J. Phys. Chem. B 2008, 112, 4561-4570
4561
Energy Transfer and Amplified Spontaneous Emission in Temperature-Controlled Random Scattering Media I-Yin Sandy Lee* and Honoh Suzuki Department of Chemistry, UniVersity of Toyama, 3190 Gofuku, Toyama 930-8555, Japan ReceiVed: December 11, 2007; In Final Form: February 5, 2008
Hydroxypropyl cellulose (HPC) in aqueous solution forms nanoparticles and becomes highly scattering above its lower critical solution temperature (LCST ∼ 41 °C). Enhancement of energy transfer (ET) and amplified spontaneous emission (ASE) have been observed in turbid HPC solutions containing Rhodamine 6G (RG) as an energy donor and Kiton Red 620 (KR) as an acceptor. A detailed analysis of self-absorption, absorption saturation, and multiple scattering effects has revealed the importance of photon diffusion in shutting down the intensity leakage. A 5-fold enhancement of ET in the turbid condition is estimated. Possible factors crucial for ET and ASE in random media are discussed, such as the donor-to-acceptor ASE energy pumping, the optical path elongation by multiple scattering, and the formation of “light pipes” in the near-field of the Mie scattering. The temperature-dependent colloidal formation is found to successfully control optical processes via multiple scattering with a sharp threshold and abrupt emergence of dense scatterers.
1. Introduction Control of a reaction is a goal for many chemical studies.1 One of the recent developments toward this goal is to control the interaction of light and matter by manipulating the local opticalenvironment.Periodicdielectricstructures,2 microcavities,3-5 and microdroplets6 can be used to control spontaneous emission and energy-transfer processes of confined molecules and polymers via modification of the photonic mode density. Optically scattering, disordered media are also capable of modifying the light field in a nontrivial way and thus are under extensive research in relation to photon localization,7 photonic band gap,8 and random lasers.9 Among such examples are laser powders,10 polymer-sphere suspensions,11 and even animal tissues.12 Recent studies on random media focus on wave interference from different scattering events. If the size of strongly scattering particles (e.g., GaAs) and the interparticle distance (more precisely, the photon scattering mean free path or length, lsca) are comparable with the wavelength of light, Anderson localization of light can happen, which completely stops the photon transport by diffusion. A related concept is that of the strong scattering regime carrying evanescent waves (the Ioffe-Regel criterion).13 Multiple scattering also bears some analogy to the bouncing between two closely spaced mirrors in a microcavity, where the imposed boundary condition leads to resonant modes in which only the light of an adequate wavelength interacts constructively. Consequences of such mode-selective optical field modification are, for example, inhibition or enhancement of spontaneous emission, lasing, and radiative energy transfer processes between molecules, all of which give promise for controlling photochemical processes in general. When the scatterer is a weaker one, i.e., the refractive index of the scatterer is not much different from that of the surrounding medium (e.g., polymer particles in water), the scattering cross section is smaller and the scattering length longer. The above * To whom correspondence should be addressed. E-mail: islee@ sci.u-toyama.ac.jp.
interference effects then become less important, and multiple scattering processes can usually be described well by photon diffusion. Although this “classical” regime has a long history of research,14 it is still under active investigation especially in relation to medical optics on biological tissue.15,16 Multiple scattering in such systems has many yet unanswered problems, particularly when it comes to anisotropic scatterers beyond the Rayleigh limit.17 Weak scatterers are available in a wide variety of material, absorptivity, size, electrical charges, and surface properties, which can further be tailored to suit experimental needs. It is of interest to see whether scattering in the classical regime has the power to control optical processes in a nontrivial manner or not. In this paper, we have examined the possibility of controlling photochemical processes in solution by using a weakly scattering medium as the solvent. We have chosen to work on an aqueous hydroxypropyl cellulose (HPC) solution, because its light scattering effect can easily be turned on or off by adjusting the temperature. The HPC solution, like those of many other cellulose derivatives, undergoes a phase transition at the lower critical solution temperature (LCST ∼ 41 °C). At this temperature, water becomes a poor solvent and causes HPC chain collapse and association; the solution abruptly becomes milky white and highly scattering, owing to the formation of metastable nanosphere aggregates with a narrow size distribution, instead of precipitation.18 Amplified spontaneous emission (ASE) and random laser action have been observed in turbid HPC solution containing Kiton Red 620 (KR), where the laser output can be tuned via temperature.19 The HPC solution has the advantage of the higher solubility of laser dyes over liquid crystals that are also studied as random laser media.20 Our aim is to extend such optical-field control to intermolecular energy transfer (ET) processes in HPC solutions. Two main mechanisms governing the intermolecular excitation transfer are radiative transfer and nonradiative transfer.21 Radiative transfer involves the photon emission from donors, which is later absorbed by acceptors (the so-called “trivial mechanism”). It occurs when the average distance between
10.1021/jp7116505 CCC: $40.75 © 2008 American Chemical Society Published on Web 03/27/2008
