1. INTRODUCTION

In 1931 Maria Göppert-Mayer predicted that an atom or a molecule could absorb two photons simultaneously in the same quantum event.[1] The first experimental evidence of this phenomenon came 30 years later when Kaiser and Garret demonstrated two-photon excitation in a CaF₂:Eu²⁺ crystal.[2] Several techniques have been used to determine two-photon excitation cross sections of various materials for more than two decades.[3]–[5] Direct measurement of two-photon absorption cross sections is usually difficult because only a small fraction of photons is absorbed in the two-photon process. Two-photon fluorescence excitation is an alternative approach to determining two-photon fluorescence excitation (TPE) cross sections, provided that the material is fluorescent and that its fluorescence quantum efficiency η₂ is known. Although the fluorescence method provides much better sensitivity, accurate determination of absolute two-photon cross sections is still hampered by the strong dependence of absorption rates on the temporal and spatial coherence of the excitation light.[6],[7]

Denk et al. succeeded in applying two-photon-excited fluorescence to laser scanning microscopy. Because the excitation rate is proportional to the square of the illumination intensity, two-photon laser scanning microscopy (TPLSM) gains its advantage over conventional confocal fluorescence microscopy from its intrinsic three-dimensional resolution and the absence of background fluorescence.[8] This discovery has stimulated new interest in the physics of two-photon excitation. Previously, two-photon spectroscopy was used mainly to study the electronic structure of excited states in molecules,[9] for which it was useful mainly because its selection rules are different from those for one-photon absorption. Less effort has been devoted to accurate quantitative studies of common fluorophores widely used in TPLSM. In addition, substantial disagreement between published values of two-photon cross sections often exists.[5] The lack of knowledge of TPE cross sections and spectra for common fluorophores used in biological studies has been a significant obstacle to the use of TPLSM. Although the TPE cross sections of Rhodamine B and Fluorescein from 750 to 840 nm were recently published by Fisher et al.,[10] the uncertainties in the data were large (1 order of magnitude) and, notably, substantial deviations from the power-squared dependence of two-photon-excited fluorescence were observed. In this paper we report TPE spectra and absolute values of TPE cross sections for 11 commonly used fluorophores in the Ti:sapphire tuning range from 690 to 1050 nm. In disagreement with the data obtained in Ref. [10], we observed no deviation from the fluorescence square-law dependence at the same wavelengths and comparable peak intensities. Our primary motivation has been to establish a reliable data base of TPE cross sections in the visible and the near IR for two-photon fluorescence microscopy. However, these data, especially with their wide spectral range, should also be valuable to molecular spectroscopists, particularly as a guide for predicting TPE spectra. Our methods should facilitate future TPE measurements, and the absolute cross-section values reported here can be used for calibration purpose.

Section 2 discusses the relevant parameters in TPE cross-section measurements and provides the necessary analysis of TPE using diffraction limited focus with both pulsed and cw light sources. The experimental setup and sample preparations are presented in Sections 3 and 4, respectively. In Section 5 we present our results in detail and discuss several key issues related to TPE cross-section measurements, such as power-squared dependence, pulse-width dependence, and fluorescence emission spectra. A comparison of our results with the existing data is presented in Section 6.

2. ANALYSIS OF TWO-PHOTON EXCITATION OF FLUOROPHORES

The crucial problem in TPE cross-section measurements is that the absorption rates strongly depend on the spatial and temporal coherence of the excitation light. Knowing the relationship between the experimentally measured fluorescence power and the excitation power is essential in TPE cross-section measurements. In this section we first examine the relationship in general and then discuss the relevant parameters in any TPE cross-section measurements. A detailed analysis that specifically applies to our experiment follows.

Because TPE is a second-order process, the number of photons absorbed per molecule per unit time by means of TPE is proportional to the two-photon absorption cross section δ and to the square of the incident intensity I.[9] In a particular experiment, the total number of photons absorbed per unit time N_abs is also a function of dye concentration C and the illuminated sample volume V:

In the absence of saturation and photobleaching, C(r, t) can be assumed to be a constant. Moreover, in the case when one can separate time and space dependence of the excitation intensity, we have

where S(r) and I₀(t) describe the spatial and temporal distribution of the incident light, respectively; i.e., I(r, t) = S(r)I₀(t). We chose S(r) to be a unitless spatial distribution function.

Assuming no stimulated emission and self-quenching, the number of fluorescence photons collected per unit time (F) is given by

where η₂ and ϕ are the fluorescence quantum efficiency of the dye and the fluorescence collection efficiency of the measurement system, respectively. Here the factor 1/2 simply reflects the fact that two photons are needed for each excitation event. In practice, only the time-averaged fluorescence photon flux 〈F(t) 〉 is measured:

We note that 〈F(t) 〉 is proportional to 〈I₀²〉, not 〈I₀〉². Because most detectors give only a signal that is proportional to 〈I₀(t) 〉, we rewrite Eq. (4) in terms of the average intensity:

where g = 〈I₀²(t) 〉/〈I₀(t) 〉² is a measure of the second-order temporal coherence of the excitation source.[11]

Equation (5) presents the relevant quantities in a TPE cross-section measurement. Hence experimental determination of δ involves the characterization of four parameters: the spatial distribution of the incident light [∫_VS²(r)dV ], the degree of the second-order temporal coherence (g), the fluorescence collection efficiency of the system (ϕ), and the fluorescence quantum efficiency (η₂).

