Determination of the two-photon absorption (2PA)cross section of fluorescent molecules in the femtose-cond (fs) regime is necessary in order to determinewhich molecule is suitable for the specific applicationof multiphoton fluorescence laser scanning micro-scopy [1]. A 2PA cross section in the fs regime alsodetermines the adequate excitation wavelength. In2PA microscopy the use of laser systems workingat a high repetition rate (HRR) enables determina-tion of the best signal-to-noise ratio at low incidentpowers. Direct measurements of the 2PA cross sec-tion using ultrashort fs pulses with HRR lasersare therefore preferred to using nanosecond (ns) orpicosecond (ps) pulsed lasers. It is well known thatthe HRR lasers generate an accumulative thermalgradient, which induces changes in the refractive in-dex of the medium [2]. This thermal gradient pro-duces an accumulative temperature distributionaround the optical propagating axis, generating a
spatial distribution of the refractive index or thermallensing (TL) [3,4]. This accumulative TL diffracts thewavefront of the laser beam in the far field, whichcan be measured as the transmittance changethrough a small aperture placed on the optical axis.
TL has been previously used to measure 2PA crosssections in nonfluorescent and fluorescent samplesusing ns and ps pulse lasers [5–8]. Some 2PA-measurement approaches have been devised to date,however, despite the fs regime’s intrinsic high sensi-tivity. Among them, Falconieri [2] has shown the pos-sibility of measuring the 2PA coefficient using theaccumulative TL effect, although the theoreticalmodel of this technique is limited to nonfluorescentmolecules. Another approach is based on interfero-metry [9] and was designed to measure the 2PA crosssections in both nonfluorescent and fluorescent mo-lecules from the photothermal phase shift induced inthe sample.
The purpose of this work is to demonstrate that theabsolute 2PA cross section can be estimated withgood sensitivity and relatively low uncertainty usingthe accumulative TL induced by absorption of one
and two photons. We perform TL experiments with atunable HRR laser system in the fs pulse regime, andwe verify the relevance of the approach measuringthe 2PA cross section of classical well-known dyemolecules.
2. Theoretical Model
Let us consider a TL induced by an HRR laser systemin a fluorescent liquid sample. The sample is placedat a distance z from the waistw0 of a Gaussian beam,and its thickness L is smaller than the Rayleighlength (z0 ¼ πw2
0=λ) [4]. The heat generated througha q-photon absorption process, the radial energy flowinto the sample in a unit of volume and a unit of time,can be described as
QðrÞq ¼ αqIðrÞqf q; ð1Þ
where
αq ¼ qðNA½C�ÞðhνÞq−1δq; ð2Þ
IðrÞ ¼ 2P expð−2r2=w2Þ=πw2; ð3Þ
f q ¼ 1 −
Φλqqhλeiq
; ð4Þ
and q indicates the number of photons absorbed dur-ing the transition. αq is the fraction of absorbed lightby the medium, NA is Avogadro’s number, [C] is theconcentration of the sample, δq is the q-photon ab-sorption cross section, h is Planck’s constant, and νis the frequency of the excitation light. IðrÞ is theaverage incident intensity of the laser beam in thesample, where w is the radius of the Gaussian beaminto the sample and P is the average incident power.Because of the fact that q-photon upconvertedfluorescence occurs beside the q-photon absorptionprocess, the expression f q is considered in our calcu-lations. This expression is the fraction of absorbedlight converted in emitted light and is deduced byconsidering the number of photons absorbed andemitted per second when the q-photon process is in-duced. The parameter Φ is the quantum yield, λq isthe excitation wavelength of the laser beam, andhλeiq ¼ R
EðλÞqdλ=R ½EðλÞq=λ�dλ is the average wave-
length of the emitted light: where EðλÞq is the q-photon upconverted fluorescence spectrum of thesample. The ratio λq=qhλei takes account of the Stokesshift, and the factor 1=q is due to the q photon beingnecessary to produce one fluorescence process.
