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Abstract
We show theoretically that the third order coherence at zero delay can be obtained by measuring the second and third order autocorrelation traces of a pulsed laser. Our theory enables the measurement of a fluorophore’s three-photon cross-section without prior knowledge of the temporal profile of the excitation pulse by using the same fluorescent medium for both the measurement of the third order coherence at zero delay as well as the cross-section. Such an in situ measurement needs no assumptions about the pulse shape nor group delay dispersion of the optical system. To verify the theory experimentally, we measure the three-photon action cross-section of Alexa Fluor 350 and show that the measured value of the three-photon cross-section remains approximately constant despite varied amounts of chirp on the excitation pulses.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Since its introduction nearly a decade ago [1], long wavelength three-photon microscopy is increasingly used to visualize biological systems [2–12]. Arguably one of the most important parameters in the design of such fluorescence imaging experiments is a fluorophore’s three-photon cross-section. Because quantum perturbation theory suggests that the cross-section is on the order of 10−82 cm6s2 [13], reliable measurements use the fluorescence generated as opposed to direct absorption [14]. Nevertheless, the fluorescence method still requires ultrafast lasers and thus knowledge of the temporal structure of the pulse. Because the average power can easily be measured, this temporal knowledge is used in estimation of the third order temporal coherence at zero delay, ${g^{(3)}}({\mathbf 0}) = \langle {I^3}\rangle /{\langle I\rangle ^3}$, where I is the intensity, and $\langle \cdot \rangle $ denotes the temporal average. We note that this temporal coherence is an extension of the definition used in the multiphoton excitation literature [14–17]. However, knowledge of ${g^{(3)}}({\mathbf 0})$ is essential for the accurate determination of a fluorophore’s three-photon cross-section.
Generally, the pulse width is measured by second order autocorrelation and a pulse shape is assumed to estimate ${g^{(3)}}({\mathbf 0})$. However, even if a technique which doesn’t assume a pulse shape (such as frequency resolved optical gating [18]) is used, the measurements are generally performed outside the sample. That is, it assumes that the pulse characteristics in the measurement apparatus are the same as those in the sample. This can present a problem if large differences in group delay dispersion (GDD) are present between the optical paths which bring the pulse to the sample and the measurement device, and these differences are not accounted for correctly. Thus, measuring ${g^{(3)}}({\mathbf 0})$ in the sample directly (i.e., in situ) would be preferred. We report on a method that allows for direct measurement of ${g^{(3)}}({\mathbf 0})$ in the sample, which takes advantage of the fact that the sample fluorescence can be used to measure the second and third order autocorrelation, and that the resulting autocorrelation traces provide sufficient information to extract ${g^{(3)}}({\mathbf 0})$. Although third order autocorrelation has been explored in the literature (see for example, [19–23]), to our knowledge it has not been used to extract ${g^{(3)}}({\mathbf 0})$. Our method can also be used to determine the three-photon cross-section from the same data generated to determine ${g^{(3)}}({\mathbf 0})$, without any assumptions about the temporal profile of the excitation pulse. We demonstrate this by measuring the three-photon action cross-section of Alexa Fluor 350 and show that the measured value of the three-photon cross-section remains approximately constant despite varied amounts of chirp on the excitation pulses.
2. Methods
2.1 Theoretical method for obtaining ${g^{(3)}}({\mathbf 0})$ and the three-photon cross-section
The method used to measure ${g^{(3)}}({\mathbf 0})$ is an extension of Xu et al. [21] which was used for direct measurement of ${g^{(2)}}({\mathbf 0})$. As shown in Fig. 1, the output of a Michelson interferometer is focused into a fluorescent sample. Critically the sample contains a mix of two fluorescent dyes; one which undergoes two-photon excitation and another which undergoes three-photon excitation. Assuming a spectral separation of the emission, the fluorescence from each dye can be collected by a separate detector. Thus, the setup simply acts as a simultaneous second and third order autocorrelator, with the sample fluorophores serving as the quadratic and cubic signal generators.
