Probing Multiphoton Photophysics Using Two-Beam Action

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Probing Multiphoton Photophysics Using 2-Beam Action Spectroscopy Nikolaos Liaros, Sandra A. Gutierrez Razo, and John T. Fourkas J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b04463 • Publication Date (Web): 19 Jul 2018 Downloaded from http://pubs.acs.org on July 19, 2018

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The Journal of Physical Chemistry

Probing Multiphoton Photophysics using 2-Beam Action Spectroscopy Nikolaos Liaros,† Sandra A. Gutierrez Razo,† and John T. Fourkas*, †,‡,§ †

Department of Chemistry & Biochemistry, ‡Institute for Physical Science and Technology, §

Maryland NanoCenter, ¶ Center for Nanophysics and Advanced Materials, University of Maryland, College Park, 20742, United States

1 Environment ACS Paragon Plus

The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ABSTRACT: Multiphoton absorption (MPA) is an enabling technology for many applications. However, due to the low probability of MPA processes, their accurate characterization remains a challenge. Here we introduce a new technique, 2-beam constant emission intensity (2-BCEIn) spectroscopy, that offers substantial advantages over other existing methods that use the generation of optical emission for the characterization of absorptive nonlinearities. We use 2-BCEIn to study nonlinear absorption in solutions of crystal violet lactone (CVL) over a range of excitation wavelengths in which the dominant nonlinear absorption process transitions from 2-photon absorption (750 nm) to 3-photon absorption (830 nm). At an excitation wavelength of 800 nm, both 2-photon absorption and 3-photon absorption contribute substantially to the nonlinear fluorescence excitation (NFE) signal, although the dynamic range of the NFE data is not sufficient to quantify the contributions of each process. 2-BCEIn spectroscopy enables the direct measurement of the local exponent at each emission intensity. 2-BCEIn measurements made at several different emission intensities demonstrate unambiguously that the nonlinear excitation of CVL at 800 nm cannot be described solely as the sum of a 2-photon process and a 3-photon process. A kinetic model that includes intrapulse excited-state absorption reproduces the features of the 2-BCEIn measurements, and enables the determination of the ratio of the 3-photon absorption cross section to the 2-photon absorption cross section. Such information cannot easily be extracted from conventional NFE measurements. These results demonstrate the power and versatility of 2-beam action spectroscopies for elucidating the complex photophysics of multiphoton absorption processes.


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Introduction Multiphoton absorption (MPA) is an important spectroscopic tool for a broad range of techniques in physical chemistry.1-2 Furthermore, materials that can undergo efficient MPA are becoming increasingly important in many applications, including 3D fluorescence imaging,3-5 3D micro- and nanofabrication,6-8 optical data storage,9-12 and optical limiting.13-14 Although MPA is a wellstudied phenomenon, its characterization in many cases remains challenging.15 MPA processes are generally weak enough that their direct measurement is difficult. MPA is often therefore characterized by an indirect measurement in which another observable, such as induced fluorescence, is used as a proxy for absorption. This type of “action spectroscopy” measures the product of an MPA cross section and another physical quantity or quantities that must also be determined. Whether a direct technique or an indirect technique is used, the accurate determination of an MPA cross section requires knowledge of the predominant order or orders of absorption of a material at a given wavelength and irradiance. The order of absorption is typically determined by making a logarithmic plot of the observable as a function of the excitation irradiance.15 The slope of such a plot is taken to be indicative of the number of photons absorbed in the dominant absorption process. The accuracy of this type of measurement suffers from a number of different inherent problems, however. First, because MPA is generally weak, it is often not possible to obtain data over the several orders of magnitude of irradiance and/or signal that are required to determine the order of the absorption process accurately. This issue is related to the problem of the reliable determination of exponents in power-law distributions.16 Second, this type of logarithmic plot is often not linear, which can indicate the presence of contributions of two or more orders of absorption and/or the presence of competing phenomena, such as saturation or nonlinear



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scattering. If multiple orders of absorption are involved in generating the observable, then even greater dynamic range in irradiance is required to determine their relative contributions. Third, even when the logarithmic plots are linear, the slope is often not an integer. A non-integer slope may indicate that two or more orders of nonlinear absorption contribute to the signal.17-18 However, a non-integer slope may also often be related to the breakdown of the implicit assumption that the parameters that contribute to an action cross section are independent of irradiance. We have recently introduced 2-beam action (2-BA) spectroscopies, a new class of approaches for the determination of the order of absorptive optical nonlinearities.19-20 As shown in Figure 1, the essential element of a 2-BA spectroscopy is that absorption is driven by two trains of pulses of the same repetition frequency. The two pulse trains are typically collinear, but are interleaved in time (i.e., if the time between pulses in one pulse train is τ, then the second pulse train is delayed by τ/2 relative to the first pulse train). The average power of each pulse train can be adjusted independently. The average power that is required to reach some chosen value of an observable generated by absorption is first measured for each individual pulse train. The average power of the first pulse train is then set at a number of values below that required to reach the selected value of the observable, and for each of these average powers the average power of the second pulse train required to obtain the same value of the observable is determined. For an m-photon absorption process, we then have that20 𝑃𝑃�1𝑚𝑚 + 𝑃𝑃�2𝑚𝑚 = 1 ,

