Efficient non-degenerate two-photon excitation for fluorescence microscopy.

Non-degenerate two-photon excitation (ND-TPE) has been explored in two-photon excitation microscopy. However, a systematic study of the efficiency of ND-TPE to guide the selection of fluorophore excitation wavelengths is missing. We measured the relative non-degenerate two-photon absorption cross-section (ND-TPACS) of several commonly used fluorophores (two fluorescent proteins and three small-molecule dyes) and generated 2-dimensional ND-TPACS spectra. We observed that the shape of a ND-TPACS spectrum follows that of the corresponding degenerate two-photon absorption cross-section (D-TPACS) spectrum, but is higher in magnitude. We found that the observed enhancements are higher than theoretical predictions.


Introduction
Two-photon microscopy [1] has had an enormous influence on biological studies providing a tool for high-resolution imaging hundreds of microns within live tissues. In neuroscience, two-photon imaging has been employed to trace neuronal connections [2], monitor biochemical signaling pathways [3], map protein distributions within individual cells and across cell populations [4], report gene expression [5,6], image oxygenation [7,8] and blood flow [9], trace neuronal circuits and image neuronal activity [10], and more. These studies relied on protein-based or synthetic fluorescent reporter molecules (hereafter referred to as fluorophores) suitable for two-photon excitation (TPE) [11].
In conventional two-photon microscopy, a fluorophore is excited by the simultaneous absorption of two photons of the same energy within the near-infrared (NIR) spectrum, derived from the same pulsed laser beam. This is called degenerate two-photon excitation, D-TPE. Alternatively, the same energy needed for the transition to the excited state can be delivered via absorption of two photons of different energy (i.e., different color) [12]. This non-degenerate two-photon excitation (ND-TPE) regime has been explored in two-photon excitation microscopy aiming to extend the excitation wavelength range [12][13][14], reduce out-of-focus fluorescence [15][16][17][18], increase spatial resolution [19,20], and increase penetration depth in scattering media [21,22].
The probability of two-photon excitation, quantified as the excitation cross-section, is an important parameter that, for a given fluorophore concentration and imaging conditions, translates into the laser power required for illumination [23]. However, a systematic study of non-degenerate two-photon absorption cross section (ND-TPACS) spectra of biologically relevant fluorophores, as a function of different combinations of excitation wavelengths, is missing. In addition, for deep imaging in biological tissue (e.g., live brain imaging), the choice of excitation wavelengths needs to be informed by considering the spectrum of water absorption and associated tissue heating.
Here, we performed ND-TPE spectroscopy of two fluorescent proteins (enhanced green fluorescent protein (EGFP) [24] and monomeric Kusabira orange 2 (mKO 2 ) [25]) and three organic dyes (Fluorescein [26], Coumarin 343 [27] and Sulforhodamine 101 (SR101) [28]) using a sensitive fluorescence excitation technique described in our recent study [29]. Further, we explored water absorption and tissue scattering as additional considerations for the selection of the excitation wavelengths for efficient ND-TPE, and compared tissue heating under different excitation regimes using heat conduction simulations.

Results
We quantified the ratio of the non-degenerate two-photon absorption cross section (ND-TPACS) to the best achievable degenerate two-photon absorption cross section (D-TPACS) within the equivalent range of the total photon energy (referred to thereafter as peak D-TPAC). In this way, we mitigated the need to measure the collection efficiency of our system at the expense of quantifying the absolute value of the ND-TPACS (see Appendices A, B, and C for experimental setup, spectroscopy protocol, and sample preparation). Plotting ND-TPACS normalized by peak D-TPACS as a function of near-infrared (NIR) and infrared (IR) wavelengths, λ NIR and λ IR respectively, generated a 2-dimensional ND-TPACS spectrum for each examined fluorophore within the available tuning range of the system ( Fig. 1(a)). The highest ND-TPACS was obtained when the sum of NIR and IR photon energies matched the ground-to-excited state transition energy. In Fig. 1(a), the excitation energy isocline, a line connecting points with equal total energy (E NIR + E IR ) on the 2-dimensional spectrum plot corresponding to the transition energy, is overlaid on the ND-TPACS spectrum for each fluorophore.
