Measurement of the relative non-degenerate two-photon absorption cross-section for fluorescence microscopy.

In non-degenerate two-photon microscopy (ND-TPM), the required energy for fluorescence excitation occurs via absorption of two photons of different energies derived from two synchronized pulsed laser beams. ND-TPM is a promising imaging technology offering flexibility in the choice of the photon energy for each beam. However, a formalism to quantify the efficiency of two-photon absorption (TPA) under non-degenerate excitation, relative to the resonant degenerate excitation, is missing. Here, we derive this formalism and experimentally validate our prediction for a common fluorophore, fluorescein. An accurate quantification of non-degenerate TPA is important to optimize the choice of photon energies for each fluorophore.

spectrometer relies on sensitive detection of the two-photon fluorescence excitation, taking advantage of the high sensitivity of modern photodetectors which can detect single photons as long as the background signal is virtually zero. For non-degenerate TPACS (ND-TPACS), several prior studies employed the Z-scan or pump-probe techniques which measure transmittance [14-17, 22, 23], i.e., a laser intensity decrease due to absorption. However, since TPA is a low probability nonlinear optical process, Z-scan two-photon absorption spectroscopy requires significantly higher laser intensities, compared to two-photon fluorescence excitation spectroscopy, for detecting the inherently low probability two-photon absorption process. The high power requirement of the Z-scan technique is even more problematic for quantification of fluorescent molecules, particularly those used in biological research, which rapidly photobleach under high illumination power. Thus, a more sensitive, low excitation power method for quantification of ND-TPACS is needed.
Toward this end, we extend the two-photon fluorescence excitation technique [20] to allow for ND-TPE. We derive the formalism for the quantifying fluorescence signal generated by ND-TPE of a fluorophore solution illuminated by two ultrashort pulsed lasers. The beams are spatially overlapped, and the fluorescence signal is calculated as a function of the temporal delay between the pulses. We assume that the fluorophore solution is homogeneous, the pulses are transformlimited Gaussian pulses, the excitation beams are undepleted and there is no photobleaching and no saturated absorption. We demonstrate that the ND-TPACS can be calculated from the temporal convolution of the two laser pulses, and that these values can be normalized to the best achievable D-TPACS. The normalization of ND-TPACS values to the corresponding best achievable D-TPACS makes it possible to quantitively compare the ND-TPAC to D-TPACS of the sample and eliminates the necessity for measuring collection efficiency of the system. Finally, we experimentally validate our theoretical prediction: we apply the non-degenerate two-photon fluorescence excitation technique to measure ND-TPACS of a well-characterized fluorophore, fluorescein [20,21], throughout our available tuning range and compare them to the corresponding D-TPACS. We observe a high correlation between D-TPE and ND-TPE spectra, as expected from the original two-photon excitation theory [24], we also observe that for some wavelength combinations the values of normalized ND-TPACS are larger than D-TPACS, showing enhancement of the TPACS under ND-TPE. We also confirm that the fluorescence generated by ND-TPE is linearly proportional to the excitation power of each beam [8,25].