(1)

where the overbar indicates normalization to the average power required to attain the selected value of the observable with that pulse train alone. Thus, a plot of 𝑃𝑃�2 as a function of 𝑃𝑃�1 can be

used to determine m for any desired value of the observable. The ability of 2-BA spectroscopies to determine m accurately at any value of the chosen observable gives these techniques a


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significant advantage over methods that rely on logarithmic plots to characterize the order(s) of nonlinear absorption.

Figure 1 Schematic of a 2-beam action spectroscopy. Two pulse trains with independently adjustable intensities are interleaved in time and focused into a sample to generate an observable via linear and/or nonlinear absorption. In this example, the observable is fluorescence. BS = beam splitter, λ/2 = half-wave plate, P = polarizer, M = mirror, f = repetition rate, PBC = polarizing beam cube, DCM = dichroic mirror, OBJ = objective lens, PD = photodiode, PC = personal computer.

The first 2-BA technique developed was 2-beam initiation threshold (2-BIT) spectroscopy,20 which is an in situ method for determining the effective order of nonlinear absorption in photoresists used for multiphoton absorption polymerization.6-8 The observable in 2-BIT is the threshold for photoinduced crosslinking, which is single-valued. More recently, we demonstrated a 2-BA technique called 2-beam constant-amplitude photocurrent (2-BCAmP) spectroscopy,19 which was used to study the linear and nonlinear absorption of a GaAsP photodiode. Because photocurrent is a continuous observable, the order of the absorption could be determined at multiple photocurrent values. Under some conditions, the photocurrent arose from a combination of linear and 2-photon absorption, yielding a non-integer value of m. The dependence of this exponent on photocurrent was used to determine the relative contributions of both orders of absorption as a function of irradiance.19



Here we extend the 2-BA spectroscopy concept to the case in which luminescence is the observable, in a method that we call 2-beam constant emission intensity (2-BCEIn) spectroscopy. We focus on crystal violet lactone (CVL), a species that, in 2-BIT experiments, was found to act as an effective 3-photon radical photoinitiator when 800 nm light is used for excitation, but whose linear absorption spectrum suggests that it could also exhibit 2-photon absorption at this or nearby wavelengths.20 We demonstrate that the multiphoton photophysics of CVL is complex, depending strongly on both the excitation wavelength and the irradiance employed. Because 2-BCEIn allows for the determination of m as a function of fluorescence intensity and wavelength, we are able to develop new, quantitative insights into the photophysical behavior of CVL under nonlinear excitation that would be difficult to extract from conventional nonlinear fluorescence excitation (NFE) data using a logarithmic plot.

Experimental For NFE and 2-BCEIn measurements, a tunable, pulsed Ti:sapphire oscillator (Coherent Mira 900F) was used as the excitation source. The repetition rate of the laser was 76 MHz, and the pulse duration was approximately 150 fs. The spatially filtered beam was chopped at 1 kHz. For the 2beam experiments, the beam was then divided in two equal parts by a beam splitter. A combination of a motorized half-wave plate and a Glan-Taylor polarizer was used in each beam to control the average power. Each beam was passed through a separate variable beam expander to allow for the adjustment of the spot size at the back aperture of the objective. The lengths of the two beam paths were adjusted so that consecutive pulses arrived at the sample with an approximately equal time spacing (ca. 6.5 ns), giving an effective repetition rate of 152 MHz. The two beams were combined with a polarizing beam cube and made collinear, passed through a quarter-wave plate, then sent


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through the reflected-light illumination port of an inverted microscope and focused into the sample using an oil-immersion, 1.45 NA, 100×, infinity-corrected, microscope objective (Zeiss, α PlanFLUAR). Although there is no requirement that the focal spots of the two beams in 2-BCEIn overlap one another, in practice this overlap helps to ensure that the fluorescence signal is identical for each beam if the average power of each beam is the same. CVL and 1,6-dicyanohexane were purchased from Sigma-Aldrich. CVL was used as received. The solvent, which was yellow when received, was fractionally distilled under reduced pressure. Before distillation, the receiving flasks were dried at 110 °C for 1 hour. The distillation apparatus was flushed with dry nitrogen. The still pot was stirred vigorously and the distillation was monitored continuously to prevent choking due to the high solvent viscosity. Once the temperature stabilized at approximately 150 °C, the first distillate was discarded. The receiving flask was then switched, and the second distillate was collected as a colorless liquid and used for the experiment. The remaining brown contents of the still pot were discarded. The CVL concentration in the sample used for these experiments was 2 wt%. The sample (volume ~200 μL) was placed in a custom-made Teflon cell with #0 glass coverslips for windows. The cell was mounted on a 3-axis piezoelectric stage for fine sample positioning in all dimensions. The piezo stage was attached to a motor-driven stage for coarse sample positioning. The movement of the piezo stage was controlled using a LabVIEW program. To avoid photobleaching during the fluorescence experiments, the piezo stage was moved constantly in a circle with a radius of 20 μm at a velocity of 50 μm/sec. This speed is fast enough to prevent immediate photobleaching, and slow enough to permit diffusion away from the beam path between subsequent scans over the same spot.



