We describe a simple and efficient means of using a white LED source to illuminate an optically levitated aerosol droplet to enable study of broadband Mie scattering. The backscattered resonances are imaged through a spectrograph and CCD which show high resolution Mie scattering intensity distributions across a spectral range of 480 to 700 nm. The wide spectral range allows assignment of resonance mode numbers and mode orders using conventional Mie theory calculations. Accurate droplet sizing, within ±2 nm, is possible for water–based droplets with radii between 2 µm and 8 µm. We additionally demonstrate that the refractive index dispersion can be determined from a single refractive index value at known wavelength. Finally, morphological droplet dynamics are presented showing non-linear droplet evaporation behaviour at a temporal resolution of 100 milliseconds.
© 2008 Optical Society of America
Understanding the behaviour of aerosols underpins a number of scientific disciplines including atmospheric science, engine combustion and drug delivery. In the study of aerosols, the basic chemical and physical properties, such as droplet size, refractive index, evaporation dynamics and chemical compositions are of great importance [1,2]. The optical levitation of individual aerosol droplets is a well-established technique that is increasingly used in these studies and provides the advantage of a non-intrusive and precise means of aerosol manipulation and characterisation [3-8].
The realisation of mesoscopic aerosols behaving like optical microcavities has led to the development of optically based aerosol characterisation techniques [9-11]. For example, the size and refractive index of a droplet can be determined from the cavity resonance structure of the elastically scattered light. In most cases, the scattered light originates from tuneable laser sources with narrow (<50 nm) wavelength ranges. The limited spectral range makes identification of the resonance modes difficult due to their repetitive appearance. The monitoring of dynamic processes is constrained by the time required to scan the laser source. Guillon and Stout  have used a different approach to overcome these limitations by imaging 90° scattered light from a blue LED with a spectral range of ~60 nm. Zardini et al.  also reported observation of droplet resonances in a backscattering geometry across a spectral range of ~50 nm when using a yellow LED. However, the resolution of the detector only allowed resonances of the same order to be observed which hindered mode assignment.
Recently, a broadband white light continuum (WLC) laser has been used for optical trapping of dielectric particles and aerosol droplets [14,15]. The WLC can be utilised as a spectroscopic light source and this offers significant benefits as the spectral range is much larger than that is possible from tuneable lasers. The studies using a WLC have shown that scattering of broadband light occurs in both forward and backward geometries. In backscattering geometry , broad spectral resonances have been used to determine droplet sizes.
Extensive research has been undertaken in the study of cavity enhanced Raman scattering and fluorescence processes within droplets, from which it is also possible to extract size and the chemical composition of the droplet [1,16-25]. For stimulated Raman scattering, the light is much weaker than elastically scattered light and scattering normally occurs across the spectral region of the Raman water-band. In fluorescent spectroscopy, the droplet must either be fluorescent or doped with fluorescent materials [17-20] and significantly greater detail of the resonance behaviour is available. However, complications may arise from the enhancement and inhibition of resonances in Raman scattering and fluorescence governed by cavity quantum electrodynamics (QED) .
Here we describe a simple and efficient means of using an LED-based white light broadband source for droplet property characterisation in combination with optical levitation. The technique provides high resolution (0.1 nm) Mie scattering intensity distributions across a spectral range of 480 to 700 nm. The experimental data provides sufficient information to allow assignment of resonance mode numbers and mode orders from conventional Mie theory calculations. Accurate droplet sizing, within 2 nm, is possible for water–based droplets when narrow first order resonances are present for droplets with radii between 2 and 8 µm. Refractive index dispersion can be determined from a single refractive index value at known wavelength. In addition, an example of morphological droplet dynamics is presented showing non-linear droplet evaporation behaviour at a temporal resolution of 100 milliseconds.