4562 J. Phys. Chem. B, Vol. 112, No. 15, 2008 donors and acceptors is larger than the wavelength. In contrast, nonradiative transfer proceeds without photon emission; it is typically described in terms of dipole-dipole interaction (Fo¨rster mechanism). A spectral overlap between the donor-acceptor pairs is critical for both processes. Nonradiative transfer is also called resonance energy transfer (RET), as it is based on the resonant coupling between donor-acceptor transitions. It has widely been used to probe changes occurring at distances of a few tens of angstroms in biomolecules and supramolecular assemblies.22 In this study, we have observed fluorescence from HPC solutions containing Rhodamine 6G (RG) as an energy donor and Kiton Red 620 (KR) as an acceptor. When the temperature is elevated and the solution becomes turbid, the energy transfer from RG to KR and the amplified spontaneous emission (ASE) of KR are both enhanced. We have analyzed the result to elucidate the enhancement quantitatively. Our approach is based on the simplified photon diffusion picture, which is fairly tractable for the experimental data analysis. It should be noted that, as ASE is (at least partially) a coherent process, an extensive onset of ASE will break down the photon diffusion picture; we thus restrict our analysis to the data where ASE is at most weak and discuss ASE in a rather qualitative manner. 2. Experimental Methods Commercially available hydroxypropyl cellulose (HPC) with an average molecular weight of ca. 370 000 (Aldrich) was used without further purification. Fluorescent dyes, Rhodamine 6G (RG) and Kiton Red 620 ()sulforhodamine B; KR) (Exciton), were added to aqueous HPC solutions of 3.3 g/L - 26.4 g/L. The dye concentrations were kept low (7.3 × 10-6 M) in order to avoid dimer formation and aggregation. During laser-induced fluorescence measurements, the temperature of the solution was controlled with a precision of (0.02 K by using a thermoelectric (Peltier) cuvette holder (Quantum Northwest TLC 50F). Absorption spectra were obtained with a Shimadzu UV-160 spectrophotometer. All of the solutions were prepared with highly deionized water from a reverse-osmosis-electrodeionization water purifier (Millipore Elix-3). The excitation source was a frequency-doubled, Q-switched Nd:yttrium-aluminum-garnet (YAG) laser (λ ) 532 nm) with the pulse duration of 5 ns and the pulse energy of 75 mJ (Continuum Surelite I-10). The collimated beam has a nearGaussian spatial profile with the spot size (radius) of 1.1 mm. The backscattering fluorescence was collected by an optical fiber that was closely aimed at the front (illuminated) surface of the 1-cm glass sample cell, in the nearly antiparallel direction to the incident excitation beam (θ ∼ 175°). This backscattering geometry was chosen because of the high turbidity of the HPC solutions above the phase-transition temperature. An optical notch filter (CVI Optics RNF-532.0) was used to eliminate the excitation wavelength from the observed light. Strong coherent backscattering caused by weak photon localization of the incident light, if any, is usually observed within a very narrow cone of the antiparallel direction (θ > 179°)23 and is expected to have little effect in our fluorescence measurements. The fiber was connected to a CCD spectrometer (Ocean Optics CHEM2000) for recording time-integrated fluorescence spectra. For timeresolved, single-wavelength emission measurements, a monochromator (Optometrics DMC1-03), a high-speed photomultiplier module (Hamamatsu H6780-04), and an oscilloscope (Tektronix TDS3032) were used. The overall time resolution of the detection system was limited by the time constants of the photomultiplier (0.65 ns) and the oscilloscope (300 MHz,
Lee and Suzuki
Figure 1. Absorption and fluorescence spectra of Rhodamine 6G (RG) and Kiton Red 620 (KR) aqueous solutions: solid (left), RG absorption; solid (right), KG absorption; dashed, RG fluorescence.
2.5 G samples/s) and estimated to be ca. 1.2 ns, based on the observation of a single-bubble sonoluminescence signal which is known to have an extremely short pulse duration. Both the CCD spectrometer and the oscilloscope were triggered with a delay generator (Stanford Research Systems DG535) which was synchronized with the laser. A home-built electronic circuit with ultrahigh-speed logic ICs was used to enable single-shot laser triggering during the continuous operation of the YAG flashlamp excitation at 10 Hz. It monitored the flashlamp output and fired a single Q-switch trigger at a precisely controlled delay time. In this manner, the influence of jitters in firing the flashlamp was eliminated, and the thermal lensing effect in the laser rod was stabilized, so that the drift and jitter of the output laser pulses were minimal. 3. Results and Discussion 3.1. Emission from HPC Solutions. The efficiency of energy transfer, either radiative or nonradiative, depends on spectral overlaps between the donor emission and the acceptor absorption. Figure 1 shows the absorption and fluorescence spectra of Rhodamine 6G (RG) and Kiton Red 620 (KR) aqueous solutions. The extensive overlapping between the RG fluorescence and the KR absorption over 530-580 nm has a strong potential for intermolecular energy transfer. For these dyes, the peak positions and band shapes remain practically intact in HPC solutions of varied concentrations up to 26.4 g/L; that is, there is little solvent effect, at least below LCST. Also, the absorption spectra of the RG-KR dye mixture in aqueous and HPC solutions show simple additivity; there is no indication of strong intermolecular interactions such as donor-acceptor complex formation, hydrophobic stacking, or aggregation. The absence of strong interactions may not be surprising, if we consider the low concentration of the dyes as well as the amphiphilicity (and thus the strong hydration) of both HPC and the ionic dyes. The reported formation constant of RG dimers (1.7 × 103 L mol-1 in water at 22 °C)24 indicates that 99% of RG dyes stay as monomers in water at the concentration of our experiments; monomers are further favored in aqueous surfactant solutions and water-organic solvent mixtures.25 Fluorescence of RG in HPC solutions is strikingly dependent on temperature. Figure 2a shows backscattering emission spectra of RG in HPC solution (26.4 g/L) at temperatures below and above LCST, which demonstrates the strong intensity enhancement and spectral narrowing in the course of phase transition; in aqueous RG solutions without HPC, no such strong dependence is observed within the same temperature range (data not shown). The peak intensity (Figure 2b) starts growing slightly below LCST and keeps increasing up to 50 °C, which coincides
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J. Phys. Chem. B, Vol. 112, No. 15, 2008 4563
Figure 3. Emission spectra of the RG-KR mixture in HPC solution (26.4 g/L) at temperatures: 35.0, 37.5, 40.0, 42.5, 45.0, 47.5, and 50.0 °C (bottom to top).
