Multiphoton Cascade Absorption in Single Molecule Fluorescence

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Multiphoton Cascade Absorption in Single Molecule Fluorescence Saturation Spectroscopy Pascale Winckler and Rodolphe Jaffiol* Laboratoire de Nanotechnologie et d’Instrumentation Optique, Institut Charles Delaunay, UMR STMR CNRS 6279, Université de Technologie de Troyes, 12 rue Marie Curie, CS 42060, 10004 Troyes cedex, France S Supporting Information *

ABSTRACT: Saturation spectroscopy is a relevant method to investigate photophysical parameters of single fluorescent molecules. Nevertheless, the impact of a gradual increase, over a broad range, of the laser excitation on the intramolecular dynamics is not completely understood, particularly concerning their fluorescence emission (the so-called brightness). Thus, we propose a comprehensive theoretical and experimental study to interpret the unexpected evolution of the brightness with the laser power taking into account the cascade absorption of two and three photons. Furthermore, we highlight the key role played by the confocal observation volume in fluorescence saturation spectroscopy of single molecules in solution.

F

increasing irradiance within the observation volume.7,9 In our work, we have not detected any manifestation of photobleaching through the decrease of the fluorescence signal during the acquisition; moreover, our FCS investigations revealed that N and τd strongly increase with the laser excitation. Consequently, all these parameters clearly suggest that either photobleaching does not influence the recording or it does not occur at all. A simple explanation can be given to explain the lack of apparent photobleaching, just by comparing the diffusion and bleaching typical times. With a usual FCS setup, the lateral size of the diffraction limited observation volume is very small: the beam waist is about 0.2−0.25 μm. For small molecules, the corresponding diffusion time τd is between 20 and 50 μs. In contrast, the typical bleaching time τB (given by the inverse of the bleaching rate constant) of many fluorescent molecules is 1 or 2 orders of magnitude higher than the diffusion time (for example, τB ≈ 0.1−1 ms for rhodamine 6G).7,9 Consequently, in the case of small dye molecules, the effects of photobleaching on FCS measurements could not be visible if the diffusion time through the observation volume is shorter than the photobleaching time. On the other hand, it is well-known that laser dye molecules under intense light excitation can easily absorb photons from the first excited molecular states to higher excited states, as shown in Figure 1.7,10 Such transitions can give rise to several undesirable effects which are until now partly explained, such as photobleaching or any other photochemical reactions. Previous interesting studies of rhodamine 6G in ethanol report that the excited state absorption cross-section σS1n, for the S1→Sn transition, and the ground state absorption cross-section σS01,

luorescence saturation spectroscopy associated with confocal fluorescence correlation analysis was initially proposed to study the triplet state dynamic of dye molecules in solution.1,2 The association of these two techniques offers to a large extent a powerful tool to determine many photophysical parameters of single fluorescent dyes. Thus, this experimental approach appears to be one of the most promising ways to decipher in plasmonics the interactions between dye molecules and metallic nanostructures, for example, to understand and quantify the enhancement process. In this framework, J. Wenger et al. have recently combined fluorescence correlation spectroscopy (FCS), lifetime measurements, and fluorescence saturation spectroscopy (FSS) to determine the rate constants of all radiative and nonradiative transitions of rhodamine 6G in interactions with a metallic nanoaperture.3,4 Although one can find several works where FCS and FSS were both implemented, the behavior of the fluorescence brightness (i.e., the fluorescence signal per molecule) under high laser excitation is rarely addressed. Thus, the aim of this paper is to show the appearance of an unexpected and so far unexplained behavior when the laser power is increased. Indeed, when focused to a few milliwatts through a high numerical objective, we observed first a normal increase of the fluorescence brightness, which is followed at a certain power threshold by a fast decrease rather than the usual saturation process. This present work proposes to highlight this surprising behavior but without any photobleaching.5−8 Indeed, there are three experimental fingerprints of the photobleaching in FCS under continuous-wave laser excitation. The first one happens during the photon counting acquisition and is characterized by an exponential time decay of the fluorescence signal under irradiation.6 The two others are the decrease of the mean number of molecules N and the decrease of their transit time, denoted by their diffusion time τd, for an © 2013 American Chemical Society

Received: February 11, 2013 Accepted: March 22, 2013 Published: March 22, 2013 4735

dx.doi.org/10.1021/ac400454f | Anal. Chem. 2013, 85, 4735−4744

Analytical Chemistry

Article

magnification of our microscope, M = 120, a multimode optical fiber with a core diameter of 62.5 μm is used as point detector for the confocal detection. The fluorescence signal is then divided in two different detection channels with a 50/50 beam splitter and recorded with two similar avalanche photodiodes (PerkinElmer APD, SPCM-ARQ-15), connected to a PicoHarp 300 single photon counting module. The acquisition time is between 10 s and 5 min, depending on the fluorescence count rate. The laser excitation power was measured with a power meter (Coherent FieldMate OP-2VIS) at the back pupil of the objective. The power was adjusted with two wheels, which contain different optical neutral density filters. FCS measurements were performed in diluted solution (nanomolar concentration) of rhodamine 6G in ethanol (Invitrogen, R634) and Alexa488 in water (Invitrogen, A20000). All measurements were performed at room temperature (23 °C) in a hermetic cuvette to avoid solvent evaporation.