Theoretical framework
In this section, we extend the formalism given in [20] for fluorescence generated by D-TPE, to allow for ND-TPE in order to derive the relation between fluorescence generated under ND-TPE and the ND-TPAC of the sample. We begin by assuming two pulsed laser beams are chosen within the near infrared window of 700 − 1000 nm and the infrared window of 1000 − 1700 nm, hereafter referred to as the near infrared (NIR) and infrared (IR) beams, respectively. These spectral windows are relevant for ND-TPE spectra of fluorophores that are excitable via D-TPE within the 850 − 1000 nm spectral window. The derivation presented here is more general and would equally apply to any other wavelength combination. When the fluorescent sample is illuminated with the NIR and IR beams that are temporally delayed by offset τ, the number of molecules that are excited via D-TPE and ND-TPE per unit time and unit volume is given by where r = (x, y) is the distance from the optical axis, λ NIR and λ IR are the wavelengths of the NIR and IR beams, C(r, t) is the space-and time-dependent distribution of the fluorophore concentration, I NIR and I IR are the space-and time-dependent photon fluxes of each beam (in cm −2 s −1 ), σ (2) D (λ i ) and σ (2) ND (λ i , λ j ) are the D-TPACS and ND-TPACS. Note that the ND-TPACS and D-TPACS become equivalent when two wavelengths are equal, σ (2) ND (λ i , λ j ) = σ (2) D (λ i ) [17]. We assume that both the NIR and IR excitation beams are undepleted for all time and relative pulse delays. The factor of 1/2 in front of the first two terms of Eq. (1) reflects the fact that in the D-TPE process two photons of the same color are needed to excite one molecule [20]. There is no factor of 1/2 in front of the last term because in every non-degenerate absorption process only one photon of each color is needed to excite one molecule [18], and the additional factor of 2 in front of the last term comes from the fact that the two absorbed photons are distinguishable while σ (2) ND (λ NIR , λ IR ) = σ (2) ND (λ IR , λ NIR ) [25,26]. We approximate each beam as a Gaussian pulse with a Gaussian cross section in the paraxial approximation, which allows us to separate the temporal and spatial components of the photon flux: where the foci of NIR and IR beams are assumed to be overlapped in space, and The pulsed beams are assumed to have transform-limited pulses with peak photon flux density I (λ) 0 (in cm −2 s −1 ) and temporal standard deviation Γ λ , and where w 0 (λ) is the beam waist, z 0 (λ) is the Rayleigh length and NA is the numerical aperture of the optical system. Inserting these expressions into Eq. (1) and separating the spatial and temporal terms we find that the number of excited molecules per unit time in the illuminated sample volume, V, is given by We assume that the fluorophores are distributed homogeneously and are stable over the course of each measurement, i.e., C(r, t) = C, and can therefore be moved in front of the integrals. By assuming the fluorescence spot size is much smaller than the volume of the fluorophore solution and that numerical apertures of the beams are roughly equal, we find that for the general case of two beams with wavelengths λ 1 and λ 2 the spatial overlap integral is For the degenerate case, the wavelengths are equal, λ 1 = λ 2 = λ NIR or λ IR , and the spatial overlap integral result simplifies to λ 3 \ 4πNA 4 . For the non-degenerate case, λ 1 = λ IR and λ 2 = λ NIR .
To relate the fluorescence photon flux to N ex , we follow Xu and Webb [20]: assuming no stimulated emission, photo-bleaching or saturated absorption, the fluorescence photon flux is where φ is the collection efficiency of the optical system, and η is the fluorescence quantum efficiency. Substituting Eq. (7) and Eq. (8) into Eq. (9) we find the following expression for the fluorescence photon flux We recognize that the signal measured by our fluorescence detector is the time averaged photon flux, not the instantaneous photon flux; as a result, we need to compute the time average of Eq. (10). To compute the time average, we solve the time overlap integral for the general case with two wavelengths λ 1 and λ 2 and non-zero temporal offset: where f is laser repetition rate. If we assume pulse width and delay time are much smaller than 1/ f , which is valid for an 80 MHz femtosecond pulsed laser, we can approximate the limits of the time overlap integral to be infinite: This expression is equivalent to the temporal convolution of our laser pulses. By substituting Eq. (4) into Eq. (12) we see that as a function of delay time, we have the convolution of two Gaussian functions, which is typically computed by application of the convolution theorem: where h * g denotes the convolution of functions h and g and F{} is the Fourier transform. Therefore, the temporal convolution of the Gaussian pulses is where Γ 2 x = Γ 2 λ 1 + Γ 2 λ 2 is the standard deviation of the convolution. For the degenerate case, wavelengths, temporal standard deviations, and peak photon flux densities are equal and τ = 0, therefore, the temporal overlap integral simplifies to πI 2 0 Γ. To complete the derivation, we rewrite the time-averaged fluorescence in terms of the average laser power measured with a power meter (in mW). Because we assumed transform limited pulses with a Gaussian spatial profile, we can write the power in terms of the photon flux where is the total photon flux at the focus, and E λ is the energy per photon. Consequently, we obtain the following equation for the laser photon flux where c is the speed of light, and h is Plank's constant. Therefore, assuming: the fluorophore solution is homogeneous; the paraxial beam approximation; an approximately constant numerical aperture; transform limited Gaussian pulses; and, undepleted excitation beams, the time averaged fluorescence as a function of relative pulse delay is The first two terms of Eq. (17) are the fluorescence from D-TPE which is constant as a function of 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 the fluorophore concentration, fluorescence quantum efficiency and collection efficiency are known. Note that for the limit that λ NIR = λ IR , our setup becomes an interferometric autocorrelator, see Appendix E, and the measured fluorescence is the interferometric autocorrelator trace [27] which is very sensitive to the optical path difference between the two beams.
As described above, 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, is to compare ND-TPACS to the corresponding D-TPACS. Therefore, we calculate the ratio of the maximum non-degenerate two-photon excited fluorescence (τ = 0) to the degenerate two-photon excited fluorescence at a chosen wavelength, λ D , given by consequently, the ND-TPACS that is normalized by D-TPACS at wavelength λ D is Our normalization procedure assists us with quantifying the ND-TPACS relative to the best achievable D-TPACS of the same fluorophore and does not rely on previously published D-TPACS of reference fluorophores.