Figure 2. Temperature-dependent RG fluorescence in HPC solution (26.4 g/L): (a) backscattering fluorescence spectra, (b) the peak intensity, and (c) the peak position (filled circles; to the left axis) and the spectral width (FWHM) (open circles; to the right axis), as a function of the temperature (a: from bottom to top, 35.0, 37.5, 40.0, 42.5, 45.0, 47.5, and 50.0 °C).
with the emergence of milky-white colloid formation. It is noted, however, that the observed change is not just due to a simple optical scattering effect, because the band shape becomes narrower and the peak position (Figure 2c) is red-shifted above LCST. Similar changes have been reported in HPC solution with a high KR concentration, which are an indication of the onset of amplified spontaneous emission (ASE).19 ASE is the most common form of mirrorless laser behavior, with output characteristics being intermediate between a true coherent laser and an incoherent thermal source. It is often observed in very high-gain dye laser amplifiers, in which the spontaneous emission from a distribution of inverted laser chromophores is amplified along a long cylinder of the laser medium.26 In the case of our HPC solution above LCST, the extensive formation of HPC nanoparticles, which have a size comparable to the excitation wavelength, makes the solution strongly scattering. This random medium acts as an ensemble of microscopic mirrors. The multiple scattering between them results in a countless collection of very long optical paths, leading to the ASE buildup. The spectral narrowing due to the finite bandwidth of the gain medium is also one of the well-
known characteristics of ASE. Above 50 °C, ASE degrades, possibly owing to the aggregation of HPC nanoparticles. It should also be noted that the formation of HPC nanoparticles does not always lead to ASE; it sometimes counteracts and prevents ASE, depending on the dye concentration and the pumping power of the excitation laser pulses.19 Emission spectra from the RG-KR mixture in HPC solution depends on temperature in a more complicated way (Figure 3). The intensity enhancement of the RG fluorescence (560 nm) is rather suppressed, and its spectral narrowing is insignificant; instead, a new emission band at 590 nm builds up with moderate narrowing. The origin of the latter band is mostly KR fluorescence, with an additional contribution from the longerwavelength tail of the RG fluorescence. The emission band shape at shorter wavelengths (610 nm) where the self-absorption is absent, I′em almost coincides with the spectrum below LCST after the correction (RIex) (curve c). It is significant because there is no adjustable scaling factor in the correction f. This has prompted us to analyze the spectrum in terms of the following equation:
I′em )
RIex ≡ f′RIex 1 + µem/µ′ex
(2)
which is an optically thick version of eq 1 and has the desired limiting behavior (I′em ∼ RIex if µem ∼ 0); µ′ex is now considered to be an adjustable parameter. This equation requires a linear relationship between µem and (RIex/I′em) with the intercept of unity, which is indeed the case (Figure 4, inset). The least-square fitting gives the intercept of 1.02 and the correlation coefficient of 0.99, indicating that it is hardly accidental. After the correction (I′em/f′), the spectrum (curve d) reproduces the intrinsic one. It is noted that, when µem/µ′ex is small, eq 2 approaches a familiar form of exponential decay, f′ ) exp(-µem/µ′ex); however, µem/µ′ex spans from 0 to 1.4 in our data, and eq 2 is found to give a better fit than the exponential model. It is surprising that eq 2 can describe the observed fluorescence above LCST so well. Because the extensive multiple scattering obviously dominates the optical property of the turbid solution, the applicability of eqs 1 or 2 is, at most, highly questionable. Although the solution is now optically thick, with the scattering coefficient µs > 50 cm-1 (see below), it is not the scattering but the absorption that counts in fluorescence emission, and simple replacement of the absorption coefficients with the scattering ones, of course, cannot be justified. Also, the numerical fitting value of µ′ex ) 1.44 does not coincide with the absorption coefficient (µex ) 0.74 cm-1) below LCST. Accordingly, eq 2 apparently suggests that, above LCST, the intensity leakage from the rear surface is suddenly turned off, with µex doubled and µem intact. We need a theory that can explain eq 2 for turbid media.
ET and ASE in Random Scattering Media 3.3. Multiple Scattering Effect. Optical scattering in turbid media is an important topic in photonics, astronomy, and medical applications, and there are a number of existing theories.14 The most rigorous form of multiple scattering theory is usually too difficult to be applied to practical problems, and alternative approaches called radiative transfer and photon transport theories have been developed. Although such approaches usually ignore interference and thus cannot handle, for example, coherent backscattering (weak localization), they have been proven to be successful in many random media where the photon diffusion picture is an adequate approximation. Among the diffusion theories with various extents of rigor and applicability, the simplest approach is probably Kubelka-Munk (KM) theory, which is well-known and widely used in reflectance spectroscopy29 and the paint industry.30 The phenomenological KM theory assumes photon fluxes in the incident and opposite directions and solves a system of two differential equations, each representing one-dimensional photon transport with absorption and scattering into the other direction. In our case of fluorescence due to collimated illumination, a suitable KM formulation is the so-called four-flux theory, in which both the excitation and fluorescence fluxes are considered. The result for the optically thick condition is
I′em ) RIexkex
J. Phys. Chem. B, Vol. 112, No. 15, 2008 4565
Figure 5. Angular distribution (relative scale) of the 532-nm light scattered by a single HPC sphere in water: the scattered irradiance is shown as a polar plot for the incident light polarized parallel (ipara) and perpendicular (iperp) to the scattering plane.