The relative position of the two focal spots in the sample was determined by monitoring the NFE signal of each beam using a CCD camera. To avoid any inner filter effects, the beams were focused near the surface of the cell, and an epifluorescence geometry was used to collect the emission. The fluorescence was detected by a Si photodiode (Thorlabs SM1PD1A), the output of which was sent to an analog low-noise preamplifier (Stanford Research Systems, SR560), and then to a lock-in amplifier (Stanford Research Systems, SR810) referenced to the chopping frequency. To prevent any excitation laser light from being sent to the photodiode, appropriate short-pass filters were used. The average power values cited here were measured with a power meter at the back aperture of the objective with the chopper running. The additional loss from the objective was ~30%. No autofluorescence was observed from solvent blanks at any excitation wavelength used here.

Results and Discussion One of the most common methods of characterizing MPA is NFE.15,21-22 In this technique, the fluorescence intensity of the species studied is measured as a function of irradiance at the wavelength of interest. The quantity that is measured is the MPA fluorescence action cross section, which is the product of the MPA cross section σn, where n is the number of photons involved in the absorption process, and the fluorescence quantum yield, Φ.22 It is generally assumed that, so long as all other aspects of the measurement remain identical, Φ is the same for linear and multiphoton excitation. Thus, the MPA fluorescence action cross section can be used to determine

σn, provided that the order n of the MPA is known and that the instrument has been calibrated carefully with a fluorophore of known Φ and known σn at the same order n.23 The order n is generally determined by making a logarithmic plot of the fluorescence intensity as a function of the irradiance.24


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Figure 2 shows logarithmic NFE plots for CVL in 1,6-dicyanohexame at three different excitation wavelengths: 750 nm, 800 nm, and 830 nm. The data in Figures 2A and 2B were obtained at laser repetition rates of 76 MHz and 152 MHz, respectively. In all cases, it was only possible to obtain data over less than one order of magnitude in irradiance with the photodetector used. At

lower

average

powers

the

fluorescence signal was too weak to detect reliably, and at higher average powers the immersion oil for the objective lens degraded and/or the solvent itself exhibited MPA. Using a

Figure 2 Nonlinear fluorescence excitation data for CVL in 1,6-dicyanohexane at different wavelengths at laser repetition rates of (A) 76 MHz and (B) 152 MHz. The error bars for both the average excitation power and the signal level are on the order of the symbol size, and so are not shown.

more sensitive detector could extend this dynamic range. With 750 nm excitation, the slope of the plot is 2.05 at 76 MHz and 2.03 at 152 MHz. With 800 nm excitation, the slope of the plot is 2.21 at 76 MHz and 2.08 at 152 MHz. With 830 nm excitation, the slope of the plot is 2. 98 at 76 MHz and 2.89 at 152 MHz. We note that we are reporting the values of n to three significant figures only for the sake of qualitative comparison, as the limited dynamic range of the data does not justify the use of this degree of precision. Indeed, the plots at 800 nm are clearly not linear, particularly in the case of excitation at a repetition rate



of 76 MHz. At each wavelength, the exponents at the two repetition rates are the same within statistical error, although there is a systematic trend across the wavelengths that the exponent is smaller at the higher repetition rate. We

can

gain

a

qualitative

understanding of this behavior from the absorption spectrum of CVL in 1,6dicyanohexane. CVL is known to exhibit solvent-dependent dual emission,25-28 a phenomenon that has been ascribed to the existence of two different singlet excited states that are semi-localized on different portions of the molecule.26 Figure 3A shows the absorption spectrum from 300 nm to 475 nm, along with the effective 2-

Figure 3 Absorption spectrum of CVL in 1,6-dicyanohexane for (A) its first absorption band and (B) its second distinct absorption band. The vertical lines indicate the effective linear absorption wavelengths corresponding to (A) 2-photon absorption and (B) 3-photon absorption at the three wavelengths for which NFE data were obtained. Note that the line for 750 nm is all the way at the left in (B). The chemical structure of CVL is also shown in (A).

photon excitation wavelengths (λ/2) for 750 nm, 800 nm, and 830 nm light. The absorption peak at the longest wavelength (corresponding to absorption to S1) is thought to be associated with the dimethylamino phthalide portion of the molecule.26 There is essentially no linear absorption at 415 nm, so 2-photon absorption of 830 nm light would be expected to be minimal. The linear absorption is relatively weak at 400 nm, but is substantial at 375 nm. We therefore might expect strong 2-photon absorption at 750 nm, and weaker 2-photon absorption at 800 nm.