Sample preparation
Fluorescein (SIGMA-ALDRICH) was dissolved in 1X phosphate-buffered saline to reach ∼ 100 µM concentration. The pH of the solution was measured after dissolving the fluorescein (pH ∼ 7.4). The true concentration, 88 µM, was measured using single-photon absorption spectroscopy (NanoDrop 2000 spectrometer, ThermoFisher) and previously published molar extinction coefficient (75000 M −1 cm −1 ). Figure 1 depicts the experimental setup for ND-TPACS measurements using the non-degenerate two-photon fluorescence excitation technique. The primary light source was a Coherent Chameleon Ultra II laser, which generates 140 − 190 fs pulses with an 80 MHz repetition rate in the 680 − 1080 nm spectral range and is referred as the NIR beam. The Chameleon was used to pump an APE Chameleon Compact Optical Parametric Oscillator (OPO), which has a signal tuning range of 1000 − 1600 nm (referred as the IR beam). Since the OPO could only be pumped with an NIR beam within the 740 − 880 nm range, the actual available tuning range of the NIR beam for simultaneous sample excitation with NIR and IR beams is smaller than the entire tuning range of the Chameleon Ultra II laser. A built-in bypass beam splitter diverted 15% of the NIR beam to be used as one of the fluorescence excitation beams. The second beam was the signal from the OPO. The power of each beam was controlled using a half wave plate/linear polarizer combination and monitored using an uncoated quartz glass slide (for beam sampling) and a power meter (Melles Griot, broadband power meter, 13PEM001) during the experiment. The laser power after the objective was measured for each wavelength and a table for the ratio of the powers measured in the setup and power after objective was generated, in other words, the transmission spectrum of the optical path after the beam sampler was measured for each laser. This was a necessary step since the transmittance of optical elements are slightly different for different wavelengths. The path length difference between the two beams was tuned using a custom optical delay line in the NIR path, which utilized a Thorlabs LTS150 150 mm linear translation stage with repeatable incremental motion of 2 µm, corresponding to a minimum time delay resolution of 13 fs. Both beams were expanded to overfill the microscope objective pupil aperture and co-polarized using halfwave plates. Subsequently, the two beams were combined and made collinear using a dichroic beam splitter with a cutoff wavelength at 1000 nm (Thorlabs, DMSP 1000). See Appendix A for a note on beam alignment. The resulting combined beam was then directed into the microscope objective (Olympus, LMPLN-10X-IR, NA=0.3), which focused the beam into a rectangular Quartz cuvette (WPI, 2 mm path, 0.7 mL volume) filled with sample solution.

Experimental apparatus
The emission was collected in epi-fluorescence mode, reflected off two dichroic mirrors with cutoffs of 678 nm (Semrock, FF678-Di01-25x36) and 735 nm (Semrock, FF735-Di01-25x36) and subsequently passed through a set of fluorescence bandpass filters (three Semrock FF01-550/200-25 and one Semrock FF01-525/50-25) to extinguish the excitation source and lower the scattered pump signal below the electronic noise in the collection path. Finally, the fluorescence was detected using a photomultiplier (PMT; Hamamatsu, H11461-03). We verified that the fluorescence emission spectrum is identical under ND-TPE and D-TPE (Appendix B), and therefore the collection path does not need to change for these two types of excitation. The PMT signal was amplified by a transimpedance amplifier (Hamamtsu C9999). The signal from the amplifier and the laser power meter readings were acquired using a data acquisition board (National Instruments). Data acquisition and analysis were performed using custom-written software (National Instruments, LabVIEW). Please see Appendix C for the spectroscopy protocol.