Following the theoretical procedure described in[4], we solve the heat conduction differential equa-tion with the heat source, given in Eq. (1). Thus,we obtain an expression for the temperature distri-bution induced by an accumulative q-photon absorp-tion thermal lens effect:
ΔTðr; tÞq ¼�
qαqPqf q2πκðπw2Þq−1
�1tc
Zt
0
11þ 2qt0=tc
× exp�−2qr2=w2
1þ 2qt0=tc
�dt0;
ð5Þ
where κ is the thermal conductivity of the sample,tc ¼ w2=4D is the characteristic relaxation time ofthe TL, and D is the thermal diffusivity of the med-ium. This temperature distribution induces an addi-tional phase shift in the laser beam wavefront, whichis given by φðr; tÞq ¼ ð2πL=λqÞΔnðr; tÞq, whereΔnðr; tÞq ¼ ðdn=dTÞΔTðr; tÞq and (dn=dT) representsthe thermo-optic coefficient. In order to evaluate themagnitude of the phase shift induced by a q-photonabsorption process, we consider the phase variationaround the optical axis of the laser beam. This isdone by calculating the q-photon photothermal phaseshift:
ΔϕðtÞq ¼ ϕðr; tÞq − ϕð0; tÞq¼ θq
tc
Zt
0τ½1 − expð−2qτuÞ�dt0; ð6Þ
where τ ¼ 1þ 2qt0=tc, u ¼ r=w, and the phase shiftamplitude is
θq ¼�
qαqLPqf qλqκðπw2Þq−1
��dndT
�: ð7Þ
Equation (6) is the phase shift added to theGaussian complex electric field Uiðu; z; tÞ of the laserbeam at the exit surface of the sample:
Ue ¼ Uiðu; z; tÞ expð−jΔϕqÞ: ð8Þ
In this approach we do not consider contributions tothe phase shift from the nonlinear refractive index[ΔφNL ¼ 2πLγIðrÞ=λ]. This consideration is reason-able if the average incident intensity in the sampleis so weak that the nonlinear phase shift [ΔφNL ¼2πLγIðrÞ=λ] can be neglected with respect to thephotothermal phase shift. By considering the smallphase shift (Δφq ≪ 1) and the far field condition(d ≫ z0, d is the distance from the sample to the de-tector), the amplitude of the complex electrical fieldEq. (8) at the detector plane can be calculated by con-sidering the Fresnel diffraction theory [4]. Thisyields
UðtÞq ¼ BZ
∞
0ð1 − jΔϕqÞ exp½−ð1þ jVÞu�du; ð9Þ
where B is a constant and V ¼ z=z0 is the relative po-sition of the sample with respect to the beam waist.By carrying out the integration over u and t0, we ob-tain the TL intensity variation at the center of thelaser beam on the detector plane:
Here all θq terms of the order greater than 1 areneglected. For the case when the absorbed energylight is completely converted in heat f q ¼ 1 (nonfluor-escent samples), Eq. (10) is equivalent to the one pre-viously reported in [2].
When the sample is fixed at the plane z ¼ z0, thegeometrical parameter V and the characteristic re-laxation TL time tc remain constants. Using the stea-dy-state condition of the TL (t ≫ tc) in order torewrite Eq. (10), we define the TL signal as
Sq ¼ Ið∞Þ − Ið0ÞIð0Þ ¼ θq
qarctan
�2qV
1þ 2qþ V2
�: ð11Þ
If the TL signal is generated by a 2PA process(q ¼ 2), we can see that the value of the 2PA crosssection is very sensitive to the following experimen-tal parameters: (1) the location of the sample withrespect to the focal plane, (2) the radius of the laserbeam into the sample, (3) the incident average power,(4) the fraction of absorbed light upconverted in theemitted light [Eq. (4)], (5) fluctuations of the laserwavelength, and (6) values of the photothermal con-stants used in Eq. (7). Some of these experimentalparameters can be eliminated by comparison witha TL signal induced by the one-photon (q ¼ 1) ab-sorption process.
By using a bismuth borate (BIBO) crystal, one cangenerate the second harmonic (SH) from the funda-mental laser beam. Thus, the SH beam can be usedas an excitation source for the one-photon absorptionprocess. In this case we can evaluate the ratio S2=S1from Eq. (11) by considering two TL effects simulta-
neously and separately induced in the sample byusing two different excitation wavelengths:
S2
S1¼ α2L
ACP22
P1; ð12Þ
C ¼ arctanf4V2=ð5þ V22Þg
2πw22 arctanf2V1=ð3þ V2
1Þg: ð13Þ
Here the linear absorbance of the sample at the SHwavelength is A ¼ α1L. V1 and V2 are the relativedistances from the sample to focal planes of the SHand fundamental laser beams, respectively. Theseequations are valid under the following assumptions:(1) the values of κ and (dn=dT) remain constants atthe wavelengths λ1 and λ2, (2) the wavelength of λ1corresponds to the SH of λ2 (λ2 ¼ 2λ1), (3) the fractionof absorbed light upconverted in emitted light isequal for both processes (f 1 ¼ f 2), and (4) only smallincident average powers P1 and P2 are considered inorder to keep the linearity between S2=S1 and P2
2=P1.Equation (12) shows that the 2PA coefficient α2 can
be deduced from the slope of the experimental curveS2=S1 ¼ f ðP2
2=P1Þ if the absorbance, path length, andexperimental geometrical parameters are known.
