a Time-ordered graph for the absorption of two photons of the same mode, in which the graph time flows upwards, the vertical line represents the change taking place in the molecule during the process, and the wave lines represent the photons with the mode (??, k) (?0 represents the time interval between the arrival of two photons, ?? denotes the lifetime of virtual state, i,j,k stands for different moments, o,r,m represents different moments within the molecule). b Schematic representations of the energy-level diagram of the two-photon absorption process exciting an electron from the ground state to an excited state passed through an intermediate virtual state under the irradiation of photons with wavelengths of 400 nm and 517 nm, respectively (Δ?~?? represents the energy difference, g denotes the initial state, i denotes the intermediate state, and e denotes the final excited state). c The relationship between intrinsic lifetime of the virtual state (??), and the energy difference between the photon energy and the stationary energy of the appropriate low-lying allowed singlet state according to Eq. (1).

The virtual state does exist; however, it does not remain populated long enough for the second photon to interact with the molecule before the virtual state “decays”. Therefore, the molecule can absorb two photons arriving at different times simultaneously when the time interval t₀ between the arrival of the two photons is less than the virtual state lifetime, ??, as illustrated in Fig. 1a. An estimation of the intrinsic lifetime of the virtual state, ??, can be obtained using Heisenberg’s uncertainty principle and the single intermediate state approximation³⁴,

where h is Planck’s constant and Δ?~?? is the energy difference between the photon energy of ?? and the stationary energy of the lowest excited state of the molecule (E₁). The values of ??, calculated using Eq. (1) with the absorption cut-off wavelength of the photoresist AR-N-7520 utilized in this study, were confirmed to be 0.8 fs and 0.3 fs for incident laser wavelengths of 400 nm and 517 nm, respectively, as depicted in Fig. 1b. The relationship between ?? and Δ?~?? is illustrated in Fig. 1c, clearly demonstrating that the values of ?? are significantly dependent on Δ?~??.

The probability of TPA is critically influenced by the spatial and temporal distribution of incident photons from a tightly focused femtosecond laser pulse. Each photon can be treated as a discrete event, with its position adhering to a Poisson distribution within the focal spot. Figure 2a illustrates a model depicting the interaction between a tightly focused photon stream, characterized by a repetition frequency f, pulse width Γ, and photosensitive molecules within the photoresist. Considering the time-dependent mechanism of TPA, we hypothesize that two photons, each with energy ?ν where ?? < E₁ ≤ 2??, continuously interact with the same molecule within the time interval ?? (Fig. 1a). This interaction facilitates the effective TPA (eTPA) of the molecule. The time-averaged number of eTPA at position r and time t is given by

where ??,? and ??,? are the single-photon absorption coefficients from g to i states and i to e states, respectively. ?(?,?) and ?(?,?−?0) represent the energy flow density separated by time t₀ at position r. ?(?,?)/(??/?)=?(?,?) is defined as the photon flow density. The precise spatial position of a photon as an individual particle colliding on the focal plane is uncertain. However, the distribution of incident photons can be represented by the light intensity distribution of a tightly focused laser spot, as calculated by the point spread function (PSF) (for details on the light intensity distribution, see Supplementary Note 2 and Supplementary Fig. 1). Similarly, the exact timing of an individual photon reaching the focal plane within the pulse duration Γ is also uncertain, though the hyperbolic secant function (HSF) can describe the temporal profile of the femtosecond laser pulse. Consequently, the potential distribution of eTPA under few-photon irradiation with a femtosecond laser pulse must incorporate both spatial and temporal uncertainties associated with the photons in the pulse.

Fig. 2: Spatial distribution of fpTPA.

a Schematic representations of two-photon absorption of molecules irradiated by the photon flux of a pixel light field from femtosecond pulses (? denotes the repetition rate, and Γ represents the pulse width). b Relationship between both numbers of triggerable eTPA (N_eTPA) and inputted photons (N_pulse) for a single pulse. c Relationship between N_eTPA and pulse width of laser (Γ). Calculated distributions of eTPA on the focused laser spots using the wavelengths of 400 nm (d) and 517 nm (e) after 700 pulses irradiated with N_spp = 6000. The calculated distribution of eTPA density in the ring located at a distance ‘r’ from the center within the focused spot with wavelengths of 400 nm (f) and 517 nm (g) (N_spp = 6000, N_pulse = 700). The blue-shaded region indicates the spatial region with a radius of r = 8 nm.