Figure 1. Molecular five-level model. The black arrows represent the radiative transitions, whereas the dashed arrows represent the nonradiative ones. All the parameters are detailed in the text.

for the fundamental transition S0→S1, have the same order of magnitude,11 which means that the two photons cascade absorption could be effective within such often used molecules. It is also possible to find some molecules where σS1n exceeds σS01, for example, in reverse saturable absorber applications.12,13 Among recent studies relating the behavior of single molecules in regards to an increase of the laser power source, cascade absorption to higher excited state is too often neglected, whereas it is possible to extract very interesting photophysical information from it. Moreover, when a five-level molecular model is used, the intersystem crossing kISC ′ between Sn and Tn is always neglected. To the contrary, our work shows that this nonradiative transition plays an important role in FSS. In the same way, although since many years some papers have highlighted the size variation of the confocal observation volume with the degree of saturation of the dye,14 this important effect is mostly ignored in fluorescence correlation spectroscopy experiments. Indeed, the theory of confocal fluorescence imaging and FCS widely assumed a linear relationship between the emitted fluorescence signal and the light excitation, thus neglecting any nonlinear effect, such as saturation or photobleaching. However, few recent papers clearly show that with a simple molecular model (two or three-level system) the diffusion time and the observation volume size in FCS will drastically increase with laser excitation power.15−17 In practice, such effect is responsible to one of the established artifacts in diffusion measurements by FCS.18 In this paper, we report a comprehensive theoretical and experimental study to interpret the evolution of the brightness with the laser irradiance, without any photobleaching, highlighting the key role played by the confocal observation volume in FSS/FCS combined experiments. We focused our work on two widely used dye molecules, namely, rhodamine 6G in ethanol and Alexa488 in water. One of the main purposes of this work is to show, in the simple case of a single molecule in solution, the consequences of consecutive absorption of two and three photons within a five-level electronic molecular system in saturation spectroscopy.

■

THEORETICAL BACKGROUND Saturation Curve. The electronic states of a fluorescent molecule can be modeled with five different electronic levels, as shown in the Jablonski diagram in Figure 1. S0 denotes the ground state; S1 and Sn are, respectively, the first and the higher excited singlet states, and T1 and Tn are the first and the higher triplet states. We consider only one luminous state: S1 and the product of its population pS1 to its radiative decay rate constant kr, giving the emitted fluorescence. The detected fluorescence signal of a single molecule, denoted ε, is defined as follows: ε = ηdpS k r

(1)

1

where ηd represents the collection efficiency of the optical microscope and ε (also known as the brightness of the fluorescent molecule) is defined in photons per second (ph·s−1). The saturation curve will be the variation of the fluorescence signal per molecule ε, in regards to the increase of the laser irradiance, Ie (W·cm−2), or the laser photons flux, ϕe (ph·s−1·cm−2). The relation between irradiance and photons flux is: ϕe = Ie/hv, where hv is the laser energy. ε can be determined from FCS measurements by dividing the detected fluorescence count rate CR (in Hz) by the total number of molecules N. Molecular System. Figure 1 shows the electronic energy diagram of a dye modeled by a five-level closed system. kS01 is the excitation rate and corresponds to the number of photons absorbed per second through the S0→S1 transition. Thus, kS01 is given by:

k S01 = σS01

Ie = σS01ϕe hν

(2)

where σS01 is the absorption cross section of this radiative transition. The definitions of kS1n and kT1n are similar and include the absorption cross section of the S1→Sn and T1→Tn transitions: k S1n = σS1nϕe

■

and

k T1n = σT1nϕe

(3)

The nonradiative depopulation of S1, Sn, T1, and Tn is described by the rate constants kIC (the internal conversion), kISC (the intersystem crossing from S1 to T1), kSn1 (the relaxation from Sn to S1), k′ISC (the inter system crossing from Sn to Tn), and kT1O and kTn1, the respective lifetime of T1 and Tn. The lifetime τ0 of the fluorescent state S1 is:

EXPERIMENTAL SECTION Our experimental setup is detailed in previous publications.19 Briefly, our homemade confocal optical device for FCS measurements is based on an inverted Olympus microscope (IX70). A 488 nm laser (Sapphire, Coherent, 10mW) provides the excitation light. The laser beam is strongly focused through a high numerical aperture water immersion objective (NA = 1.2, 60×, Olympus), and the fluorescence emission is collected through the same water objective. According to the total

τ0 = 4736

1 1 = k0 k r + kIC + kISC

(4)



Article

This nonlinear dependence of the brightness on the laser flux means that the fluorescence signal will not increase without bound but instead becomes saturated at a maximum value proportional to kr/κ. Saturation Curve in a Five-Level System. High laser excitation may induce consecutive absorption of two or three photons, initially to reach one of the first electronic levels S1 or after the intersystem crossing, T1, and then second, to reach higher levels Sn or Tn (Figure 1). Consequently, by introducing two new electronic levels, the rate equations which give the temporal evolution of the population states and so govern the molecular dynamics of the dye molecules can be written as:

Saturation Curve in a Three-Level System. In the case of low laser excitation, a simplified three-level system is often used to establish the saturation of dye molecules. In this case, the well-known steady state expression of the detected fluorescence signal per molecule is: ε = ηdσS01

k r ϕe k 0 1 + ϕe ϕ

(5)

S

where ϕS is the saturation level, which is equal to: ϕS =

k0 σS01κ

κ=1+

with

kISC k T10

(6)

0 0 ⎞⎛ p ⎞ k r + kIC k T10 ⎛ pS ⎞ ⎛−k S01 ⎟⎜ S 0 ⎟ ⎜ 0⎟ ⎜ ⎜ 0 0 ⎟⎜ pS ⎟ k S01 −(k r + kIC + kISC + k S1n) k Sn1 ⎜ pS1 ⎟ ⎜ ⎟⎜ 1 ⎟ d ⎜p ⎟ ⎜ 0 0 kISC k Tn1 ⎟⎜ pT1 ⎟ −(k T10 + k T1n) ⎜ T1 ⎟ = ⎟⎜ ⎟ dt ⎜ ⎟ ⎜ ′ ) 0 ⎟⎜ pSn ⎟ ⎜ pSn ⎟ ⎜ 0 0 k S1n −(k Sn1 + kISC ⎟⎜ ⎟ ⎜⎜ ⎟⎟ ⎜ ⎜p ⎟ p ⎜ T ′ ⎝ n⎠ ⎝ 0 0 k T1n kISC −k Tn1 ⎟⎠⎝ Tn ⎠

where pXi denotes the population of the level Xi. Then, assuming a closed system for which pS0 + pS1 + pT1 + pSn + pTn = 1, the

steady state solutions of eq 7 can be easily obtained. The equilibrium population of the luminous state becomes: k S01

pSeq =

⎧ kT k 0 + k S01⎨κ + (κ − 1) k 1n + Tn1 ⎩

1

⎛ ⎜1 + ′ ⎝ k Sn1 + kISC k S1n

By replacing this expression in eq 1, we obtained the new expression of the fluorescence signal per molecule within a fivelevel system: ε = ηdσS01

kr k0 1 +

ϕS1

+

ϕe 2 ϕS2

+

ϕe3 ϕS3

(9)

+

′ kISC k T10

+

′ ⎛ kISC ⎜1 k Tn1 ⎝

+

k Tn1 ⎞⎞⎫ ⎟

⎟⎬

k T10 ⎠⎠⎭

⎛ κσS ⎞−1 ′ σS1nkISC ⎟ = ϕS1 = ⎜⎜ 01 + ′ ) ⎟⎠ k 0(k Sn1 + kISC ⎝ k0 1+

The quadratic and cubic term appearing in the denominator depicts the multiphoton consecutive absorption. It is interesting to note that if kISC ′ = 0, the cubic term disappears. Next, three

ϕS2

′ kISC k S01

(8)

different saturation levels appear in eq 9: one corresponds to the absorption of one photon (ϕS1) (close similar to the saturation level in a three-level system), the second one to the consecutive absorption of two photons (ϕS2), and the third one to the cascade absorption of three photons (ϕS3). The expressions of these three saturation levels are:

ϕe ϕe

(7)

ϕS ′ σS1nkISC ′ )σS κ(k Sn1 + kISC 01

(10)

⎫−1 ⎧ ⎛ σS1nσS01 ⎪ σS σT (κ − 1) ′ ′ ⎞⎪ kISC kISC 01 1n ⎟ ⎜ ⎬ 1+ =⎨ + + ⎪ ′ ) ⎜⎝ k 0k Tn1 k 0(k Sn1 + kISC k T10 k Tn1 ⎟⎠⎪ ⎭ ⎩ =ϕs

′ ) κk Tn1k T10(k Sn1 + kISC ′ (σT1nk T10(κ − 1) + σS1n(k Tn1 + k T10)) k T10(σT1nk Sn1(κ − 1) + σS1nk Tn1) + kISC

⎛ ⎞−1 ′ σS1nσS01σT1n κk k (k ′ + k Sn1) kISC ⎟⎟ = ϕ T10 Tn1 ISC ϕS3 = ⎜⎜ S ′ )⎠ ′ σS1nσT1nkISC ⎝ k 0k T10k Tn1(k Sn1 + kISC −2

(11)

Fluorescence Saturation in Confocal Microscopy. In fluorescence confocal microscopy, the sample is excited with a focused laser beam and the fluorescence, which is commonly detected with the same objective used for the laser irradiation, is spatially gated according to the use of a small pinhole. It rejects the out of focus signal and provides axial resolution (as a first approximation, the lateral resolution is given by the focused laser beam). More details about confocal microscopy can be found in the recently published book by J. Mertz.20

(12)

−4

The units of ϕS2 are (s ·cm ); the units of ϕS3 are (s−3·cm−6), and those of ϕS1 are (s−1·cm−2). Figure 2 shows an example of a saturation curve given by eq 9, for different values of ϕS1, ϕS2, and ϕS3. It is very interesting to note that ε, in a five-level molecular system, exhibits a maximum. 4737