3. Experimental Setup and Measurements Principle
The experimental setup is shown in Fig. 1. Thesource light consists of a mode-locked Spectra-Physics Tsunami laser system, which delivers pulsesof 100 fs at 80MHz and is tunable in the broad rangefrom 700 to 1000nm (350–500nm by using an SHgeneration crystal). The laser beam is modulatedby a mechanical chopper placed at the exit of the la-ser system, and its frequency is set at 10Hz. Afterpassing the laser beam through a cubic beam splitter(CBS) (50R/50T), the transmitted beam is directedto the BIBO crystal where the frequency-doubled
Fig. 1. (Color online) Experimental setup used to measure 2PA cross section of the classical dyes. M, mirror; Ch, chopper; F, filter; D3, D4,detectors; S, sample.
radiation is generated. This SH beam (h1i) is focusedby means of the lens L1 (f¼10 cm), which is mountedon a linear translation stage in order to adjust therelative distance V1 from the focal plane of beam h1ito the sample. The reflected beam from the CBS (h2i)is used to induce the two-photon TL effect in the sam-ple. We use another lens of 10 cm of focal distance tofocus this beam near the sample. We measure the ra-dius variations of both beams on their respective op-tical axes by setting a beam profiler in the sampleholder position. Scanning the beam profiler throughthe focal plane of the beam h2iwith a precision linearstage, the measured waist beam at λ2 ¼ 800nm isw02 ¼ 25 μm, given a Rayleigh range of Z02 ¼2:5mm. We thereby fix the sample position at theplane z ¼ z02, approximately. Maintaining this posi-tion for the beam profiler, we scan L1 in order to mea-sure the Rayleigh range of beam h1i (z01 ¼ 3:5mmat 400nm) and determine the value of parameterV1. The uncertainty in these measurements isapproximately 5%.
Plate beam splitters BS (8R/92T) are used to re-flect the SH and fundamental beams for monitoringof incident powers on the sample. The optical at-tenuators OA1 and OA2 obtain the adequate levelsof these incident powers, which are adjusted to avoidthermal aberrations in the sample. During the mea-surements, OA1 remains fixed, yielding a constantTL signal S1, while OA2 is rotated in order to gener-ate a range of TL signals S2.
Samples consist of a fused silica cuvette with 1mmof path length, filled with classical fluorescent dyes:Rhodamine B (RhB) and Rhodamine 6G (Rh6G) inmethanol solutions and Fluorescein (Fl) in alkalinewater (pH ¼ 11). Concentrations are directly con-firmed (f½C�RhB ¼ 1:05; ½C�Rh6G ¼ 2:9; ½C�Fl ¼ 1:01g×10−4 M) by measuring linear absorption spectra withan Ocean Optics spectrophotometer (USB4000).One- and two-photon upconversion fluorescencespectra are also obtained with an Ocean Optics spec-trometer using the fundamental and SH generationlaser beams. All these measurements are performedat room temperature.
In the far field S1 and S2 TL signals are monitoredusing apertures (P) and photodiodes D1 and D2,respectively. The TL signals are simultaneouslyand separately recorded by a four-channel digital os-cilloscope, where they are measured after averaging512 signals in order to obtain good signal-to-noise ra-tio. To avoid overlapping between beams h1i and h2iinto the sample, they are separated 1 cm from eachother. The wavelength of beams h1i and h2i are diag-nosed by means of the built-in spectrometer in theTsunami laser system and the Ocean Opticsspectrometer.
4. Results and Discussions
Typical UV-visible spectra of the dyes measured atthe concentrations previously indicated are shownin Fig. 2, indicating that the molecules have no linearabsorption between 750 and 850nm, the region se-
lected to induce the 2PA photothermal process. Simi-larly, the UV-visible spectra also show that, in thespectral region 375–435nm, samples exhibit smallabsorbance (see inset of Fig. 2), which are consideredto produce one-photon TL signals. From this spectralregion, the absorbance A at one specific wavelengthis directly measured.
Comparison of the one- and two-photon upcon-verted fluorescence spectra is plotted in Fig. 3 for theRhB sample. Each spectrum was obtained by block-ing the SH beam and exciting the sample with thefundamental beam. Thereafter, the SH beam is usedto collect the one-photon fluorescence spectrum.From these spectra the average wavelength for eachexcitation process is calculated yielding hλei1 ¼579nm and hλei2 ¼ 578nm. With values λ2 ¼ 2λ1 ¼800nm and Φ ¼ 0:7, the ratio f 2=f 1 yields a valueof 1.002. That the fluorescence spectra induced by ex-citation of one or two photons are basically the sameand are independent of the excitation wavelengthhas been reported [10,11]. Similar results areobtained for the samples of Rh6G and Fl, confirmingour assumption f 1 ¼ f 2.
Fig. 2. (Color online) Absorbance spectra of the dyes used in thiswork. The inset shows the absorbance in the spectral region of375–435nm.
