## Abstract

We have developed a fluorescence saturation technique for accurate measurements of the absolute molecular two-photon absorption (TPA) cross-section of fluorescent dyes. We determine the TPA cross-section both from measurements at excitation intensities well below saturation onset (in the square power-law regime) and from data obtained near the onset of saturation. The two estimates have different sensitivities to potential sources of errors. Using the square power-law regime requires calibration of the overall collection efficiency of the detection channel, including the quantum yield of the dye. In the saturation regime, the two key requirements are a good knowledge of the excitation profile and an adequate model of the two-photon excitation transition. To fulfill the former requirement, we developed diagnostic tools to characterize the tightly focussed excitation beam. To satisfy the latter requirement, we included the correct polarization dependent averaging over molecular orientations in our model. We measured the TPA cross-section of Rhodamine B (RhB) and Rhodamine 6g (Rh6g) in methanol at 798 nm for linear and circular polarization. For RhB we observed excellent agreement between the TPA cross-section estimate 〈σ_{2}〉 obtained from the square power-law regime and that obtained from the saturation regime, 〈σ_{2}〉^{sat}. For the case of linear polarization we found: 〈σ_{2}〉 = 12 ± 2 GM and 〈σ_{2}〉^{sat} = 10.5 ± 2 GM. For the case of circular polarization we obtained: 〈σ_{2}〉 = 8.4 ± 2 GM and 〈σ_{2}〉^{sat} = 7.5 ± 2 GM. The results obtained with linear polarization are in good agreement with previously published non-linear transmission data (δ = 2σ = 20.4 GM at 800nm). For Rh6g the difference between 〈σ_{2}〉 and 〈σ_{2}〉^{sat} is larger, but still considerably smaller than the variance of σ_{2} values found in the literature.

© 2006 Optical Society of America

## 1. Introduction

The techniques commonly used for the determination of the two-photon cross-section can be divided into two groups. One group includes techniques based on nonlinear transmission measurements [1–4], e.g. the so-called z-scan technique. These techniques directly yield the two-photon absorption (TPA) cross-section. The second group involves fluorescence methods [1], [5–13] which measure the two-photon excitation (TPE) cross-section, which is simply the TPA cross-section multiplied with the quantum yield of the dye. The results obtained with both groups of techniques are still subject to controversy [1]. Methods that detect the fluorescence are highly sensitive but there are uncertainties involved in the calibration of the system, which includes determining the collection efficiency and quantum yield. On the other hand, nonlinear transmission techniques are considerably less sensitive and therefore require rather high dye concentrations and high peak incident intensities, which may easily produce artifacts. In the present paper we developed a novel technique using the onset of excited-state saturation, which exploits the sensitivity of fluorescence detection but directly yields the TPA cross-section. The idea to use the saturation was already proposed in [13, 14, 15] as well as in our own preliminary work [16]. One detects the fluorescence signal but the information about the TPA cross-section is obtained not only from the square power-law regime but also from the onset of excited-state saturation. The two regimes complement each other with respect to potential sources of errors. Therefore, their combination allows a highly accurate estimation of the TPA cross-section. The relevant theory with models for the detected fluorescence signal including the position averaging and orientational averaging of the excitation of a dye molecule is outlined in Section 2. The saturation regime demands precise knowledge of the space-time profile of the exciting femtosecond laser beam. On the other hand, the square power-law regime requires the calibration of the detection channel. These issues are addressed in Section 3. Finally, in Section 4 we present the results of measurements on Rhodamine B (RhB) and Rhodamine 6g (Rh6g).

## 2. Theory

In our calculation of the expected photon yield from a fluorescent dye solution, two-photon excited by a short and strongly focused laser pulse, we make the following assumptions:

- The solution is confined in a layer thin enough to neglect beam attenuation.
- Stimulated emission is neglected. This implies that the relaxation of the primary two-photon excited-state is fast compared with the pulse duration [17].
- The lifetime of the fluorescent excited-state τ
_{F}is much longer than the pulse duration τ. - The lifetime of the fluorescent excited-state τ
_{F}is much longer than the Brownian rotation relaxation time τ_{R}of the dye molecule in the solvent. - The rotational relaxation time τ
_{R}is much longer than the duration of the light pulse τ. - Fluorescence emission takes place from the same set of excited states and with the same efficiency as with one photon excitation.