Experimental results
The fluorescence intensity of fluorescein was measured as a function of the temporal delay between the excitation pulses, resulting in a signal which was dependent on the temporal convolution between the NIR and IR pulses ( Fig. 2(a)). The NIR laser beam generated degenerate two-photon fluorescence which appeared as a constant fluorescence background upon which the convolution signal from ND-TPE was added (Eq. (17)). For the selected tuning range, the contribution of the IR to D-TPE was insignificant relative to the NIR beam contribution, because of the extremely low D-TPACSs in this range. The fluorescence intensity curves, i.e., the convolution signals, were generated for each combination of NIR and IR wavelengths with 740 nm < λ NIR < 870 nm and 1030 nm < λ IR < 1400 nm, producing a 2-dimensional ND-TPACS spectrum. The ND-TPACS was extracted from the convolution signal as described in previous sections. We can normalize ND-TPACS spectra to any spectral feature of interest. We chose to normalize the ND-TPACS data by the peak D-TPACS at λ NIR = 920 nm, meaning that we normalized ND-TPACS by the corresponding best achievable D-TPACS for the given fluorophore. The resulting normalized 2-dimensional ND-TPACS spectrum is shown in Fig. 2(b). For some of the wavelength combinations, less than 5% of points, the convolution curve significantly deviated from the predicted Gaussian profile. For these data points, we chose to interpolate the data, please refer to Appendix D for details. As expected, the highest fluorescence flux was obtained when the sum of NIR and IR photon energies matched the transition energy. In Fig. 2(b), the isocline corresponding to a transition energy of 2.7 eV, corresponding to the 920 nm peak in the D-TPE spectrum of fluorescein, is overlaid on the fluorescein ND-TPE spectrum. In Fig. 2(c) we show the normalized ND-TPACS spectra plotted versus the degenerate wavelength equivalent to every combination of IR and NIR wavelengths used to excite the sample, i.e., 2/λ Eq = 1/λ NIR + 1/λ IR . Since all NIR and IR wavelength combinations along one specific isocline correspond to the same equivalent wavelength, we report several ND-TPAC values for each equivalent wavelength. Therefore, variation of ND-TPACS for each equivalent wavelength is not caused by measurement errors as each point represents an individual measurement at a different wavelength combination.
We measured the D-TPACS using two methods: either by blocking the IR beam and illuminating the sample with the NIR beam (independent measurement) or by using the baseline offset of the convolution signal (dependent measurement); please refer to Appendix C for details.  17)). b) Color-coded ND-TPACS spectrum normalized by the D-TPACS peak at 920 nm, extracted from independent measurements of fluorescein D-TPACS, for fluorescein showing dependence of the ND-TPACS on NIR and IR wavelengths. The 2.7 eV isocline is overlaid as black dashed line. c) Normalized ND-TPACS of fluorescein, as a function of the equivalent degenerate wavelength λ Eq where 2/λ Eq = 1/λ NIR + 1/λ IR is shown in red circles. Since all NIR and IR wavelength combinations along one specific isocline in panel b correspond to the same equivalent wavelength, we report several ND-TPAC values for each equivalent wavelength. Independently measured D-TPACS normalized by its peak value at 920 nm is shown as blue line. Dependent D-TPACS extracted from the fluorescence intensity curve background (see panel a) and normalized by the peak value of independently measured D-TPACS at 920 nm is shown in black circles. The dashed line shows the position of the peak at 920 nm.
We normalized both dependent and independent D-TPACS by the independent D-TPACS at λ NIR = 920 nm. We also overlay the ND-TPACS spectra with both dependent and independently measured D-TPACS of the dye. We observe that the dependent and independent D-TPACS measurements agree (Fig. 2(c)), which verifies the integrity of the ND-TPACS spectrum. We also see that the shape of the linearized ND-TPACS spectra is highly correlated with the D-TPACS spectra. We observe that for almost all wavelength combinations, the ND-TPACS are larger than the equivalent D-TPACS. This observation is explained by the phenomenon known as "resonance enhancement" [17]: the increased probability of ND-TPE caused by a longer virtual state lifetime as compared to degenerate two-photon excitation.   Fig. 3. Non-degenerate two-photon fluorescence excitation dependence on a) IR excitation power (P NIR = 5 mW), and b) NIR excitation power (P IR = 15 mW). The power dependence was tested for two different wavelength combinations: λ NIR = 740 nm, λ IR = 1230 nm and λ NIR = 850 nm, λ IR = 1150 nm.
For both D-TPE and ND-2PE measurements, the laser powers were kept below 30 mW for the IR laser and 10 mW for the NIR laser. In our experimental setup, 1 mW of average excitation power corresponds to ∼ 10 28 photon/(cm 2 s) laser photon flux. Xu and Webb [20] have demonstrated that for this excitation intensity range, the degenerate two-photon excited fluorescence signal is merely generated by D-TPE and other non-linear phenomena such as stimulated emission and excited-state absorption are absent. From Eq. (17) we expect that the fluorescence generated by ND-TPE scales linearly with excitation power of each beam [8,13]. To verify that for ND-TPE the generated fluorescent signal is solely due to ND-TPE, we tested the linear power dependence of the generated fluorescence on each laser power for the applied excitation power range for multiple wavelength combinations (Fig. 3). We observe that for both NIR and IR beams the fluorescence caused by ND-TPE is linearly scaled with excitation power of each beam. This behavior is observed for all wavelength combinations where we show two of them in Fig. 3, verifying the assumptions of our model.

