Accurate and efficient determination of the radius, refractive index, and dispersion of weakly absorbing spherical particle using whispering gallery modes
Thomas C. Preston and Jonathan P. Reid, "Accurate and efficient determination of the radius, refractive index, and dispersion of weakly absorbing spherical particle using whispering gallery modes," J. Opt. Soc. Am. B 30, 2113-2122 (2013)
We present an algorithm that can be used to simultaneously determine the radius and the refractive index (with dispersion) of a spherical, homogeneous particle. This is accomplished by fitting characteristic resonances calculated using Mie scattering coefficients to the measured whispering gallery mode resonances. The advantage of this algorithm over those that have been presented previously is that a large portion of the search can be reduced to two dimensions (a search that includes radius and refractive index with dispersion will always be at least three dimensions). Using this algorithm, we analyze two large sets of cavity-enhanced Raman spectra from optically trapped aerosol particles. The speed of the algorithm allows for best fits to be found in real time. Precision is found to be limited by the resolution of the spectrograph.
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Size Parameters at Which Resonances Occur in a Homogeneous Sphere with for Both TE and TM Polarizations Across the First 10 Orders of Mode Numbers , 50, and 75a
The tabulated resonances are either the solutions to Eq. (7) or the real part of the solutions to Eq. (8). See Section 2 for a more detailed description of the two methods used to calculate the resonances.
Table 2.
Size Parameters at Which Resonances Occur in a Homogeneous Sphere with for Both TE and TM Polarizations of Mode Numbers , 50, and 75a
Mode orders are listed up until the point at which Eq. (12) () no longer provides a satisfactory value of in the sequence in Eq. (11).
Table 3.
Uncertainties Determined by Fitting a Set of Simulated Modes Using the Algorithm from Section 3.Ca
(nm)
(nm)
()
0.0010
0.114
0.000033
0.0032
0.0020
0.218
0.000064
0.0057
0.0050
0.589
0.000172
0.0160
0.0100
1.128
0.000330
0.0327
0.0200
2.229
0.000652
0.0651
0.0500
5.552
0.001621
0.1658
0.1000
10.718
0.003133
0.3281
The parameters used to generate the simulated mode set were , , and . Only resonance positions for and modes between 635 and 665 nm were used. This gave a set of 12 modes. For each standard deviation of the Gaussian noise , a table containing 1000 mode sets was constructed and Gaussian noise was added to each resonance. Each set in this table would then be fitted using the algorithm discussed in Section 3.C. The standard deviation of , , and were determined from the resulting fits.
Tables (3)
Table 1.
Size Parameters at Which Resonances Occur in a Homogeneous Sphere with for Both TE and TM Polarizations Across the First 10 Orders of Mode Numbers , 50, and 75a
The tabulated resonances are either the solutions to Eq. (7) or the real part of the solutions to Eq. (8). See Section 2 for a more detailed description of the two methods used to calculate the resonances.
Table 2.
Size Parameters at Which Resonances Occur in a Homogeneous Sphere with for Both TE and TM Polarizations of Mode Numbers , 50, and 75a
Mode orders are listed up until the point at which Eq. (12) () no longer provides a satisfactory value of in the sequence in Eq. (11).
Table 3.
Uncertainties Determined by Fitting a Set of Simulated Modes Using the Algorithm from Section 3.Ca
(nm)
(nm)
()
0.0010
0.114
0.000033
0.0032
0.0020
0.218
0.000064
0.0057
0.0050
0.589
0.000172
0.0160
0.0100
1.128
0.000330
0.0327
0.0200
2.229
0.000652
0.0651
0.0500
5.552
0.001621
0.1658
0.1000
10.718
0.003133
0.3281
The parameters used to generate the simulated mode set were , , and . Only resonance positions for and modes between 635 and 665 nm were used. This gave a set of 12 modes. For each standard deviation of the Gaussian noise , a table containing 1000 mode sets was constructed and Gaussian noise was added to each resonance. Each set in this table would then be fitted using the algorithm discussed in Section 3.C. The standard deviation of , , and were determined from the resulting fits.