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Resonant scattering of light by spherical particles is often referred to as "morphology-dependent resonances" (MDRs) or whispering-gallery modes.
The previous page demonstrated that such resonances can be extremely dependent on:
This page examines Mie resonances using (a) Mie calculations and (b) Debye series calculations. It is based on the paper Analysis of Mie resonances using the Debye series published in J. Opt. Soc. Am A, 38, 1357 (2021). You can download a copy of that paper here.
Mie resonances are clearly visible in Fig. 1 below which plots |S1(θ)| for the arbitrary scattering angle θ = 150° as a function of size parameter x = 2πr/λ, where r is the radius of the homogeneous sphere with refractive index m = 1.3333. As Fig. 1 covers the range 98 ≤ x ≤ 100, this would correspond to 10.138 μm ≤ r ≤ 10.345 μm if we assume that the wavelength of the incident light λ = 0.65 μm. Resonances can be characterized by n (the partial wave number) and by l (the number of radial modes), as indicated at the top of Fig. 1. Some of the resonances (e.g. at x = 98.1312 and x = 98.1932) coincide with local maxima of |S1(θ)|, whereas other resonances (e.g. at x = 99.2567) coincide with local minima. The resonances at x = 98.4732 and x = 99.7511 seem to have a local minimum and a local maximum.
The Mie results shown in Fig. 1 have been re-plotted in Fig. 2 below as a parametric curve plotting |S1(θ)| together with the phase φ of S1(θ). The parametric curve includes several strange nearly-circular features, each of which seems to be associated with one of the Mie resonances shown in Fig. 1. Another oddity is that each of the resonances is positioned at the extreme left or extreme right of these nearly-circular features. Fig. 2 reinforces the idea that some Mie resonances can appear as local maxima, local minima or both. Following the curve in Fig. 2 starting at x = 98, the first resonance occurs at x = 98.1312, slightly before the red dot which indicates a local maximum. The next resonance at x = 98.1932 occurs just before another red dot at x ≈ 98.19321 corresponding to a local maximum. The next resonance at x = 98.4732 is near a green dot corresponding to a local minimum. Similarly, the resonance at x = 98.99 coincides with a local maximum, whilst the resonance at x = 99.2567 is very near to a local minimum. However, the resonance at x = 99.7511 is accompanied by a local minimum at x = 99.7444 and a local maximum at x = 99.7594.
It has long been known that TM and TE resonances of order n are respectively associated with the an and bn terms in the Mie calculations. This is illustrated in Fig. 3 below which shows that the TE resonance at x = 98.1312 (n = 105, l = 4) is due to the b105 term, whereas the TE resonance at x = 98.1932 (n = 110, l = 3) is due to the b110 term.
Fig. 4 below confirms that the resonances in Fig. 3 coincide with the imaginary part of bn being zero (in both cases, the value of bn → 1).
The Debye series defines the values of an and bn terms in the Mie calculations in a form that isolates specific scattering mechanisms of order p, where:
As Mie resonances seem to be the combined result of many terms in the Debye series (rather than being caused by a single value of p), it is not immediately obvious how the Debye series can be used to analyze Mie resonances. One approach is shown in Fig. 5 below, which compares the Mie results (shown in red) with the sum of the Debye series contributions (shown in blue) for p = 0 through p = pmax for various values of pmax. If the Mie results exactly matched the Debye series results, the red lines would not be visible because they would have been overwritten by the blue lines. Fig. 5(a) shows that the match is far from perfect for pmax = 100: the blue line reproduces the general trend of the red line in the vicinity of the broad resonance at x = 98.1312, but it entirely misses the narrow resonance at x = 98.1932. Figs. 5(b)-(f) demonstrate that increasing the value of pmax for the Debye series gives progressively closer matches to the Mie result. In particular, Fig. 5(b) shows that the broad resonance at x = 98.1312 can be approximated by the first 200 terms of the Debye series, but Fig. 5(f) suggests that the narrow resonance at x = 98.1932 requires slightly more than 10,000 terms of the Debye series.