First we consider a single dye molecule at a position **r** having a certain orientation Ω with respect to the laboratory frame. By virtue of assumptions 2 and 3, the treatment is quite simple. One does not even need to solve the difficult rate equations with arbitrarily shaped excitation pulses. Upon the exposure to a laser pulse, the molecule is found to be excited with a certain probability *P*_{E}
(**r**). Subsequently, one may detect a fluorescence photon with the probability *P*_{D}
(**r**). Thus, the overall probability *P*(**r**) to detect a photon from the given molecule per excitation pulse is:

Once *P*_{E}
(**r**) and *P*_{D}
(**r**) are known, the expected number of detected fluorescence photons per laser pulse is given by

Here *c̅* is the macroscopic dye concentration and 〈⋯〉_{Ω} denotes the averaging over all orientations of the dye molecules in the fluid, whereas the integration represents the averaging over their positions.

The excitation probability *P*_{E}
(**r**) is complementary to the probability that the given dye molecule is *not* excited. It can be expressed in terms of the molecular TPA cross-section σ_{2} and the local irradiance *I*(**r**, **t**) as

Under our experimental conditions the space and time dependence of the irradiance can be factored as

where *Q* represents the number of photons per pulse, *a*
_{E0} the 1/e-beam waist radius and *X*(*t*) the temporal pulse profile. The dimensionless spatial excitation profile *Y*_{E}
(**r**) is, in our case, well approximated by a Gaussian beam:

where ρ is the distance from the beam axis, and ${a}_{E}^{2}$(*z*) = ${a}_{E0}^{2}$(1 + *z*
^{2}/${z}_{E}^{2}$); *z*_{E}
= *k*_{e}
${a}_{E0}^{2}$ is the Rayleigh range and *k*_{E}
the wavenumber of the excitation. Upon carrying out the time integration, Eq. (3) can be rewritten as

where *I̅*
^{2} denotes the mean square intensity:

τ is the characteristic pulse width which depends on the pulse profile:

In the case of a Gaussian pulse profile with 1/e half-width τ′, the characteristic pulse width is τ = √2πτ′. Since we are only interested in the onset of excited-state saturation, we expand Eq. (6) in a series

The detection probability *P*_{D}
(**r**) is proportional to the quantum yield of the fluorescence *q* and the detection efficiency η. If we let the emission dipole of the excited dye molecule point in the direction **â** and the receiver polarization along **ê**
_{D}, we can write the probability *P*_{D}
(**r**) of observing a fluorescence photon emitted at position **r** as

Here, *Y*_{O}
(**r**) is the observation beam profile and Φ_{O} is the fractional solid angle of observation:

where *n* is the refractive index of the sample and Ω_{O} is the effective solid angle of observation [18]. In general, the exact value of Ω_{O} must be determined experimentally. Here, we only note two extreme cases: in the case of confocal observation through a single-mode fiber Ω_{O} = λ^{2}/π${a}_{O0}^{2}$ and in the case of full aperture observation through an objective Ω_{O} = π*NA*
^{2} (in the paraxial approximation).

Our receiver actually accepts two polarization states **ê**
_{x}, **ê**
_{y}. Thus, we write the total detection probability as

Furthermore, *P*_{D}
(**r**) also depends on spatial characteristics of the observation scheme. In the case of confocal observation with a single-mode fiber receiver, the observation profile *Y*_{O}
(**r**) is a Gaussian beam

where *a*
_{O0} is the 1/e-beam waist radius of the observation beam. In the case of full aperture observation through an objective, which can be implemented with a multimode fiber receiver, we set *Y*_{O}
(**r**) = 1. We neglect the depth of the sample cell as it is much larger than the Rayleigh range of the tightly focussed excitation beam. Upon carrying out the spatial integration of Eq. (2) (position averaging), the expected number *N*_{F}
of detected fluorescence photons per laser pulse is given by

where ω_{o} = 3/2 (**â** ∙ **ê**
_{x})^{2} + (**â** ∙ **ê**
_{y})^{2} and *V*_{n}
are the overlap volumes