A. Spatial Dependence

We use the intensity distribution resulting from an aperture diffraction-limited focus in these experiments. We first consider the three-dimensional intensity distribution at the focal spot of a diffraction-limited objective lens with uniform illumination. In practice we achieve this by expanding the incident laser beam so that the beam diameter is much larger than the back aperture of the focusing lens. The lens’s properties are then determined by its point-spread function. Let z be the distance along the optical axis, ρ be the distance away from the optical axis, n be the refractive index of the sample medium, and λ be the wavelength in vacuum. The dimensionless distance from the optic axis v and the distance from the in-focus plane u are given by

where N.A. = n sin θ and θ is the half-angle of collection for the lens. We use the paraxial form of the normalized intensity point-spread function (h²[u, v]) for a diffraction-limited lens[12]:

The paraxial approximation is adequate for sin θ < 0.7, or all N.A. of <1.0 with an oil immersion objective.[13] The intensity distribution near the focal point will be of the form

where I₀ is the intensity at the geometric focal point (u = v = 0). The relation between instantaneous incident power P and I₀ follows from energy conservation:

We define pulse intensities as the instantaneous intensities averaged over the pulse duration. However, S²(r) = h⁴(u, v) cannot be integrated analytically. In thick samples for which the sample thickness is much greater than the focal depth, numerical calculations[14] show that

Substituting Eq. (9) and relation (10) into Eq. (5), we get the time-averaged detected photon flux:

Although the collection efficiency ϕ is dependent on the N.A. of the collecting lens, we note that the total fluorescence generation is independent of the N.A. of the focusing lens in thick samples. A similar form for 〈F(t) 〉 can also be obtained for Gaussian intensity distribution.[15]

B. Temporal Dependence

As Eq. (5) shows, for a quantitative measurement of δ one needs to monitor not only the average but also the temporal fluctuations of the input photon flux, i.e., the degree of second-order temporal coherence g. We consider two cases that are used in our experiment: pulsed excitation and single-mode cw excitation.

1. Pulsed Excitation

The focused intensity obtained from a mode-locked laser is the periodic function in time:

where f is the pulse repetition rate. Let t = 0 be at the peak of one excitation pulse. Because of the periodic nature of the pulse train, one needs to calculate g for only one cycle. Defining τ as the excitation pulse width (FWHM) and fτ as the duty cycle, we can express g in terms of the ratio of the dimensionless quantity g_p, which depends only on the shape of the laser pulse and the duty cycle:

For pulses with a Gaussian temporal profile one finds that g_p = 0.664, whereas for a hyperbolic-secant square pulse one finds that g_p = 0.588.

Combining Eqs. (11) and (13), we get

The numerical value of g = g_p/(fτ) for a mode-locked Ti:sapphire laser is approximately 10⁵ (f ∼ 100 MHz and τ ∼ 100 fs). Consequently, mode-locked pulsed excitation can excite dyes at very low average laser power.

2. Single-Mode cw Excitation

An ideal single-mode cw laser has g = 1. Measuring two-photon excitation cross sections by using such lasers requires only the value of the average excitation intensity. However, single-mode cw excitation requires 10² – 10³ ( $\sqrt{g}$ of a mode-locked Ti:sapphire laser) times more average power than pulsed excitation to yield the same rate of TPE, because of the much smaller value of g. Nevertheless, cw excitation is feasible for dyes with relatively large cross sections. Setting g = 1 in Eq. (11) yields the appropriate equation for single-mode cw excitation.

3. EXPERIMENTAL METHODS

A. Pulsed Excitation

The experimental apparatus is shown schematically in Fig. 1. A mode-locked Ti:sapphire laser (Spectra-Physics Tsunami) pumped by an argon laser is used for pulsed excitation. The pulse width is continuously monitored by an intensity autocorrelator (Spectra-Physics 409) and the pulse repetition rate is measured with a fast photodiode. A monochromator measures the excitation wavelengths, and a CCD camera continuously monitors the dispersed pulse spectrum. This is essential because cw radiation can be produced along with the sub-picosecond pulses.[16] Four different intracavity mirror sets are necessary to cover the spectral range from 690 to 1050 nm. Customized special coatings are used to reach the wavelength ∼690 nm. A half-wave plate and a Glan–Thompson polarizer are used to control the illumination intensity. Linear polarized light is used throughout the experiment. A 5× beam expander expands the laser beam to approximately 1–1.5 cm in diameter (1/e²) to overfill the back aperture of the microscope objective, thereby forming an aperture diffraction-limited focal spot. A long-pass dichroic mirror (custom-made by Chroma) with >95% reflectivity for λ < 610 nm separates the fluorescence light from the excitation light. An achromatic microscope objective (Zeiss Neofluoar 0.3N/A 10×) serves as both the focusing lens and the collecting lens. Aberration caused by index mismatching at the interfaces is negligible for such a low-N.A. lens.[17] Another lens recollimates the beam onto a calibrated power meter (Molectron PM3) to measure the excitation power at the sample. The transmission of the lens is measured to be 92% throughout the laser tuning range. For most experiments a square capillary tube (Vitro Dynamics) with inner dimensions of approximately 0.9 × 0.9 mm and wall thickness 0.18 mm is used as the sample cell. An average excitation power of 1 mW at the sample in such a focusing geometry corresponds to a pulse intensity of ∼10²⁸ photons/(cm² s). Two-photon-excited fluorescence is confined to the focal region.[8] The focal depth in this experiment is estimated to be ∼0.04 mm. Because the sample thickness is much greater than the focal depth, relation (11) can be used without correction. The advantage of working with thick samples is that the generated fluorescence power is insensitive to the size of the focal spot, assuming that there is no aberration. Hence a small variation in the laser beam size (with excitation power monitored after the objective lens) would not affect the results of this measurement. Fluorescence is detected through a liquid barrier filter (2-cm path-length 1-M CuSO₄ solution) with O.D. > 10.0 for wavelengths >690 nm to exclude excitation illumination. A photomultiplier tube (Hamamatsu R1924) is used as the detector and is read by a computer-controlled photon counter (Stanford Research Systems SR400). Excitation powers of 1–10 mW at the sample are typically used.