Conclusion
We reported the theoretical and experimental details of the non-degenerate two-photon fluorescence excitation technique for sensitive measurement of ND-TPACS. Assuming a homogeneous fluorophore solution, the paraxial beam approximation, an approximately constant numerical aperture, transform limited Gaussian pulses, and undepleted excitation beams, we calculated the time averaged fluorescence signal as a function of relative pulse delay. By doing this, we have extended the previously published theoretical framework for fluorescence generated under D-TPE [20] to account for ND-TPE. Using two femtosecond pulsed, synchronized independently tunable lasers, we measured the non-degenerate two-photon excitation spectra of fluorescein. We observe that the shape of the ND-TPE spectrum closely follows that of the D-TPE spectrum. We also found that the ND-TPACs are consistently higher than the D-TPACs for equivalent excitation energies. This enhancement in absorption probability is consistent with the resonance enhancement phenomenon [17]. We demonstrated that within the employed range of laser powers used for the non-degenerate two-photon fluorescence excitation technique, the generated fluorescence scales linearly with the excitation power of each beam showing that the ND-TPE is the only mechanism generating the fluorescence. By introducing the sensitive method for ND-TPACS measurement of fluorescent molecules, we hope to assist future studies on application of ND-TPE in fluorescence microscopy.

Appendix A: Beam size and alignment characterization
The NIR and IR beams were aligned by maximizing the signal generated by a fluorescent sample: steering mirrors were used for in-plane alignment; a telescope was used for axial alignment; a delay line was used for temporal alignment. In order to confirm the beam overlap in space when aligned using the described method, we performed knife-edge beam profiling. In Fig. 4 we show the profiles of NIR, λ NIR = 740 nm, and IR, λ IR = 1230 nm, beams for three different axial positions (z = 41, 42, 43 mm). We fit the measured data by the equation where P is the normalized power, P 0 is the background power, and erf is a standard error function; we obtain a value for the beam radius w and center x 0 for each measured profile. We find that the center of the two beams are aligned almost perfectly in all three transverse planes for both NIR and IR beams (x 0 = 6.5 ± 0.1 mm). The relationship between beam radius and the distance from beam waist z 1 is given by where w 0 is the beam waist and z 0 is the Rayleigh range. By using the known w values for the three z positions, we can obtain the z 1 , z 0 , and w 0 values of each beam. For NIR beam we find w 0 = 0.7 ± 0.1 µm, z 0 = 1.8 ± 0.1 µm, z 1 = 27.8 ± 0.5 mm, and for the IR beam w 0 = 1.2 ± 0.1 µm, z 0 = 3.8 ± 0.1 µm, z 1 = 27 ± 0.5 mm , which indicates that the beams are aligned in the axial direction. Furthermore, from the obtained numbers we see that the w 0 = λ/(πNA) and z 0 = πw 2 0 /λ relations are valid for the beams that overfill the back aperture of an objective lens with numerical aperture NA. We also confirm the validity of our assumption of constant NA for different wavelengths (NA = 0.34 ± 0.1 for λ NIR = 740 nm and NA = 0.33 ± 0.1 for λ IR = 1230 nm).

Appendix B: Fluorescence emission spectra under ND-TPE and D-TPE
It has been shown that fluorescence generated by single-photon excitation and D-TPE has the same emission spectrum [20]. For fluorescence emission spectrum measurements under D-TPE and ND-TPE, we added a spectrometer (HR4000CG High Resolution Fiber Optic Spectrometer) to the collection path of the setup depicted in Fig. 1. As shown in Fig. 5, we observe that the fluorescence emission spectra under ND-TPE and D-TPE are identical.   Fig. 6. Data interpolation. a) Fluorescence intensity versus delay time between IR and NIR pulses where λ NIR = 740 nm and λ IR = 1100 nm. The "double peak" effect caused by irregular IR beam profile is clear in the plot. b) Color-coded ND-TPE spectrum normalized to the corresponding D-TPACs at λ Eq = 920 nm, for fluorescein before the interpolation process. c) Same normalized spectrum shown in panel b after removing the "double peak" data points and interpolating the data. The 2.7 eV ( λ Eq = 920 nm) isocline is overlaid as black dashed line in (b) and (c).