The overlap volumes *V*_{n}
can be explicitly calculated for an arbitrary *n*. In the case of single-mode observation, the resulting expressions are too lengthy to be reproduced here, but in the case of multimode observation, we obtain

It remains to carry out the averaging over all orientations of the dye molecule. Because of assumptions 3-5 (τ ≪ τ_{R} ≪ τ_{F}), the orientational averaging can be performed separately for the detection and excitation. Comparing the typical value of the rotational relaxation time of xanthene dyes of about 100 ps with the fluorescence lifetime of several ns, one might expect that the fluorescence is completely depolarized, i.e.〈ω_{O}>〉_{Ω} ≈ 1. However, caution is advised because two-photon excitation of collinear molecules greatly enhances the polarization selection: the initial distribution (**ê**
_{E} ∙ **d̂**)^{4} is quite sharp [19]. To estimate the magnitude of the polarization effect, we assume that the rotational relaxation is isotropic and that the emission dipoles are initially sharply oriented along the x-axis. Using Favro’s classical results on rotational Brownian motion [20], we obtain after a simple calculation:

where τ_{F} is the fluorescence lifetime and τ_{R} the rotational relaxation time given by τ_{R} = *νV*_{R}
/*kT, ν* being the solvent viscosity and *V*_{R}
the rotational hydrodynamic volume. Using *ν* = 0.544 ∙ 10^{-3} Pa∙s, τ_{F} ≈ 3 ns and *V*_{R}
≈ 1000 Å^{3} as the typical value for xanthene dyes (see e.g. [21]), we find that the polarization effect amounts to less than 2%. Thus, we finally write for the detection probability:

The orientational averaging of the excitation is more complicated. Two-photon excitation involves at least one virtual state that is dipole connected to both the initial and final states of the excitation process [22, 23]. Another mechanism, involving the permanent dipoles of the initial and final states and the transition dipole connecting these states is also discussed in the literature [24]. In order to perform the orientational averaging of ${\mathrm{\sigma}}_{2}^{\left(n\right)}$, we use a simplified description of the excitation process via a single virtual state. The orientational average of the *n*-th moment of the TPA cross-section is then given by

Here *μ*
_{2} is the maximum TPA cross-section of a perfectly oriented collinear molecule ((**ê**
_{E} ∥ **d̂**
_{0i} ∥ **d̂**
_{i1}). The geometry and orientation are included in ω = |(**ê**
_{E} ∙ **d̂**
_{0i}) (**d̂**
_{i1} ∙ **ê**
_{E})|^{2} , where **ê** is a polarization vector of the excitation. With the beam propagation axis aligned in the *x*-direction we set **ê**
_{E} = (0,0,1) for linear polarization and **ê** = (0,*i*, 1)/√(2) for circular polarization. The unit dipole moments for the transitions from the initial state 0 to the virtual state *i* and from the virtual state *i* to the final state 1 are denoted **d̂**
_{0i} and **d̂**
_{i1}, respectively. The geometry is outlined in Fig. 1. The Euler angles *θ*, *ϕ* and *ψ* specify the orientation of the molecule in the laboratory frame. The molecular frame is aligned such that **d̂**
_{0i} points in the *z*′ direction. Since the molecules are randomly oriented, a single parameter, namely the angle α, is sufficient to define the second dipole moment **d̂**
_{i1}. For an isotropic sample, the calculation of 〈ω^{n}〉 is straightforward [25]. For example, for α = 0 one obtains 〈ω〉_{α=0} = 1/5 and 〈ω^{2}〉 _{α=0} = 1/9 for linear polarization, whereas values of 〈ω^{2}〉_{α=0} = 2/15 and 〈ω^{2}〉_{α=0} = 8/315 are found for circular polarization of the excitation beam. The orientationally averaged version of Eq. (14) now reads

Equation (20) contains two unknown parameters *μ*
_{2} and *α* which, in principle, can be estimated using a standard nonlinear least-square fitting method. However, our simple model of the two-photon transition may be oversimplified. To be less dependent on the theory, we prefer to estimate two other parameters: *p*
_{1} in the term in front of the brackets (prefactor) and *p*
_{2} in the saturation term:

Thus, we fit the data using

where *P* is the average incident laser power and *C*_{i}
are known coefficients. Once *α* is estimated (see below), we introduce higher order terms in Eq. (23) to re-check the quality of the fit.

