponds to the low density air medium. Thus, the spherical features in Fig. 2(c) correspond to pockets of voids that were formed when the resist cured around the bubbles of uncured resist; voids were formed when the part was chemically developed to remove the uncured resist. For fully enclosed bubbles, a partial void is generated in the developed part due to condensation of the vapor trapped inside the bubbles. On the left panel of Fig. 2(c), the designed cavities on the right-top and right-bottom are visible because these cavities are connected to the exterior surface through a series of bubbles. During development, these two cavities were drained of the unpolymerized resist leading to the formation of air–polymer interfaces at the cavity walls. In contrast, the cavity on the left-center is not visible in the X-ray CT image, because it is still filled with the unpolymerized resin. This is because the bubbles formed at the walls of this cavity were not large enough to connect it to the exterior. Thus, material interfaces were absent at the cavity walls; without these interfaces, the walls could not be imaged in the phase contrast mode. The existence of the left-center cavity in the printed part can be indirectly verified by the presence of the bubbles formed due to proximity effects at the four corners of the expected location and orientation of the cavity.

Laser damage occurs when the energy absorbed by the resist exceeds the energy required for photo-polymerization. This could occur either when the laser power is too high or when the writing speed is too low. Thus, the damage threshold power must be characterized for a specific writing speed. The effect of writing speed and incident laser power on the quality of features is illustrated in Fig. 3. Laser damage is observed at low writing speeds and at high laser powers. In addition, the line width reduces with an increase in the writing speed or a decrease in the laser power.

The laser damage threshold power at a particular writing speed is the laser power at and above which damage in the part is guaranteed. We have evaluated this threshold power by recording the generation and growth of bubbles during printing of line features. To quantify the damage threshold for widely spaced lines, we have written lines separated by 10 μm and at varying laser power levels. During printing, we have recorded the power level at which bubbles were observed. The bubbles were observed in the real-time live images of the printing process as captured via a video camera. Representative images of bubble formation are provided in Fig. 4. The undamaged lines can be identified from the optical images as those lines that are continuous without any breaks, whereas the damaged lines are either discontinuous or have permanent solidified bubble features.

An accurate damage threshold was evaluated by performing these tests in two iterations. In the first iteration, a set of ten lines was written at the same writing speed but with the average laser power level progressively increasing by 5 mW across each line starting from the 5 mW level. The data from this test identify the threshold within 5 mW of the actual value. In the second iteration, another set of ten lines were written within the 5 mW range identified in iteration 1. The laser power level was varied by 0.5 mW across each line in iteration 2. The data from the second set accurately identify the threshold within 0.5 mW of the actual value. The results of the second iteration for different writing speeds are summarized in Fig. 4(a). The damage threshold power varies from 20.5 mW at a writing speed of 0.1 mm/s to 56 mW at 10 mm/s. As expected, the damage threshold power increases with the writing speed; this is because the net energy absorbed by a particular material point decreases with an increase in the writing speed. No laser damage is expected during writing of widely spaced lines when the laser power level is chosen to be below this damage threshold. The damage threshold power for speeds higher than 10 mm/s was not observable as the damage threshold for such speeds was higher than the maximum system power of 60 mW. It is important to note that these measured damage thresholds are higher than those observed at the writing conditions for the part imaged via X-ray CT (Sec. 2.1). This is because these two sets of experiments were performed before and after optics realignment that altered the laser beam shape while maintaining the average laser power. Despite this quantitative mismatch between the two data sets (Secs. 3.1 and 3.2 versus Sec. 2.1), there is an unequivocal evidence for the existence of proximity effect in both writing conditions.

To quantify the proximity effect, we have empirically evaluated the laser damage threshold power during writing of log-pile structures composed of alternating layers of overlapping line features (“logs”). The in-plane feature spacing was varied across the test structures; however, the feature spacing within a single structure was maintained uniform. The out-of-plane feature spacing was held constant at 0.6 μm in all of the test structures. A schematic of the test structure is shown in Fig. 5; the structures were square pillars of size 10 μm × 10 μm × 5 μm. The same two-step iteration technique was used to accurately identify the damage threshold within 0.5 mW of the actual value. The effect of in-plane feature spacing on the damage threshold is summarized in Fig. 5(a). The proximity effect is evident from the reduction in damage threshold while writing closely spaced overlapping lines. The damage threshold reduces by as much as 47% during writing of overlapping lines as compared to writing of widely spaced lines. Such overlap line writing is necessary during writing of dense features or solid structures. Thus, during writing of dense parts, the laser power and writing speed must be carefully selected so that the damage threshold is not exceeded for any feature in the part.