B. Single-Mode cw Two-photon Excitation

To calibrate further the data obtained with the femtosecond pulsed excitation, we also obtain fluorescence excitation data by using a single-mode cw Ti:sapphire laser (Coherent 899-21 ring laser with three internal étalons) from 710 to 840 nm to illuminate the same sample. (Quasi-cw illumination obtained by reflected feedback into a mode-locked laser retained sufficient rms noise to prevent its use as a stable cw source.) The specified linewidth of this laser of less than 0.5 MHz is maintained by intracavity étalons. We confirmed this stable cw mode by measuring the linewidth, using a supercavity optical spectrum analyzer (Newport SR-100). An excitation power of ∼100 mW at the sample is necessary for an adequate fluorescence signal. The excitation and detection scheme is the same as in the pulsed excitation case. However, several additional problems are encountered. Because of high cw illumination intensity and low fluorescence output, background light became significant. Substitution of sample cells that consist of deep well slides (Fisher Scientific) and #1½ cover slips reduced this background to insignificance.

C. System Collection Efficiency (ϕ) and Fluorescence Quantum Efficiency (η₂)

System collection efficiency (ϕ) is a function of the collection efficiency of the objective lens, transmission of the optics, and photocathode quantum efficiency. We use two independent approaches to determine ϕ in this experiment.

In the first approach, ϕ is calculated for each measured fluorophore based on its fluorescence emission spectrum, the measured N.A. of the objective lens, transmission of the objective, and the transmission of the filters. The photomultiplier response was obtained from the manufacturer’s catalog.

In the second approach, one-photon-excited (OPE) fluorescence is used directly to measure ϕ for Fluorescein and Rhodamine B. Direct comparison with OPE fluorescence also gives the absolute values of TPE cross section, assuming equal emission quantum efficiencies.[5] OPE fluorescence is achieved with the 488-nm line from an air-cooled argon laser (Ion Laser Technology ILT 5000). To ensure that the collection efficiency for OPE and two-photon-excited fluorescence are the same, we use rectangular capillary tubes with inner dimensions of 0.1 mm × 2.0 mm. The much shorter path length (0.1 mm) ensures uniform fluorescence collection efficiency throughout the sample, which is necessary because OPE fluorescence occurs throughout the entire sample thickness (i.e., it is not localized to the focal region as in TPE). Here we have assumed that OPE and two-photon-excited fluorescence emission spectra are the same. As discussed below, we observe no exceptions to this assumption.

To test the accuracy of the calculated collection efficiencies we compare the values of ϕ obtained by the above two methods. Assuming that the one-photon fluorescence quantum efficiencies η₁ of Fluorescein and Rhodamine B are 0.9 and 0.7, respectively,[18] collection efficiencies obtained with OPE fluorescence agree with our calculated values to within 30%. The consistency between the two entirely different approaches demonstrates that the calculated collection efficiencies are reasonably accurate.

A direct measurement of the two-photon-excited fluorescence quantum efficiency η₂ is difficult. However, η₂ and η₁ are expected to be the same if the same excited state is reached. Here we have assumed that fluorescence quantum efficiencies are constant over the entire spectral range and that η₂ = η₁ for each fluorophore used in this experiment. Support for these assumptions is given in Section 5.

D. Fluorescence Emission Spectra

The excitation setup remains the same as that described in Subsection 3.A. Modifications in the detection setup accommodate a spectrograph with a liquid-nitrogen-cooled CCD detector (Spex 270M). Fluorescence is coupled into a 50-μm-diameter multimode optical fiber, and the output of the fiber is then imaged onto the entrance slit of the spectrograph. A motorized dual grating turret with 150- and 1200-line/mm gratings is used. With the low-density grating a spectral range of 600 nm is obtained with a resolution of 1 nm.

4. SAMPLE PREPARATION

Sample preparations are listed in Table 1. All the samples were purchased from either Eastman Kodak or Molecular Probes.

Stock solutions at 1–2 mM are formulated and then diluted to ∼100 μM. We further verify the dye concentrations by measuring one-photon absorption (OPA) spectra in a spectrophotometer (HP 8451A diode array spectrometer).

5. RESULTS AND DISCUSSION

A. Two-Photon Excitation Spectra

Figure 2 presents all the TPE spectra reported here. We obtain absolute cross-section values by using relation (15) (for pulsed excitation) and relation (11) (for cw excitation), assuming that η₁ = η₂. We determine the pulse-shape factor g_p in relation (15) by comparing cw and pulsed measurements and set it to be 0.41 throughout the entire wavelength range (see Subsection 5.E for details). Because the spectral width of a 100-fs pulse in the near IR is approximately 5–10 nm, TPE spectra are taken in steps of approximately 10 nm, except near the end of the tuning range. Blank controls are carried out at several different wavelengths. At the excitation intensity level for measuring cross sections no significant background is found. For comparison, we also indicate in Fig. 2 the wavelengths that correspond to twice the OPA maxima. Cross-section values at some selected wavelengths are also listed in Table 2. In the cases when the quantum efficiency η₁ is unavailable in the literature, TPE action cross sections (defined as η₂δ) are given. The main contribution to the estimated uncertainties in Table 2 is in the determination of the system’s collection efficiency (ϕ). The level of relative uncertainties also applies to the data presented in Fig. 2.