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normalized expression of the point-spead-function and PSF(r,z) = |h(r,z)|2, where the function h(r,z) is given in the ref 21. The key point, to model the changes induced by the saturation process within the confocal observation volume, is that the excitation level of each molecule depends on its location along the focused laser beam. In other words, the expression of the fluorescence signal emitted by one molecule in the sample needs to take into account the relative position of the molecule with respect to the laser spot center. Hence, we replace ϕe in eqs 5 and 9 by ϕe × PSFn(ro,zo), where ϕe corresponds now to laser photon flux at the geometrical focal point (ro = 0, zo = 0). Thus, the total fluorescence signal in the sample can be written as: fo (ro , zo , t ) = ε(ro , zo)C(ro , zo)

where C(ro,zo) is the local concentration of the fluorescent species. Next, eqs 13, 14, and 15 can be rewritten to exhibit the observation volume W(ro,zo), also called molecular detection efficiency function, and the collection efficiency function cef(ro,zo), which depicts the detection volume created by the pinhole:

Figure 2. Brightness of a single molecule as a function of the exciting laser flux (in log scale). In black, the plot of the usual saturation curve for a three-level system from eq 5 (ηd = 0.001, kr/k0 = 0.9, σS01 = 1 × 10−16 cm2, ϕS = 2.5 × 1024 s−1·cm−2). In red, using eq 9, evolution of the saturation curve according to a five-level molecular system (ηd = 0.001, kr/k0 = 0.9, σS01 = 1 × 10−16 cm2, ϕS1 = 2.5 × 1024 s−1·cm−2, ϕS2 = 5 × 1049 s−2·cm−4, ϕS3 = 3 × 1075 s−3·cm−6). In blue, the ′ = 0 (ηd = 0.001, kr/k0 = evolution of the brightness in the case of kISC 0.9, σS01 = 1 × 10−16 cm2, ϕS1 = 2.5 × 1024 s−1·cm−2, ϕS2 = 5 × 1049 s−2·cm−4, ϕS3 = 0).

F (t ) =

∫P dΩiP(ri)fi (ri , z i , t ) i

∫S dΩoW (ro , zo)C(ro , zo , t ) o

W (ro , zo) = ε(ro , zo)cef n(ro , zo)

To determine the theoretical expression of the fluorescence signal recorded with a confocal microscope, we closely followed the 2D scalar approach previously published.21 The fluorescence signal recorded just after the pinhole, noted F(t), can be written as a fraction of the total signal in the pinhole plane, denoted f i(ri,zi,t) (we use the subscript i for the image space): F (t ) =

∫P dΩiP(ri)PSFn(ri − ro , z i − zo)

cef n(ro , zo) = cn

i

(16) (17)

(18)

cn is a constant of normalization. More details about the cef(ro,zo) are given in ref 21. It clearly appears in eq 17 that the observation volume in confocal fluorescence microscopy is defined by the laser illumination profile (through the PSF), the saturation degree of the molecule (through ε), and the collection profile function (through the cef). Finally, let us introduce the normalized brightness profile, denoted θ(ro,zo), in order to highlight the influence of the saturation process on the observation volume:

(13)

P(ri) is the pupil function of the pinhole, z corresponds to the optical axis, and dΩi is the volume element. In the scalar description of incoherent image formation, the image can be written as the convolution of the fluorescence intensity in the sample with the point-spead-function (PSF) of the microscope objective:

θ(ro , zo) =

fi (ri , z i , t ) = (f0 *PSFn)(ri , z i , t )

ε(ro , zo) ε (ro , zo) max

(19)

Thus, the expressions of θ(ro,zo) in a 3-level and 5-level molecular system can be written as follows:

∫S dΩofo (ro , zo , t )PSFn(ri − ro , z i − zo , t )

=

o

(14)

θ3l(ro , zo) = (1 + R )

where * denotes the convolution and the subscript o is the signal in the sample (in the object space). PSFn is the Θ5l(ro , zo) = cn′

(15)

PSFn(ro , zo) 1 + R PSFn(ro , zo)

(20)

PSFn(ro , zo) 1 + R1PSFn(ro , zo) + [R 2 PSFn(ro , zo)]2 + [R3PSFn(ro , zo)]3

where cn′ is a normalization factor, which can be only obtained numerically for each value of R1, R2. and R3. R = ϕe/ϕS is the ratio of the laser photons flux ϕe at the geometrical focal point over the saturation level. Similarly, R1 = ϕe/ϕS1, R2 = ϕe/ (ϕS2)1/2, and R3 = ϕe/(ϕS3)1/3. An example of the normalized brightness profile is shown in Figure 3a, for ϕe→0 (i.e., R = R1 = R2 = R3 = 0). In this case, the brightness pattern is exactly the same as the laser excitation spot in the sample. As summarized in eq 17, the normalized observation volume can be easily calculated

(21)

by multiplying the normalized brightness θ(ro,zo) with the normalized cef(ro,zo), as illustrated in Figure 3. The brightness profile and the collection efficiency function plotted in Figure 3a,b take into account the numerical aperture (NA) of the objective, the index of the solution, the excitation and emission wavelength, and also the accurate diameter (according to the magnification of the microscope) of the pinhole used in our FCS experiments. Fluorescence Correlation Spectroscopy. FCS is a powerful experimental technique particularly well suited to 4738