It is important to note here that the proximity effect is lower at low writing speeds, i.e., the change in absolute damage threshold values at spacings greater than 1 μm versus spacings less than 100 nm is lower at low writing speeds than at high writing speeds. Piezo stages with a maximum writing speed of ∼100 μm/s have been the most commonly used motion stages in past implementations of TPP. Due to this, we suspect that one of the reasons why laser damage proximity effect has not been reported in the past is because of the lack of empirical data in the high writing speed regime, i.e., at speeds approaching 10 mm/s and higher. With the advent of galvanometric motion stages in TPP systems with writing speeds exceeding 10 mm/s, accounting for proximity effects will become even more important to prevent laser damage.

As laser damage occurs due to the accumulation of excess energy, the proximity effect could be either due to an increase in the energy absorption at the exposed spot around pre-existing features or due to a decrease in heat flow out of the exposed spot. Out of these two, the second source is unlikely to be a dominant source of increased thermal damage, because the thermal conductivity of the cured resist is expected to increase with the degree of polymerization [16]. This suggests that the increased laser damage arises primarily due to the increase in absorption of energy in the vicinity of pre-existing features. The absorbed energy near pre-existing features is higher because the polymerized and cured resin absorbs single-photon near IR radiation. We have verified this hypothesis of change in IR absorptivity of the resist due to curing by performing a set of experiments that decouple the single- and two-photon absorption modes.

To verify that the near IR single-photon absorptivity of the cured resin is non-negligible, we have exposed the prepolymer component of the resist that does not contain any photoinitiator to IR laser in the vicinity of cured features and away from these features. The cured features were fabricated with IP-DIP photoresist, whereas subsequent exposures were performed in trimethylolpropane triacrylate (TMPTA) prepolymer that does not contain any photoinitiator. No laser damage was observed when regions away from the cured feature were exposed, whereas laser damage was observed in the prepolymer when the laser beam was focused on the top surface of the cured features. These experiments were performed on two different sets of cured features: (i) features that were exposed to diffuse UV light for 12 h after fabrication (bleached) and (ii) features that were not exposed to UV light (unbleached). The damage threshold for these two cases is shown in Fig. 6. As the bleached sample does not contain any appreciable amount of active photoinitiator, the observed laser damage occurs due to conventional single-photon absorption. In addition, the contribution due to the two-photon absorption mode in the unbleached sample is non-negligible and causes a reduction in the damage threshold with respect to that in the bleached sample. It was also observed that the sensitivity of damage threshold power to writing speed is higher for the single-photon absorption mode. This sensitivity to writing speed is discussed later in Sec. 4.1 within the context of the scaling laws for damage threshold.

To explain the proximity effect behavior, we have developed a first-order analytical model that captures the essential physics of the damage process in terms of scaling laws for the proximity effect. Fundamentally, laser damage due to boiling of resist occurs when the energy delivered to the resist during laser writing exceeds the energy required for boiling. The transfer of energy from the laser to the resist is determined by laser–matter interactions, and the subsequent spatiotemporal distribution of this energy in the resist is determined by the heat flow conditions in the resist. As it is not possible to generate a closed-form analytical model that fully captures these physical phenomena, we have made several simplifying approximations to model heat accumulation through lumped parameter models. Although these approximations do not capture the full complexity of these phenomena, they are sufficiently accurate to elucidate the scaling of the damage threshold proximity effect during writing of overlapping features.

Here, P is the time-averaged transmitted power of each pulse, r is the in-plane radial position, z is the out-of-plane axial position with the origin at the focal plane, and σ is the beam size. The beam size is the radial position at which the intensity falls to 1/e² times the intensity at the beam axis.

Here, the material property parameter β is the two-photon absorptivity of the material and is dependent on the concentration of the photoinitiator, type of the photoinitiator, and the wavelength of the laser.

Here, t_p is the duration of a single pulse, f_p is the pulse repetition rate, v is the laser scan speed (i.e., the writing speed), and σ is the beam radius at the axial position z away from the focal plane.

Here, σ_o is the beam radius at the focal plane, the parameters $α¯$ and $β¯$ are the area-averaged single-photon and two-photon absorptivity, and the parameter δ_a is the characteristic absorption length along the axial direction.

