S18_Whispering Gallery Modes

Whispering Gallery Modes in Fluorophore Doped Microspheres

Jacob Buatti & Benjamin Vaaler

University of Minnesota

Methods of Experimental Physics II, Spring 2018

Introduction

Whispering Gallery Modes, or WGMs, are the specific resonant wavelengths of light emitted from a spherical cavity. Light produced within the cavity is able to reflect continuously and internally along the circumference of the sphere. Some of this light overlaps onto itself causing both destructive and constructive interference. Which wavelengths constructively interfere or destructively interfere depends on the radius of the sphere, wavelength of light, and indices of refraction of the sphere and environment. Wavelengths that do not constructively interfere are increasingly diminished due to destructive interference. Because these resonators are very sensitive to their environments and have high quality factors, they have various applications to single molecule biosensing, photon storing, and more.

Figure 1: Whispering gallery fluorescence pattern in a silica microsphere [2].

Theory

Whispering gallery modes are produced from total internal reflection within a cavity, resulting in larger amplitudes due to constructive interference. The phenomena of WGMs occurs when light of certain wavelengths internally reflects within the sphere and overlaps with itself. When the path lengths of light are half and whole integers of the wavelength, we observe constructive and destructive interference.

In order for light to continuously and internally reflect, illumination from within the sphere is required. According to Snells law, reflection and refraction depend on the indices of refraction of the material used in the sphere and the surrounding environment, and the angle of incidence:

The spheres used in this experiment are composed mostly of polystyrene, which has an index of refraction ns ≈ 1.6 at the wavelengths where WGMs are observed, and are surrounded by water, which has an index of refraction ns ≈ 1.34. The difference in indices makes it impossible for outside light to refract inside the sphere at angles small enough for internal reflection, called the critical angle. All light from the outside will either refract at angles too large for internal reflection or be reflected off of the sphere without entry.

Figure 2: Geometric representation of a Whispering Gallery Mode of mode 8 (due to 8 internal reflections)

To counter this issue, we used microspheres doped with fluorophores. Fluorophores are fluorescent compounds that can be excited by photons. These compounds re-emit light at longer wavelengths depending on the type of compound and other environmental factors. Fluorophores solve the issue of reflection and refraction by producing light from within the sphere. Fluorophores near the edge of the spheres release a Gaussian profile of different wavelengths of light (shown below) in all angles including ones small enough to elicit total internal reflection within the sphere.

Figure 3: Emission and absorption spectra of the fluorescent microspheres. The bold line indicates the emission spectra, the dashed line indicates the absorption spectra. [7]

WGMs are theorized using the geometry of the sphere, the materials, and the transverse electric (TE) and transverse magnetic (TM) wave theory of light. The equations for theoretical modes are given below.

where R is the radius of the sphere, ns is the index of refraction of the sphere, m is the ratio of sphere index to environment index , and v is the mode number plus a half, l+0.5.

The index of refraction is a major factor in interpreting WGMs. The ratio of sphere index to environment index, m from the theoretical equations above, determines the spacing between transverse electric and transverse magnetic mode pairs.

In short, the refractive index for the polystyrene spheres is not known. The manufacturer fails to provide data on how the spheres were made or their effective indices of refraction. Since the spheres are doped with the fluorescent compounds, fluorophores, the index is not that of pure polystyrene.

Additionally, indices of refraction are not absolute and “change” based on the wavelength of light. This is called dispersion. A refractive index calculator shows that the index of refraction of pure polystyrene decreases with increasing wavelength.

Though we do not know the actual index of refraction, our spheres are at least partially polystyrene and behave in a similar way. The theoretical mode equations do not take this into account. Because WGMs of the microspheres are so sensitive to their environment, dispersion needs to be taken into account for the analysis.