Fig. 1. Layout of the system. The output of a mode-locked Ti:S oscillator is sent though a prism pair (made of two SF11 prisms) and then is directed through a half waveplate (10RP52-2B, Newport) and polarizer (GT10, Thorlabs), which serve as a power control, to a Michelson interferometer. The interferometer consists of a 50/50 beam splitter (BSW29R, Thorlabs) and motorized translation stage (Z825B, Thorlabs), and the output is directed through a 715 long pass (LP) filter (715 LP, FF01-715/LP 25, Semrock), and focused by an aspheric lens (#5722-B, New Focus), into a cuvette (CV10Q35EP, Thorlabs) containing AF350 and AF594. Fluorescence is epi-collected by the lens and directed by a LP dichroic mirror (685 LP, FF685-Di02, Semrock) through a short pass (SP) filter (720 SP, FF01-720/SP-25, Semrock). The fluorescence is split into two channels by a LP dichroic mirror (573 LP, 573-Di01, Semrock). Each channel consists of a series of color glass and interference filters to block the excitation light and separate the fluorescence emission of the two dyes (850 SP, FESH0850, Thorlabs; FGS900A, Thorlabs, 800 SP, FESH0800, Thorlabs; FGS600, Thorlabs, 650/60, FF01-650/60, Semrock; 650 SP, FESH0650, Thorlabs; BG18, Newport; 600 SP, FESH0600, Thorlabs; 450 SP, FESH0450, Thorlabs), and a photomultiplier tube to detect the fluorescence (PMT, HC125-02, Hamamatsu). The mirror in the AF350 path between the 573 LP and 650 SP is a dielectric mirror (DFM1-E02, Thorlabs). The PMT counts are recorded by two separate SR430 photon counters (Stanford Research Systems).
For simplicity, consider one excitation pulse entering the interferometer. In the absence of saturation and assuming no variation of the temporal pulse characteristics within the focal volume in the sample, the n-photon fluorescence detected by the system is,
Integrating Eq. (2) with respect to $\tau $ over a range $2T$ much longer than the pulse width will eliminate the oscillatory terms (if the order of integration is exchanged). Additionally, since terms of the form of ${P^{n - 1}}(t)P(t - \tau )$ and $P(t){P^{n - 1}}(\tau )$ are near zero for $|\tau |> T$, the integration region can be extended to ${\pm} \infty $. Thus, for n = 2 we obtain,
For $\tau $ much greater than the pulse width the detected fluorescent signal is simply,
Thus, collecting the second and third order autocorrelation traces simultaneously provides enough information to yield ${g^{(3)}}({\mathbf 0})$. In fact, our method can be extended to measure the nth order temporal coherence at zero delay, and a general case for obtaining ${g^{(n)}}({\mathbf 0})$ is shown in Supplement 1.
Once ${g^{(2)}}({\mathbf 0})$ and ${g^{(3)}}({\mathbf 0})$ are measured it is also possible to obtain the action cross-section by considering the amount of fluorescence generated for long (infinite) delay from one of the half beams (i.e., ${F_{n,\infty }}/2$) versus the fluence from one of the half beams (i.e., $\mathop \smallint \nolimits_{ - \infty }^\infty \textrm{d}tP(t)$). This is equivalent to putting Eqs. (3) and (5) (or Eqs. (4) and (5)) together to obtain,
Equations (9) and (10) provide a way to measure the two- and three-photon cross-sections at the same time. A general case for the n-photon action cross-section is presented in Supplement 1.
2.2 Experimental methods
To demonstrate this method experimentally, we built the experimental setup shown in Fig. 1, and used a mode-locked Titanium:Sapphire (Ti:S) oscillator (Tsunami, Spectra Physics, repetition rate of 81.97 ± 0.1 MHz) at 921 nm to perform simultaneous two- and three-photon excitation of a solution of ∼3.8 µM of Alexa Fluor 594 NHS ester (AF594, Fisher Scientific) and ∼700 µM of Alexa Fluor 350 NHS ester (AF350, Fisher Scientific) in water. AF594 undergoes two-photon excitation and has an emission peak ∼617 nm while AF350 undergoes three-photon excitation and has an emission peak ∼445 nm. Concentrations were measured using an absorption spectrometer (Cary 300, Agilent) and the molar extinction coefficients provided by the manufacturer. The spectrum of the Ti:S oscillator was measured with a spectrometer (CCS175, Thorlabs). A pair of SF11 prisms was used to compensate for the GDD of the optical system, as well as change the amount of GDD by translating the prisms into the beam. We chose three prism positions for demonstration, one which made the pulse close to transform limited (position 1), and two more with increasing amounts of GDD (positions 2 and 3).