The unique resonance structure in both elastic and inelastic scattering by aerosols is commonly referred to as a Whispering Gallery Mode (WGM) [1,4,13] or Morphology-Dependent Resonance (MDR) [16,18,19,27]. It can be understood as light rays circling around a droplet via total internal reflection at the liquid-air interface. When the wavelength of the light matches a resonance mode of the cavity, two counter propagating waves form a standing wave around the droplet. For each resonance mode, a mode number n is commonly used to describe the number of maxima around the circumference of the cavity. A mode order l is employed to specify the number of maxima existing in radial intensity dependence. The resonance mode can be a Transverse Electric (TE) mode or a Transverse Magnetic (TM) mode. A TE mode has no radial dependence on the electric component of the field and a TM mode has no radial dependence on the magnetic component of the field . Mie theory provides analytical descriptions for the intensity of the electromagnetic field scattered by mesoscopic particles.
Mie scattering coefficients a n and b n, often referred to as partial wave amplitudes, are used in calculating the intensity of the scattered field [10,28]. These coefficients are associated with TM and TE modes respectively, and can be evaluated by applying the boundary conditions of the dielectric sphere . The results are in the forms
where m is the relative refractive index of the sphere to the surrounding medium, x=2πa/λ is the size parameter functions corresponding to vacuum wavelength λ. ψ n and ξn are the Riccati-Bessel functions of the first and second kind of order n respectively, which are defined as,
For a sphere with real refractive index, Mie resonances occur when the imaginary parts of the coefficients a n and b n vanish and the real parts equal unity. The exact locations of these resonances can be calculated by solving Eqs. (5) and (6) for C n(x,m)=0 or D n(x,m)=0 from the definition of a n and b n using Im(a n)=0 or Im(b n)=0 [11,28,29], where Im() indicates the imaginary components of the argument. The real roots produced by the two methods should be identical . For each mode number of a resonance, an infinite series of roots will be obtained for solving the above equations. The first root is labeled with l=1, the second root is labeled l=2, where l is the mode order of the resonance. For a fixed mode number n, as the mode order l increases, the spectral width of the resonance peaks also increases. For a fixed mode order l, the spectral width of the resonance peaks decreases as n increases. As C n(x,m)=0 and D n(x,m)=0 are both transcendental equations, in this paper, Im(a n)=0 and Im(b n)=0 are solved numerically to identify the mode order and mode number of the observed resonances .
The spacing of the successive peaks in size parameter (Δx≡x l n-x l-1 n) for the same mode order can be calculated without prior knowledge of the mode order and mode number. It is given by 
This is approximation holds within 1% accuracy when x≫1, and n-x<4. Using the definition of size parameter and knowledge of the relative refractive index of the droplet for Eq. (7), the size of the droplet can be estimated by,
where λl n and λl n-1 are the wavelengths of the consecutive resonances.
Accurate sizing of the droplet requires fitting of experimental scattering data to theoretical positions of Mie scattering coefficients a n and b n calculated using the relative refractive index dispersion of the droplets to the surrounding media across the spectral range. There are a number of alternative methodologies used to enable accurate fitting of theoretical Mie scattering positions to experimental data [11,20,30,32]. The goal is to assign the experimental peak positions with resonance mode number and order and then extract information on physical parameters such as droplet radius and refractive index. However, there are factors arising from both the droplet and the apparatus that influence the Mie resonance positions. The factors include; droplet radius, relative refractive index, collection angle, acceptance angle, refractive index dispersion and secondary components, such as temperature, that may alter one or more of the above parameters. From a computational perspective it is necessary to place some sensible constraints on the range of values that are practical for these factors. In practice, variation in droplet size and refractive index are used to generate candidate spectra whose resonance positions can be compared to experimental data by inspection or by using a fitting quality algorithm . Assignment is greatly facilitated when a large number of modes and orders are present. If required, a detailed analysis of the scattering intensity distribution in the vicinity of the resonances can also be used to determine characteristic features of the spectra which aid resonance assignment. Once the initial assignment has been made iterative variation in the other factors can be used to optimize the fitting quality.
An estimate of the droplet size is obtained from either optical imaging of the droplet (with a variation of ±10%), or using experimental resonances and an approximation of the refractive index as described in Eq. (8). The refractive index of the droplet material is typically taken from bulk solution (with variation of <1%). However, for the broadband scattering over the large wavelength ranges encountered in this study the dispersion of the refractive index has a significant influence  and knowledge of the dispersion range is essential.