In Fig. 1(b), normalized D-TPACS of the five measured fluorophores along with ND-TPACS normalized to the peak D-TPACS are plotted as a function of the equivalent degenerate wavelength λ D , where 2hc/λ D = hc/λ NIR + hc/λ IR (h is the Planck constant, c is the speed of light in vacuum, and we have assumed equal refractive indices). In this way, we generated a linearized ND-TPACS spectrum that can readily be compared to the corresponding D-TPACS spectrum. Independently measured D-TPACS spectra were similar to the previously reported spectra of the fluorophores [23,[30][31][32]. Several combinations of IR and NIR wavelengths correspond to the same equivalent degenerate wavelength (energy isoclines). Therefore, we report several ND-TPACS values (red dots) for each equivalent wavelength in Fig. 1(b). For each measured fluorophore, the shape of the linearized ND-TPACS spectrum was highly correlated with the D-TPACS spectrum for the longer wavelength region ( Fig. 1(b)). However, we observed different amounts of enhancement, as indicated by different range of ND-TPACS values across different fluorophores. The measured fluorescent proteins have a larger spread in ND-TPACS values compared to the measured organic dyes. This might be because fluorescent proteins have a more complex molecular structure. Further studies are required to identify the mechanism behind this phenomenon.
In all examined cases, we observed an enhancement in ND-TPACS (red dots) compared to D-TPACS (black curves); a few red dots below the black curves fall well within our experimental measurement error (15% relative error, as shown in Appendix B). The observed enhancement in ND-TPACS, compared to D-TPACS, is partially due to the resonance enhancement effect [33][34][35][36]: when the energy difference between the high-energy incident photon ( ω NIR in our experiment, where ω NIR = 2πc/λ NIR is the frequency of the NIR beam) and that of the nearest showing the dependence of the normalized ND-TPACS on NIR and IR wavelengths. The ND-TPACS at each combination of wavelengths was normalized by the peak D-TPACS value of the fluorophore. The isoclines corresponding to the ground to excited state transition energy (hc/λ NIR + hc/λ IR = 2.66 eV for EGFP, 2.6 eV for mKO2, 2.7 eV for Fluorescein, and 2.76 eV for SR101) are overlaid as dashed black lines. b) ND-TPACS normalized by the peak D-TPACS as a function of the equivalent degenerate wavelength 2/λ D = 1/λ NIR + 1/λ IR are shown in red. Along any energy isocline, multiple excitation wavelength combinations within our tuning range sum to the same total energy and, therefore, the same equivalent wavelength. Thus, for each λ D value we report several values of ND-TPACS (red dots). Independently measured D-TPACS normalized by its peak within the equivalent range of the total photon energy (2.3 − 2.9 eV) are shown in black. The black dashed vertical line indicates the position of the peak D-TPACS used for the normalization procedure (930 nm for EGFP, 950 nm for mKO2, 920 nm for Fluorescein, 860 nm for Coumarin 343, and 900 nm for SR101). These wavelengths correspond to the energy isoclines shown in panel (a). one-photon allowed state is decreased, the probability of ND-TPE increases due to a longer lifetime of the reached virtual state as compared to the virtual state corresponding to D-TPE. In other words, the probability of two-photon absorption is proportional to the lifetime of the virtual intermediate excited state, which is inversely proportional to the energy difference between the closest one-photon allowed state and the virtual state [33][34][35][36]. This energy difference is called detuning energy ∆, which is maximal in the case of D-TPACS, where the virtual state is in the middle between ground and first excited state. Therefore, for excitation to a specific excited state, D-TPA has a lower probability of occurrence compared to ND-TPA. The detuning energy, in case of excitation into the lowest excited state, is equal to the energy of the low-energy incident photon ω IR , where ω IR is the frequency of the IR beam. Since all five measured fluorophores are non-centrosymmetric, two-photon absorption is allowed to the lowest excited state. Therefore, we are particularly interested in the case where one of the incident photons approaches the resonance with the lowest one-photon allowed state, known as intermediate state resonance enhancement (ISRE). Using the 2-level essential-state approximation of the sum-over-states (SOS) model, we find that the relation between ∆ = ω IR and ISRE 2 for non-centrosymmetric fluorophores is given by the following formula (see Appendix D): where we have assumed that 2 ω D = ω NIR + ω IR or, in other words, ISRE 2 is the fractional change of ND-TPACS relative to D-TPACS at the equivalent degenerate frequency. From Eq. (1), one finds that ISRE 2 increases