One general property conserved in all measured TPE spectra is that the TPE peak wavelengths appear blue shifted and never red shifted relative to twice the OPA peak wavelengths. As shown in Fig. 2, molecules of dyes such as Cascade Blue, Coumarin 307, Lucifer Yellow, and Indo-1 have very similar one- and two-photon excitation spectra. This suggests that the same excited states are reached regardless of the excitation mode. An examination of the molecular structures shows that these dye molecules have no center of symmetry. Because parity restrictions may be relaxed in such cases, the similarity between one-photon fluorescence excitation spectra and TPE spectra is not unexpected. In fact, for molecules having no center of symmetry it has been predicted that one-photon-allowed transitions are likely to show nonnegligible two-photon intensity if the excitation changes the dipole moment.[21]

For TPE spectra that are blue shifted, such as for Rhodamine B, Fluorescein, and DiI, one explanation is that some higher excited singlet states are reached with greater probability by TPE than by OPE fluorescence.[22],[23] Thus parity restrictions imply much larger TPE cross sections at the blue-shifted wavelengths than at twice the OPA wavelengths. Our experiments have confirmed this behavior for Fluorescein, Rhodamine B, and DiI. However, in addition to the parity restriction, other factors may affect the TPE spectra: for example, vibronic coupling may alter the inversion symmetry and partially allow the parity forbidden bands.[24]

Values of TPE cross sections depend on the polarization of the excitation light.[9] The ratio of δ values for circular and linear polarization, δ_(cir)/δ_(lin), falls in the range 0–1.5. The polarization ratios of Rhodamine B, Fluorescein, and DiI were tested at 768 nm, and δ_(cir)/δ_(lin) were all found to be 0.6 ± 0.1. All the data in Fig. 2 and Table 2 are obtained with linearly polarized excitation light and without polarization analysis of the fluorescence.

B. Power-Squared Dependence of Two-Photon-Excited Fluorescence

We tested the power-squared dependence of two-photon-excited fluorescence for most of the dyes listed in this paper at one or several excitation wavelengths. Pulse intensities at the geometric focal point are calculated with Eq. (9), and a tenfold range of excitation intensities is typically tested. Results for Rhodamine B and Fluorescein are summarized in Fig. 3. Other results are listed in Table 3. In all cases we recorded nearly perfect power-squared dependence of two-photon-excited fluorescence.

Theoretically, two-photon-excited fluorescence should obey the square-law dependence at low excitation intensity. However, because of various factors such as stimulated emission,[23],[27] excited-state absorption,[22] excited-state saturation, intensity-dependent TPE cross section, and lack of corrections for one-photon excitation, significant deviations from the square-law dependence of two-photon-excited fluorescence frequently have been observed. Harmann and Ducuing,[23] using nanosecond pulses, found that Rhodamine B showed linear dependence at 694 nm at pulse intensity levels above 2 × 10²⁵ photons/(cm² s). They suggested that the deviations were caused by stimulated emission from the fluorescence state. Bradley et al.,[22] using picosecond pulses, found that Rhodamine B at 1064 nm and Rhodamine 6G at 694 nm had similar deviations from square dependence at an intensity level of ∼10²⁸ photons/(cm² s). They suggested stimulated emission and excited-state absorption as the cause for Rhodamine 6G and excited-state absorption for Rhodamine B. In Fig. 3(a) we show results of our tests of intensity-squared dependence for Rhodamine B excited at 702 nm for intensity levels from 10²⁷ to 10²⁸ photons/(cm² s), using 100-fs pulses. Despite the fact that we used 2-orders-of-magnitude higher pulse intensity than that in the experiment of Hermann and Ducuing,[23] we find that the fluorescence obeys an intensity-squared dependence. Assuming that the deviation from square-law dependence in Ref. [23] was indeed caused by stimulated emission alone (as those authors suggested), the discrepancy between the two experiments can be explained by the following argument. Stimulated emission is a one-photon process and is proportional to the total number of photons incident upon the excited fluorophore.[27] Although a femtosecond light source provides a high pulse intensity, the average intensity is quite low. Therefore little stimulated emission is expected in our case. However, if pulses with much longer durations are used, as in the cases described in Refs. [22] (picoseconds and nanoseconds) and [23] (nanoseconds), stimulated emission could occur with significant probability at high intensity levels.

Fischer et al.[10] recently reported a significant departure from the fluorescence square-law dependence for Rhodamine B and Fluorescein at 770 and 825 nm at intensities of ∼10²⁷ – 5 × 10²⁸ photons/(cm² s). They further concluded that the resolution of two-photon microscopy using these fluorophores is worse than expected because of the subsquare-law dependence. We tested the power-squared dependence of two-photon-excited fluorescence for Fluorescein and Rhodamine B at the same intensity levels at 770 and 825 nm. In disagreement with the results of Fischer et al.,[10] we observed no deviations from the square-law dependence for both fluorophores [Figs. 3(b) and 3(c)]. To exclude solvent effects as a source of this discrepancy, we performed the same test for Rhodamine B, using ethanol as the solvent. Again, we observed no deviations from the square-law dependence.