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Figure 3. Calculated 2D profiles. (a) Brightness profile θ(ro,zo) within the sample, in the case of ϕe→0 (ON = 1.2, λlaser= 488 nm). (b) cef(ro,zo) according to a pinhole diameter of 62.5 μm, a microscope magnification of 120 and λfluo = 550 nm. (c) Corresponding observation volume W(ro,zo).

investigate diffusion process in biological membrane,22,23 but this technique also gives access to the accurate number of molecules within the observation volume, which constitutes the key parameter discussed in this paper. Fluorescence correlation spectroscopy is based on the temporal autocorrelation of the fluorescence signal F(t), emitted by dyes diffusing through the observation volume. The autocorrelation G(τ) is evaluated as: G (τ ) =

⟨F(t )F(t + τ )⟩ ⟨F(t )⟩2

rhodamine 6G in ethanol and the other with Alexa488 in water. In both cases, we observe a drastic diminution of the autocorrelation amplitude with the laser power, as shown in Figure 4A. The autocorrelation curves were fitted according to eq 23, and the evolution of the mean number of molecules and the diffusion time are plotted as a function of the laser power (Figure 4C,D, respectively). The laser power indicated in Figure 4 is the effective power within the sample, given by the power measured at the entrance of the objective corrected by the transmission of the objective (≈90% at 488 nm). To quantify accurately the effective volume in FCS, Veff, we plot the mean number of molecules N as a function of the concentration of Alexa488 solution at three different excitation powers (Figure 4B). Since N = C × Veff, it is easy to extract the accurate value of Veff from the slope of the plot. The slope strongly increases with the irradiance, yielding an effective volume increase, from Veff = 0.39 μm3 at 21.6 μW to 0.94 μm3 at 165.2 μW. However, the evaluation of Veff over a broad range of laser power by this procedure can quickly become a tedious task. Fortunately, we can directly evaluate the relative increase of N just by dividing the value obtained from the nanomolar solution by the smallest value of N at very low laser irradiation, as shown in Figure 4C. Finally, since the concentration of the solution is fixed, the relative increase of N and Veff are exactly the same. Consequently, the increase of the effective volume can be directly found in Figure 4C, and the response differs completely between the rhodamine 6G and Alexa488 molecules. In fact, the Veff relative increase reaches 1 order of magnitude with ≈3 mW for Alexa488 and only a factor 3 with the same power for rhodamine 6G. Nevertheless, for both cases, the phenomenon is evidently huge and reveals the size variation of the effective volume in FCS as a function of the excitation power. Concerning the diffusion time, Figure 4D shows its dependence with the laser excitation. Similar observations have been previously reported in the literature.25 Before interpreting this unusual increase of the diffusion time, we first examine the asymptotic behavior at low irradiation of the two experimental curves. Indeed, at very low laser excitation, the shapes of the observation volume and focused laser spot are close similar (Figure 3); hence, the smallest lateral size of the observation volume and the focused laser beam, denoted ωL, are equal. Thus, we can evaluate precisely the size of the laser beam waist, which is constant when the laser power increases, just by evaluating the asymptotic minimum value of τd. Next, according to a Gaussian

(22)

where the brackets ⟨ ⟩ denote temporal averaging. Assuming a 3D Gaussian-like profile of the observation volume, like in Figure 3c, and molecules freely diffusing in 3D, eq 22 has a simple analytical solution of the following form:24 1+ 1⎛ B ⎞2 G (τ ) = 1 + ⎜1 − ⎟ N⎝ F⎠ 1 + τ τ

(

d

τ p e− τp 1−p

)

1+

τ S 2τd

(23)

N appears to be the total number of molecules (in bright state and in dark state). B and F, are respectively, the mean background noise (i.e., detector dark count and Raman scattering of the solvent) and the mean photocounts detected. The term p depicts the fraction of molecules in a nonfluorescent state, such as the triplet state. The characteristic time related to such a nonfluorescent process is τp. The diffusion time τd is related to lateral waist ωr of the Gaussian-like observation volume (ωr is the lateral radius which the molecular detection efficiency has dropped by a factor of e2) and the diffusion coefficient D through τd = ωr2/4D. S characterizes the axial elongation of the observation volume (i.e., the ratio of axial to radial dimensions of W(ro,zo)); see Figure 3c. In FCS, N is the average number of molecules residing inside an effective probe volume, denoted Veff, given by (in our 2D approximation): V eff =

(∫ d ΩW (r , z))2

∫ d ΩW 2(r , z)

(24)

and, if C is the concentration of the dye solution,

■

N = C × V eff

(25)

RESULTS AND DISCUSSION Increase of the Observation Volume. Two series of FCS experiments were performed on nanomolar solutions: one with 4739