asymptotically as the energy of the IR photon approaches zero, ω IR → 0, or, equivalently, when the energy of the NIR beam approaches the transition energy, ω NIR → 2 ω D . In Table 1, we show experimental and theoretical ISRE 2 values calculated using Eq. (1) for four fluorophores, where the measurements were performed along the isoclines corresponding to the respective lowest excited state (see Fig. 7 for the graphical representation of Table 1). Assuming a constant total transition energy, ISRE 2 increases by decreasing the detuning energy, as expected. We observe that, within our estimated 15% relative experimental error , most of the experimental values of the ISRE are higher than those predicted by 2-level approximation of the SOS model given by Eq. (1). Adding more intermediate states to the essential-state approximation might increase the predicted enhancement. In Appendix D, we derive the formalism for the 3-level approximation of the SOS model (ISRE 3 given by Eq. 11). An explicit calculation ISRE 3 would require knowledge of the transition and permanent dipole moments of the fluorophores that currently are unknown. Using the 2-level model, as the simplest approximation, is the limitation of our approach for calculating a theoretical prediction of the enhancement. Additional enhancement of ND-TPACS may also be due to the participation of low-lying vibrionic transitions, having very low transition dipole moments associated with them and thus not readily observable in the linear absorption spectra. Within our available tuning range, we achieve enhancement in the ND-TPACS of up to ∼ 75% for mKO2 ( Fig. 1). Higher enhancement values would be achieved by further decreasing the detuning energy, as long as that energy is larger than the natural bandwidth of the first one-photon excited state transition [34]. At the limit where the energy of the incident high-energy photon is within the natural bandwidth of the transition, fluorescence from single photon excitation dominates over multiphoton excited fluorescence, negating all advantages of multiphoton excitation. The knowledge of ND-TPACS alone may be enough to guide the choice of wavelengths for imaging relatively thin and transparent samples. For deep imaging in live biological tissues, however, this choice is affected by other considerations, such as light scattering, water absorption, and the associated heating of the sample. In Fig. 2(a), we show the effects of water absorption, and tissue scattering for different wavelengths (adapted from [21] with permission). Longer wavelengths undergo lower scattering in tissue; however, as shown in Fig. 2(a), water absorption increases significantly at wavelengths near 1500 nm and beyond 1800 nm. Higher water absorption at longer wavelengths will result in greater tissue heating. In ND-TPE, the IR beam has longer wavelengths compared to D-TPE, and excessive tissue heating is a potential concern for ND-TPE, even when equal total excitation power is used. However, it has been shown that the increase in tissue temperature is much smaller than the increase of the water absorption coefficient [37] and therefore, the increase in tissue heating for longer wavelengths is not as drastic as expected. This is also evident in tissue heating simulations shown in Fig. 2(b), where we plot the maximum temperature change of the sample when illuminated by both NIR and IR beams, ∆T NIR+IR = ∆T NIR + ∆T IR , relative to the maximum temperature change upon illumination with a single beam at the equivalent degenerate wavelength, ∆T D , assuming equal total powers (see Appendix E for the description of the simulation). For example, the ratio of the water absorption coefficient at 1340 nm to the absorption coefficient at 920 nm is ∼ 30 (red plot in Fig. 2(a) and Table 3), but the ratio of the temperature change between D-TPE with λ D = 920 nm and ND-TPE with λ NIR = 700 nm and λ IR = 1340 nm and equal excitation power is ∼ 3. This result agrees with the idea that most of the incident light is absorbed before escaping the thick tissue sample and therefore, the variation in the absorption coefficient of different wavelengths does not drastically affect the rate of tissue heating [37].  [21]): photon fraction at depth of 1 mm, considering both absorption and scattering, for average brain tissue optical properties is shown as blue line. Percent of the photons absorbed by brain tissue in 1 mm is shown as red line. b) Simulation results for tissue heating by a scanned focused light: ratio of the maximum tissue temperature change under simultaneous excitation by NIR and IR beam to maximum temperature change under excitation with a single beam with equivalent degenerate wavelength, 2hc/λ D = hc/λ NIR + hc/λ IR and λ D = 920 nm, versus λ IR assuming equal total excitation power P NIR + P IR = P D = 100 mW at focal point (250 µm below surface).