We employ the Monte Carlo method to simulate the spatial and temporal stochastic processes of photons within a femtosecond pulse, coupled to a focusing system with a numerical aperture (NA) of 1.49 (for detailed quantum model, see Supplementary Note 3 and Supplementary Fig. 2). Each focused photon beam originates from a pixel in the graphics generator, such as DMD, and the sampling area on the focal plane is defined as 3 nm × 3 nm in the simulation. Subsequently, the number of eTPA (N_eTPA) is calculated by integrating the spatial and temporal methods as described above (see Supplementary Note 4-7 for details).

The triggerable N_eTPA increases quadratically with the N_spp at wavelengths of 400 nm and 517 nm (Fig. 2b), consistent with the I² relationship. The shorter pulse width of the femtosecond laser enhances N_eTPA (Fig. 2c) by temporally increasing the probability of effective second photon absorption. Our simulations indicate that achieving reliable eTPA a single pulse requires approximately 1000 and 1800 photons (N_spp) at wavelengths of 400 nm and 517 nm, respectively, with a pulse width of 238 fs. When the pulse width narrows to 100 fs, the required N_spp decreases to 400 and 1000, aligning with the temporal distribution of photons in an ultrashort pulse laser. The N_eTPA at wavelengths of 400 nm is nearly seven times greater than at that of 517 nm when using the same pulse width (refer to Supplementary Fig. 3 and Supplementary Table 2). Reducing the pulse width from 238 fs to 100 fs results in a 2.4-fold increase in N_eTPA. This suggests that shorter pulse widths significantly enhance the probability of triggering eTPA. The efficiency of photon conversion to N_eTPA depends on ??; longer ?? values yield higher N_eTPA. This correlation with Δ?~?? (Fig. 1c) implies that incident photon energy closer to E_lowest likely increases N_eTPA due to extended ??.

The spatial and temporal stochasticity of photons at the focal spot determines the potential distribution of N_eTPA. Under few-photon irradiation, the randomness in the distribution of eTPA decreases as N_spp increases, as illustrated in Supplementary Fig. 4. A lower N_spp results in a higher variance coefficient for the probability of eTPA occurrence (Supplementary Table 1), attributable to quantum random noise (Supplementary Fig. 4). Nonetheless, while pulse accumulation increases N_eTPA, it does not affect the efficiency of triggerable eTPA for each pulse (Supplementary Fig. 5). Next, we focus on the spatial distribution of eTPA. Typical examples with an N_spp of 6000 and an irradiated pulse number (N_pluse) of 700 are shown in Figs. 2d and 2e (and Supplementary Fig. 6) for wavelengths of 400 nm and 517 nm, respectively. The statistical densities of N_eTPA (d_eTPA) are presented in Figs. 2f and 2g. We calculated the N_eTPA within a circular belt of 4 nm width and divided it by the area of the belt. The d_eTPA sharply decreases from the center of the focal spot, reaching approximately half its value at a radius of 8 nm, independent of wavelength (Fig. 2f-g). This result indicates that the resolution of TPA under few-photon irradiation can significantly surpass the resolution of the employed wavelength.