WGMs are notable for their large quality factors, or Q-factors, which measures the exponential decay of the energy inside a resonator. Inside the spherical cavity, destructive interference quickly leads to the loss of energy inside the resonator, while constructive interference preserves energy for much longer. The Q-factor, then, depends highly on the resonant wavelength, and is given as:

Where power loss is the energy lost by the resonator every cycle, ω is the resonance frequency at which the Q-factor is measured, and ωFWHM is the full-width-half-maximum measurement of the resonance peak. In WGMs, power loss is minimized due to constructive interference, so energy is lost comparatively slowly and Q-factors can reach values of up to 107

Apparatus

Figure 4: Experimental Setup

Light from a 405 nm 4.5 mW laser was shone into a Ziess Axiovert 35 inverted phase contrast microscope at angles near parallel to the optical bench.

A dichroic mirror selectively reflects light at specific wavelengths [5]. In this case the dichroic mirror is placed at about 45 ◦ above the horizontal to reflect the blue light (405nm) into either a 40X or 20X microscope objective. Approximately 90% of the blue light is reflected, reducing the intensity of the light slightly.

The reflected light then passes through either a 40X or 20X microscope objective and onto the microscope slide containing the fluorescent microspheres. The blue laser light excites the fluorophores within the microspheres. The excited cells will emit light at a lower energy and larger wavelength (green) than the laser.

The resulting green light emitted from the spheres passes back through the objective and onto the dichroic mirror. Because the dichroic mirror is tuned to absorb or reflect light with a lower wavelength (i.e. 405nm), the light from the spheres passes through the dichroic mirror.

The small portion of blue light from the laser that remains and the green light from the spheres is then shined onto a FLG435 Barrier Filter. The FLG435 Barrier filter blocks all blue laser light emitted that went through the dichroic filter and allows the green light to pass through. From there, the output is directed to the Moticam 1 PiX camera attached to the microscope, the eyepiece, or the Ocean Optics Flame-T spectrometer to measure and record the emission wavelength.

Results

Whispering Gallery Modes were observed in emission spectra from single fluorescent microspheres. The spectra were analyzed by fitting the theoretical modes to the peaks in the emission spectrum. Since the effects of dispersion are not known, a small slice of each spectrum was fitted at a time (Figure 5). Dispersion has a noticeable effect across a range greater than 10 nm. Therefore, two neighboring WGM pairs have a similar index of refraction and can be fitted using one value. This allows for the determination of a radius.

Figure 5: WGM peak pairs for mode number 92 and 91, fitted without accounting for dispersion

The index of refraction, n_s determines the spacing between the TE and TM modes of the the same pair. Smaller indices of refraction were found to reduce the spacing between mode pairs, while changing the radius resulted in significant theoretical mode translation. These general guidelines made it possible to guess a refractive index and radius solution to fit the data. The table below shows the radius and index of refraction combination for the first two mode pairs for spheres 4, 5, and 6.

Table 1: Radius and index of refraction for mode 92 and 91 for spheres 4, 5, and 6.

Since the refractive index depends on wavelength, WGMs within a slice differ due to different radii (see figure below). At 500 nm, the peaks are not in the exact same position. This is due to different radii of the spheres.

Figure 6: Sphere 4, 5, and 6 spectra from 490-530 nm.

The effects of dispersion are noticeable in aligning the theoretical modes. Small changes in the index of refraction result in slight translations in the mapped TE and TM modes. To realign modes, small changes were made to the index of refraction at larger wavelengths to realign the modes to their pairs. The following are the different indices of refraction for different mode numbers, corresponding to different wavelengths

Table 2: Indices of Refraction for each mode number

The above modes have been mapped onto the emission spectrum of sphere 4 to show alignment.

Figure 7: Dispersion corrected WGM peaks using different indices of refraction (Table 2). Leftmost pair of peaks are mode number 92, which decreases going to the right.