The optical tweezing equipment has been described in previous publications and therefore will only be briefly outlined here [33,34]. A schematic diagram of the experimental setup is shown in Fig. 1. In this study we have used two separate laser systems for optical trapping of droplets. These were (i) a cw Nd:YAG laser (Laser 2000) operating at its fundamental frequency (1064 nm) and (ii) a cw Ar-ion laser (Coherent Innova 308C) operating at 514.5 nm. The Nd:YAG laser is used for all of the initial size characterisation studies primarily because water has a low imaginary refractive index at this wavelength which is crucial for droplets to stabilise. In addition, using the 1064 nm trapping laser avoids the excitation of spontaneous Raman resonances across the water band, which are readily observable with a 514.5 nm trapping laser . Therefore, we eliminate possible confusion of the resonance structure in the water O-H Raman band with the resonant structure in the back-scattered white light, and also the possibility of seeding white light resonances with the Raman resonances.
The trapping laser beam is passed through two sets of lenses to permit optical manipulation of the droplet and to expand the laser beam diameter to match the back aperture of the microscope objective . The beam is then directed into an inverted microscope (Leica, DM-IRB) and reflected by a dichroic mirror (CVI Technical Optics Ltd) to be focussed using a ×63 water immersion objective with numerical aperture of 1.2 and cone angle of 60°. The focal point of the laser is matched to the imaging plane of the microscope by careful translation of the last lens in the optical train. The trapping power was approximately 2 mW for 1064 nm and 10 mW for 514.5 nm.
A similar setup is employed for the broadband LED white light source (Comar Instruments 6V, 20 mA, 3.0 cd). The LED light is directed into the microscope using broadband 50:50 beam splitter and is brought to focus with the same objective lens as the trapping laser. A pin-hole is placed in a focal plane of the LED optical path, such that an image of the pin-hole is formed at the optical focus, parfocal with the laser trap and the optical microscope imaging plane. The size of this spot is 10 µm as imaged and measured using the microscope. Backscattered light is collected with the same optics and is directed to a spectrograph (Acton SP2500i) and CCD (Princeton Instrument Spec10:400 BR). Spectra are collected from a 300 groove/mm grating and a 1200 groove/mm grating to allow both broadband collection and the ability to resolve fine resonance structures. The spectrograph and CCD were calibrated across the spectral range using a Hg:Ne discharge lamp. The resolution of the spectrograph was 0.2 nm (300 groove/mm) and 0.03 nm (1200 groove/mm). The spectrograph slit width was 15 microns.
Aerosol droplets, generated by an ultrasonic nebuliser (Schill Medical), are circulated into an enclosed aerosol cell which is mounted on the microscope. A mass flow control unit provides a near-saturated environment (approximately 99% RH) in the cell by means of passing dry nitrogen through water, thus reducing evaporation from the droplet. Prior to this study a range of sodium chloride concentrations were used to produce droplets which were trapped and observed for size changes due to evaporation. The presence of sodium chloride reduces the vapour pressure due to the large curvature of droplets, which allows the droplets to be stable and trapped for hours in this near-saturated environment . At 20 gl-1 sodium chloride solution the droplets, once trapped, were observed to stay at constant size with in the limits of the optical imaging resolution. Calculation using Köhler theory supports the observation that droplets with a 20 gl-1 sodium chloride concentration in the radius range of 0.2 µm to 5.5 µm and a relative humidity of approximately 99% will equilibrate with the surroundings. We have assumed that the concentration of salt in these droplets is very similar to that which is nebulised. Stable trapping of droplets of 1 to 8 µm in radius is achieved. The ambient temperature of the laboratory was 22 °C.
The relative position of the droplet to the white light illumination can be manipulated by moving the steering mirrors in either optical train for x-y steering or lenses for z control as described above. Although higher signal to noise ratios may be obtained with edge illumination geometry, here we will only discuss centre illumination.