Fig. 6 displays the Debye series results in an entirely different way: each dot plots the value of S1(θ) in terms of its magnitude |S1(θ)| and phase φ for x = 98.1 and θ = 150° for every value of p from p = 0 to p = 20,000. This diagram also shows the Mie result at |S1(150°)| = 32.9 and φ = 357.9°. The dominant contributions to the Mie result are from the p = 2, p = 0 and p = 7 terms in the Debye series. The dots are also colour-coded in accordance with the scale shown to the right of Fig. 6. The arrangement of the colours of the dots in the diagram suggests that, when p is large, the amplitudes of the contributions generally decrease as p increases, but careful examination indicates that there are many exceptions to this rule. Although Fig. 6 shows Debye series results for values of p ≤ 20,000, Fig. 5(b) shows that pmax = 200 is sufficient to reproduce the Mie result when x = 98.1. Looking at Fig. 6, it is clear that the phases of the high-order terms are distributed between 0° and 360°. If they were distributed uniformly in terms of phase, destructive interference would occur because a contribution with phase φ would be cancelled by another contribution of similar amplitude with phase φ ± 180°. In such cases, the high-order terms would have no effect on the sum.
The value of x = 98.1 used in Fig. 6 was selected because it is not near a Mie resonance. What happens near Mie resonances? Fig. 7 is a set of diagrams similar to Fig. 6 illustrating the effects on the amplitudes and phases of the Debye series contributions for 0 ≤ p ≤ 20,000 as x is increased from 98.191 to 98.1938. The amplitudes do not seem to vary much, but the individual phases change dramatically near the (n = 110, l = 3) resonance at x = 98.19319 where the phases of the Debye terms seem to “congregate” together. Very few of the dots are in the left side of Fig. 7(d) (i.e. with phases between 90° and 270°) compared with those in the right side (i.e. with phases between 270° and 360° or between 0° and 90°). Examination of Fig. 7(d) suggests that this concentration is very pronounced for the yellow dots (representing Debye series terms with 4,000 ≤ p ≤ 5,999), the green dots (6,000 ≤ p ≤ 7,999) and the cyan dots (8,000 ≤ p ≤ 9,999).
The animation in Fig. 8 below confirms that the coloured dots congregate on the right side of these diagrams near the resonance at x = 98.19319, as well as the change from clockwise spiral to counter-clockwise spiral.
The phases of the Debye series contributions for x = 98.19319 have been plotted in Fig. 9(a) below as a function of p. This diagram shows a startling absence of red dots in the zone centred on φ = 180° and p ≈ 6,000. As demonstrated in Fig. 3(d), the b110 term is responsible for this particular resonance. The phases of the b110 contributions are plotted as blue dots in Fig. 9(a) appearing as a blue straight line very close to 360°. The amplitudes of the Debye series contributions are plotted as a function of p in Fig. 9(b). The red dots show no obvious pattern as a function of p, but the straight blue line shows that the contributions from the b110 term reduce with each successive value of p. However, Fig. 9(b) also shows that the contributions from the b110 term are dominant when p ≈ 6000. As shown by the blue line in Fig. 9(a), all of the b110 contributions have phases close to 360°, thus explaining why the red dots in Fig. 9(a) are clustered around 360° (or 0°) when the b110 contributions are dominant.
Fig. 10 below explores the effects of the b110 term on the Debye series results at this resonance. Fig. 10(a), is very different to Fig. 10(b) where the b110 term has been set to zero. The uneven distribution in Fig. 10(a) is obviously caused by the b110 term.
The results shown in Fig. 10 have been combined in Fig. 11 below by plotting the cumulative contribution made by each term p in sequential order starting from p = 0 though p = 20,000. Results for the full Debye series are given in Fig. 11(a), which includes an extraordinary horizontal line pointing towards the Mie theory result. In essence, Fig. 11(a) is the sum of the two parts of Fig. 11(b) – demonstrating that the horizontal line in Fig. 11(a) is caused by the b110 term, corresponding to the partial wave n = 110. The dominant p = 0, 2, 3, 6 and 7 terms are identical in both parts of Fig. 11. The very high-order p terms are responsible for the horizontal line in Fig. 11(a). On the other hand, the upper part of Fig. 11(b) shows that terms with p > 200 make no significant contribution to the pattern when b110 = 0. The fact that the multi-coloured lines in Fig. 11 (a) and (b) are straight when p is large suggests the scattering contributions from different values of p are in phase. As these lines are also horizontal, it seems that the average phase of these contributions is very close to 0°.