Fig. 3. (Color online) Fluorescence emission spectra of the RhBsample induced by absorption of one and two photons.
Simultaneous measurements of the TL signals as afunction of the incident powers are obtained from theselected dyes. Typical experimental curves, collectedat λ2 ¼ 800nm, are shown in Fig. 4 (open squares).Here the value of P1 (5mW) is fixed, and the incidentpower P2 varies in the range from 50 to 300mW, ap-proximately. Figure 4 represents the linear variationof S2=S1 versus the one of P2
2=P1 measured in the ap-proximation θq < 1; nonlinear variations of S2=S1 ¼f ðP2
2=P1Þ are removed from the experimental dataand are not considered in the calculations. The lineardependency observed in these curves indicate thatthe measured TL signals have no thermal convection[12] into the selected power range, and therefore thephotothermal effects are predominantly induced bylinear and nonlinear absorption.
Equations (2) and (12) show that the 2PA cross sec-tion δ2 is proportional to the slope of the experimen-tal curve. To estimate δ2 for the RhB at λ2 ¼ 800nm,first we calculate α2 by using the slope m ¼0:029W−1, the geometrical parameter C ≈ 3:23 ×
108 m−2 (measured at V1 ¼ V2 ≈ 1), the absorbanceARhB400nm ¼ 0:0624, and the path length L ¼ 1mm in
Eq. (12) (α2 ¼ mA=LC). After that, we use the calcu-lated value of α2 and Eq. (2) (δ2 ¼ α2=2NA½C�hν) toobtain the 2PA cross section of the RhB δ2 ¼ ð110�10ÞGM at 800nm. Similarly, evaluation of the sam-ples of Rh6G and Fl obtain results of δ2 ¼ð58� 6ÞGM and δ2 ¼ ð31� 3ÞGM, respectively. InTable 1 we summarize results for δ2 for each sampleand the those reported elsewhere; our absolute mea-surements correlate within the error margin of the δ2previously published [10,13,14]. The lower values ofδ2 for RhB and Rh6G reported in [15] however dis-agree with this study, possibly due to spurious non-linear effects such as stimulated-Rayleigh scatteringfrequently observed at high intensity levels.
Finally, an interesting application of Eq. (12) con-sists in determining the value of C for a sample witha well-known and accepted value of δ2. Assumingthat α2L=A is known and constant, the slope ex-tracted from S2=S1 ¼ f ðP2
2=P1Þ curve determines C.This calibration measurement is effectuated onceand is valid for all consecutive measurements atthe same wavelength. Once C is calculated, the2PA cross section of the sample under test (δ2) canbe estimated by Eqs. (12) and (2). We test this bytaking the δ2 values of RhB reported in [14] inthe excitation wavelength range of 750–850nm.Experimental results from the Rh6G sampleplotted in Fig. 5 correlate with those reported in
Fig. 4. (Color online) Typical experimental curves showing the linear dependence between S2=S1 and P22=P1. The standard error is
indicated by the error bar on each curve.
Fig. 5. (Color online) Experimental 2PA spectra of the Rh6G ob-tained after experiment calibration is done with respect to RhB.The data plotted with open triangles correspond to [14].
Table 1. 2PA Cross Section of the Classical Dyes Measuredat 800 nm of Excitation Wavelength Using TL Effect
2PA Cross Section (GM)
RhB Rh6G Fl
This work 110 58 31[10] 140 - 38[13] - 40 36[14] 120 65 36[15] 50 26 -
[14] considering our experimental error. With thesekind of measurements possible, systematic errorscommitted during the estimation of the waist andthe position of the sample are eliminated, whichare the main error sources in our experiment.
5. Conclusions
Measurement of δ2 in fluorescent molecules based onthe accumulativeTLeffect and theFresnel diffractionmodel is accomplished by simultaneous measure-ment of the TL signals induced by the one- and two-photon absorption process in the fs regime.Because ofthe HRR of the laser, a large accumulative TL is in-duced, requiring lower levels of incident power thantechniques such as nonlinear transmittance methodsrequire. Hence, the photobleaching of the sample isavoided, and consequently a more accurate 2PA spec-trum for a particular chromophore is obtained.
Simultaneous TL measurement allows the elimi-nation of constants that increase the experimentalerror and reduces the statistical dispersion of theexperimental data due to fluctuations in the powerand wavelength of the laser. Working with a well-characterized reference sample, fewer experimentalparameters have to be measured in order to estimatethe value of δ2. The validity of the approach is demon-strated from 2PA cross-section measurements inRhB, Rh6G, and Fl, and the results agree well withvalues obtained from luminescent methods.
The authors are grateful for the financial supportprovided by theMasdar Institute of Science andTech-nology and the support of the Dean of EngineeringProf. Khraisheh. We would like to thank Dr. KevinGarvey for reading and improving the manuscript.
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