4. Results and discussion
4.1 Droplet sizing and the dispersion relation
Elastic backscattering of several droplets with sizes ranging from 1 to 8 µm radius were obtained. A spectrum of structured resonances was obtained with the exposure time of 1 s as shown in Fig. 2. Although it has been reported that the scattering efficiency is very low when the incident light is a broadband light source , the signal to noise ratio obtained with the apparatus described here does not encounter these problems. We attribute this to the pin-hole in the optical path of the LED confining illumination to only the droplet of interest. The broad background envelope is associated with the intensity profile of the white light source. Superimposed on this broad feature, the characteristic two-peak Mie resonance structure, which is related to TE and TM modes, can be clearly seen. The normalized spectrum is obtained by dividing the raw spectrum with the background. The relatively high noise level at long wavelength is due to the low absolute intensity of the white light in that region, which is associated with a reduced efficiency of the dichroic mirrors.
The spectrum is acquired using a 300 groove/mm grating thus the width of the peaks observed is limited by the resolution of the grating. Droplet size can be estimated from the spacing of consecutive TE and TM modes with 1% accuracy using Eq. (8) , where the refractive index for a 20 gl-1 salt solution is explicitly used with the dispersion relation calculated using the algorithm reported by Millard and Seaver .
Figure 3(a) shows the experimentally obtained resonance peaks for a different droplet using the 1200 groove/mm grating. The spectrum consists of seven contiguous spectra produced by scanning the grating across the wavelength range. First order resonances are now apparent at longer wavelengths, although these become difficult to resolve at shorter wavelengths. An estimate of the droplet size is first calculated from the spacing of the resonance modes using Eq. (8) and comparing this value to the microscope images (Fig. 4). The refractive index dispersion was assumed a priori to be that of a 20 gl-1 salt solution and the dispersion relation, calculated using the algorithm reported by Millard and Seaver , is then used in generating a series of theoretical scattered spectrum as described in section 2. The theoretical spectra are varied for droplet sizes within ±10% of the estimated size (3.50±0.35 microns) until a best fit of the experimental resonance peaks is obtained as shown in Fig 3(b). At the optimum radius, 3.523 microns, the average distance between experimental and theoretical peaks was found to be less than 0.1 nm. The refractive index dispersion was then varied at this radius for changes in salt concentration (20 gl-1±2 gl-1) and temperature (22 °C±2 °C). Further iterations of radius to match these changes did not improve the theoretical to experimental peak fitting. Single refractive index values could not generate sensible theoretical spectra for fitting as water based droplets have a particularly high dispersion, i.e. 0.010, across the wavelength range.
The theoretical resonances were calculated assuming a 180° backscattering geometry. The calculations appeared to generate satisfactory peak positions but the scattering intensity distribution was not ideal. The illumination geometry as described is unusual for Mie scattering studies as the light source is focused with a cone angle approaching 60°. Thus, whilst scattered light is collected in a 180° backscattering geometry, there are Mie scattered light contributions that originate from all angles within this cone. The overall effect is analogous to integrating the Mie scattered light over a definite solid acceptance angle around the backscattered direction . The best fit for the scattered intensity distribution is for integration between 120° and 180° as shown in Fig. 3(c). Such a fit significantly improves the intensity distribution from that of 180° backscattering geometry whilst the resonance positions are identical.
Calculated TM and TE modes using the method described in the theory section are labeled in Figs. 3(a) and 3(b). Fine alignment of both first order and second order peaks across a broad spectral range minimizes the possibility of identifying the modes with wrong mode numbers. If only narrow spectrum (<50 nm) is obtained, the periodic sets of resonance peaks make the mode number and mode order difficult to assign. If the calculated spectrum is displaced by one mode order towards the red or blue end of the spectrum, the “minimum” error in the peak alignment will increase from 0.1 nm to more than 1 nm. The assignment shows that the narrow peaks between 540 nm and 700 nm are first order resonances. As wavelength decreases, which corresponds to an increase in the size parameter, the lowest order resonances become increasingly narrow and eventually cannot be resolved by the spectrometer. The narrowest peak observed (a48,1) shows a FWHM of approximately 0.1 nm, which is also the step size used in generating the theoretical scattered spectrum. Therefore as shown in Fig. 3(b), the theoretical scattered spectrum has approximately the same resolution as the spectrometer. The true line width of this resonance is however much narrower than it appears . The narrowing of the resonances as the wavelength decreases can also be observed with the second order resonances. At long wavelength, a34,2 and b34,2 overlap and are too broad to be resolved as separate resonances. As wavelength decreases, the resonances become narrower and eventually appear as two well defined resonances between 480 nm and 540 nm.