It is important to note that 〈ω^{2}〉 ≠ 〈ω〉^{2} and therefore *p*
_{2} ≠ 〈σ_{2}〉. Theoretically, 〈σ_{2}〉 can be obtained from *p*
_{2} as 〈σ_{2}〉^{sat} = *p*
_{2}/*r*(*α*). The correction factor *r*(*α*) = 〈ω_{2}〉 /〈ω〉 for linear (LP) and circular polarization (CP) is shown in the left-hand side of Fig. 2. For collinear molecules with *α* = 0, we obtain *r*_{LP}
= 25/9 for linear polarization, whereas for circular polarization the ratio *r*_{CP}
= 10/7 does not depend on *α*.

To estimate the unknown angle *α*, we use the polarization data: As shown in the right-hand side of Fig. 2, the polarization ratio *γ*
_{1} (*α*) = 〈ω〉_{CP} / 〈ω〉_{LP} is quite sensitive to *α*. As a check of consistency, we also employ the second-order ratio *γ*
_{2}(*α*) = [〈ω〉^{2})_{CP} 〈ω^{2}〉_{LP}] / [〈ω〉_{CP}〈ω^{2}〉_{LP}] . The polarization ratios are *γ*
_{1} = 2/3 and *γ*
_{2} = 12/35 for α = 0, respectively.

Examining Eq. (20) one finds that deviations of the two-photon excited fluorescence signal *N*_{F}
- from the square power-law at the onset of excited-state saturation depend only on 〈σ^{2}〉^{sat}, the two-photon fluence *I̅*
^{2}τ and properties of the excitation- and observation beam which are included in *V*_{n}
. Neither η Φ_{O} nor *q* are required for the estimation of 〈σ_{2}〉_{Ω} from the saturation terms. However, ${a}_{E0}^{4}$ which occurs in *I̅*
^{2} (refer to Eq. (7)) must be determined with great accuracy. On the other hand, the prefactor in Eq. (20) exhibits practically no dependence on the profile of the excitation beam since the relevant factor ${a}_{E0}^{4}$ appears in the denominator of Eq. (7) as well as in the numerator of Eq. (16). However, the estimation of 〈σ_{2}〉_{Ω} from the prefactor requires the calibration of the overall collection efficiency *q*ηΦ_{O}, including the quantum yield *q* of the dye. Thus, the cross-section appears in two terms with different sensitivities to potential sources of errors. A comparison of the two estimates of 〈σ_{2}〉_{Ω} - from the square power-law regime and from the onset of saturation - provides a perfect test of the quality of the estimate.

## 3. Material and methods

The experimental setup is shown in Fig. 3. A regenerative amplifier system (RegA 9000, Coherent) with a pulse repetition rate of *f* = 250 kHz, pumped by a 10 W frequency-doubled Nd:YAG laser (Verdi V10, Coherent), is used to amplify the pulse energy of a mode-locked Ti:Sapphire laser (Mira 900, Coherent), pumped by a 6 W frequency-doubled Nd:YAG laser (Verdi V6, Coherent). The wavelength of Ti:Sapphire amplifier (800±5 nm) is measured with an NIR-Spectrometer (Ocean Optics).