The ratio between two-photon-excited and OPE fluorescence is proportional to the excitation intensity. Thermally assisted OPE fluorescence could become significant at low intensity levels if the wavelength of attempted TPE overlaps the absorption band for OPE fluorescence. This causes a severe departure from the square-law dependence. We have observed nearly linear power dependence with Rhodamine B at 705 nm, using a single-mode cw laser at an excitation intensity of ∼10²⁴ photons/ (cm² s). This is in good agreement with results of Hermann and Ducuing[23] at 694 nm. Using the same cw laser illumination, we also find significant deviation from square-law dependence for Rhodamine B at intensity levels of approximately 10²⁴ – 10²⁵ photons/(cm² s) for λ < 730 nm.

Two-photon-excited fluorescence saturation causes deviation from square-law dependence at high pulse intensity. Assuming that δ ∼ 10⁻⁴⁸ cm⁴ s/photon and τ₀ ∼ 100 fs, simple estimation shows that only 0.001% of the molecules at the focal spot are excited per laser pulse at pulse intensities of ∼10²⁸ photon/(cm² s). Hence saturation has no effect at the intensity level used in our experiment. We have, however, monitored the approach to saturation of some of the same fluorophores at pulse intensities of approximately 10³¹ - 10³² photons/(cm² s). Knowledge of fluorescence decay times and saturation conditions can be used for independent determination of two-photon-excited fluorescence quantum efficiency η₂ and TPE cross-section δ.[28],[29]

C. Dependence of Two-Photon-Excited Fluorescence on Pulse Width

Under pulsed excitation, g = g_p/(fτ) is inversely proportional to the excitation pulse width. Therefore TPE is expected to increase linearly with 1/τ under the condition that the pulse shape (i.e., g_p) is constant. A prism pair[30] was added (to Fig. 1) to chirp the laser pulse and enables us to vary the pulse width from 60 fs to ∼1.2 ps to test this expectation. Pulse width τ is measured before the objective lens by an autocorrelator and assuming a sech² pulse shape.[31] Figure 4(a) shows the fluorescence dependence on pulse width from 90 to 230 fs for Indo-1 at 770 nm. The 7% deviation from slope -1 in the logarithmic plot is probably caused by a change of pulse profile and thus of g_p. In Fig. 4(b) we compare TPE fluorescence of Indo-1 with second-harmonic generation of powder KDP particles excited by broad pulses with pulse widths of approximately 500 fs to 1.2 ps. Our results show that these particles have the same pulse-width dependence. We find significant deviation from 1/τ dependence at τ below 90 fs in these measurements because of pulse broadening after the pulse-width measurement that is caused by group-delay dispersion of the objective lens.[32],[33]

One way to distinguish simultaneous TPE from sequential TPE (two-step TPE)[34] is to study the fluorescence dependence on excitation pulse width. Two-photon-excited fluorescence is inversely proportional to τ only if τ is much longer than the intermediate state lifetime. In simultaneous TPE, virtual states with lifetimes of the order of 10⁻¹⁶ s serve as the intermediate states. Hence one expects two-photon-excited fluorescence to be inversely proportional to τ down to the 100-fs region accessible with a Ti:sapphire laser. However, because real states with lifetimes of approximately 10⁻⁹ – 10⁻¹² s serve as the intermediate states in sequential TPE, two-photon-excited fluorescence should be independent of τ when τ is approximately 100 fs. Hence, the results in Fig. 4 confirm simultaneous TPE for Indo-1 at 770 nm. Varying the time between adjacent pulses (by use of a Michelson interferometer, for example) instead of varying the pulse width provides a more convenient approach for studying the intermediate state lifetimes. Preliminary results show that intermediate state lifetimes for Rhodamine B (excited at 700 and 770 nm), Fluorescein (excited at 700 and 782 nm) and Coumarin 307 (excited at 770 nm) are less than 100 fs. Hence sequential two-photon excitation is highly unlikely for these dyes at the wavelengths tested.

D. One-Photon- and Two-Photon-Excited Fluorescence Emission Spectra Compared

In most cases reported in the literature, OPE and two-photon-excited fluorescence emission spectra were the same.[35] However, there are cases in which differences between OPE and two-photon-excited fluorescence emission spectra were observed.[36] We compared OPE and two-photon-excited fluorescence emission spectra for Rhodamine B, Fluorescein, DiI, Lucifer Yellow, and Coumarin 307, using the 488-nm line of an air-cooled argon laser to achieve OPE fluorescence. Figure 5 represents the emission spectra of fluorescein under OPE and two-photon-excited fluorescence. In all cases we found that OPE and two-photon-excited fluorescence emission spectra are identical within the resolution of the experiment. These results confirm our assumption that OPE and two-photon-excited fluorescence emission spectra are the same. Two-photon-excited fluorescence emission spectra are also obtained at different excitation wavelengths for Cascade Blue (750 and 800 nm), Rhodamine B (700, 800, and 1000 nm), and Fluorescein (700 and 800 nm). Results for Cascade Blue are presented in Fig. 6. Emission spectra are independent of excitation wavelengths. These results support the assumption that the fluorescence quantum efficiency is a constant regardless of the excitation wavelength.