Changes in the droplet size of less than a nanometer can lead to a shift of the calculated resonance modes which are readily discernable when compared to the experimentally obtained spectrum. However, due to the limitation of the CCD resolution, we estimate the error in droplet sizing is within 2 nm. The size range, for which an accuracy of 2 nm applies, is dependent on the presence of narrow, first order peaks which provide the most stringent test of fitting quality. For water based droplets, with which we are primarily concerned with in this study, the first order resonances will disappear from our detector when the radius is below 2 µm. At above 10 µm radius it will become difficult to clearly resolve the resonance positions, although there is a further practical caveat to this in that only droplets of radius 1 µm to 8 µm can be optically trapped. Thus, the practical size range for the application of this methodology is 2 µm to 8 µm for water based droplets. This range will decrease to lower sizes as refractive index increases. Analyzing the images of the droplet acquired from the CCD camera in the microscope (Fig. 4) shows the droplet size calculated from Mie scattering is slightly below that imaged by the camera under bright field illumination .
We also show that a single refractive index at known wavelength can be used to match theoretical and experimental scattering data. For example, using a fixed refractive index of 1.3360 at 598 nm, the refractive index of 20 gl-1 saltwater at room temperature , we align the first order resonance peak near 598 nm by varying droplet size within ±1 % of the calculated size. The sequential resonance modes at other wavelengths are then aligned by assuming constant size and adjusting refractive index. The refractive index variation of saltwater across the spectral range of 480 to 700 nm in ambient environment is less than 0.010 , if we impose a similar limitation we can determine if the initial alignment of the peak near 598 nm was for the correct mode. Incorrect mode assignment leads to significantly larger variations in refractive index than 0.010. In this way the dispersion relation can be experimentally obtained as shown in Fig. 5. Comparing the dispersion relation obtained with this method and with the dispersion relation calculated using Millard’s algorithm [22,37] suggests broadband Mie resonances can be used for determining the refractive index dispersion of the droplet.
4.2 Droplet size evolution
Droplet evaporation on long time scales has been reported by a number of investigators [13,14,22]. Here, we describe observation of transient droplet size variations at a temporal resolution of 0.1 second. The dependence of droplet evaporation is followed using the two different trapping laser wavelengths discussed previously. Figure 6 shows the temporal evolution of the scattered spectrum from a droplet trapped by the Nd:YAG laser with color coded intensity map. The spectral position of the high intensity regions corresponds to resonance peaks in the backscattered spectra. Over the 3000 s experimental observation time shown in Fig. 6, the resonances at the finish are blue shifted by ~1 nm in comparison with the resonances at the start, corresponding to a droplet size reduction of ~3 nm, from the initial 1.590 micron radius. However, close inspection of the spectra shows small variation of the resonances’ spectral positions of the order of several nanometers in a time scale of several seconds, which may indicate the presence of slight variations in the droplet environment.
The study with the argon-ion laser operating at 514.5 nm suggests the droplet evaporation undergoes very different dynamics on the same timescale in comparison with the Nd:YAG laser trapping methodology, as shown in Fig. 7. The time resolution between spectra is 0.1 seconds. The resonances are shifted towards the blue end of the spectrum by 43.8 nm over 680 seconds, corresponding to a droplet size change from 4.85 to 4.57 µm. The calculated size is only an approximation here as a significant reduction in droplet size may lead to a change in the dispersion relation of the droplet. As water evaporates, it is assumed that the total amount of salt within the droplet is constant. This leads to an increase in salt concentration which in turn increases the refractive index of the droplet. Future work will look into the development of a more elaborate fitting mechanism to address this issue.