Here, δ_th is the thermal characteristic length over which boiling occurs, ρ is the mass density, and H is the equivalent enthalpy of vaporization of the resist (in J/kg), i.e., it is the sum of the enthalpy change due to vaporization and the enthalpy change required to raise the temperature of the resist to its boiling point.

The model for damage threshold can be used to explain these experimental observations: (i) the damage threshold increases with an increase in the writing speed and (ii) the damage threshold is less sensitive to the writing speed when absorption is dominated by the two-photon mode (P_th,∞ ∼ δ_th^0.5v^0.5) as compared to when it is dominated by the single-photon mode (P_th,∞ ∼ δ_thv). This model prediction is supported by the experimental data summarized in Fig. 6, wherein the contribution of the single-photon mode was separately quantified. The damage threshold was observed to be more sensitive to writing speed for the bleached sample that had negligible two-photon absorption. Thus, this model qualitatively explains the observed scaling of the damage threshold to writing speed. This model overestimates the sensitivity to the writing speed (in the form P_th,∞ ∼ v^0.5 and P_th,∞ ∼ v) if the dependence of δ_th on the writing speed is neglected. This is because the thermal characteristic length (δ_th) is dependent on the heat flow conditions and is expected to weakly depend on the reciprocal of the writing speed. The dependence of δ_th on writing speed arises because heat is confined to a smaller volume when the writing speed is higher. Accounting for this effect would further reduce the sensitivity of the damage threshold power to the writing speed as observed experimentally (P_th,∞ ∼ v^0.21 and v^0.16). It is important to note that during writing of 3D structures, the assumption of zero single-photon absorptivity is not valid as there is always some overlap of the laser beam with pre-existing cured features. In such a case, both single- and two-photon absorption modes must be accounted for in the inequality (6).

As shown in Fig. 7, writing of closely spaced overlapping features differs from writing of widely spaced features due to a larger fraction of the incident power that is absorbed via single-photon absorption in the cured features.

Here, g is the in-plane overlap spacing between the previously written line feature and the axis of the laser beam, w is the width of the voxel line feature, erf(x) is the Gauss error function, the first term on the right-hand side (RHS) (⁠$α¯o$⁠) is the single-photon absorptivity due to the overlapping features in the layers below the current layer, and the second term on the RHS is the single-photon absorptivity due to the overlapping line feature that lies on the layer currently being written. As the second term on the RHS depends on the in-plane overlap line spacing (g), the proximity effect arises due to overlapping features in the current layer. The change in single-photon absorptivity due to the changes in gap between the overlapping features is shown in Fig. 8. The damage threshold (P_th,C) for this case can be evaluated by substituting for the effective single-photon absorptivity in the inequality (6) to obtain

Several deductions can be made from Eqs. (8) and (9) about the nature of proximity effect. First, when the overlap spacing (g) is large relative to the beam width, both the error functions in the second term of the RHS in Eq. (8) approach unity, and the additional absorptivity is zero. Physically, this condition represents the case when the voxel feature is outside the laser beam and does not contribute to thermal damage. Second, Eq. (8) predicts that the absorptivity due to the overlapping line feature is maximum when the spacing (g) is zero, and it reduces as the spacing increases. This is consistent with the experimental observation of increase in the damage threshold power as the overlap spacing is increased (Fig. 5). Third, the absorptivity is less sensitive to the overlap spacing when the line width is higher, i.e., the same increment in overlap spacing reduces the absorptivity by a smaller amount when the line width is high. This prediction is consistent with the experimental observation summarized in Fig. 5. During the experiments, it was observed that the proximity effect is higher at high writing speeds. As line width is inversely related to the writing speed, this observation is consistent with the prediction of Eq. (8). Thus, the scaling laws represented by Eqs. (8) and (9) can be used to predict the laser damage proximity effect.

The model presented here is a lumped parameter energy balance model that captures the proximity effect by comparing the energy required for boiling of the resist to the energy absorbed by the resist. Within this lumped parameter framework, an internal consistency test can be performed by numerically comparing these two energies. Specifically, the following two tests can be performed: (i) is the laser beam powerful enough to deliver the energy required to boil a specific amount of the resist and (ii) does the resist have a physically reasonable absorptivity value to achieve boiling by absorbing a fraction of the total incident laser energy? We have verified that the model presented here satisfies these two tests by using numerical estimates for the model parameters. These model parameters are summarized in Table 2.