The TPE setup is built around an inverted microscope (Axiovert 200, Zeiss). The excitation laser beam is coupled through a 10×0.25 NA objective (Achroplan, Zeiss) into a hollow core photonic bandgap fiber (HC-800-02, Crystal Fibre), which not only delivers the beam into the microscope but also serves as a spatial filter providing a Gaussian profile. The power is controlled in front of the fiber by a set of polarization optics consisting of a Glan-laser calcite polarizer (Thorlabs) between two half-wave plates. The second half-wave plate is used to align the input polarization into the optimal fiber axis: because of the peculiar polarization properties of the hollow core fiber [26, 27], care must be taken to minimize the dispersion of the polarization modes. The output of the fiber is collimated by a high numerical aperture objective (Neofluar 32×0.5, Zeiss) and expanded with a 5× beam telescope to a beam radius which optimally fills the back aperture of the microscope objective used to focus the beam onto the sample (A-Plan 40×0.65, Zeiss). Theoretical considerations show that already a single-mode Gaussian beam with a radius of *a*
_{1/e} = *N*.*A*. *f*/2 can be focused to the diffraction limit. Working in the saturation regime, photo bleaching of the dye plays an important role and must be prevented. Therefore, the measurements are done using a flow chamber (*μ*-Slide I, Ibidi GmbH) with a window thickness corresponding to a standard cover slip. The required dye flow of about 0.5 ml/min is generated by a syringe pump and filtered through a Millex-GV 220 nm filter (Millipore). The filter has no observable effect on the dye concentration. The fluorescence is spectrally filtered (GG435 and BG39, ITOS) and collected with a receiving system, consisting of a 10×0.3 microscope objective (Neofluar, Zeiss) and an optical fiber. For the most accurate definition of the overlap volumes *V*_{n}
and for the elimination of the background (such as second harmonic from the flow chamber surfaces), confocal observation with a single-mode fiber receiver would be ideal. However, the accuracy and reproducibility of the confocal alignment turned out to be insufficient with the opto-mechanical components available to us. As a compromise between simplicity and a good selection of the observation volume we employ a 50 *μ*m multimode fiber (MMF-488-50/125, OZOptics) in the observation channel. Fortunately, the background contribution turned out to be negligible. The detector is a single-photon avalanche diode (SPCM-AQR-13-FC, Perkin Elmer). As a photon counter, we employ a time-correlated single photon counting unit (SPC-830, Becker&Hickl), which allows simultaneous determination of the fluorescence lifetime of the dye. By selecting a time window corresponding to the lifetime, we discriminate the background due to dark counts or residual ambient light. Because of the deadtime of the detector, we can only detect the *first* fluorescence photon per excitation pulse. To correct for this, we employ a standard correction for the Poisson distributed number of photons per laser pulse [28]:

where < *M*_{F}
> is the measured average number of photon counts per laser pulse.

#### 3.1. Measurement of the space-time beam profiles

Our model requires the knowledge of three experimental parameters, namely the number of incident photons per pulse *Q*, the pulse width τ′ and the beam waist radius *a*
_{E0} of the excitation beam in the sample. *Q* = *P̅*
_{L}/(*hν*_{E}*f* ) is determined by measuring the average laser power *P̅*
_{L} and using the known laser repetition rate *f*. The power meter consists of a 4-inch integrating sphere (LPM-040-SF, Labsphere) equipped with a silicon detector and a controller for a power range from 15 *μ*W to 15 W. Placed on the microscope stage above the sample, this sphere allows precise diagnostics of illuminating beams with divergences of up to *NA* = 0.65. The pulse width (τ′ = 438 ± 10 fs) is measured directly in the principle focus on the microscope stage using an adapted intensity autocorrelator (Carpe, APE). The two-photon excited sample itself is used as the second-order nonlinear medium for the pulse width measurement.

The beam waist radius *a*
_{E0} is determined at the excitation wavelength of 798 nm by off-focus profiling, based on the propagation theory of a nonparaxial Gaussian beam [29]. Several
far-field images of the laser beam at different z-positions after the microscope objective (viz. Fig. 4) are taken and assembled into a 3-dimensional far-field map of the Gaussian beam. A nonparaxial beam propagation model is then fitted to the data. The images are acquired with an adapted CMOS-camera (PL-A741-R, Pixelink), equipped with a fiber optic taper bonded onto the CMOS chip (Schott, Proxitronic). The taper allows taking images in planes sufficiently close to the focus, so that the whole profile of the divergent beam can be captured. Moreover, the taper prevents the formation of disturbing interference fringes. The radius of the single-mode excitation beam, obtained with this method, amounts to *a*
_{E0} = 342±10 nm.