2βex 1 1 1 + βex 1 + βem κex + κem (κ ≡ [k(k + s)]1/2, β ≡ [k/(k + 2s)]1/2) (3)
where k and s are empirical parameters representing “effective” absorption and scattering in the model and usually taken to be twice the value of the corresponding coefficient (µ).29 This factor of 2 comes from directional averaging over the angular distribution of the flux under the assumption of ideal (isotropic) diffusion. A close examination of eq 3, however, reveals that it is not compatible with our experimental result. Specifically, (i) it does not give the asymptotic behavior (I′em ∼ RIex if µem ∼ 0) and (ii) under the dominant scattering condition (k , s), βem , 1 and κem ∼ (kem sem)1/2, so that it predicts a linear relationship between (µem)1/2 and (RIex/I′em), which is excluded on the basis of a log(µem) vs log(RIex/I′em - 1) analysis giving a slope of 0.98. An attempt to fit the data with the square-root model also indicates large deviations (Figure 4, Inset, dashed curve). The KM theory is known to work well in the Rayleigh limit (isotropic scattering), i.e., in the case of scatterers that are small compared with the optical wavelength, but often found to be inadequate where scatterers are larger and the scattering is highly anisotropic. Therefore, the major cause of the above failure may be traced back to the anisotropic scattering. In our solution above LCST, the typical radius (r) of the HPC nanoparticles is ca. 270 nm.18 If we assume the refractive index (n) of the nanoparticles to be the mesophase average value (1.43),31 the corresponding size parameter x ) 2πnr/λ is 4.56 at λ ) 532 nm and 4.1 at λ ) 590 nm, which indicates that the particles are in fact large enough to cause the KM approach to break down. Angular distribution of the light scattered by a single spherical HPC particle in water is calculated by the standard Mie theory32 and shown as a polar plot in Figure 5. It is clearly seen that the scattering is strongly peaked in the forward direction as expected; for r ) 270 nm, the backscattering at θ ) 180° is less than 0.1% of the forward scattering (θ ) 0°). This leads us to the following intuitive picture: since the photons scattered by a single HPC particle are mostly falling into the forward direction, the scattering is virtually transparent and has little
Figure 6. Photon diffusion picture of multiple scattering in a random medium, which consists of HPC particles (circles) embedded in water. Highly anisotropic (forward) scattering is depicted as arrows. Four length scales (mean free paths) that characterize the photon diffusion are also schematically shown: scattering length (lsca), transport length (ltr), inelastic length (linel), and absorption length (Labs).
effect in small length scales (µm); the excitation beam thus penetrates more deeply into the solution than expected from the conventional scattering coefficient (µs). On the other hand, numerous multiple scatterings in large length scales (cm) effectively hinder the photons to reach the rear face of the sample, preventing the intensity leakage. Multiple scattering in random media is conveniently characterized by several length scales, i.e., photon mean free paths (Figure 6).33 Suppose that the medium is strongly scattering but weakly absorbing (µs . µa). When a collimated incident beam penetrates into the medium, it is gradually removed from the incident direction and converted into scattered photons (called diffuse intensity). The scattering length (lsca) characterizes the exponential decay of the coherent beam. It is usually equal to the inverse of the scattering coefficient µs ) NCsca, where N is the number density of scatterers and Csca the scattering cross section of a single scatterer. (Note that it is different from the actual interparticle distance.) If the scattering is anisotropic, as is the present case, it takes more than a few scattering events for the direction of photons to be fully randomized. The transport length (ltr) is thus defined as a distance by which a scattered photon loses its directional memory. It is the inverse of the reduced scattering coefficient, µ′s ) (1 - g)µs [i.e., ltr ) lsca/(1 - g)], where g ) 〈cos θ〉 is the asymmetry parameter (average cosine of the scattering angle).32 In the case of isotropic
4566 J. Phys. Chem. B, Vol. 112, No. 15, 2008 scattering (g ) 0), the two lengths coincide; however, if the scattering is highly anisotropic in the forward direction (g ∼ 1), ltr can be very long. It is known that the diffuse intensity can be described by photon diffusion theory and the Boltzmann equation.13 Notably, the total diffuse transmittance through a slab of nonabsorbing scatterers does not decay exponentially but follows Ohm’s law, i.e., it is inversely proportional to the slab thickness.34 After many scattering events, absorption becomes significant. The diffuse intensity decays exponentially, now owing to absorption. The inelastic (or bulk absorption) length, linel ≡ µa-1, characterizes the absorptive decay; however, it proceeds not along a straight line but following a zigzag path. If linel . ltr, this random walk can be described by a diffusion process (Brownian motion), so that the (diffuse) absorption length, Labs ≡ (linel ltr/3)1/2, can be defined as the average (rootmean-square) begining-to-end distance (i.e., penetration depth)35 with which the diffuse intensity decays down by a factor of e-1. These length scales are useful in understanding the optical properties of our HPC solution above LCST. The scattering length (for cHPC ) 26.4 g/L) at λ ) 532 nm has been obtained by assuming an exponential attenuation of line-of-sight (coherent) transmittance T in a thin cuvette with the path length d: lsca ) [-ln(T)/d]-1 ) 170 µm. If we use the Mie scattering cross section of a single HPC sphere with the radius of 270 nm (Csca ) 3.7 × 10-2 µm2), the number density of HPC nanoparticles is estimated to be N ∼ 0.16 µm-3. This number is in reasonable agreement with the value (0.24 µm-3) calculated from the polymer concentration and the aggregation number in a 270-nm nanoparticle (naggr ∼ 1.8 × 105 molecules/particle).18 It is noted that, since HPC nanoparticles are weak scatterers in water, lsca is much longer than the average interparticle distance (1.8 µm). The highly anisotropic scattering (g ) 0.88 calculated from Mie theory) makes the transport length even longer: ltr ) 1.4 mm. If short-range interparticle correlations exist, ltr tends to be even longer, but the diffusion approximation is still applicable with modification by the interparticle structure factor.36 HPC is also a weak absorber in the visible, and the absorption is dominated by the dye chromophores in our solution. The inelastic and absorption lengths are calculated as: linel ) 4.8 mm and Labs ) 1.5 mm for the RG absorption and linel ) 13 mm and Labs ) 2.5 mm for the KR absorption. It is evident that multiple scattering makes the penetration depth (Labs) much shorter than that in a