F17_WGM

Whispering Gallery Modes of Fluorescent Microspheres

Pedro Angulo-Umana and Kai Wang

University of Minnesota

Methods of Experimental Physics II, Fall 2017

Introduction

Whispering Gallery Modes, or WGMs, the resonant wavelengths of dielectric spheres [1][2]. They are caused by repeated total internal reflection of light as it traverses the inside circumference of a sphere. Those wavelengths that fold back onto themselves undergo constructive interference, while all other wavelengths undergo increasing amounts of destructive interference the more times light reflects inside of the sphere. As such, these types of resonators have very high Q-factors, which yields many potential applications, including single-molecule biosensing and high-quality lasing[1][3]. In the case of fluorescent microspheres, these resonant wavelengths are visible as spikes in the sphere's fluorescent curve. The positions these resonant peaks are dependent on the sphere's refractive index and radius. Thus we intend to observe and measure the positions of these resonances, and use them to accurately determine the sphere's refractive index and radius[2].

Where n=n_s/n_ambient is the sphere's relative index of refraction, P has a value of n for TE modes and 1/n for TM Modes, and α₁ is the first zero of the Airy function Ai(-z). The greek letter v = l+1/2, where l is the mode number of the resonance; the mode number is essentially the number of reflections undergone by light inside the sphere.

Fluorescent microspheres are polysterene spheres filled with a fluorescent dye. This dye gives the microspheres distinct absorption and emission spectra, as shown in Figure 2.

Figure 2: The absorption (dashed) and emission(solid)

spectra of the microspheres used in the experiment. Image from [6]

As can be seen, the spheres emit over a wide bandwidth. Within this bandwidth, the resonant wavelengths will resonate inside the sphere, causing sharp resonant peaks in the sphere's spectra on top of the flourescent curve. By measuring these wavelengths, the resonance equation given above can be fitted for R and n_s.

Methods

Figure 3: A schematic diagram of our experimental

apparatus. Image by authors.

A 405 nm blue laser was focused through a microscope objective onto a prepared sample of 10μm yellow-green fluorescent microspheres. The micropsheres come from the manufactured suspended in an aqueous ambient, and have a nominal refractive index of about 1.5. The light emitted by these spheres was collected and directed into either a spectrometer for analysis of the spectral data, or a microscope-mounted CMOS camera so that the spheres could be imaged and measured independent of the fitting algorithm.

The spheres were measured by imaging them with the CMOS camera while lit by a halogen lamp. This was done so that the boundaries of the sphere were clearly visible. The program ImageJ was used to measure the sphere radius. This measurement was then compared to value found by fitting our resonance equation.

Figure 4: Left: A microsphere as seen with the CMOS camera. Note

that the lighting is from a halogen lamp so that the sphere is

not excited. Right: The same sphere being measured with

ImageJ's measuring capability. Image by authors.

We were also able to measure the quality factor (Q-factor) of these resonators by fitting Lorentzian peaks to the resonance curves and measuring the full width at half maximum to calculate the Q-factor.

Figure 5: A peak is fitted to measure FWHM and thus

calculate the Q-factor for this resonantor. Image by authors.

Results

Figure 5: The spectra of the three spheres measured. Note the sharp resonance curves overlaid on the

wider fluorescence curve. The spheres are labeled with their radii as measured with ImageJ. Image by authors.

Table 1: A summary of the findings of the fitting algorithm. As can be seen, the fitted radii closely match the radii

measured using the camera. The refractive indices are smaller than the value stated by the manufacturer. It is possible

that this is due to the spheres being exposed to air without being suspended in water for extended periods of time .

Figure 6: The observed peaks and their predicted positions (solid red and green lines). As can be seen, the fit is not very good. Mode

number assignations were done by eye based off of which predicted resonance position most closely matched the position observed.

Image by authors.

Table 2: A summary of the Q-factors measured for each sphere. These values were in accordance

with the expected value of 1000 for spheres like those used here [1].

Conclusions and Further Work

We were able to determine the radius and refractive index of fluorescent microspheres by studying the positions of their WGM resonances. The spheres were also observed to be high-quality resonators, with Q-factors on the order of 1000.

Further lines of investigation can include:

• Causes for the qualitatively poor fit shown in Figure 6

• Effects of higher-order radial modes on the spectra and positions of resonances

• Study of this procedure for spheres of other radii

Acknowledgments

We would like to thank Professor Joachim Mueller for his guidance and support during the course of this project.

References

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