Two-Photon Optical Projection Nanolithography

To evaluate the effectiveness of our proposal concept and spatiotemporal model, we conducted TPDOPL using a femtosecond pulse laser and a DMD (Fig. 3a, Supplementary Fig. 7). The DMD, with a megapixel-resolution projection layout of arbitrary features, is irradiated by a flat-top beam and focused onto a photoresist film on a cover glass using an oil-immersion objective lens (Nikon, 100×, NA 1.49). The optimal placement angle was determined based on insights gained from our previous studies³⁵ on the diffraction efficiency of the DMD (Supplementary Fig. 7b). We chose a commercially available non-chemically amplified (non-CA) negative photoresist (AR-N 7520) (phenolic resin acts as the matrix, with bisazide compounds serving as cross-linking agent) because its degree of polymerization, driven by the stepwise photopolymerization mechanism, is easily controllable and quantifiable for N_eTPA under few-photon irradiation. This resist has an absorption peak at 323 nm and an absorption cut-off wavelength of 353 nm (Supplementary Fig. 8), ensuring that only TPA occurs when using femtosecond pulse lasers at both 400 nm and 517 nm wavelengths. Note that the number of excitable molecules in the photoresist by TPA should be less than the calculated N_eTPA from the proposed model. The calculated N_eTPA predicts the possible opportunity and distribution of eTPA under photon irradiation from the viewpoint of incident photons, but it ignores the molecular concentration, distribution, and quantum yield of TPA in the photoresist. Furthermore, the conversion efficiency from excited molecules by TPA to the practically initiated coupling reaction between molecules should be considered. The number of reaction sites of the photosensitive molecules is ultimately limited by the final absorbed N_eTPA and their quantum yield to initiate the reaction.

Fig. 3: Line width narrowing mechanism for achieving nanoscale resolution.

a Schematic of the TPDOPL system with few-photon irradiation including simulation of photon distribution, N_eTPA distribution and reaction site number of the photosensitive molecule distribution under low-photon irradiation (N_spp = 1.25 × 10³, N_pulse = 1 × 10⁴). b Schematic for the narrowing mechanism of the stepwise photopolymerization under femtosecond pulse irradiation. The polymerization degree is modulated by pulse number and photon density. c SEM image of the polymer line irradiated by N_spp = 5.14 × 10³ (1.97 fJ/(pulse·pixel)) with N_pulse = 8.5 × 10⁷. d The simulated distributions of eTPA with N_spp = 5.14 × 10³ and N_pulse = 8.5 × 10³. e SEM images of the polymer lines irradiated under different N_spp with N_pulse = 6 × 10⁷. The magnified SEM image is the polymer line irradiated with N_spp = 6.52 × 10³ (2.51 fJ/(pulse·pixel)) and N_pulse = 6 × 10⁷. The smallest feature size is 26 nm and the average line width is 43 nm with a standard deviation of 7 nm and roughness of 3-4 nm. f The relationship of the line width with a single pixel array as a function of photon flux density. g The relationship of the line width exposure with a single pixel array of irradiated pulse numbers. Error bars represent mean ± SD based on 10 independent measurements for each data point.

We utilized incident light with an average power of 1 mW after passing through the objective lens as an illustrative example, specifically with parameters N_spp = 1250 (0.48 fJ/(pulse·pixel)) and N_pulse = 1 × 10⁷. The distribution of N_eTPA for a line composed of one pixel demonstrates a notable 24% reduction in full width at half maximum (FWHM) compared to the photon distribution (Fig. 3a). According to photopolymerization theory^36,37, the relationship between the concentration of photosensitive molecules (M) excited by TPA and the photon distribution is nonlinear. As the reaction step is repeated, the molecular weight at the center of the exposure field increases exponentially due to the chemical cross-linking reaction. The crosslinking degree of the monomers is controlled by the exposure dose^38,39. The concentration distribution of photosensitive molecules involved in the reaction is illustrated in Fig. 3b (Supplementary Note 3 for more details). The exponential increase in molecular weight leads to faster gelation at the exposure field’s center, forming insoluble polymer networks more quickly than in the surroundings. Consequently, the superposition and coordination of optical and chemical nonlinearity can effectively reduce the feature size in TPDOPL under few-photon irradiation.

To investigate the effectiveness of the proposed spatiotemporal model for TPDOPL under few-photon irradiation, we fabricated separate lines using our TPDOPL system with a femtosecond laser wavelength (λ) of 517 nm and pulse width of 238 fs. Using a single-pixel DMD layout, a line with an average width of 41 nm and a minimum feature size of 28 nm (Fig. 3c) was achieved under the irradiation of a total incident photon number per pixel of 4.37 × 10¹¹ (0.167 μJ) with accumulation N_pulse of 8.5 × 10⁷ pulses containing N_spp of 5.14 × 10³ (1.97 fJ/(pulse·pixel)). Correspondingly, we calculated the eTPA distribution using the same photon flux as the experimental result in Fig. 3c but only performed 8.5 × 10³ pulses in simulation. The simulation result shown in Fig. 3d indicates that the eTPA distribution is concentrated in a central area of about 30 nm. This validates the effectiveness of our spatiotemporal model for predicting the feature size of TPDOPL under few-photon irradiation.