transparent solution (linel); it qualitatively explains the shutting down of the rear-side leakage above LCST. On the other hand, it still fails to be quantitatively consistent with eq 2: if we tried to use Labs-1 simply as an effective absorption coefficient, the (µem)1/2 dependence of (RIex/I′em) would again show up, which is not the case. This is because ltr ∼ Labs, and thus, the Brownian motion picture is an oversimplification in our HPC solution. The above argument can be quantitatively refined by resorting to recent, more elaborate theories on multiple scattering in biological tissue, where the scattering is both extensive and highly anisotropic. According to the photon migration theory,37 light propagation in turbid media is described with randomwalking photons, which travel in paths with discrete photonparticle interaction events of absorption, scattering, and fluorescence emission. By counting all of the escaping probabilities with the corresponding weights (photon survival fractions), the backscattering fluorescence in the strongly scattering limit is shown to be proportional to the product of the intrinsic
Lee and Suzuki fluorescence (RIex) and the effective diffuse reflectance at the emission wavelength (R′em)
I′em ) qRIexR′em
(4)
where q is the proportionality constant that depends on the excitation wavelength. When the dependence of scattering on the wavelength is weak, R′em is close to the total diffuse reflectance Rem, which is in turn described by
Rem )
1 1 + Bµem/µ′s
(5)
where B is a constant to be determined by experiments or Monte Carlo simulations; a typical value is 14.38 This expression for the diffuse reflectance is generally in good agreement with tissue experiments, Monte Carlo simulations, the path-integral model,39 and the radiative transport theory.40 It is easily seen that eq 4 combined with eq 5 is equivalent to our experimental result eq 2, provided that µ′s/B ) µ′ex and q ) 1. The former condition suggests µ′s ∼ 20 cm-1, and also µs ∼ 170 cm-1, if we take g ) 0.88. This value of µs is close to, but somewhat larger than, the experimental scattering coefficient (60 cm-1). On the other hand, whether the latter condition q ) 1 is satisfied or not in our case is unclear and should be left as an open question. In our treatment, q is defined as a relative value compared with the non-scattering, transparent solution in a cuvette (with refractive index mismatch at air-glass and glasssolution interfaces), so that comparison with the tissue theory would need a careful examination. One version of the theory37 seems to suggest q ∼ (1 - Rex), where Rex is the diffuse reflectance of the excitation beam due to scattering; hence q ∼ 1 means that the leakage of excitation photons scattered away from the front side is significantly reduced via some additional mechanism, such as strong coherence or highly forward-directed scattering near the incident surface, which might offer a possible explanation for our result. In any case, we have reached a heuristic interpretation of eq 2 with the aid of the tissue theory: the incident excitation photons penetrate into the medium, get scattered multiple times, and most of the light are effectively absorbed inside it. The “effective” penetration depth, (µ′s/B)-1, seems insensitive to the absorption coefficient at the excitation wavelength and mostly determined by the scattering coefficient; it is ca. 7 mm for our HPC solution. The fluorescent emission comes from that depth, on average, and is characterized by the albedo (scattering-to-total-extinction ratio): f′ in eq 2. 3.4. Enhancement of Energy Transfer Process in Random Media. We are now at the position to analyze the emission from the equimolar mixture of RG and KR in HPC solution (Figure 3). We focus on the 590-nm band that is mostly KR emission. In order to separate contributions from the RG fluorescence tail, the KR direct excitation, and the RG-to-KR energy transfer, the following assumptions are made: (i) At our excitation fluence, the absorption by RG is completely saturated, whereas that by KR is unsaturated and can be described with the conventional (small-signal) absorption coefficient. (ii) The emission of RG has the same band shape as that of the intrinsic fluorescence, i.e., ASE is not operative for RG in the mixture. (iii) In the turbid solution above LCST, the absorption correction is described by eq 2 for both RG and KR. On the basis of (i), the self-absorption correction for RG in the transparent solution is described by a modified form of eq 1
ET and ASE in Random Scattering Media
Iem )
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Rµex,RGIsat,RG {1 - exp(-µemL)} ≡ f′′RIsat,RG (6) µem
in which the absorbed energy density (µexIex) in eq 1 is replaced with the saturation limit of RG (µex,RGIsat,RG); µex,RG is the smallsignal absorption coefficient of RG and Isat,RG the saturation intensity. Attenuation of the excitation beam does not appear in this equation because it is uniformly absorbed regardless of the penetration depth. Attenuation of emission is represented by the absorption coefficient (µem). The choice of this parameter is a subtle question: if RG is still at the S1 state during its encounter with emitted photons, the attenuation is caused by KR only (µem ) µem,KR); otherwise, the attenuation is caused by both RG and KR (µem ) µem,RG + µem,KR). In practice, however, the RG absorption is small at longer wavelengths (>560 nm), and this choice hardly affects our result. The absorption saturation in the mixture is schematically shown in Figure 7, where the absorbed energy density is represented by rectangular areas, i.e., products of µex (horizontal side) and Iex (vertical side). The corresponding fluorescence intensity (ΦµexIex where Φ is the quantum yield) is also depicted by dark-gray areas. It is seen that, in spite of the saturation, the higher absorption coefficient and the higher fluorescence quantum yield of RG (Φ ) 0.89)41 can result in a fluorescence intensity comparable with that of KR. Since the saturation behavior is a soft transition without a sharp threshold, this scheme is an oversimplification. Nevertheless, it is expected to give a reasonable description of our dye mixture; it allows us to simplify the analysis tremendously and avoid complication of, for example, simulation by partial differential equations that describe local excitation populations. With the assumptions (i) and (ii), we can evaluate the contribution from the RG fluorescence tail below LCST by simple extrapolation. We start with the intrinsic RG fluorescence spectrum, and deform it by multiplying the wavelengthdependent self-absorption factor (f′′) in eq 6; finally, the deformed spectrum is scaled so that the intensity coincides with the emission from the mixture at wavelengths where RG is the only contributor (550-560 nm). The contribution at 590 nm is then calculated. We can also check the consistency of our analysis by comparing the