#### 3.2. Calibration of the detection efficiency

Since one-photon cross-sections of dyes are easily measurable, the most straightforward approach is to estimate the overall observation collection efficiency of the microscope by using one-photon excitation in the same setup [7]. In doing so, however, one has to take into account the large difference between one-photon and two-photon excitation profiles. Moreover, it is not easy to control the bleaching of the dye in the highly divergent one-photon excitation profile. Therefore, we prefer to split the calibration task into two independent steps. It should be noted that the collection efficiency η is actually a shorthand notation for

where *S*_{F}
(λ) is the normalized fluorescence spectrum, η
_{M}
represents the optical path of the microscope and hD the remaining components of the detection channel. η
_{M}
includes the objective, the barrier filter, the dichroic beam splitter and the side-port prism. Since the response of these components is quite flat in the wavelength range of interest, η
_{M}
is nearly constant and can be taken out of the integral. In the first step of the calibration, we simply launch a 532 nm laser beam through the observation fiber and measure the overall attenuation upon its exit from the objective. The resulting value η
_{M}
= 0.38 agrees reasonably well with the values calculated from the manufacturer’s estimates of the partial attenuations. At the same time, we also determine the observation solid angle Ω_{O} for a given fiber-objective combination from measurements of the far-field profile of the divergent beam exiting the objective.

In the second step, we measure independently the factor *qη*_{D}
= *q* ∫ *S*_{F}
(λ)*η*_{D}
(*λ*)*dλ*, resulting from the quantum yield of the dye and the overlap of the fluorescence spectrum with the transmission curves of the filters and the spectral response of the detector. The underlying formula is a one-photon and CW version of the first term in Eq. (20), namely

where *P*_{F}
is the measured fluorescence photon count rate, 〈σ_{1}〉 is the orientationally averaged one-photon absorption cross-section, *c̅* is the dye concentration and *P* is the excitation power (in photons/s). The fractional observation solid angle Φ* _{O}* is defined in the context of Eq. (10) and

*V*is the excitation-observation overlap volume

_{O}The following scheme guarantees the best possible accuracy of the overlap volume *V _{O}* and of Φ

_{O}[18]: The sample cell is a standard rectangular fluorescence cuvette. The excitation beam, collimated to a beam radius

*a*

_{E0}= 275

*μ*m, is launched through a single-mode fiber to assure a perfect Gaussian excitation profile. The observation is realized in a 90° geometry, using a single-mode fiber receiver with an observation beam expanded to a radius of

*aO*

_{0}= 1014

*μ*m. The measurements are done with the 488nm Ar:ion laser line. To assure the absence of bleaching, we check the linearity of the response with increasing incident power, applying a dead time correction whenever appropriate. The single-photon absorption cross-section 〈σ

_{1}〉 is measured on a standard spectrometer (Perkin Elmer). Note, that the calibration factor

*qη*

_{D}contains the quantum efficiency

*q*of the dye. Our procedure offers a possibility to cross-check the calibration: We calibrate the detector response by using the same technique to measure on toluene, a well characterized scattering standard (see e.g. [30] and references therein). Combining the detector data with the fluorescence and absorption data, we can estimate the quantum yield of the dye.

Since the fluorescence spectrum and the quantum yield depend on the solvent, the procedure must be repeated for each dye-solvent combination to be used in σ_{2} determination. Strictly speaking, one should also consider the effect of concentration quenching, but the dye concentrations used in the present experiments are sufficiently low to neglect the concentration effects. Our measurements were done on Rhodamine 6g (Acros Organics) and Rhodamine B (Acros Organics) dissolved in methanol (spectroscopy grade, Merck) at concentrations *c̅* of 4.3 ∙ 10^{-7}M and 1.0∙ 10^{-6}M.