The energy required for boiling of the resist is represented by the right-hand side of Eq. (6). To numerically estimate the energy, one requires the thermal properties of the resist. Although the exact composition of the commercial resist used here (IP-DIP) is unknown, we have estimated the thermal properties of the resist by considering the properties of the prepolymer components of the commonly used viscous alkyl acrylate resists; these prepolymers are pentaerythritol triacrylate (PETA) and pentaerythritol tetraacrylate. Representative values for the relevant properties are: (i) boiling temperature of 205  °C for PETA, (ii) density of 1.18 g/cm³ for PETA, and (iii) specific heat capacity of 1870 J/kg-K for pentaerythritol tetraacrylate [18]. As the prepolymer PETA is unstable and potentially explosive at temperatures close to its boiling point, we can evaluate the equivalent enthalpy of vaporization (H) in Eq. (6) as the heat required to raise the temperature of the resist from room temperature to its boiling temperature and neglect the heat capacity of the liquid-to-vapor phase change. With a room temperature of 22  °C, the parameter H in Eq. (6) is evaluated as H = 342.2 kJ/kg.

Here, λ is the wavelength, and NA is the numerical aperture of the objective lens. By substituting for these values from Table 1 into Eq. (10), the beam waist is obtained as 340 nm.

The thermal characteristic length (δ_th) appearing in Eq. (9) depends on the heat flow conditions around the writing zone that may vary based on the writing conditions. For example, during writing of single widely spaced lines printed directly at the resist–substrate interface, the heat can conduct across the surface in addition to conduction to the surrounding resist. This may be particularly relevant for coated glass slides such as those used here. An accurate estimation of the thermal length requires complex spatiotemporal 3D modeling of the exact set of boundary conditions in the system. Nevertheless, we can estimate the limits of the thermal characteristic length between a set of realistic upper and lower bound. A lower bound estimate can be obtained by considering that boiling occurs only within the Rayleigh length of the beam, i.e., within the depth of focus. This is an arbitrary lower limit but corresponds to a length scale at which proximity effect would still be observed at overlap spacings equal to the spot size, i.e., at feature overlap spacings of 340 nm and higher. The Rayleigh length is evaluated as π$σo2$ /λ = 466 nm.

For a writing speed of 1 mm/s and a thermal diffusivity (α_t) of 10⁻⁷ m²/s for polymers, the upper bound for δ_th is 33 μm.

Thus, the energy density required for boiling can be estimated from Eq. 6 as: 0.06 nJ/μm <  $U¯$ < 4.53 nJ/μm.

Out of all the material points that lie on the focal plane, the ones located at the axis of the beam are exposed to the laser for the longest duration. This duration of exposure is given by: 2σ_o/v. The total number of pulses that are incident can be evaluated from the laser pulse repetition rate as: 2f_pσ_o/v. For writing speeds varying from 1 mm/s to 10 mm/s, the number of laser pulses lies between 54,440 and 5440. As the average laser power lies between 20 mW and 60 mW, the pulse energy can be obtained as: 0.25–0.75 nJ (=average power/repetition rate). Thus, the total incident energy per exposure varies from 1.36 μJ to 40.8 μJ.

To verify whether the laser is powerful enough to deliver sufficient energy within the absorption zone, we consider an absorption length of ∼1 μm. This is the axial length over which the incident energy is absorbed by the material. The total incident energy density is then 1.36–40.8 μJ/μm. Thus, the incident laser energy density is several orders of magnitude higher than the energy density required for boiling of the resist (<5 nJ/μm). This satisfies the first internal consistency test for the model.

To verify whether the single-photon absorptivity of the material is physically reasonable, we have evaluated the fraction of energy that must be absorbed by the resist to achieve boiling assuming that all of the heating of the resist occurs only via single-photon absorption. To correspond to a strongly absorbing resist, we consider the case when the energy density for damage is 4.53 nJ/μm, and the number of pulses is 54,400 with a pulse energy of 0.38 nJ corresponding to a writing speed of 1 mm/s (obtained from the threshold power in Fig. 5). Under these conditions, 0.022% of the incident light is absorbed by the resist over a distance of 1 μm. This results in an absorptivity α = 2.22 cm⁻¹. Using the Beer–Lambert's law, this absorptivity corresponds to a transmission of 64% for a 2-mm thick slab of the resist material. Comparing this with the ∼90% transmission for a 2 mm slab of clear acrylic, it can be verified that the single-photon absorptivity of the polymerized resist obtained from this model is indeed physically reasonable and lies in a range that corresponds to transparent/translucent polymer materials.