Conclusion
In conclusion, we showed that, within our measured excitation range, the ND-TPACS spectrum of all fluorophores tested follow their corresponding D-TPACS in shape and exhibit enhancement in magnitude. We showed that the amount of the observed enhancement is higher than the theoretical prediction given by the 2-level essential-state approximation of the SOS model. Different fluorophores exhibit different amounts of resonance enhancement and we observed up to ∼ 75% enhancement in the ND-TPACS compared to the corresponding D-TPACS. We also showed that the increase in tissue heating, due to excitation with longer wavelengths, is not as large as water absorption increase for longer wavelengths. This increase in heating may be an acceptable trade-off for leveraging enhanced ND-TPACS and other advantages of ND-2PE [22] in deep tissue imaging applications.

Appendix A: Experimental setup
Detailed description of the spectroscopy setup is found in our recent work [29]. In brief, two temporally synchronized and spatially aligned pulsed femtosecond laser beams, one with near infrared (NIR) and another with infrared (IR) energies, were derived from a Ti:Sapphire laser and optical parametric oscillator (OPO), respectively. The beams, linearly polarized in the same plane, were combined on a dichroic mirror and focused by a microscope objective (Olympus, LMPLN-10X-IR, NA=0.3) into the sample solution ( Fig. 3(a)). The NIR and IR wavelengths were scanned in 10 nm increments in the range of 740 − 870 nm and 1100 − 1400 nm, respectively. We measured and recorded the fluorescence intensity and the following parameters for each laser beam: wavelength, spectral pulse bandwidth, average power, and repetition rate. The NIR optical path included a delay stage, so that the time-averaged intensity of the fluorescence could be measured as a function of the temporal delay between the NIR and IR pulses (Fig. 3(b)). All measurements were conducted at sufficiently low incident powers, ensuring that two-photon excitation was the only mechanism that generated the fluorescence signal and other non-linear phenomena such as stimulated emission and excited-state absorption are absent [29]. To show that the ND-TPE was the only fluorescence excitation mechanism, we plotted the fluorescence signal value at the peak of the temporal overlap (fluorescence generated by both ND-TPE and D-TPE) minus the fluorescence signal at no temporal overlap (fluorescence generated by D-TPE) as a function of the excitation power of each individual beam as shown in Fig. 4. The linear dependence of the fluorescence signal generated by ND-TPE on the excitation power of each beam is the characteristic signature of the ND-TPE [22,29].

Appendix B: Spectroscopy protocol
The fluorescence intensity measured as a function of the delay-line position ( Fig. 3(b)), resulted in a cross-correlation curve that was generated for each combination of NIR and IR wavelengths ( Fig. 3(b)). When NIR and IR pulses arrived at different times (Temporal offset >0), the fluorescence signal was generated by degenerate two-photon excitation (D-TPE) due to absorption of two photons of equal energy coming from either the NIR or IR beam. An increase in the fluorescence signal was detected as laser pulses from the two beams overlapped in time indicating the additive effect of the non-degenerate two-photon excitation (ND-TPE) occurring due to simultaneous absorption of one photon from each NIR and IR beam. To isolate the ND-TPE signal, for each combination of NIR and IR wavelengths, we subtracted the D-TPE contribution from the fluorescence intensity at zero delay between NIR and IR pulses. Next, we compared the probability of two-photon absorption between D-TPE and ND-TPE. In D-TPE, the time-averaged fluorescence signal can be expressed as the following equation: where C is the concentration of the fluorophore, P is the laser power, λ is the laser beam wavelength, η is the quantum efficiency of the fluorophore, φ is the collection efficiency of the measurement system, σ (2) D (λ) is the degenerate two-photon absorption cross-section (D-TPACS) at wavelength λ, f is the laser repetition rate, c is the speed of light, h is the Planck's constant, and Γ is the temporal width of the laser pulse [29]. When excited with NIR and IR laser beams, the time-averaged fluorescence signal is a sum of fluorescence generated by ND-TPE and fluorescence