## 4. Results and discussion

The calibration factors *qη* were estimated to be 0.030 ± 0.001 for RhB and 0.063 ± 0.002 for Rh6g. Apart from a slight influence of the different fluorescence spectra, the difference is mainly due to quantum yield. The estimated quantum yields of *q*_{RB}
= 0.63 ±0.04 and *q*
_{R6g} = 0.92 ± 0.04 are in agreement with the standard values from the literature [31]. The error bars reflect the uncertainty of the detector response curve, taken from manufacturer data and calibrated at only a single wavelength of 488nm. The fluorescence lifetimes of both dyes (*τ*_{R6G}
= 3.88±0.05ns, *τ*_{RB}
= 2.30±0.05 ns) were found to be independent of the excitation intensity and the polarization state, indicating that fluorescence always takes place from the same set of excited states. The fluorescence decay curves taken at two different power levels are shown in Fig. 5. The dependence of the dead-time corrected florescence signal
*N*_{F}
*f* on the incident average laser power *P̅*
_{L} is presented in Fig. 6 for Rh6g and RhB. Measurements at higher average power levels that extend further into the saturation regime are avoided because of possible unwanted nonlinear effects [1]. To make sure that the solvent does not contribute to the signal, the measurements were repeated in pure methanol at zero dye concentration. The signal from the solvent was found to be negligible throughout the entire range of average laser power used. The results of the data analysis, performed following the procedures outlined in Section 2, are shown in Tab. 1. The first column contains the orientationally averaged TPA cross-sections 〈σ_{2}〉 obtained from the parameter *p*
_{1} (i.e. from the square power-law regime). These results are based on the calibration of the detection channel. The data in the last column are model-based estimates 〈σ_{2}〉^{sat} obtained from the parameter *p*
_{2} (i.e. from the saturation regime). These estimates were obtained as follows: First we calculated the values *γ*
_{1} and *γ*
_{2} (defined in the caption of Tab. 1) from the parameter *p*
_{1} (Eq. (21)) for circular and linear polarization. The polarization ratios γ_{1RB} = 0.68 and *γ*
_{1R6g} = 0.67 are in good agreement with previously reported values [7]. From these ratios we estimate the azimuthal angle *α* between the two transition dipole moments, using the right-hand plot of Fig. 2. The angles for Rhodamine B are determined to be 39° and 30° from *γ*
_{1} and *γ*
_{2}, respectively. Since the uncertainty in *γ*
_{1} is smaller than that in *γ*
_{2}, we use the former value of *α* to determine the correction factors *r*(*α*) from the left hand plot of Fig. 2. For linear polarization, the variation of this factor across the relevant angle range is relatively small. For circular polarization, *r* is independent of *α*. The TPA cross-section extracted from the onset of saturation can now be estimated using 〈σ_{2}〉^{sat} = *p*
_{2}/*r*(*α*).

Now we compare these two estimates 〈σ_{2}〉 and 〈σ_{2}〉^{sat}. As discussed in Section 2, they respond in different ways to potential sources of errors: for 〈σ_{2}〉 we need to know the overall collection efficiency including the quantum yield and the detector response, whereas 〈σ_{2}〉^{sat} requires a good knowledge of the excitation profile and requires the validity of the model of the two-photon excitation process. Thus, if 〈σ_{2}〉 = 〈σ_{2}〉^{sat} we have a strong indication that all experimental and theoretical requirements are fulfilled. We see in Tab. 1 that for Rhodamine 6g 〈σ_{2}〉^{sat} is somewhat lower than 〈σ_{2}〉. This indicates that the simple model of the excitation process is not quite adequate for Rh6g, which results in imprecise values of the correction factor *r*. Therefore the data from the square power-law regime (first column in the table) 〈σ_{2}〉_{LP} = 6.4 ± 0.9 GM are more likely to be valid. On the other hand, for Rhodamine B the agreement between 〈σ_{2}〉^{sat} and 〈σ_{2}〉 is excellent for both polarizations. Thus, Rhodamine B suggests itself as a good reference dye for future TPA cross-section measurements.

To compare our results with previously published data, we must first recall the variety of existing definitions of two-photon ”cross-section”. Our definition is based on the rate equation for the concentration *c*
_{0} of the absorbing molecules in the ground state, namely on *dc*
_{0} = -σ_{2}
*c*
_{0}
*I*
^{2}
*dt*. Other authors prefer to regard the absorption process from the point of view of the impinging photons, employing a ”quadratic Beers-Law”: *dI* = -δ*c*
_{0}
*I*
^{2}
*dx*. We see that δ = 2 σ_{2}, since two photons are absorbed in each excitation event. Furthermore, researchers working with fluorescence techniques, often include the quantum yield *q* in their definition of the so-called two-photon excitation cross-section δ_{TPE} = 2*q*σ_{2}. Another issue is the polarization dependence of the TPA cross-section. In the literature the incident polarization state is often not stated.