emission from the mixture with that from RG solution within the wavelength region of RG emission (550560 nm; Figure 8). After the appropriate correction for each, they both should represent the intrinsic RG emission (RIsat,RG) and thus coincide, except for the possible reduction in the mixture if nonradiative energy transfer from RG to KR is operative. It turns out that RIsat,RG in the mixture is actually not lower but similar to (555 nm) than that in the RG solution. This indicates the limitation and the extent of errors in our estimation of the RG fluorescence tail. It also implies that contribution from nonradiative ET is negligible, at least below LCST, which is consistent with the low concentration of the dyes in our solution and the sixth-power dependence of the nonradiative transfer efficiency on the donoracceptor distance.21 Evaluation of the contribution from the directly excited KR fluorescence below LCST is straightforward. Because of the saturation, RG is practically “transparent” and does not contribute to absorption, so that we can use eq 1 with the absorption coefficients for KR. This is exactly the same expression for the KR solution. The intrinsic KR fluorescence (RIex, Figure 4) should also be the same, as we have used the same optical geometry and detection system. Accordingly, we can simply
Figure 7. Comparison of saturation of absorption by RG and KR (not to scale). The vertical and horizontal axes represent intensities and absorption coefficients, respectively. The effects of relative absorption coefficients (µem,RG ∼ 3µem,KR) and the saturation of Iex (for RG) on the absorbed energy density (light and dark gray areas) and the fluorescence intensity (dark gray area) are shown. Note that Iex for KR is a function of the penetration depth (z).
Figure 8. Self-absorption correction of the emission spectra in HPC solutions below LCST in the range 550-560 nm. Solid line, observed from the RG solution [Iem(RG)]; dashed line, corrected [Icorr(RG) ) Iem/f′′]; dotted line, observed from the RG-KR mixture [Iem(mix)]; dashdot-dash, corrected [Icorr(mix)].
take the emission intensity from the KR solution as the contribution from direct excitation of KR in the mixture. The assumption (iii) is for the analysis above LCST. We follow the same procedure as above, except that we use eq 2 for the absorption correction. The extrapolation for the evaluation of the RG tail also requires assumption (ii), the validity of which can be questioned here; the onset of ASE (560 nm) is obvious for the RG solution above LCST (Figure 2). It is not as prominent in the mixture, being obscured by the strong KR absorption (Figure 3). The ASE above LCST in the mixture is certainly less than that in the RG solution, but it is possible that the band shape of the RG emission assumes narrowing to some extent. Therefore, the RG tail estimation based on the assumption (ii) should be considered as the upper bound; it can be smaller if the ASE narrowing occurs. The result of our analysis is summarized in Figure 9. The contributions from the RG tail and the KR direct excitation are approximately twice as large above LCST as those below LCST, which is caused by the difference in the absorption correction factor (f and f′), and mainly due to the shutting down of the rear-side intensity leakage above LCST. In contrast, the residual
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Figure 9. Emission intensity at 590 nm from the RG-KR mixture in HPC solution below and above LCST. The estimated contributions from RG fluorescence tail, KR direct excitation, and RG-to-KR energy transfer (which may also include ASE effect) are shown.
intensity (ET) is enhanced by a factor of 5 above LCST. This intensity comes from the energy transfer from excited RG to KR chromophores and may also contain contribution from ASE of KR above LCST, which becomes evident in the emission spectra of the mixture at higher temperatures (Figure 3). The fact that the KR-only solution does not show ASE, either below or above LCST, and that ASE of RG is comparatively suppressed in the RG-KR mixture both imply the key role of the RG-to-KR energy transfer in promoting ASE for KR. Now the question is: how can the HPC nanoparticles enhance the energy transfer and ASE? 3.5. Energy Transfer Mechanism above LCST. In the transparent solution below LCST, there is no indication of strong RG-KR interaction or nonradiative energy transfer (ET), and all the observed ET contribution is ascribed to “trivial” radiative transfer, which is the same thing as the inner-filter effect of KR on RG fluorescence. On the other hand, one may argue that the situation can be different above LCST: because of the onset of microscopic inhomogeneity, RG and KR chromophores might come close to each other by, e.g., adsorption to HPC nanoparticles or higher partition into hydrophobic phases.42 To examine the possibility of nonradiative ET above LCST, we have measured the lifetime of emission from RG and from KR (Figure 10). Generally speaking, nonradiative transfer shortens the decay of donor emission, but radiative transfer leaves it unchanged. We have used 5-ns excitation pulses and a detection system with the time resolution of ∼1 ns and fitted the temporal emission profiles with a single-exponential decay without deconvolution. Accordingly, the results must be considered to be of qualitative nature; thus, whereas the measured lifetime of RG emission in water is in good agreement with the literature value (5.5 ns),43 that of KR emission in KR-HPC solutions is likely to be limited by the excitation pulse duration and the instrumental response, being longer than the reported value in aqueous KR solution (1.2 ns).44 Nevertheless, we can still see a distinct trend: the RG lifetime is similar in the RG solution and in the mixture, i.e., KR has little effect on it, at each temperature. This suggests an absence of nonradiative ET both below and above LCST. The significant difference between the two temperatures is possibly due to the viscosity effect of the solution. Other factors that are likely to play a role, such as pulse shortening effect of ASE26 and delay caused by multiple scattering,45 are not clear in our data. On the other hand, the KR lifetime shows a contrasting trend; the temperature dependence is small, but RG makes the KR lifetime substantially longer. It can be explained in terms of radiative ET, by which photons are supplied to KR with a delay determined by the RG
Figure 10. Emission lifetime of RG at 550 nm (upper figure) and KR at 590 nm (lower figure) as a function of HPC concentration (dye concentration ) 1.5 × 10-8 M). Upper figure: solid-square, RG at 35 °C; solid-circle, RG at 45 °C; open-square, RG-KR mixture at 35 °C; open-circle, RG-KR mixture at 45 °C. Lower figure: solid line (lower), KR at 35 °C; broken line (lower), RG at 45 °C; solid line (upper), RG-KR mixture at 35 °C; broken line (upper), RG-KR mixture at 45 °C.