The Quality Factor of each sphere was determined by using Origin to find the FWHM of each WGM peak, averaged across each spectra for sphere 4, 5 and 6 respectively. In order to find an accurate measurement, the base emission spectra (as indicated by the bold line in Figure 3) was subtracted from the WGM peak spectra and then fitted to a Gaussian. After subtracting the spectrometer contribution to the FWHM determined by calibration with a mercury lamp, a Q-Factor was found for each sphere. As seen in Table 3, Q-Factors were found on the order of 10³ consistently.

Table 3: Q-Factors determined for spheres 4, 5, and 6

Figure 8: A single peak fitted with a Gaussian, used to find the Q-Factor of the cavity

Conclusion

WGMs of microspheres are important for their ability to sense small changes in their environment. Radius and index differences on the order of thousandths have measurable changes to the position of theoretical WGMs on the graph. Changes this small have the same, measurable effect to WGMs of the fluorescent microspheres, which are excellent detectors.

The measuring techniques used to determine radius from images is not near precise enough for the experiment performed. The results obtained from the graphing “guess and check” method are within 1σ of the imaged results. This confirms that the sphere sizes are within the manufacturer’s description, but fails to be precise enough for comparing to observed spectra to sphere size. It is possible to have obtained a different result that is also within the error of the imaged radius measurements.

Additionally, there is a significant issue in not knowing the index of refraction of the sphere or the amount of dispersion. The methodology used to determine a radius and refractive index via graphing has an obvious bias, where the index can be manually changed to fit exactly what you want it to be. Since we do not know ns, the dispersion found in the analysis is a simply made up to fit our solution. The only way ns is helpful, is with determining spacing between a TE and TM mode of the same pair.

Q-Factors determined are on the order of 10³, which is comparable to established results. [1], [6] Adding a thin layer of gel had no significant impact on the observed Q-Factor or spectra, although results were inconclusive.

Since microspheres can detect very small changes in environment, radii and refractive in- dices need to be well measured and known in or- der to draw meaningful conclusions. The analysis method used for the obtained results is a messy estimate for a solution near the real and exact solution. Each microsphere has its own, special emission spectrum containing WGMs. The exact solution cannot be determined with- out the proper algorithm, which would be able to determine a more accurate radius and refractive index of the sphere.

Future MXP Projects

Reducing the effect of bleaching would be of great benefit to data taking. In our project, strong WGMs were only visible for fractions of a second. A neutral density filter and/or a larger wavelength of light may aid in this. Specifically, a new 450 nm laser received by the lab might allow for multiple measurements of the same sphere without risking bleaching.

Gel Layers did not have a noticeable effect during data analysis, but only so many trials were conducted, and the Q-Factor of the spheres varies greatly between trials. A more comprehensive experiment and analysis might reveal a trend that does show that the gel layer has a substantial effect that this experiment could not show due to lack of data.

Adding sugar to the water surrounding the spheres would change the external index of refraction. Observing the shift in WGM peaks and attempting to match this to the new index could test the sensitivity of WGMs to environmental change. Similarly, running the same experiment with a heat source could be used to determine minute changes in sphere radius due to thermal expansion.

Acknowledgments

We would like to thank Professor Mueller for his advice and guidance during this project.

References

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[3] Tabata, K., Braam, D., Kushida, S., Tong, L., Kuwabara, J., Kanbara, T., . . . Yamamoto, Y. (2014). Self-assembled conjugated polymer spheres as fluorescent microresonators. Scientific Reports,4(1). doi:10.1038/srep05902

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[6] Angulo-Umana, P., Wang, K. (2017). Whispering Gallery Modes of Fluorescent Microspheres, https://sites.google.com/a/umn.edu/mxp/student-projects/fall_2017/f17_wgm

[7] Pang,S; Beckham, R.; Meissner, K. Quantum dot-embedded microspheres for remote refractive index sensing, Appl. Phys. Lett. 2008, 92 221108 2008

[8] Snell's Law and Refraction. (n.d.). Retrieved February 21, 2018, from http://www.animations.physics.unsw.edu.au/jw/light/Snells_law_and_refraction.htm