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Figure 4. (A) Typical evolution of the autocorrelation curves with the cw laser power, for a nanomolar solution of rhodamine 6G in ethanol. The cw laser power indicated in the figure is the effective power within the sample. (B) Plot of the mean number N of molecules against the concentration C of Alexa488 solutions. The red curve is the linear fit according to the general equation, independent of the shape of the observation volume, N = C × Veff. The effective volumes are: Veff = 0.39 ± 0.04 μm3 for a laser power of 21.6 μW (a), Veff = 0.56 ± 0.04 μm3 for a laser power of 59.6 μW (b), and Veff = 0.94 ± 0.04 μm3 for a laser power of 165.2 μW (c). (C) Relative increases of the mean number of molecule N in effective volume, obtained from a nanomolar solution, as a function of the laser power. (D) Also with a nanomolar solution, increases of the diffusion time τd with the laser power (the fitting curve is explained in the text).

lateral profile of the laser spot, we can transform the laser power within the sample, PL, to a laser photon flux, as follows: ϕe =

Ie PL = 1 hν hν 2 πωL 2

Table 1. Asymptotic Value of the Diffusion Time at Low Excitation, τdo, and Waist of the Focused Laser Beam, ωL rhodamine 6G in ethanol Alexa488 in water

(26)

To get the asymptotic value of τd at low laser excitation, the two curves of the Figure 4D were fitted with a saturation-like function: τd = τd 0 +

PL PS

D (μm2·s−1)

ωL (nm)

28.2 ± 0.5 23.6 ± 0.3

343−355a 435b

198 ± 2 203 ± 1

a

Calculated from refs 32 and 33 and according to the ratio of viscosity between water (0.89 mPa.s) and ethanol (1.074 mPa.s) at 25 °C. b Taken from ref 33.

αPL 1+

τd0 (μs)

to two distinct effects at high laser power in fluorescence correlation spectroscopy. The first one is the depletion of the dye concentration, which leads to a decrease of N. The second one is the decrease of the apparent diffusion time, only if the bleaching time is similar or shorter than their residence time in the observation volume. Experimental Saturation Curves. Figure 5 shows the fluorescence brightness (the count rate per molecule) of rhodamine 6G in ethanol and Alexa488 in water. The laser excitation flux was determined according to eq 26 and Table 1. Figure 5 clearly reveals the drastic decrease of the brightness ε at high laser excitation. A comparison with Figure 2 suggests a role of the consecutive absorption of two and three photons in this behavior. Thus, the two experimental saturation curves

(27)

where τd0 is the asymptotic minimum value of the diffusion time and PS and α are two adjustable parameters. The relevance of such fitting function will be discussed later. Since the diffusion coefficient of Alexa488 and rhodamine 6G is nowadays wellknown, we can easily determine ωL, according to the relation ωL ≈ (4Dτd0)1/2, as shown in Table 1. For Figure 4C,D, it appears that N and τd gradually increase with the laser excitation, which clearly means that our observations were not affected by dye photobleaching. As previously explained in the introduction, photobleaching would give rise 4740



(1.4 ± 0.1) × 10 (5 ± 1) × 1024 (6.4 ± 0.2) × 10 (9.3 ± 0.1) × 1023 rhodamine 6G in ethanol Alexa488 in water

b All the experimental data were obtained with a cw laser with excitation at 488 nm, except for the lifetime measurements (τref 0 ). This reference value of ϕS is calculated according to eq 6, including the d ref c experimental values of kISC, kT10, σS01, and the reference value of the lifetime τ0 . See in refs 11, 13, 34, and 35. See in refs 6 and . eMeasured with a picosecond laser at 532 nm. fFrom Invitrogen Web site (excitation wavelength is not indicated). a

3.8e 4.1f 1−10 1−10 10 10−16/−17

kSn1, kTn1d (ps−1)

(2.5 ± 0.1) × 10 (1.41 ± 0.02) × 1023

2.33 × 10 1.48 × 1023

0.3 ± 0.1 1.0 ± 0.1

2.1 ± 0.1 0.21 ± 0.01

1 × 10 2.8 × 10−16

−16/−17

σS1n, σT1nc (cm2) σS01 (cm2) kT10 (μs−1) kISC (μs−1) b ϕref (ph·s−1·cm−2) S

−16 25

(ϕS3)1/3 (ph·s−1·cm−2) (ϕS2)1/2 (ph·s−1·cm−2) ϕS1 (ph·s−1·cm−2)