lifetime. The insensitivity of the KR lifetime on temperature also suggests that the local environment of KR is intact above LCST. As nonradiative ET is also excluded above LCST, we are left with radiative ET as the only possibility. It is, however, also unlikely that the simple inner-filter effect could account for all of the enhancement. Our analysis indicates that the RG emission at 560 nm is attenuated roughly down to the half by the KR absorption, both below and above LCST, which is short of explaining the observed factor-of-5 enhancement. A possibility of the quantum electrodynamical (cavity field) effect can be excluded, because of the interparticle distance and the transport mean free path that are much longer than the optical wavelength. Accordingly, ASE of RG seems essential in supplying extra optical energy to KR via radiative ET. We infer the following mechanism: above LCST, HPC nanoparticles induce ASE of RG, which enables RG to absorb and radiate photons rapidly via stimulated processes. It thus transfers more energy to KR, beyond its ordinary saturation limit. It then eventually causes KR to build up its own ASE in the RG-KR mixture. 3.6. Amplified Spontaneous Emission in Random Media. Now we turn to several factors that may be important for the onset of ASE in our random media. The intensity of ASE through a cylindrical gain medium depends on the population inversion, the cross section, and the medium length (L) in an exponential form: IASE ∼ exp(∆N σem L), where σem is the stimulated emission cross section.26 In the case of laser dyes in solution, vibrational relaxation within the S0 and S1 manifolds is so fast that the inverted population density (∆N) can be set equal to the S1 population density (N1).25 When one or more of
ET and ASE in Random Scattering Media
J. Phys. Chem. B, Vol. 112, No. 15, 2008 4569
the parameters (N1, σem, and L) become sufficiently large, ASE will grow exponentially. Because ASE is a nonlinear process with a soft threshold, a quantitative treatment is rather difficult; we only attempt a 1-order-of-magnitude estimate for the onset condition here. According to the standard four-level population model, the relationship between N1 and the pump intensity (Iex) is given by26
N1/Ntotal )
ηWpτrad 1 + ηWpτrad
(7)
where η is the quantum efficiency and τrad the radiative decay time; Wp ) σexIex/hν is the pump transition probability, with σex being the corresponding absorption cross section. For our experimental condition at 532 nm, ηWpτrad ∼ 2 × 103 for RG, which is consistent with the strongly saturated behavior. Therefore, we may replace N1 with the total number density of the chromophores. With a typical value of σem ∼ 1 × 10-16 cm2, the threshold conditon (N1 σem Lth ∼ 1) gives a path length that is necessary for the onset of ASE: Lth ∼ 2 cm. Thus, the RG solution is very close to the threshold. In the case of KR, however, the lower quantum yield, the smaller absorption cross section, and the shorter excited-state lifetime reduce the product (ηWpτrad < 80), which may decrease further if the loss due to excited-state absorption is taken into account. It suggests that the KR solution is below saturation, and N1 can be considerably less; σem is also reduced by the low quantum yield (η < 0.5). Accordingly, Lth is longer, making it more difficult to achieve the threshold. In the HPC solution above LCST, an optical path length can become much longer than the actual cavity (cuvette) length by multiple scattering. This is probably the major cause for the ASE of RG, where absorption is saturated and N1 is already large enough; a longer L can easily trigger ASE. The spatial coherence is (of course) no good in random media, but once an acceptable path is established, ASE starts up. Once it happens, however, the stimulated process tends to shorten τrad and thus to decrease N1 again, so that there is a subtle balance between ASE and the population. This sort of ASE in strongly scattering gain media is often observed as a precursor to the random laser action.46,47 It is also noted that the condition for efficient pumping over scattering loss (linel/ltr < 100) in amplifying random media48 is satisfied in our solution. On the other hand, KR may still have difficulty: it is below the saturation, and thus has a smaller N1 and also a smaller σem. Therefore, in the case of KR, a longer L caused by scattering is not a sufficient condition for ASE, and scattering may even inhibit ASE by intensity attenuation. Indeed, a subtle effect of scattering on ASE has been observed in HPC solutions containing a higher concentration of KR, where both emergence and disappearance of ASE are reported above LCST, depending on the experimental condition.19 Accordingly, in our dilute solution, strong up-pumping via the enhanced energy transfer from RG is essential for KR to build up its ASE. This mechanism also suggests feasibility of random mixed-dye lasers, in which the external pumping light can be efficiently converted to the emission of a largely different wavelength via energy transfer between chromophores. Another factor that also deserves attention is the optical nearfield effect. When a nanoparticle scatters light, it can also act as a sort of microscopic lens that deforms the electromagnetic field and concentrates the optical energy density in its vicinity. This effect is usually ignored in photon diffusion theory of
Figure 11. Near-field intensity around a HPC sphere (r ) 270 nm) in water calculated by Mie theory. Upper left: the profile of normalized intensity along the incident direction (z axis). Upper right: 2D map of the intensity field. Lower: 3D map. The span corresponds to the interparticle distance (1.8 µm).