Photon irradiance density and the accumulated pulse numbers critically influence the line width of TPDOPL. By decreasing N_spp from 1.12×10⁴ (4.30 fJ/(pulse·pixel)) to 6.52 × 10³ (2.51 fJ/(pulse·pixel)), the average line width of the polymer line was reduced from 164 nm to 43 nm under the accumulation of 6 × 10⁷ pulses, achieving a minimum feature size of 26 nm (1/20 λ), as shown in Fig. 3e. The N_eTPA under different N_spp irradiations can be observed in Supplementary Fig. 10. The relationship between the polymer line width and photon irradiance density is depicted in Fig. 3f, indicating that the feature size can be reduced by decreasing the photon irradiance density. However, lower photon irradiance density may increase line roughness due to quantum noise, which can increase edge roughness for fine lines (Supplementary Fig. 11). On the other hand, increasing the accumulation of pulses with a fixed photon flux density leads to a widening of the line width, as shown in Fig. 3g.

Another significant aspect pertains to periodic lines in photolithography, which determine the potential feature density achievable in device applications. Our setup corresponds to “the general Sparrow criterion” for parallel projection lithography, where d = λ/NA. Further detailed in Supplementary Note 13 and Supplementary Fig. 12. Generally, the minimum distinguishable period between adjacent lines is dependent on the wavelength and determined by the equation HP (half pitch) = 0.5 λ/NA, following the Sparrow criterion⁴⁰. When the design pattern period is less than the minimum resolvable distance between two lines, double patterning lithography (DPL) can overcome this problem⁴¹. For instance, at λ = 517 nm and NA = 1.45, the criterion yields an approximate value HP_limit of 174 nm. We designed a line array using the DMD pixel period of 7.56 µm combined with 2 pixels on and 1 pixel off periodically (Supplementary Fig. 13a), corresponding to a period of 226.8 nm. Using irradiation conditions with N_pulse = 6 × 10⁷ and N_spp = 1.52 × 10⁴ (5.84 fJ/(pulse·pixel)), the lines were indistinguishable (Supplementary Fig. 13. c). We efficiently utilized the flexibility of TPDOPL by using a DMD as a digital mask, enabling in-situ digital multiple exposures (iDME) to print dense features without being constrained by the diffraction limit. It’s a digital dual-exposure technique that eliminates the need for alignment to enhance lithographic resolution in TPDOPL. Using computer-controlled DMD to generate low spatial frequency, sparse ‘digital mask’ patterns and alternating dual exposures, we double the density of nanopatterns. Since the spacing between DMD micromirrors is fixed, alignment errors are eliminated, making multiple exposures on a single photoresist coating feasible without physical mask alignment steps — surpassing the diffraction limit achievable with single exposures in traditional lithography. Exploiting DMD characteristics, two split layouts with a period of 2’p’ are sequentially loaded in situ for double exposure, achieving an exposure result with a period of ‘p’, as depicted in Fig. 4a. Under twice alternating exposure of N_pulse = 6 × 10⁷ and N_spp = 8.53 × 10³ (3.28 fJ/(pulse·pixel)), we successfully achieved a dense line array with a period of 210 nm (0.41λ, HP = 105 nm ≈ 0.3 λ/NA< HP_limit), a linewidth of 150 nm, and a gap spacing of 60 nm, as shown in Fig. 4b, surpassing the diffraction limit. Further detailed in Supplementary Note 13 and Supplementary Fig. 12.

Fig. 4: In-situ Digital multiple exposures (iDME) and the applications for dense structure patterning.

a Schematic diagram of in-situ digital double exposure. b SEM image of dense line patterns with a period of 210 nm via iDME. c Original mask layout (Mask 0) of chip metal layer. The local layout period is smaller than the diffraction limit, i.e. hp < λ/2NA. d Photoresist pattern by one exposure used Mask 0. In two hotspot areas, the proximity effect is obvious and adjacent lines cannot be distinguished. e Two independent mask layouts (Mask 1, light green; Mask 2, light blue) of chip metal layer by disassembling the original mask in (c). There is no forbidden period (hp < λ/2NA) in either layout. f Photoresist pattern by double exposure. The adjacent lines are distinguishable because proximity effects can be avoided by multiple exposures.