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Figure 3B shows the absorption spectrum of CVL in 1,6-dicyanohexane between 250 nm and 300 nm. Because this absorption band is substantially stronger than that shown in Figure 3A (by a factor of approximately 22; see Figure S1), to prevent saturation the concentration of CVL was lowered in the sample used to obtain the spectrum in Figure 3B. This absorption band is thought to involve transitions to higher excited states of both the dimethylaniline and dimethylamino phthalide groups in the molecule,26 so we denote the final state Sq. Based on the linear absorption spectrum, we might expect substantial 3-photon absorption of 830 nm light and 800 nm light, with weaker 3-photon absorption of 750 nm light. All other things being equal, 2-photon absorption would generally be expected to be substantially stronger than 3-photon absorption when both processes are possible. Indeed, in the case of 750 nm excitation, both processes can occur, but the fact that the slope of the NFE plot is close to 2 indicates that 2-photon absorption dominates at this wavelength. Conversely, at 830 nm the linear absorption spectrum suggests that there is little chance for 2-photon absorption, so n is close to 3. At 800 nm, both 2-photon and 3-photon absorption contribute substantially, so n is between 2 and 3. From the NFE plots in Figure 2, it is also evident that the nonlinear fluorescence signal depends on the repetition rate. A steady-state kinetic model can be used to determine how the NFE plot should change if the laser repetition rate is doubled, under the assumption that the doubled repetition rate is low enough that each pulse has an effect that is independent of that of its predecessor. When this condition is met, doubling the repetition rate at constant pulse energy will double the fluorescence signal, regardless of the order of the effective nonlinear absorption. On the other hand, the average excitation powers required to achieve the same fluorescence signal at



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the two different repetition rates can be related directly to the order of the effective nonlinear absorption. We assume that the fluorophore has only a ground state, S0, and an excited state, S1. The average irradiance is I, and the pulses are assumed to be square in time. The peak irradiance is then I/D, where D is the duty cycle (i.e., the pulse duration divided by the repetition time). The average rate at which molecules are excited by an n-photon process is knD(I/D)n, where kn is the n-photon rate constant and the extra factor of D appears because the laser field is only on for this fraction of the time on average. The rate of change of the population of the ground state is given by −

𝑑𝑑[𝑆𝑆0 ] 𝑑𝑑𝑑𝑑

𝐼𝐼 𝑛𝑛

= 𝑘𝑘𝑛𝑛 𝐷𝐷𝑛𝑛−1 [𝑆𝑆0 ] − 𝑘𝑘𝑓𝑓𝑓𝑓 [𝑆𝑆1 ] ,

(2)

where kfl is the rate constant for fluorescence. We have ignored non-radiative relaxation here, because this process does not change the fundamental dependence of the signal on the repetition rate. We also assume that either the fluorescence lifetime is shorter than the repetition time or, more generally, that the excited-state population created by one laser pulse is not affected by subsequent laser pulses. Combining Eq 2 with the conservation requirement that at any time t [𝑆𝑆0 ]0 = [𝑆𝑆0 ]𝑡𝑡 + [𝑆𝑆1 ]𝑡𝑡

(3)

and then solving for the fractional steady-state population of the excited state (which is proportional to the steady-state fluorescence rate), we find [𝑆𝑆1 ]𝑆𝑆𝑆𝑆 [𝑆𝑆0 ]0

=

𝑘𝑘𝑛𝑛 𝐼𝐼𝑛𝑛 𝐷𝐷𝑛𝑛−1 𝑘𝑘 𝐼𝐼𝑛𝑛 𝑘𝑘𝑓𝑓𝑓𝑓 + 𝑛𝑛 𝐷𝐷𝑛𝑛−1

.

(4)

So long as the system is far from the saturation regime, this equation reduces to [𝑆𝑆1 ]𝑆𝑆𝑆𝑆 [𝑆𝑆0 ]0

= 𝑘𝑘

𝑘𝑘𝑛𝑛 𝐼𝐼 𝑛𝑛

𝑓𝑓𝑓𝑓 𝐷𝐷

𝑛𝑛−1

.