nonabsorbing scatterers but can be important in mixed systems that are composed of scatterers and chromophores in which they can come close to each other. Our HPC solution typically has one nanoparticle and 1.8 × 104 dye molecules in every 4 µm3; a volume that is equal to the nanoparticle volume would contain ca. 400 dye molecules. These dye molecules expose themselves to the near-field of the scatterer. The near-field intensity around a typical HPC sphere in water is calculated by Mie theory (Figure 11).32,49 The electric field (absolute squared) normalized by the incident intensity is plotted in the figure for a span that corresponds to the average interparticle distance (1.8 µm). It can be seen that the intensity is nearly doubled near the rear surface of the sphere. It also extends into the surrounding external volume on the rear side and further reaches the limit of the span. This suggests that the high-intensity volumes can connect to each other to form a long path. Within such a path, N1 may greatly be enhanced and, together with the long L, leads to ASE more easily. In this manner, the forward-directed anisotropic scattering is advantageous in building up “light pipes” (or virtual waveguides) for ASE. They are especially relevant for cases where the interparticle distance matches the extent of the high-intensity nearfield, and for processes in which the optical path length is crucial, e.g., ASE and random lasers. Conclusions We have successfully controlled the optical environment of the scattering media by adjusting the temperature to observe the enhancement of energy transfer (ET) and amplified spontaneous emission (ASE). The detailed analysis has revealed the important role of photon diffusion in shutting down the intensity leakage, and a 5-fold enhancement of ET is estimated. Possible factors that are crucial for ET and ASE in random media, such as the donor-to-acceptor ASE energy pumping, the optical path elongation by multiple scattering, and the formation of light pipes in the near-field of the Mie scattering, are discussed. Temperature-dependent colloidal formation in polymer solutions has advantages in controlling multiple scattering easily with a sharp threshold and in realizing a relatively high density of scatterers (and thus a short interparticle distance). An externally controllable “Sunny Side of the Street” in the near-field may
4570 J. Phys. Chem. B, Vol. 112, No. 15, 2008 be an attractive concept for optical, medical, and engineering applications, such as temperature-sensitive micro dye lasers. Acknowledgment. This work is supported by the Japanese Ministry of Education, Science and Culture (Grant Nos. 15750115 and 17510086) and by the University of Toyama (Grant for Young Researchers 2007). We thank Prof. Koichi Nozaki for discussion on fluorescence lifetime measurements. References and Notes (1) Zare, R. N. Science 1998, 279, 1875. (2) Martorell, J.; Lawandy, N. M. Phys. ReV. Lett. 1990, 65, 1877. (3) Hopmeier, M.; Guss, W.; Deussen, M.; Go¨bel, E. O.; Mahrt, R. F. Phys. ReV. Lett. 1999, 82, 4118. (4) Andrew, P.; Barnes, W. L. Science 2000, 290, 785. (5) Lidzey, D. G.; Bradley, D. D. C.; Armitage, A.; Walker, S.; Skolnick, M. S. Science 2000, 288, 1620. (6) Arnold, S.; Holler, S.; Druger, S. D. J. Chem. Phys. 1996, 104, 7741. (7) Wiersma, D. S.; Bartolini, P.; Lagendijk, A.; Righini, R. Nature 1997, 390, 671. (8) John, S. Phys. ReV. Lett. 1987, 58, 2486. (9) Lawandy, N. M.; Balachandran, R. M.; Gomes, A. S. L.; Sauvain, E. Nature 1994, 368, 436. (10) Noginov, M. A.; Egarievwe, S. U.; Noginova, N.; Wang, J. C.; Caulfield, H. J. J. Opt. Soc. Am. B 1998, 15, 2854. (11) Rojas-Ochoa, L. F.; Mendez-Alcaraz, J. M.; Saenz, J. J.; Schurtenberger, P.; Scheffold, F. Phys. ReV. Lett. 2004, 93, 073903. (12) Siddique, M.; Yang, L.; Wang, Q. Z.; Alfano, R. R. Opt. Commun. 1995, 117, 475. (13) Lagendijk, A.; van Tiggelen, B. A. Phys. Rep. 1996, 270, 143. (14) Ishimaru, A. WaVe Propagation and Scattering in Random Media; Wiley-IEEE Press: New York, 1999. (15) Zhang, Q.; Mu¨ller, M. G.; Wu, J.; Feld, M. S. Opt. Lett. 2000, 25, 1451. (16) Mu¨ller, M. G.; Georgakoudi, I.; Zhang, Q.; Wu, J.; Feld, M. S. Appl. Opt. 2001, 40, 4633. (17) Lemieux, P.; Vera, M. U.; Durian, D. J. Phys. ReV. E 1998, 57, 4498. (18) Gao, J.; Haidar, G.; Lu, X.; Hu, Z. Macromolecules 2001, 34, 2242. (19) Lee, K.; Lawandy, N. M. Opt. Commun. 2002, 203, 169.
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