The currently accepted value for TPA cross-section for Rhodamine B in methanol at 800 nm for linear polarization is σ_{2} ≈ 75 GM (δ = 150 ± 56 GM), measured by a fluorescence technique (Xu et al. [7]). In view of our results, this value seems to be an overestimate. Likely reasons for such a high value of σ_{2} are: i) underestimation of the collection efficiency because of dye bleaching or detector saturation during the calibration, ii) underestimation of the pulse width in the sample or iii) underestimation of the excitation intensity (hot spots in the beam). Interestingly, the only reported independent measurement of Rhodamine B in methanol at 803 nm by Oulianov et al. [1] yielded σ_{2} = 10.2 GM (δ = 20.4 GM), in excellent agreement with our result. This measurement was done using the non-linear transmission technique (NLT), but discarded by the authors because of the large discrepancy with the reference value from [7]. The first TPA cross-section measurement in the saturation regime (Wang et al. [14]) excited at 840 nm and neglected the orientational averaging. We reevaluated the result from this study by applying the appropriate correction factor 9/25 and extrapolating to 800 nm. In this way, we obtained 〈σ_{2}〉 of about 25 GM. The observed discrepancy is probably due to the attempt to fit data obtained using high intensities, too far into the saturation regime, and therefore using an inadequate model.

A similar situation is found for Rhodamine 6g in methanol at 800 nm. Whereas Albota et al. [8] reported σ_{2} ≈ 20 GM (δ = 40 GM) from fluorescence, Oulianov et al. measured a NLT-value of σ_{2} = 6.4 GM (δ = 12.8 GM), again in agreement with our result 〈σ_{2}〉 = 6.4 ± 0.9 GM. This value is further supported by Tian and Warren [4] (σ_{2} = 7.6 GM, δ = 15.3 ±2 GM, NLT) and Sengupta [3] (σ_{2} = 8.1 GM, δ = 16.2 ± 2.4 GM, NLT). Note that the discrepancy is considerably smaller than in the case of Rhodamine B. Bleaching during the calibration of the fluorescence technique could explain the observed deviations: as Rhodamine 6g is much more photostable than Rhodamine B [32], this would lead to larger discrepancies in the Rhodamine B measurements than in the Rhodamine 6g measurements.

## 5. Conclusion

We have investigated the capabilities of the fluorescence saturation technique for measurements of the absolute molecular TPA cross-section 〈σ_{2}〉 of fluorescent dyes. We have shown that information about the TPA cross-section can be obtained from two different regimes of excitation intensities with different sensitivities to potential sources of errors. The determination of the TPA cross-section in the square power-law regime requires a calibration of the overall collection efficiency *qη* Φ_{O} including the quantum yield *q* of the dye, while the excitation beam profile has little influence. On the other hand, the crucial requirements for estimation of the TPA cross-section in the saturation regime are a good knowledge of the excitation profile and a theoretical model of the two-photon excitation process. The requirement for an accurate characterization of the spatial and temporal beam profiles is fulfilled by our beam diagnostic tools. The model we use includes the polarization and averages ;σ_{2} over the orientational distribution of the molecules. Our measurements on Rhodamine B in methanol at 798 nm show excellent agreement between the two TPA cross-section estimates obtained from the two different regimes. Note that our data agree very well with the values obtained by NLT-techniques. (This is the first time, to our knowledge, that such an agreement between NLT and fluorescence has been found.) For Rhodamine 6g the agreement between the two estimates is less good, but still satisfactory considering the usual spread of ;σ_{2} values found in the literature. However, the role of permanent and transition dipole moments in TPA is a rather complex issue and further studies have to be done. With its simple linear structure, DPH would be an interesting model molecule. A mayor advantage of our technique is it’s superior sensitivity compared with NLT and other techniques. This will allow us to determine two-photon cross-sections *in-situ*, for example within a labeled biological cell.

## Acknowledgement

We would like to thank René Nyffenegger and Res Friedrich for technical assistance. This work was financially supported by the Swiss National Science Foundation.

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