(5)

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Thus, if the repetition rate (and therefore the duty cycle) is doubled, achieving the same steadystate excited-state population requires that the irradiance be increased by a scaling factor of 𝑠𝑠(𝑛𝑛) =

2𝑛𝑛−1⁄𝑛𝑛 . Although, within the constraints of this derivation, Eq 5 is only rigorously correct when a single order of nonlinear absorption exists, this expression is also approximately correct when n is not an integer. In Figure 4 we show for each excitation wavelength the NFE data at 76 MHz and 152 MHz, as well as the 152 MHz data predicted by scaling the average excitation power for the 76 MHz data by s(n), where the value of n is that determined from the 76 MHz data. In all cases, the predicted 152 MHz data are in qualitative agreement with the actual data. However, there is also a clear systematic trend of the predicted scaling factors ranging from being too large at the shortest wavelength to too small at the longest wavelength. To test whether this trend arises from some sort of interpulse excited-state absorption (ESA) process, we obtained NFE data with the two pulse trains not spatially overlapped, such that the two sets of pulses arrived at the two different focal spots, each with a 76 MHz repetition rate, even though the overall repetition rate was 152 MHz. Because fluorescence is not a cumulative observable, the fluorescence intensity is not expected to be influenced by the overlap of the beams when interpulse ESA is not present. At each wavelength, we indeed observed that the exponents derived from the 152 MHz NFE data were the same within experimental error whether or not the beams were spatially overlapped. This observation is in agreement with the previous results of Li and Maroncelli in similar solvents.27 There are some additional puzzling aspects of the NFE data. First, despite the reasonable agreement between the predictions and the data in Figure 4, as noted above there is also a clear



trend that the measured exponent at each wavelength decreases with increasing repetition rate. Second, the logarithmic plot is obviously nonlinear in the case of 800 nm excitation, particularly at a repetition rate of 76 MHz. Additionally, we have found previously that radical photoinitiation with CVL is a relatively efficient 3-photon process at 800 nm,20 which means that n might be expected to be higher than what we have measured from NFE data for excitation at this wavelength. We

next

turn

to

2-BCEIn

measurements. Shown in Figure 5 are representative wavelengths

data studied.

at The

all

three

measured

exponents m are 2.02 for 750 nm excitation (at an average power of 7.1 mW for one beam, corresponding to a 21.5 µV signal), 2.13 for 800 nm excitation (at an average power of 12.6 mW for one beam, corresponding to a 6.4 µV signal), and

Figure 4 Experimental nonlinear fluorescence excitation data at 76 MHz (red), 152 MHz (blue), and predicted for 152 MHz (black) for excitation at (A) 750 nm, (B) 800 nm, and (C) 830 nm.


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2.90 for 830 nm excitation (at an average power of 14.3 mW for one beam, corresponding to a 2.6 µV signal). These results are in qualitative agreement with the NFE measurements, although the 2-BCEIn measurements offer greater precision, with uncertainties ranging from ±0.02 at 750 nm to ±0.05 at 830 nm. Additionally, the 2BCEIn measurements give the effective exponent at a given fluorescence intensity

Figure 5 2-BCEIn data for CVL in 1,6-dicyanohexane for excitation at 750 nm, 800 nm, and 830 nm. The lines are fits to Eq 1.

(lock-in reading) as opposed to requiring determination of the slope in a logarithmic NFE plot to determine an average exponent.

A

non-integer

2-BCEIn

exponent is not equal to the slope of a logarithmic plot at that fluorescence intensity, but the slope can be derived from the exponent.19 The ability to determine m at any fluorescence intensity is of particular

Figure 6 2-BCEIn data for CVL in 1,6-dicyanohexane with excitation at 800 nm at different average powers for single beam. The lines are fits to Eq 1.

value when two or more orders of absorption make a substantial contribution. As an example, in Figure 6 we show 2-BCEIn data for CVL in 1,6-dicyanohexane with 800 nm excitation at four different values of the average power for one beam, 12.6 mW, 13.6 mW, 14.5 mW, and 16.0 mW



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(corresponding to signals of 6.42 µV, 8.05 µV, 10.0 µV, and 13.0 µV, respectively). Despite the fact that the irradiance changes by only a factor of 1.27 for these data, there is an unmistakable increase in m over this range that could not be measured easily with an NFE plot. The data from these 2-BCEIn plots are summarized in Table 1.

Table 1. Single-beam average excitation powers (P), corresponding lock-in readings, measured 2-BCEIn exponent (m), and relative amplitude of 2-photon absorption (a) in conjunction with 3photon, 4-photon, or 5-photon absorption. The uncertainties in the relative amplitudes are ±1 standard error from fits to Eq 7. P (mW) 12.6 13.6 14.5 16.0

signal (µV) 6.42 8.05 10.0 13.0

m 2.13±0.03 2.20±0.03 2.31±0.04 2.37±0.04

a2,3 0.848±0.006 0.779±0.008 0.68±0.04 0.58±0.03

a2,4 0.910±0.003 0.870±0.005 0.81±0.02 0.75±0.02

a2,5 0.930±0.003 0.900±0.004 0.85±0.01 0.81±0.02

It would appear reasonable to assume that CVL undergoes a combination of 2-photon and 3-photon absorption at 800 nm. Indeed, the NFE data in Figure 4B can be fit well to the sum of a term that is quadratic in the irradiance and one that is cubic in the irradiance. However, an equally good fit can be obtained for a quadratic term and a quartic term. As we have shown previously,19 when the two contributing exponents differ by one, going from the NFE signal arising 90% from the lower exponent to the signal arising 90% from the higher exponent requires a change in irradiance of a factor of 81. When the exponents differ by two, a factor of 9 change in irradiance is required for the same transition. NFE data do not generally have sufficient dynamic range to capture this changeover, and so it is difficult to distinguish the contributions of two different absorption orders (or even which absorption orders contribute) in NFE plots that are curved or that have a slope that is not an integer.