E. TPE Spectra Excited by cw and Femtosecond Pulsed Lasers Compared

As discussed in Section 2, determination of the second-order temporal coherence (i.e., g) is crucial for precise cross-section measurements. TPE cross sections measured with femtosecond pulses have the uncertainty of the pulse-shape factor g_p. To determine g_p as a function of wavelength in our experiment, we measured TPE spectra of Cascade Blue and Fluorescein with a single-mode cw Ti:sapphire laser from 710 to 840 nm. Because of the significant amount of OPE fluorescence (as discussed in Subsection 5.B), the TPE spectrum of Rhodamine B is obtained by cw excitation only from 750 to 840 nm. For each data point, we record the fluorescence signals at several different excitation powers to verify the square-law dependence under cw excitation. Single-mode cw operation of the laser and thus g = 1 are confirmed by the measured narrow linewidth of 0.5 MHz. Figure 7 shows the TPE spectra excited by the single-mode cw laser for Cascade Blue and TPE spectra from pulsed excitation for comparison. As Fig. 7 shows, there is good agreement. The absolute value of g_p estimated from Fig. 7 is 0.41 ± 0.09. This small standard deviation (∼20%) shows that g_p of the mode-locked Ti:sapphire laser does not change significantly from 710 to 840 nm. Although the absolute value of g_p differs significantly from the expected theoretical value for a sech² pulse profile, the absolute TPE cross-section values are not affected because we have made no assumption of g_p in obtaining δ. The good agreement between TPE spectra from cw and pulsed measurements suggests that the considerable spectra spread of ultrashort pulses (5–10 nm in this experiment) does not alter the excitation spectra significantly.

An experimental determination of the pulse shape in the femtosecond region is difficult. Kennedy and Lytle[26] have proposed the use of Bis-MSB as a power-square sensor and obtained Bis-MSB TPE spectra from 537 to 694 nm with a single-mode ring dye laser. Jones and Callis[37] measured dispersion of second-order coherence (relative values of g), using second-harmonic generation from KDP powder. Both techniques require some known two-photon standards in the same wavelength range. Without using any two-photon standards we have developed a new technique to determine simultaneously the absolute values of g and fluorophore TPE cross sections by directly obtaining the second-order autocorrelation functions, using the fluorophore as the quadratic medium. Details of this new technique are reported elsewhere.[38]

6. COMPARISON OF ABSOLUTE TPE CROSS-SECTION VALUES WITH EXISTING DATA

Although a considerable amount of TPE cross-section data exists in the literature,[39] most of the early experiments provided TPE cross-sections at only one or a few discrete wavelengths because of the limited tunability of the excitation source. With tunable dye lasers several fluorescent molecules have been studied over a relatively wide spectral range.[39] However, because of the lack of an excitation source, few data exist in the spectral range from 690 to 1050 nm. Rhodamine B two-photon cross sections were obtained at both N:YAG (1064-nm) and the ruby (694-nm) frequencies.[22],[23] With ring dye lasers, Bis-MSB cross sections were measured from 547 to 693 nm.[26] In all these studies the authors made the same assumption that OPE and two-photon-excited fluorescence quantum efficiencies are the same. In Table 4 we compare cross-section data for Rhodamine B, Fluorescein, and Bis-MSB obtained by three other groups with our own results. The discrepancies are within the uncertainties of the experiments.

7. CONCLUSION

Using a mode-locked Ti:sapphire laser, we obtained the TPE spectra of 11 dyes in the spectral range from 690 to 1050 nm. The TPE spectra for Cascade Blue and Fluorescein were also obtained with a single-mode cw Ti:sapphire laser from 710 to 840 nm. By comparing the spectra obtained with the two methods above, we conclude that a mode-locked solid-state laser can provide reasonably accurate TPE spectra despite the uncertainty in the pulse shape. The good agreement between cw and pulsed measurements also indicates that the considerable spectral spread of ultrashort pulses does not alter the excitation spectra significantly.

The peak wavelengths in the measured TPE spectra for Rhodamine B, Fluorescein, and DiI are considerably shorter than twice their OPA peak wavelengths. A desirable consequence of such significant blue shifts is that two-photon microscopy using these fluorescent labels should have a substantially higher resolution than previously predicted.[40] The blue shifts also permit utilization of conveniently available laser sources to excite popular fluorophores.

A wide range of fluorophores with diverse molecular structures was selected for these measurements and, we hope, is fairly representative. The peak TPE cross sections for the fluorophores were measured between ∼10⁻⁵⁰ and ∼10⁻⁴⁸ cm⁴ s/photon. A large TPE cross section is clearly desirable in TPLSM and some other fields. It would be useful if one could find rules for predicting molecular TPE cross sections. Ultimately one may also be able to design and synthesize dyes with very large nonlinear coefficients.[41],[42]

ACKNOWLEDGMENTS

This research was carried out in the Developmental Resource for Biophysical Imaging and Opto-electronics with funding by the National Science Foundation (DIR8800278) and the National Institutes of Health (RR04224 and RR07719). The authors thank A. L. Gaeta and M. Lanzerotti for providing the laser and essential assistance on the single-mode cw measurements. They acknowledge fruitful discussions with F. Wise, A. L. Gaeta, J. Mertz, J. Shear, W. Denk, and J. B. Guild.

Figures and Tables

 

Fig. 1 Schematic drawing of the experimental setup: RG630, red-pass filter (>630 nm); BE, 5× beam expander; DC, dichroic mirror; PMT, photomultiplier tube.

Download Full Size | PDF

 

Fig. 2 Two-photon fluorescence excitation spectra of 11 dyes. Spectra are excited with linearly polarized light. η₂, two-photon fluorescence quantum efficiency (see text for details). δ, two-photon absorption cross section. An arrow indicates twice the wavelength of the OPA maximum. Four different intracavity mirror sets, ●, +, ▲, and ▼, are used to cover the entire tuning range of a mode-locked Ti:sapphire laser from 690 to 1050 nm. No averaging or normalization is done for the overlapping points. Spectra excited by a single-mode cw Ti:sapphire laser (×) are also plotted for Cascade Blue, Fluorescein, and Rhodamine B. See Table 2 for estimated uncertainties in the measurements.