Table 2. Photophysical Parameters of Rhodamine 6G and Alexa488a 4741

24

have been fitted according to the theoretical expression of the brightness in a five-level molecular system in eq 9 (Figure 5). The excellent agreement between the fitting curve and the experimental data highlights that the intramolecular dynamic of these two molecules need to include absorption from the first excited molecular states (S1,T1) to higher excited states (Sn,Tn), as suggested in Figure 1. Table 2 gives the different experimental saturation levels of each molecule, ϕS1, ϕS2, and ϕS3, obtained through the fitting procedure. To progress in the discussion, a reference value of the one-photon saturation level, denoted ϕref S and given by the eq 6, has been calculated. Thus, we have measured with a spectrophotometer the absorption cross section at 488 nm for each molecule (Table 2). Next, as previously developed in refs 1 and 2, we have determined the rate of intersystem crossing and triplet depopulation. These transition rates have been obtained through the same FCS measurements presented in Figure 4, by analyzing the evolution of p and τp with the laser excitation flux (see Supporting Information). The reference fluorescence lifetime (τref 0 ) of rhodamine 6G in ethanol was measured with a pulsed laser excitation at 532 nm. On the contrary, for Alexa488 molecules in water, τ0 is taken from the literature (Invitrogen). It is assumed that the nonradiative depopulations Sn→S1 and Tn→T1 are in the subpicosecond or picosecond time scale.6,26 Moreover, for such dye molecules, the absorption cross-section σS1n and σT1n are about 10−16 to 10−17cm2.11,13 Consequently, k′ISC can be estimated according to the ratio between the two first saturation levels, ϕS2/ϕS1. Since we do not have the accurate values of σS1n, σT1n, kSn1, and kTn1, we have only obtained the order of magnitude of k′ISC = 109−1010 s−1 for rhodamine 6G in ethanol and 109−1011 s−1 for Alexa488 in water. The direct consequence of this evaluation is that ϕS1 ≈ ϕS in eq 10, which is confirmed by the fact that the estimated reference values ϕref S are similar to ϕS1, as indicated in Table 2, with less agreement for the rhodamine 6G. These results confirm quite clearly that our theoretical and experimental approach give quite good values of the saturation levels. More interestingly, by reversing eq 6, the experimental values of ϕS1, σS01, and κ can provide the fluorescence lifetime of the dye. We have calculated τ0, thus obtained for a cw laser excitation at 488 nm: 3.5 ns for rhodamine 6G in ethanol and 4.2 ns for Alexa488 in water.

24

Figure 5. Saturation curves of Alexa488 in water and rhodamine 6G in ethanol. The experimental values (marks) were fitted according to the theoretical expression of the brightness in a five-level molecular system (lines), eq 9.

24

τref 0 (ns)

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It is very interesting to note that the lifetimes provided from the saturation level measurements under cw excitation are similar to the reference ones, obtained with a pulsed excitation, Table 2. The role of k′ISC is crucial in these saturation investigations. Indeed, it allows a depopulation of the fluorescent state S1 with the increase of the laser irradiance. On the contrary, if we neglect the intersystem crossing between the two higher excited states (which is commonly assumed in the literature), the molecules will not stay long enough in the excited state Sn to allow an efficient depletion of the brightness. Moreover, if kISC ′ ≈ 0, then ϕS3 = 0 and the eqs 9, 10, and 11 will be less complex. It is interesting to note that the two experimental saturation curves in Figure 5 will also be well fitted with the simplified eq 9 (data not shown). However, in this approximation, the estimated ratio ϕS2/ϕS based on reference values is strongly different from the experimental one (about 3 orders of magnitude), which definitively rejects the approximation for k′ISC. Numerical Simulations. Let us now consider the computational simulation of the molecular detection efficiency W(ro,zo) from eqs 17 to 21. According to our experimental configuration (see Experimental Section), all W(ro,zo) simulated patterns are evaluated for the same pinhole diameter of 62.5 μm according to a microscope magnification of 120 for the same filling of the back aperture of the microscope objective (with a numerical aperture of 1.2) with a laser excitation wavelength of 488 nm and with a mean fluorescence emission wavelength around 550 nm. Figure 6 presents several theoretical 2D profiles of the observation volume W(ro,zo) and its dependence with the laser excitation. Since rhodamine 6G and Alexa488 molecules have different saturation levels, two separate evaluations of the observation volume were made, according to their different ratios R1/R2 = (ϕS2)1/2 /ϕS1 and R1/R3 = (ϕS3)1/3 /ϕS1 for Alexa488 and for rhodamine 6G (Figure 6). As expected, these simulations confirm the size increase of the confocal observation volume, whatever the molecular systems chosen. An interesting way to compare the experimental and theoretical observation volume variation with the laser excitation power is to plot the relative increase of N (which is also equivalent to the relative increase of Veff) versus R1, as presented in Figure 7. Similarly, since the diffusion coefficient of each molecule seems to be constant with the excitation power (Table 1),27 the relative increase of the lateral waist of the observation volume ωr was also plotted versus R1 (inset in Figure 7). ωr appears to evolve similarly for both molecules, whereas Veff increases are definitively different. In this plot, the effective volume progresses more rapidly for rhodamine 6G than for Alexa488. This behavior is directly related to the ratio R1/R2 and R1/R3, which is weaker for rhodamine 6G than for Alexa488. As a consequence, the size variation of the probe volume in fluorescence saturation spectroscopy strongly depends on the dye molecule used and its photophysical parameters ϕS1, ϕS2, and ϕS3. N and Veff will increase faster and faster as ϕS2 and ϕS3 tend to be close to ϕS1. Moreover, according to eq 24, the Veff calculations confirm the experimental results, as highlighted in Figure 7. We can deduce from these simulation two important points. On the one hand, the three-level molecular model is definitively not suited for explaining the two different relative increases of Veff recorded for both molecules. Indeed, in this model, the Veff behavior will be the same, regardless of the molecules studied. On the other hand, even if the agreement between the two

Figure 6. Evolution of the molecular detection efficiency profiles W(ro,zo) as function of the laser excitation for different photophysical models. The right column corresponds of what occurs in a 3-level molecular system, and the two left ones are for a 5-level molecular system. Concerning the 3-level molecular model, it is very interesting to note that the maximum size of W(ro,zo), which corresponds to R→∞, is exactly equal to the cef(ro,zo) profile. Concerning the 5-level molecular model, R1/R2 = 2.54 and R1/R3 = 5.62 for rhodamine 6G, and R1/R2 = 6.6 and R1/R3 = 37.7 for Alexa488. 4742



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calculation. We are currently working on the improvement of our numerical code to take into account a nonanalytic expression of the PSF.