Power dependence of the fluorophores (i.e., AF350 is undergoing three-photon excitation and AF594 is undergoing two-photon excitation) is tested by measuring the photomultiplier tube (PMT) counts as a function of laser power when the delay was set to be much larger than the pulse width. The counts were recorded on an SR430 photon counter (Stanford Research Systems) as the average counts of 1024 10.486 ms counting bins. The power was recorded before the interferometer by a power meter (S132C, Thorlabs), and the power (from both half beams) after the focusing lens is ∼2.5x less than this value. Dark counts of ∼30-40 s-1 were corrected for.
We measured the fluorescence as a function of delay, using a custom MATLAB program to drive a motorized translation stage on one arm of the interferometer, and simultaneously recorded the fluorescence as a function of time to produce two interferometric autocorrelation traces. We used 124.4 mW, 159 mW, and 194.5 mW, for prism positions 1, 2 and 3 respectively, before the interferometer and a bin size of 327.68 µs. Dark counts were not corrected for here since the average dark counts per bin are insignificant. Because the resulting traces will be integrated over, resolving fringes is not necessary (so long as the apparatus measures Eq. (2) without the oscillatory terms), but we used it here as it provides a way to calibrate the delay axes and check the quality of the optical alignment. Additionally, the intensity autocorrelation (Eq. (2) without the oscillatory terms) can be recovered digitally via Fourier transforms [24]. Fitting the intensity autocorrelation to the resulting function which arises due to assuming a Gaussian or sech2 shape is used to determine the full width at half maximum (FWHM) pulse width. The fitting functions can be found in Supplement 1.
For determining the action cross-section, ${V_2}$ and ${V_3}$ were estimated by measuring the beam’s spatial profile before the aspheric lens and then using the lens’s focal length to estimate the spatial profile at the focus. The collection efficiency was estimated by considering the transmission (or reflection) of the fluorophore’s emission spectrum through the various optical elements in the collection path and the PMT quantum efficiency (similar to what is done in [14,15,25]).
3. Results and discussion
The laser spectrum (Fig. 2(A)) and power dependence for all three prism positions (Fig. 2(B)-(D)) are shown in Fig. 2. The laser spectrum is fit to a sech2 (gray line in Fig. 2(A)) and carrier and FWHM bandwidth (BW) are found to be 921 nm and 17.1 nm, respectively. A transform limited pulse for this spectrum will have a pulse FWHM of ∼52 fs (assuming a sech2 shape). Refitting the spectrum to a Gaussian shape gives the same carrier and a FWHM BW of 18.2 nm, suggesting a transform limited pulse FWHM of ∼69 fs (assuming a Gaussian pulse shape).
Fig. 2. A. Spectrum of the Ti:S oscillator. The raw data from the spectrometer is shown in black and a fit of the spectral density vs. frequency shown in gray. The numbers on the upper left are the carrier and FWHM BW found from the fit. B-D. Power dependence of AF350 and AF594 fluorescence for when the prisms are in (B) position 1, (C) position 2, and (D) position 3. The circles are data points, and the black dotted lines are least squares fits to the log-transformed data. Data points were repeated at ∼50 mW when the prisms were in position 1 and ∼100 mW when the prisms were in positions 2 and 3 to ensure that the fluorescence counts are stable over time. The slopes of the least squares fits to the log transformed data are shown on each graph for each fluorophore.
The power dependence (Fig. 2(B)-(D)) was quantitatively tested by determining the slope of the fluorescent counts vs. power on a log-log plot. All graphs show a slope of ∼2 for AF594 and ∼3 for AF350 confirming that AF594 is undergoing two-photon excitation and AF350 is undergoing three photon excitation.
Figure 3 shows the resulting second (Fig. 3(A)-(C)) and third (Fig. 3(D)-(F)) order autocorrelation traces for when the prisms are in position 1 (Fig. 3(A) and (D)), position 2 (Fig. 3(B) and (E)) and position 3 (Fig. 3(C) and (F)). When the prisms are in position 1, we find that the pulse FWHM is ∼63.5 fs assuming a sech2 shape, and ∼70 fs assuming a Gaussian shape (average of the second and third order traces) indicating the pulse is close to transform limited. This allows us to compare ${g^{(2)}}({\mathbf 0})$ and ${g^{(3)}}({\mathbf 0})$ as calculated from Eqs. (7) and (8) with calculations based on the pulse width.
Fig. 3. Second (A-C) and third (D-F) order autocorrelation traces for when the prisms are in position 1 (A and D), position 2 (B and E) and position 3 (C and F). The black line is the interferometric trace, and the gray line is the recovered intensity trace. Insets on upper right show the interferometric data between -30 and 30 fs. The numbers at the upper left are the carrier wavelength, the pulse FWHM assuming a Gaussian and sech2 shape and ${g^{(2)}}({\mathbf 0})$ and ${g^{(3)}}({\mathbf 0})$ calculated using the method presented in this paper. Note that zero signal on the main graphs is not at the bottom.