Download Full Size | PDF

 

Fig. 3 Logarithmic plots of the dependence of relative two-photon induced fluorescence on pulse intensity for (●) a 10⁻⁴-M solution of Rhodamine B in Methanol and (+) a 10⁻⁴-M solution of Fluorescein in water. The excitation wavelength is indicated in the upper left corner of each graph. The fitted slopes are represented in the lower right corners. The excitation pulse width is ∼100 fs (assuming a sech² pulse shape). Ordinate scales of fluorescence are in unrelated arbitrary units. The estimated uncertainties of the fitted slopes are ±0.03.

Download Full Size | PDF

 

Fig. 4 Dependence of two-photon-excited fluorescence on excitation pulse width. (a) Logarithmic plot of two-photon-excited fluorescence versus excitation pulse width in a 10⁻⁴-M solution of Indo-1 (low Ca). (b) Comparison of two-photon-excited fluorescence in a 10⁻⁴-M solution of Indo-1 (low Ca) and second-harmonic generation (SHG) in KDP powder at eight different excitation pulse widths from ∼500 fs to ∼1.2 ps. Data were collected for constant average excitation power at λ = 770 nm. The size of a circle approximately represents the uncertainties of the data.

Download Full Size | PDF

 

Fig. 5 Fluorescence emission spectra of a 10⁻⁴-M solution of Fluorescein in water excited by (solid curve) one photon (488 nm) and (dashed curve) two photons (800 nm). The peaks at 488 and 800 nm are residual excitation light. The instrument wavelength resolution is 1.0 nm.

Download Full Size | PDF

 

Fig. 6 Two-photon-induced fluorescence emission spectra of a 10⁻⁴-M solution of Cascade Blue in water excited at (solid curve) 750 nm and (dashed curve) 800 nm.

Download Full Size | PDF

 

Fig. 7 Comparison of TPE spectra of a 10⁻⁴-M solution of Cascade Blue in water obtained with (♦) a mode-locked femtosecond Ti:sapphire laser and (×) a single-mode cw Ti:sapphire laser. We determined ♦ by using relation (15) and assuming that g_p = 0.41. See Table 2 for estimated uncertainties in the measurements.

Download Full Size | PDF

Table 1. Sample Preparation

View Table | View all tables in this article

Table 2. TPE Cross Sections at Selected Wavelengths^a

View Table | View all tables in this article

Table 3. Dependence of TPE Fluorescence on Excitation Pulse Intensity

View Table | View all tables in this article

Table 4. Comparison of Absolute Cross-Section Measurements with Existing Literature

View Table | View all tables in this article

REFERENCES AND NOTES

1. M. Göppert-Mayer, “Über Elementarakte mit zwei Quanten-sprüngen,” Ann. Phys. 9, 273–295 (1931). [CrossRef]  

2. W. Kaiser and C. G. B. Garrett, “Two-photon excitation in CaF₂:Eu²⁺,” Phys. Rev. Lett. 7, 229–231 (1961). [CrossRef]  

3. A. J. Twarowski and D. S. Kliger, “Multiphoton absorption spectra using thermal blooming,” Chem. Phys. 20, 259–264 (1977). [CrossRef]  

4. M. Sheik-Bahae, A. A. Said, T. Wei, D. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990). [CrossRef]  

5. J. P. Hermann and J. Ducuing, “Absolute measurement of two-photon cross sections,” Phys. Rev. A 5, 2557–2568 (1972). [CrossRef]  

6. R. Swofford and W. M. McClain, “The effect of spatial and temporal laser beam characteristics on two-photon absorption,” Chem. Phys. Lett. 34, 455–460 (1975). [CrossRef]  

7. R. R. Birge, “One-photon and two-photon excitation spectroscopy,” in Ultrasensitive Laser Spectroscopy, D. S. Kliger, ed. (Academic, New York, 1983), pp. 109–174. [CrossRef]  

8. W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990). [CrossRef]   [PubMed]  

9. W. M. McClain and R. A. Harris, “Two-photon molecular spectroscopy in liquids and gases,” in Excited States, E. C. Lim, ed. (Academic, New York, 1977), pp. 1–56. [CrossRef]  

10. A. Fischer, C. Cremer, and E. H. K. Stelzer, “Fluorescence of coumarins and xanthenes after two-photon absorption with a pulsed titanium-sapphire laser,” Appl. Opt. 34, 1989–2003 (1995). [CrossRef]   [PubMed]  

11. R. Loudon, The Quantum Theory of Light, 2nd ed. (Clarendon, Oxford, 1983), pp. 82–119.

12. T. Wilson and C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984), pp. 25.

13. C. J. R. Sheppard and H. J. Matthews, “Imaging in high-aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987). [CrossRef]  

14. J. Guild, Applied Physics, Cornell University, Ithaca, New York 14853 (personal communications, 1995). The estimated relative accuracy of the volume integration is 3%.

15. Assuming that the sample thickness is much greater than the Raleigh length of the Gaussian beam, the appropriate form for 〈F(t) 〉 is 〈F(t) 〉 = 1/2gϕη₂Cδ(nπ/λ) 〈P(t) 〉². We note that the total fluorescence generation is independent of beam waist size.