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When two different absorption orders contribute to the generation of an observable signal,19 2BA spectroscopy data can be used to determine these orders. Here we provide a brief description of this process. We first assume that the fluorescence signal, S(I), arises from j-photon and kphoton absorption, where k > j: 𝑆𝑆(𝐼𝐼) = 𝐴𝐴𝐼𝐼 𝑗𝑗 + 𝐵𝐵𝐼𝐼 𝑘𝑘 .

(6)

Here, A and B are independent of I, and in the case of fluorescence encompass factors such as the MPA cross section, the pulse shape, and the fluorescence quantum yield for the respective absorption orders.22 According to Eq 6, the ratio of the contribution of k-photon absorption to that of j-photon absorption at irradiance I is given by BIk-j/A. In the corresponding 2-BCEIn experiment, a plot at a given value of the single-beam irradiance follows 𝑗𝑗 𝑗𝑗 𝑎𝑎�𝑃𝑃�1 + 𝑃𝑃�2 � + 𝑏𝑏�𝑃𝑃�1𝑘𝑘 + 𝑃𝑃�2𝑘𝑘 � = 1.

(7)

Here, a and b are functions of the irradiance (or, equivalently, the average excitation power) for a single beam. We impose the further constraint that a + b = 1. The ratio of the contribution of kphoton absorption to that of j-photon absorption at irradiance I is given by b/a. There is no a priori means of determining the values of j and k from a single 2-BA spectroscopy plot. A set of 2-BA data can be mapped to any pair of values of j and k via the equation19 𝑗𝑗 𝑗𝑗 1 − 𝑃𝑃�1𝑘𝑘 − 𝑃𝑃�2𝑘𝑘 = 𝑎𝑎�𝑃𝑃�1 + 𝑃𝑃�2 − 𝑃𝑃�1𝑘𝑘 − 𝑃𝑃�2𝑘𝑘 � . 𝑗𝑗

(8)

𝑗𝑗

A plot of 1 − 𝑃𝑃�1𝑘𝑘 − 𝑃𝑃�2𝑘𝑘 as a function of 𝑃𝑃�1 + 𝑃𝑃�2 − 𝑃𝑃�1𝑘𝑘 − 𝑃𝑃�2𝑘𝑘 can thus be used to determine a (and therefore b) for any given pair of j and k. If the appropriate orders have been chosen, then b/a must be equal to BIk-j/A. Thus, a plot of b/a as a function of I should be linear if j and k differ by one, quadratic if j and k differ by two, and so on. In all cases the plot should pass through the origin.



We can use this property to determine whether irradiance-dependent 2-BCEIn data are consistent with a particular model of the absorption processes contributing to the signal.19 Figure 7 shows b/a values derived from Eq 8 for 2-photon absorption in conjunction with (red circles) 3-photon absorption, (green triangles) 4-photon absorption, and (blue squares) 5-photon absorption, along with best-fit lines for a linear, quadratic, and cubic dependence, respectively. The a values for these three different cases are also listed in Table 1. It is clear from this figure that neither the combination of 2photon absorption and 3-photon absorption nor the combination of 2-photon absorption

Figure 7 Values of b/a for combinations of 2-photon absorption and 3-photon absorption (red circles), 2-photon absorption and 4-photon absorption (green triangles), and 2photon absorption and 5-photon absorption (blue squares). The lines are the best linear fit to the first case, the best quadratic fit to the second case, and the best cubic fit to the third case. All fits were constrained to pass through the origin. The error bars are determined from the standard error in a.

and 4-photon absorption is consistent with the data. The correspondence with a combination of 2photon absorption and 5-photon absorption is fair, but observing this combination of absorption orders is highly unlikely. The most likely explanation for the data in Figure 7 is that the transition from 2-photon absorption to 3-photon absorption occurs much more rapidly than would be predicted by Eq 6. A number of different mechanisms might drive this phenomenon. One possibility is that each pulse encounters some excited-state population left by the previous pulse, thus leading to ESA. However, as discussed above, NFE data are identical whether or not the two pulse trains are overlapped spatially. The 2-BCEIn exponent is also identical whether or not the pulse trains are