generated by D-TPE due to absorption of either the NIR or IR beam alone and is given by the following equation: where τ is the temporal offset, P NIR and P IR are the laser power of the NIR and IR beams, λ NIR and λ IR are the wavelengths of the NIR and IR beams, σ (2) ND (λ NIR , λ IR ) is the non-degenerate two-photon absorption cross-section (ND-TPACS) at a specific combination of NIR and IR wavelengths λ NIR and λ IR , Γ NIR and Γ IR are the temporal width of the NIR and IR beams, and Γ x is the temporal width of the cross-correlation curve of the NIR and IR laser pulses [29]. The first two terms of Eq. (3) are the fluorescence from D-TPE for each one of the beams. These terms are constant with respect to pulse delay and appear as a constant fluorescence background. By acquiring the fluorescence intensity as a function of relative pulse delay, we can measure both the ND-TPACS and D-TPACS by fitting this model to the data, assuming that all other parameters including fluorophore concentration, fluorescence quantum efficiency, and collection efficiency are known [29]. Measurement of the absolute ND-TPACS would require knowledge of the fluorophore quantum efficiency η and the collection efficiency of the measurement system φ. Our goal, however, was to compare ND-TPE to D-TPE and evaluate the possibility of enhancement of two-photon absorption upon ND-TPE. Therefore, for each fluorophore, we quantified the ratio of ND-TPACS to the peak (i.e., the best achievable) D-TPACS and not the absolute value of ND-TPACS. To that end, we normalized all ND-TPACS measurements by the peak D-TPACS, which occurred at the wavelength where the photon energy matched half of the total energy needed for the transition to the excited state. This is because under D-TPE, two photons are drawn from the same beam and have the same photon energy. D-TPACS was independently measured within the same range of the total energy of two photons by blocking the IR beam. This normalization mitigated the need to explicitly measure φ and η, because these terms disappear when dividing Eq. (3) by Eq. (2). For each fluorophore, the normalized ND-TPACS spectrum was computed by fitting cross-correlation curves (Fig. 3(b)) with Eq. (3) for each pair of NIR and IR wavelengths and normalizing the obtained valued by the peak D-TPACS that is a fluorophore-specific constant. For a more detailed description of the spectroscopy protocol please refer to our recent work on the spectroscopy technique [29]. We characterized the measurement error via repetitive measurement of the ND-TPACS along one isocline (Fig. 5). The main source of error was the uncertainty in the measurement of IR beam pulse width, fluorescence signal value, and laser powers.

Appendix D: Derivation of the intermediate state resonance enhancement (ISRE)
In this section we derive the formula for ISRE 2 (Eq. (1) in the main text). We start with the expression for TPA tensor S (f ) for the transition from ground state |g to final state |f which is given by: where µ mn = m| µ|n , i.e. vectors with components (µ x mn , µ y mn , µ z mn ), are the element of the electric dipole operator µ; and e 1 and e 2 are the unit polarization vectors of the two photons with energies ω 1 and ω 2 , respectively [41][42][43]. Index 'i' in Eq. (4) runs over intermediate states |i with energies E i including the ground state (i = 0) and the final state (i = f ). The damping constant ξ i is associated with the line shape (density of states) of the intermediate state and is commonly set (in calculations) to an arbitrary chosen small value. The relation between S (f ) and ND-TPACS is given by: where ρ f (ω NIR + ω IR − ω f ) is the line shape function [41][42][43]. Let's assume, for simplicity that all transitions are polarized in the same direction and this direction is parallel to the two polarization vectors e 1 and e 2 . This assumption will not affect the generality of the conclusions. Under this assumption, the |S (f ) | 2 expression can be written in a simpler form: The full sum-over states (SOS) model considers the contribution of all excited (intermediate) states in the TPA tensor. However, in the essential-state approximation the full SOS treatment is truncated by assuming one strongly one-photon allowed state |i that acts as intermediate state for TPA into two-photon allowed state |f . The essential-state model has been successfully applied for