Taking advantage of TPDOPL-iDME, we can achieve distinguishable dense structure patterning. When the pitch is less than 5 pixels, a single exposure cannot meet the resolution consistent with the design pattern (Supplementary Fig. 14). Supplementary Fig. 9 illustrates the system’s capability for high-throughput operation, enabling uniform exposure of 80 × 100 µm² in a single exposure field and achieving a fabrication rate of 1 × 10?³ mm²/s. Figure 4c shows a typical circuit layout selected from a commercial chip design, including isolated and dense lines with a width of 3 or 7 pixels and intervals of 1 and 2 pixels between lines (Supplementary Fig. 15). We employ algorithms⁴² to strategically distribute polygons with interspacing distances below 2 pixels across distinct sub-masks, optimizing their arrangement for TPDOPL-iDME. SEM images show that direct single exposure causes indistinguishable results in dense line areas (Fig. 4d). By splitting this layout into two (Fig. 4e) and performing our TPDOPL-iDME approach, we successfully achieved the expected circuit patterning (Fig. 4f). The dense lines are clearly distinguished, and the periods agree well with the design. Furthermore, by optimizing exposure parameters and layout design for TPDOPL-iDME, line width, period, and gap distance can be controlled for finer and denser feature patterning.

Optical devices with curved and circular microstructures have been fabricated using TPDOPL, such as patterns including arrayed waveguide gratings and micro-ring resonators⁴³. The radius of the ring affects the value of the free spectral range, and the gap or spacing between the guide and the ring affects the coupling ratio between the waveguide and the ring⁴⁴. Through layout design and the TPDOPL-iDME method, we can fabricate micro-ring filters with varied radius pitches. The widths of the circular rings can be adjusted from 220 nm to 346 nm by increasing N_pulse under the irradiation of N_spp = 8.53 × 10³ (3.28 fJ/(pulse·pixel)) (Supplementary Fig. 16a). We patterned the line waveguides, followed by the fabrication of circular rings with different diameters (Fig. 5a), leveraging TPDOPL-iDME. The gap distances between the line and circular rings can be finely adjusted from 66 nm to 480 nm (Supplementary Fig. 16b), optimizing the structures and improving the properties of photonic resonance devices.

Fig. 5: Examples of TPDOPL with iDME for photonic and biological circuits.

a Photoresist pattern of microring waveguide with the adjustable gap at the scale of hundreds of nanometers (d₁ = 2d₂ = 15 μm). The insert shows several microring waveguide patterns with different gaps of 283 nm, 204 nm, 245 nm and 104 nm. b Photoresist pattern of biological microfluidic channels with the cross-scale feature structure. The structural feature scale covers the range of 120 μm to 70 nm. The maximum exposure structure size in a single field is 120 × 60 μm².

The flexibility of TPDOPL-iDME allows us to create arbitrary patterns with various sizes, shapes, and densities, applicable not only in microelectronics and microphotonics but also in microfluidics^45,46. Microfluidics in microbiology offers an in vitro platform for interactions among diverse cell types, enabling real-time observation and assessment of reaction processes⁴⁷. We designed a rectangular module to substitute the cell chamber and a circular module to replace the cell secretion chamber, with channels of varied sizes to facilitate the addition and observation of multiple cell types and their reactions⁴⁸. Figure 5b shows complex patterns of biological microfluidics fabricated by TPDOPL-iDME, with a total layout size of 120 × 60 µm². The design includes square cell incubators (3 × 3 µm²), rectangular cell chambers (2.8 × 6 µm²), and circular cell collectors with micrometer and sub-micrometer scales are connected by different channels with widths from 70 nm to 800 nm (Fig. 5b iii), effectively carrying and separating viruses of different sizes. Most biomolecular analytes are below microns in size⁴⁹, especially foreign objects such as viruses⁵⁰, which are usually 20-300 nm in size. Cross-scale biological microfluidics, from micrometer to nanometer, hold promise for providing research platforms for diagnostic and therapeutic methods for viruses like the coronavirus.