From Eqs. (7) and (8) we find that ${g^{(2)}}({\mathbf 0})$ = 128 ± 8 × 103 and ${g^{(3)}}({\mathbf 0})$ = 182 ± 7 × 108. When calculated based on the measured pulse width (assuming a sech2 shape) we find that ${g^{(2)}}({\mathbf 0})$ = 113 × 103 and ${g^{(3)}}({\mathbf 0})$ = 151 × 108. Repeating these calculations using the measured pulse width, assuming a Gaussian shape, we find, ${g^{(2)}}({\mathbf 0})$ = 115 × 103 and ${g^{(3)}}({\mathbf 0})$ = 155 × 108. These numbers are close to those obtained from our technique, consistent with the knowledge that the transform limited pulses from a mode-locked Ti:S oscillator can be approximated as having a sech2 or Gaussian shape.
We find that when the prisms are in position 2, the pulse is chirped, and the pulse FWHM is ∼107.5 fs assuming a sech2 shape, and ∼119 fs assuming a Gaussian shape (average of the second and third order traces). From Eqs. (7) and (8) we find that ${g^{(2)}}({\mathbf 0})$ = 78 ± 2 × 103 and ${g^{(3)}}({\mathbf 0})$ = 66 ± 2 × 108. When the prisms are in position 3, the pulse is even more chirped, and the pulse FWHM is broadened to ∼163.5 fs assuming a sech2 shape, and ∼180.5 fs assuming a Gaussian shape (average of the second and third order traces), and from Eqs. (7) and (8) we find that ${g^{(2)}}({\mathbf 0})$ = 462 ± 6 × 102 and ${g^{(3)}}({\mathbf 0})$ = 245 ± 5 × 107.
Knowing the values of ${g^{(2)}}({\mathbf 0})$ and ${g^{(3)}}({\mathbf 0})$ allows for the determination of the two- and three-photon cross-sections without any assumptions about the pulse width and pulse shape. While the pulse width and chirp varied dramatically in the three measurements performed in Fig. 3, we find the two-photon action cross-section of AF594 to be ∼1.12 GM, ∼1.05 GM, and ∼1.08 GM, for when the prisms are in positions 1, 2 and 3, respectively (1 GM = 10−50 cm4s, average ± standard deviation of 1.08 ± 0.04 GM), the three-photon action cross-section of AF350 to be ∼5.5 × 10−84 cm6s2, ∼4.5 × 10−84 cm6s2, and ∼4.7 × 10−84 cm6s2, for when the prisms are in positions 1, 2 and 3, respectively (average ± standard deviation of 4.9 ± 0.5 × 10−84 cm6s2). The measured value of the two-photon action cross-section of AF594 is close to that reported by Zipfel (∼0.71 GM at 920 nm) [26]. More importantly, the small variation in the cross-section values, despite the large differences in the amount of chirp and pulse widths, indicates that our technique accurately accounts for pulse width and chirp, and gives a reliable measurement of the three-photon action cross-section.
4. Conclusion
In conclusion we have presented a method that can be used to measure ${g^{(3)}}({\mathbf 0})$ in situ, which does not require prior knowledge of the pulse shape. Although it is possible to perform complete pulse characterization to estimate ${g^{(3)}}({\mathbf 0})$, our method measures ${g^{(3)}}({\mathbf 0})$ in the sample. Therefore, no calculations and assumptions about GDD in the measurement system need to be considered. By using the fluorophores as the nonlinear media, our method enables the measurement of the three-photon (and two-photon) cross-section of the fluorophores without knowledge or assumptions about the excitation pulse shape. Furthermore, the method presented here can be extended to measure ${g^{(n)}}({\mathbf 0})$ (and ${\sigma _n}$), provided autocorrelation traces up to the nth order, and therefore ${g^{(n - 1)}}({\mathbf 0})$, ${g^{(n - 2)}}({\mathbf 0}),$… ${g^{(2)}}({\mathbf 0}),$ are also simultaneously measured.
Funding
National Institutes of Health (R01EB033179); National Science Foundation (DBI-1707312).
Acknowledgments
We thank Warren R. Zipfel for the use of his absorption spectrometer and SR430.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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