degenerate TPA calculations in a variety of fluorophores [44][45][46]. Research showed that the essential state model can effectively reproduce the converged full SOS model results for ISRE of two fluorene derivatives [34]. For non-centrosymmetric molecules, there is no formal mutual exclusion between the selection rules for one-photon and two-photon transitions |i = |f ), and we have a 2-level system. Under the 2-level approximation of the SOS model the TPA tensor for non-centrosymmetric molecules can be written as: Here E 0 = 0, since this is our reference ground state energy. We can consider that the line broadening parameters ξ 0 , ξ f and ξ i are small compared to all the other energy terms. Therefore, for a resonance to the final state (E f = ω 1 + ω 2 ) we have: It is easy to show analytically (or numerically) that Eq. (8) has its minimum when ω 1 = ω 2 (assuming constant transition energy ω 1 + ω 2 ). By inserting Eq. (8) into Eq. (5) we find the following simple scaling relation for ND-TPACS Therefore, the ISRE 2 is given by: where we have assumed 2 ω D = ω NIR + ω IR , or in other words, ISRE is the fractional change of ND-TPACS relative to D-TPACS at the equivalent degenerate frequency. Using the same procedure as for the 2-level system given above, here we derived the ISRE for a 3-level system with |i as the additional intermediate state:

Appendix E: Heating comparison
To study potential heating under ND-TPE, we simulate tissue heating through scanned focused light using a MATLAB code, which was provided by K. Podgorski [37]. The code uses Monte Carlo simulations to simulate light spread in tissue and the finite differencing method for heat conduction, metabolic heating, and cooling by blood perfusion [37]. The code assumes a heterogeneous sample incorporating a 350-µm glass window and immersion water above the sample [37]. The simulated focused beam is set to scan a 1 mm 2 area at 250 µm below the surface. We used previously published values for optical parameters of brain tissue [21,47] (Table  S2). For simulation of heating under simultaneous excitation with NIR and IR beams, we used the code to simulate the heating under individual NIR and IR beams and then added the resulting tissue temperature changes to obtain the heating for the case that both beams are illuminating the sample at the same time. This is a viable approach since the superposition principle is valid for heat propagation which is described by a linear partial differential equation. Podgorski et al. showed that heating varies in different parts of the sample relative to the focal point and the temperature change is greatest deep below the brain surface and focal plane [37]. In Fig. 8(a) we show the spatial temperature profile of brain tissue in resting equilibrium and when illuminated using a NIR wavelength of 740 nm, an IR wavelength of 1230 nm and the equivalent degenerate wavelength of 920 nm (2/λ D = 1/λ NIR + 1/λ IR ). Light is focused and scanned 250 µm below the cover glass over a 1 mm 2 scan area. Power of the excitation beams were chosen such that P NIR + P IR = P D = 100 mW and P NIR = P IR at focal depth (250 µm below cover glass). For example, at λ NIR = 740 nm we set P NIR (z = 250 µm) = 50 mW at focal depth which means the input laser power at surface is P NIR (z = 0 µm) = 100 × (P NIR (z = 250 µm)/T(λ NIR = 740 nm; z = 250 µm)) = 250 mW. Tissue exhibits the highest temperature change when illuminated with the IR beam ( Fig. 8(b)) as expected given high absorption of water at this wavelength. Fig. 8. Tissue heating simulations. Simulation code was provided by K. Podgorski [37]. a-d) Simulated spatial temperature profile in brain tissue with cover glass and immersion water (temperature of the immersion water 1 mm above cover glass was kept constant at 25 • C). Profiles are shown for no illumination, 100 mW at 920 nm, 50 mW at 1230 nm, and 50 mW at 740 nm. e-g) Simulated temperature change for 100 mW at 920 nm, 50 mW at 1230 nm, and 50 mW at 740 nm. Contour lines are shown for 1 • C intervals. All powers are given for a focal plane 250 µm below the surface. Table 3. Optical parameters for average brain tissue [21,47] that were used in our heating simulations. µ a and µ s are the absorption and scattering coefficients, respectively, and T = 100 × e −(µa +µs )z is the light transmission percentage at depth z. We set z = 250µm for our tissue heating simulations.