Page 18 of 32

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overlapped spatially. These observations rule out any substantial contribution from interpulse ESA. Another possibility is that thermal effects at higher irradiance lead to spectral shifts and/or to a change in the fluorescence quantum yield. Again, the fact that the 2-BCEIn data are identical when the two pulse trains are not overlapped spatially suggests that this scenario is not plausible. Temperature-dependent absorption and fluorescence spectra further indicate that any thermal effects are minimal (Figures S2 and S3). Because our laser pulses have a duration of ~150 fs, another possibility is that intrapulse ESA occurs following 2-photon absorption.29-30 Indeed, when 2-photon and 3-photon absorption can both be driven by the same wavelength of light, a sequential (2 + 1 photon) absorption process must also be possible. To investigate this scenario, we construct a kinetic model of these processes. We assume temporally square laser pulses of duration tp, where tp is short enough that we can ignore any fluorescence during a pulse. Electronic dephasing is assumed to be rapid enough that there is no quantum interference between the simultaneous (3-photon) and sequential (2 + 1 photon) pathways. We further assume that the excited states relax completely between laser pulses. Within this model, the relevant differential rate equations for the ground state S0, the first excited singlet state S1, and a higher excited singlet excited state Sq to which 3-photon absorption occurs are − and

𝑑𝑑[𝑆𝑆0 ] 𝑑𝑑𝑑𝑑

= 𝑘𝑘2 𝐼𝐼 2 [𝑆𝑆0 ] + 𝑘𝑘3 𝐼𝐼 3 [𝑆𝑆0 ]

(9)

𝑑𝑑[𝑆𝑆1 ]

= 𝑘𝑘2 𝐼𝐼 2 [𝑆𝑆0 ] − 𝑘𝑘1 𝐼𝐼[𝑆𝑆1 ]

(10)

𝑑𝑑�𝑆𝑆𝑞𝑞 �

= 𝑘𝑘3 𝐼𝐼 3 [𝑆𝑆0 ] + 𝑘𝑘1 𝐼𝐼[𝑆𝑆1 ] .

(11)

𝑑𝑑𝑑𝑑

𝑑𝑑𝑑𝑑



Page 20 of 32

Here, k2 and k3 are the rate constants for 2-photon and 3-photon absorption from the ground state to states S1 and Sq, respectively, and k1 is the linear absorption rate constant from state S1 to state Sq. After the pulse, the population of the first excited singlet state is [𝑆𝑆1 ]𝑡𝑡𝑝𝑝 = 𝑘𝑘

𝑘𝑘2 𝐼𝐼 2 2 3 2 𝐼𝐼 +𝑘𝑘3 𝐼𝐼 −𝑘𝑘1 𝐼𝐼

[𝑆𝑆0 ]0 �𝑒𝑒 −𝑘𝑘1 𝐼𝐼𝑡𝑡𝑝𝑝 − 𝑒𝑒 −�𝑘𝑘2 𝐼𝐼

2 +𝑘𝑘

3 𝐼𝐼

3 �𝑡𝑡 𝑝𝑝

and the population of the higher excited singlet state is �𝑆𝑆𝑞𝑞 �𝑡𝑡

𝑝𝑝

�

(12)

𝑘𝑘3 𝐼𝐼 3 𝑘𝑘1 𝐼𝐼𝑘𝑘2 𝐼𝐼 2 2 3 =� 2 − � [𝑆𝑆0 ]0 �1 − 𝑒𝑒 −�𝑘𝑘2 𝐼𝐼 +𝑘𝑘3𝐼𝐼 �𝑡𝑡𝑝𝑝 � 3 2 3 2 3 (𝑘𝑘2 𝐼𝐼 + 𝑘𝑘3 𝐼𝐼 − 𝑘𝑘1 𝐼𝐼)(𝑘𝑘2 𝐼𝐼 + 𝑘𝑘3 𝐼𝐼 ) 𝑘𝑘2 𝐼𝐼 + 𝑘𝑘3 𝐼𝐼 +�𝑘𝑘

𝑘𝑘2 𝐼𝐼 2 2 3 2 𝐼𝐼 +𝑘𝑘3 𝐼𝐼 −𝑘𝑘1 𝐼𝐼

� [𝑆𝑆0 ]0 �1 − 𝑒𝑒 −𝑘𝑘1 𝐼𝐼𝑡𝑡𝑝𝑝 � ,

(13)

where [S0]0 is the initial population of the ground state. For the purposes of testing the behavior of this model, we assume that tp = 200 fs, k1 = 1.00 × 106, k2 = 3.00 × 10-29, and k3 = 2.00 × 10-35. The units of k1I, k2I2, and k3I3 are all s-1. The effective fluorescence quantum yields for states S1 and Sq are Φ1,eff and Φq,eff, respectively. The effective fluorescence quantum yield for state j is the product of the absolute fluorescence quantum yield Φj and the fluorescence detection efficiency following excitation to that state. Note that Φq,eff involves any contribution of emission arising from population that was excited initially to Sq, whether this emission is from Sq or any excited state below it. In Figure 8A we show logarithmic plots of the total amount of fluorescence as a function of excitation irradiance for Φ1,eff = 0.5 and values of Φq,eff ranging from 0 to 1, and in Figure 8B we show the corresponding slopes. For reference, we also show a plot for k1 = 0 and Φq,eff = 1. In the latter case there is no ESA, and so we expect a curve that can be fit to Eq 6 for j = 2 and k = 3, as is indeed the case. The curve for Φq,eff = 1 is virtually identical to the reference curve, because under these circumstances ESA does not diminish the signal in any way. Similar results (not shown) are observed for values of Φq,eff down to ~0.3. As Φq,eff becomes smaller, the point at which