Here, N is the density of the photoinitiator in terms of molecules per unit volume, h is the Planck's constant, and ν is the frequency of the incident light. The two-photon absorptivity for boiling can be obtained by comparing Eqs. (1) and (3) and using the relationship βI = α, where I is the intensity, and α is the equivalent single-photon absorptivity obtained from the previous single-photon only absorption analysis. The intensity can be obtained as: pulse energy/(pulse duration × focal area). Thus, the intensity of the beam for pulse energy of 0.38 nJ is 1.03 TW/cm². This results in a two-photon absorptivity of β = 2.14 cm/TW. With a 0.1 wt % concentration of photoinitiator molecules in the resist, the two-photon cross section of the hypothetical resist is obtained from Eq. (12) as: σ⁽²⁾= 27 × 10⁻⁵⁰ cm⁴s/molecule. Most UV curable photoinitiators that are used in two-photon polymerization resists have a two-photon cross section less than 10 × 10⁻⁵⁰ cm⁴s/molecule [9]. Although custom photoinitiators that are optimized for a high two-photon cross section can exceed these values, such photoinitiators are often used at a lower concentration, thereby reducing the absorptivity. Thus, it is unlikely that the two-photon absorption mode alone can generate sufficient heat to boil the resist. This analysis supports our hypotheses that laser damage occurs due to a combination of the single-photon and two-photon absorption modes, and that the proximity effect arises primarily due to the dependence of the single-photon absorption on the feature overlap spacing.

As this model is based on lumped parameters, it does not explicitly capture the spatial distribution of the temperature or the localized light absorption. Accounting for these distributions is necessary to accurately evaluate the values of the lumped parameters δ_th (thermal characteristic length) and δ_a (characteristic absorption length). Absent these distributions, scaling of the damage threshold with the writing speed is difficult to ascertain. This is because these parameters themselves depend on the writing speed. For example, in our previous empirical work, we have observed that the damage threshold power for single widely spaced lines scales as the 0.47 exponent of the writing speed [19], whereas here we have empirically observed a scaling of 0.21 (Fig. 4). The difference in scaling with power is likely due to the different substrates used for printing (glass for previous work versus indium tin oxide coated glass for this work). To accurately capture the spatial distribution of the temperature around the exposed zone, one must model the heat flow in the writing zone. Similarly, to accurately predict the absorption length, one must model the light absorption in the spatiotemporally varying laser–matter interaction zone. Such detailed models will not only improve the accuracy of these lumped parameters, but will also elucidate the accurate scaling of the damage threshold with the writing speed. Nevertheless, at a constant writing speed, the proximity of features will still adversely affect the damage threshold as predicted by Eqs. (8) and (9) under any set of heat flow conditions.

Although the model presented here is based on the log-pile architecture for assembled cuboid structures, it is also applicable for other assembled part shapes. This is because the shape of the elementary feature (voxel) does not change when the shape of the assembled structure is changed. The model is based on overlapping the elementary voxel oval shapes. As the voxels always have the same oval axial cross section in the serial writing scheme, any set of two or more overlapping features will demonstrate proximity effect. The model presented here is based on the assumption that the laser travels along a straight line path with a constant speed. This corresponds to a layout of the voxel features in a Cartesian grid. When the feature layout is changed to a non-Cartesian grid (such as a circular cylindrical grid), the velocity of the beam continuously changes. However, when the beam is moved along a nonstraight line (such as a circular arc), the piece-wise straight-line assumption with a constant speed is still valid as long as the radius of curvature is substantially higher than the laser beam spot size and the thermal diffusion length. This straight-line assumption breaks down at sharp corners with very small radii of curvature. In such instances, higher proximity effect is expected due to the enhanced overlap with pre-existing features. It is important to note that the proximity effect vanishes when the feature spacing increases to a few microns. Thus, the layout of structures (such as arrays of cuboids or pillars) does not influence the proximity effect if the structures are themselves separated by more than a few microns. As such, proximity effect is a phenomenon that is dominated by the layout of overlapping voxels. For a quantitative prediction of the proximity effect, one must account for the voxel overlap geometry.