16. J. Kafka, M. L. Watts, and J. W. Pieterse, “Picosecond and femtosecond pulses generation in a regeneratively mode-locked Ti:S laser,” IEEE J. Quantum Electron. 28, 2151–2161 (1992). [CrossRef]  

17. S. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, “Aberrations in confocal microscopy induced by mismatches in refractive index,” J. Microsc. 169, 391–405 (1993). [CrossRef]  

18. J. N. Demas and G. A. Crosby, “The measurement of photoluminescence quantum yields,” J. Phys. Chem. 75, 991–1024 (1971). [CrossRef]  

19. G. Grynkiewicz, M. Poenie, and R. Y. Tsien, “A new generation of Ca2+ indicators with greatly improved fluorescence properties,” J. Bio. Chem. 260, 3440–3450 (1985).

20. DAPI not bound to DNA. The fluorescence quantum efficiency of DAPI is expected to go up by 20-fold upon binding to DNA (from Molecular Probes Handbook 1992–1994, R. P. Haugland, ed., p. 222).

21. B. Dick and G. Hohlneicher, “Importance of initial and final states as intermediate states in two-photon spectroscopy of polar molecules,” J. Chem. Phys. 76, 5755–5760 (1982). [CrossRef]  

22. D. J. Bradley, M. H. R. Hutchinson, T. M. H. Koetser, G. H. C. New, and M. S. Petty, “Interaction of picosecond laser pulses with organic molecules I: Two-photon fluorescence quenching and singlet states excitation in rhodamine dyes,” Proc. R. Soc. London Ser. A 328, 97–121 (1972). [CrossRef]  

23. J. P. Hermann and J. Ducuing, “Dispersion of the two-photon cross-section in rhodamine dyes,” Opt. Commun. 6, 101–105 (1972). [CrossRef]  

24. L. Goodman and R. P. Pava, “Two-photon spectra of aromatic molecules,” Acc. Chem. Res. 17, 250–257 (1984). [CrossRef]  

25. D. J. Bradley, M. H. R. Hutchinson, and H. Koetser, “Interaction of picosecond laser pulses with organic molecules II: Two-photon absorption cross-sections,” Proc. R. Soc. Lond. A 329, 105–119 (1972). [CrossRef]  

26. S. M. Kennedy and F. E. Lytle, “p-Bis(o-methyl-styryl)benzene as a power-square sensor for two-photon absorption measurements between 537 and 694 nm,” Anal. Chem. 58, 2643–2647 (1986). [CrossRef]  

27. I. Gryczynski, J. Kusba, V. Bogdanov, and J. R. Lakowicz, “Quenching of fluorescence by light: a new method to control the excited-state lifetime and orientations of fluorophores,” J. Fluoresc. 4, 103–109 (1994). [CrossRef]   [PubMed]  

28. C. Xu, J. Guild, and W. W. Webb, “Two-photon fluorescence excitation spectra of calcium probe indo-1,” Biophys. J. 66, A161 (1994).

29. C. Xu, J. Guild, and W. W. Webb, “Two-photon excitation cross-sections for commonly used biological fluorophores,” Biophys. J. 68, A197 (1995).

30. J. D. Kafka and T. Baer, “Prism-pair delay lines in optical pulse compression,” Opt. Lett. 12, 401–403 (1987). [CrossRef]   [PubMed]  

31. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980). [CrossRef]  

32. Z. Bor, “Distortion of femtosecond laser pulses in lenses and lens systems,” J. Mod. Opt. 35, 1907–1918 (1988). [CrossRef]  

33. J. Guild, C. Xu, and W. W. Webb, “Measurement of group delay dispersion of high numerical aperture objectives using two-photon excited fluorescence,” submitted to Appl. Opt.

34. Two-photon transitions consist of two separate one-photon transitions.

35. P. F. Curley, A. I. Ferguson, J. G. White, and W. B. Amos, “Application of a femtosecond self-sustaining mode-locked Ti:sapphire laser to the field of laser scanning confocal microscopy,” Opt. Quantum Electron. 24, 851–859 (1992). [CrossRef]  

36. L. Parma and N. Omenetto, “Fluorescence behavior of 7-hydroxycoumarine excited by one-photon and two-photon absorption by means of a tunable dye laser,” Chem. Phys. Lett. 54, 544–546 (1978). [CrossRef]  

37. R. D. Jones and P. R. Callis, “A power-square sensor for two-photon spectroscopy and dispersion of second-order coherence,” J. Appl. Phys. 64, 4301–4305 (1988). [CrossRef]  

38. C. Xu, J. Guild, W. W. Webb, and W. Denk, “Determination of absolute two-photon excitation cross-sections by in situ second-order autocorrelation,” Opt. Lett. 21, 2372–2374 (1995). [CrossRef]  

39. W. L. Smith, “Two-photon absorption in condensed media,” in Handbook of Laser Science and Technology, J. Weber, ed. (CRC, Baco Raton, Fla., 1986), pp. 229–258.

40. C. J. R. Sheppard and M. Gu, “Image formation in two-photon fluorescence microscopy,” Optik (Stuttgart) 86, 104–106 (1990).

41. G. S. He, G. C. Xu, P. N. Prasad, B. A. Reinhardt, J. C. Bhatt, and A. G. Dillard, “Two-photon absorption and optical-limiting properties of novel organic compounds,” Opt. Lett. 20, 435–437 (1995). [CrossRef]   [PubMed]  

42. J. Zyss, I. Ledoux, and J. F. Nicoud, “Advances in molecular engineering for quadratic nonlinear optics,” in Molecular Nonlinear Optics, J. Zyss, ed. (Academic, San Diego, Calif., 1994), pp. 130–200.