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the exponent becomes 3 occurs at higher irradiances, because ESA leads to a substantial

decrease

in

overall

fluorescence. For Φq,eff = 0.001 and smaller, this effect causes the slope to drop below 2 at the onset of ESA, before increasing to 3. When Φq,eff = 0, the limiting value of the slope becomes 1, as in this case we essentially have 2-photon excitation of fluorescence followed by linear deactivation through ESA. We can use this model to determine the apparent value of BI/A in the case of a 2photon process plus a 3-photon process. In this situation, the slope of a logarithmic

Figure 8 (A) Model of fluorescence as a function of excitation irradiance for different values of the effective fluorescence quantum yield of Sq (see text for values of other parameters in the model) (B) Slopes of the plots in (A) as a function of excitation irradiance.

plot at a given value of I is 𝑑𝑑 log(𝐴𝐴𝐼𝐼 2 +𝐵𝐵𝐼𝐼 3 ) 𝑑𝑑 log(𝐼𝐼)

𝐵𝐵𝐵𝐵

= 2 + 𝐴𝐴+𝐵𝐵𝐵𝐵 .

(14)

Based on this equation, the local exponent is 2 plus the fraction of the signal that comes from 3photon absorption. We then have that 𝐵𝐵𝐵𝐵 𝐴𝐴

𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠(𝐼𝐼)−2

= 3−𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠(𝐼𝐼) ,

(15)

where slope(I) is the slope of the logarithmic plot measured at irradiance I. In the presence of intrapulse ESA, Eq 14 no longer holds. However, we can still use Eq 15 to determine the apparent value of BI/A. These values are shown in Figure 9A under the same conditions used in Figure 8.



In the absence of ESA, BI/A increases linearly with I, as discussed above. In the presence of ESA, the apparent value of BI/A is no longer a linear function of I. When Φq,eff is less than Φ1,eff, the plot of the apparent value of BI/A curves upward with increasing irradiance. For small enough values of Φq,eff, the apparent value of BI/A can even be negative, and when Φq,eff = 0 the apparent value of BI/A never becomes positive. This effect also causes the irradiance range over which the system goes from apparently having 10% 3photon absorption (I10, at a slope of 2.1) to apparently

having

90%

3-photon

absorption (I90, at a slope of 2.9) to be

Figure 9 (A) Apparent values of BI/A based on the slopes in Figure 8B. (B) Ratio of the irradiance at which the apparent contribution of 3-photon absorption is 90% to the irradiance at which the apparent contribution of 3-photon absorption is 10% for the model in Figure 8 at different values of Φq,eff. The dashed line indicates the value expected in the absence of ESA.

smaller than the factor of 81 predicted in the absence of ESA when Φq,eff < Φ1,eff (Figure 9B), due to the concave nature of the apparent BI/A curve. When the two quantum yields are equal, the ratio I90/I10 = 81 is recovered, and when Φq,eff > Φ1,eff then I90/I10 exceeds 81 due to the convex nature of the BI/A curve. Based on the fact that CVL in 1,6-dicyanohexane appears predominantly to undergo 2-photon absorption with 750 nm excitation and 3-photon absorption with 830 nm excitation, it is reasonable to assume that there is a mixture of these two absorption orders with 800 nm excitation. The


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simulated values of BI/A in Figure 9 bear a strong resemblance to the data for 2-photon absorption plus 3-photon absorption in Figure 7. We can conclude that the values of b/a found from 2-BCEIn data are only apparent values, as ESA was not taken into account in determining these values. This phenomenon can also explain the difference between the predicted and observed changes in the NFE data in Figure 4 with changing repetition rate. Effective fluorescence quantum yields are influenced by instrumental sensitivity at different wavelengths, which means that different instruments may yield different characterizations of nonlinear absorption in the same system. CVL, for instance, is known to exhibit dual fluorescence.26 Furthermore, excitation of Sq can lead to fluorescence in the ultraviolet (Figure S4), a region in which our detection system is not particularly sensitive. Based on the above analysis, the effects of ESA are amplified by the fact that our instrument leads the Sn state to have an even smaller apparent fluorescence quantum yield than it does naturally, as the effective quantum yield is influenced by factors such as collection efficiency and the spectral sensitivity of the detector. ESA is responsible for the slopes in Figure 8B dipping below 2. Eventually, k1It becomes so large that the exponentials with this argument in Eqs 12 and 13 approach zero. In other words, any population that reaches Sq via 2-photon absorption undergoes immediate ESA. As we approach this situation, we can assume that k1 >> k2I2 + k3I3, and that (k2I2 + k3I3)tp

Probing Multiphoton Photophysics Using Two-Beam Action

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