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CONCLUSION This work describes the evolution of the fluorescence signal of single molecules in solution with the laser excitation. We propose a physical model, not including photobleaching, that highlights the possible effects of the cascade absorption of two and three photons within a five-level molecular system. This multiphoton consecutive absorption leads us to reconsider the common expression of the saturation curve. A second and a third saturation intensity thus appears which, respectively, depends on the irradiance squared and cubed, which is responsible for a strong decrease of the fluorescence at high laser excitation regime. Moreover, our numerical simulations suggest that this combined one-photon, two-photon, and three-photon saturation process drastically influences the size and the shape of the confocal observation volume and, therefore, the FCS measurements. Consequently, the mean number of molecules and the diffusion time, which are the main interesting parameters in many FCS applications, will change together with the observation volume. If these FCS parameters are not under control, this can lead to artifacts, particularly during the waist calibration routine. Interestingly, we proved that it is possible to determine the fluorescence lifetime without using any pulsed laser excitation, just through the investigation of the saturation. This unusual procedure to obtain the lifetime of single molecules in solution could be an interesting alternative for many photophysical investigations. Moreover, to our knowledge, we evaluate for the first time the order of magnitude of the intersystem-crossing rate between the two higher excited states Sn and Tn (k′ISC ≈ 109−1011 s−1). Unexpectedly, this combined experimental and theoretical study shows that the plausible fingerprint of the multiphoton cascade absorption is a decrease of the brightness for high irradiance, which comes with a depletion of the fluorescence signal at the center of the observation volume. The typical laser power needed to create this depletion is about a few mW at the entrance pupil of the objective. Consequently, with such reasonable laser power excitation (just greater than the square root of the second saturation level), the conventional diffracted limited observation volume will be transformed into a doughnut-shaped volume, as it is usually achieved with a well-designed phase mask.29−31

Figure 7. Evolution of the relative increase of N and Veff as a function of R or R1. Inset: relative increase of the lateral waist of the observation volume ωr with R or R1. The blue curves correspond to simulations according to the 3-level molecular systems; the red ones and the green ones correspond to the 5-level molecular systems (red curve: R1/R2 = 2.54 and R1/R3 = 5.62; green curve: R1/R2 = 6.6 and R1/R3 = 37.7).

simulated relative increases of Veff and the experimental data is not perfect, the five-level molecular model predicts reasonably well the experimental data. Two main reasons can explain these differences: (i) we use a scalar theory of image formation in our simulation; (ii) possible aberrations of the focused laser beam are ignored. Finally, consecutive absorption of two and three photons is an important phenomenon, which cannot be neglected in fluorescence saturation spectroscopy of a single molecule. Next, for a five-levels molecular model, Figure 6 reveals interestingly that the shape of the molecular detection efficiency is strongly disturbed at high laser excitation. A depletion at the geometrical focal point (ro = 0, zo = 0) is particularly observed for R1 greater than 10 for rhodamine 6G and 35 for Alexa488. This depletion can be intuitively explained considering that the molecules which diffuse near the geometrical focal point, i.e., where the laser photon flux is maximum, will be more excited than those diffusing far from the center. Thus, in the center, the molecules will be statistically more often in a nonfluorescence state than the other ones, which gave rise to a depletion of the fluorescence signal as the laser power increases. In this framework, a recent paper proposes to break the diffraction barrier by monitoring the dark states of molecules.28 Strictly speaking, this fluorescence depletion suggests that the W(ro,zo) profile is no longer Gaussian, which means eq 22 cannot be used to fit the autocorrelation function at high excitation regime. In fact, the average number N of molecules is not strongly affected by this effect, because G(0)−1 is independent of the shape of the observation volume. In contrast, the evaluation of the diffusion time τd with eq 22 will be less and less accurate, as the laser excitation increases over the first saturation level ϕS1. Nevertheless, Figure 7 shows that the lateral waist ωr of the observation volume (and so the diffusion time τd) increases as suggested by the theory. Simulation for a 3-level molecular model also confirms that the saturation-like function used to fit the evolution of the diffusion time τd with the laser power, eq 27, is well suited. To predict more precisely the evolution of N and τd, it would be better to record the PSF of our microscope objective and then inject this experimental PSF in our

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ASSOCIATED CONTENT

S Supporting Information *

Details about kISC and kT10 determination. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: rodolphe.jaffi[email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Ligue Contre le Cancer (Comité de l’Aube), the Conseil Général de l’Aube, and the Conseil Régional Champagne-Ardenne (CELLnanoFLUO project). The authors acknowledge Davy Gérard, Cyrille Vézy, and Christophe Couteau for their careful reading of this paper. 4743



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Multiphoton Cascade Absorption in Single Molecule Fluorescence

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