The various horizontal discontinuities in Fig. 7 suggest that a rapid spectral shift of the resonance structure occurs within ~2 to 3 second time domains. A magnification of a transition near t=60 seconds shows the scattered spectrum is displaced by almost exactly one set of resonances and may contain more than one transition step (Fig. 8). Similar features are repetitively observed for other fast transitions. In Fig. 8(a) between 45 and 53 seconds, the scattered spectrum shows small temporal variations in the spectral position within 2 nm wavelength, which corresponds to a droplet size variation of ~15 nm. After a rapid transition between t=54 s and t=57 s, the droplet enters a meta-stable state for ~10 s and shows slight red shifts, which corresponds to droplet size increase. Then the scattered spectrum is again blue shifted at t=67 s and is stabilized corresponding to a droplet size of 4.78 µm and is stable within 1 nm.
The spectral evolution of one of the resonance peaks is tracked to follow a rapid shift and this is plotted in Fig. 9. The arrows indicate the times where the droplet appears to experience a discrete heating event. Spline fitting is employed to produce a curve so that differentiation can be performed. The first negative derivative of the spectral shifts shows three prominent maxima, which corresponds to three highest rates of droplet size reduction. However, care must be taken when correlating the spectral shifts with droplet size evolution. The refractive index of the droplet changes due to increasing salt concentration must be considered for accurate droplet sizing. An increase in salt concentration corresponds to an increase in refractive index, which causes the resonances to shift to the red end of the spectrum, which counters the blue shift as a result from the size reduction of the droplet to a certain degree. Therefore, the actual size of the droplet is slightly larger than that is calculated here using a fixed dispersion relation.
Since the spectrum is shifted by approximately one set of resonances, and there are three maxima in the rate of size change, we propose that the heating event may be caused by the 514.5 nm trapping laser coupling into the resonance modes. Such coupling should dramatically increase the intensity of the internal electromagnetic field and may cause a temporal temperature elevation which then induces evaporation. If this were the case as soon as the size of the droplet changes, the laser wavelength is immediately off resonance, and the droplet cools. However evaporation is still sufficient that the two remaining resonances, shown in Fig. 9, could also couple to the laser before a gap in the Mie resonances is reached and the droplet stabilises. Further studies are planned to increase the understanding of these mechanisms. The primary difficulty lies in performing detailed size analysis within the short transitions as the parameters of droplet size, temperature, salt concentration and refractive index dispersion are all subject to variation.
It is clear during our studies that rapid decreases in size occur with the 514.5 nm laser but not the 1064 nm trapping laser. This is likely to be a combination of the Mie resonances at 1064 nm being much weaker and, the laser trapping power for the 1064 nm beam being significantly lower (20%). The heating events normally occur over 1–2 s and would be difficult to follow with techniques requiring long acquisition times. The observed behaviour could play an important role (i) when trying to optimize conditions to maximise the coupling of laser light into droplets or (ii) when equilibrating a stable droplet size for reaction chemistry studies.
An inexpensive and efficient means of using a white light LED to study the broadband Mie backscattering from optically levitated aerosol droplets is demonstrated. Multiple Mie resonances are observed simultaneously across a spectral range from 480 nm to 700 nm. The mode order and mode number of these peaks are explicitly identified using Mie theory with the dispersion relation taken into account and the droplet size can be determined with an accuracy of ±2 nm. The size range over which this accuracy is valid is between a radius of 2 µm and 8 µm where narrow fist order resonances are present. A dispersion relation is also generated with the knowledge of the refractive index at a particular wavelength. The experimentally determined dispersion relation shows good agreement with the algorithm reported by Millard et al . Such calculation can also used to verify the mode assignments. We demonstrate for the first time transient non-linear droplet size variations and discuss a possible mechanism for droplet heating. The high signal-to-noise ratio indicates that this technique means can be used for monitoring much faster droplet dynamics in aerosol chemistry with potentially sub-microsecond resolution by optimization of the detector.
The technique has a wide range of potential applications, including the study of droplet deformation and droplet coagulation, as resonances are sensitive to droplet shape . Since temperature and refractive index are correlated, the technique can also be developed for temperature measurement of mesoscopic dielectric spheres including particles and emulsions in solution. Finally, absorption of resonating light by molecules within the droplets, or at the air/water interface, can significantly alter Mie scattering intensities, a feature which could be used to follow photochemistry and heterogeneous surface chemistry.
We would like to thank the Science and Technologies Facility Council for providing funding through the Strategic Initiative Program Grant (HS30635).
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