A demonstration of the applicability of the data generated here to predict proximity effect for other 3D structures is illustrated in Fig. 9. We printed a set of three truncated cones (Fig. 9(b)) using IP-DIP resist at three different laser powers (12.5 mW, 20 mW, and 25 mW) and at a constant writing speed of 2.5 mm/s. Each truncated cone was 10 μm tall with a base diameter of 30 μm and a top diameter of 20 μm. These cones were discretized by sequentially printing 2D layers of circular cross sections separated by 0.6 μm along the Z direction. In each 2D layer, the outer annulus of thickness 6 μm was discretized in the form of concentric rings spaced by a variable radial pitch (Fig. 9(c)). The pitch was varied over three levels: 90 nm, 250 nm, and 450 nm (each column in Fig. 9(a)). The central region of the cones was discretized in a log-pile Cartesian layout with a spacing of 450 nm. Such heterogeneously discretized structures with varying feature spacings and layout orientations may be required if structural anisotropy through variable surface roughness is desired. By interpolating the data in Fig. 5 for the writing speed of 2.5 mm/s, the predicted damage threshold at a spacing of 450 nm is 29.5 mW, whereas at 90 nm spacing is 20.25 mW. Thus, no damage is expected in any of the cones printed at the lowest power of 12.5 mW (bottom row in Fig. 9(a)). Similarly, no damage is expected for the 450 nm cones printed at any power (right column in Fig. 9(a)). In addition, one expects damage to occur in the densest cone (90 nm spacing) printed at a power of 25 mW. All of these expectations are supported by the experiments summarized in Fig. 9(a). Thus, the data generated here can be used to select the writing parameters to print a variety of complex 3D structures.

It is important to note that all of the experiments summarized here have been performed at the same out-of-plane layer spacing (0.6 μm). We have empirically observed (data not presented here) that the damage threshold also depends on the layer spacing, i.e., the proximity effect also exists for the out-of-plane Z spacing. Within the framework of the model presented here, this out-of-plane proximity effect has been captured by the $α¯o$-lumped parameter in Eq. (8). However, for a more generalized model, one must expand this term into a Z-spacing-dependent single-photon absorptivity term.

Although the model presented here does not capture the effect of beam shape for non-Gaussian shapes, the schematic presented in Fig. 7 can be used to qualitatively estimate the influence of beam shape on proximity effect. If a top-hat beam with uniform intensity profile is used instead of a Gaussian beam, it is expected that there would be minimal spacing-dependent proximity effect at very low and very high overlaps. This is because in such cases the entire feature would either lie within or outside a uniform intensity beam. At intermediate overlap spacings, the overlapping feature would only partially lie within the beam and would lead to a spacing-dependent proximity effect. This proximity effect behavior is substantially different from the case of Gaussian profile presented here in which the proximity effect is large at low overlap spacings. In reality, the beam profile lies in between these two extremes and must be characterized if an accurate qualitative and/or quantitative prediction of proximity effect is desired.

Herein, we have quantified the laser damage that occurs due to the proximity of features during two-photon polymerization. In addition, we have shown that the damage threshold is deterministic and demonstrates distinct trends with changes in the writing process parameters. In general, proximity effect reduces the threshold power for laser damage during writing of closely spaced overlapping features. Unfortunately, it has been experimentally observed that proximity effect is structure-dependent and varies with the writing speed. As a result, an empirical calibration of laser damage at a select few intermediate overlap spacings and/or writing speed does not provide a reliable estimate of the lower bound of the damage threshold. Instead, one must calibrate the damage threshold over a wider range of overlap spacings and writing speeds. We have identified that this proximity effect arises primarily due to an increase in the single-photon absorptivity of the cured photopolymer. In addition, we have found that these proximity effects increase with an increase in the writing speed. As scalability of manufacturing is fundamentally limited by the ability (or inability) to write at high speeds, the influence of proximity effect on part damage is an important consideration for scalability of two-photon polymerization. The empirical data and scaling model generated here can be used to properly select the writing parameters to prevent laser damage during printing of dense features, thereby enabling high-quality additive manufacturing of millimeter scale parts with submicron building blocks.

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344. Funding was available via Laboratory Directed Research and Development project #16-ERD-006 (Document #LLNL-JRNL-694037). S. K. S. thanks Dr. James Oakdale at LLNL for providing the TMPTA prepolymer and for helpful discussions on two-photon lithography and thanks Dr. Jack Campbell at the University of Nebraska–Lincoln for helpful discussions on laser damage mechanisms.