Fig. 9 demonstrated that, even at resonance, the phases of the very high-order terms for the full Debye series are typically not close to 0°. This anomaly is evident in Fig. 12 which is a much-magnified version of the "straight line" in Fig. 11(a) for values of p ≈ 18,000, demonstrating that the contributions made by individual values of p exhibit dramatic variations in amplitude and phase. By comparison, the straight line in the lower part of Fig. 11(b) is very well-behaved since all of the contributions from the p terms are precisely in phase, resulting in a smooth straight line. Despite the detailed differences in shape, the two sets of horizontal lines in Fig. 11 have the same overall length, confirming that the net effect of contributions with p > 200 is zero when the b110 term is ignored.
Recalling that Fig. 11 has been calculated for the resonance condition at x = 98.19319, Fig. 13 below shows what happens at other values of x close to this resonance. Note that the multi-coloured lines display clockwise spirals when x < 98.19319 and counterclockwise spirals when x > 98.19319. Although the shapes of the multi-coloured lines in Fig. 13 are critically dependent on x, the rest of the diagram does not appear to change with x, at least over the limited range of x used in Fig. 13. The almost-circular blue line in Fig. 13 has previously been seen in Fig. 2 in results from Mie calculations. The explanation of this feature becomes obvious in Fig. 13: it is simply the locus of the end-points of the Debye series calculations when pmax is sufficient to replicate the Mie result in the vicinity of the resonance.
In Fig. 13, the resonance at x = 98.19319 is at the extreme right of the almost-circular feature. However, Fig. 2 reminds us that some resonances (e.g. at x = 98.99) are on the extreme left of the almost-circular feature. This difference is caused by the τn (θ) term in Eq. (1): for TE resonances, increasing p causes the horizontal straight lines to extend to the right when τn (θ) > 0 and to the left when τn (θ) < 0
Fig. 11 indicates that the Mie resonances of order n are caused by the Debye series terms of order n for p > 0 being in phase with each other. In Eq. (4) for the Debye series, the expression –T21n (R121n) p - 1 T12n for p > 0 corresponds to transmission through the sphere with p – 1 internal reflections. Note that all of these coefficients (T21n , R121n and T12n ) are represented by complex numbers. To achieve resonance, each p + 1 term must be in phase with the p term, which implies that R121n must be positive and real (i.e. the imaginary part of R121n must be zero).
This differs from the widely-accepted criterion that TE resonances of order n occur at the value of x where the real part of bn is 1 and imaginary part of bn is zero. Given this discrepancy, it is worth examining some numerical results: the two multi-coloured lines in Fig. 14 show the Debye series approximations to the Mie results for b110 as p is increased from zero to 20,000 for x = 98.1931901251548 where Im(b110) = 0 and for x = 98.19319011687 where Im(R121110) = 0. Both lines start at b110 = 1.37×10-4 + i 4.64×10-5 when p = 0, but they diverge as p increases. The upper line curves towards b110 = 1, whereas the lower line is straight because all of the terms are in phase when p > 1. The vertical scale in Fig. 14 has been greatly exaggerated to highlight the differences in these curves. It also demonstrates the “horizontal” straight lines noted in Figs. 11 and 13 are not quite horizontal: in this case, the difference is 0.0053° which is caused by the product (-T21110 T12110) in Eq. (4).
In this case, there is no practical difference in the values of x predicted by the two criteria of Im(bn ) = 0 and Im(R121n) = 0. Nevertheless, the criterion based on R121n seems preferable because the concept of resonance is intrinsically linked to many terms being in phase with each other. Furthermore, as shown below, various characteristics of resonances are dependent on R121n.
Increasing p by 1 reduces the scattered amplitude by the factor of |R121n|, as indicated in Table 1 using the calculated numerical value of R121110 ≈ 0.999727 corresponding to the resonance at x = 98.19319 where n = 110 and l = 3.
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Although Table 1 shows that |(R121110) p - 1 | becomes very small for large values of p, Eq. (4) shows that the Mie result is proportional to the sum to infinity of this geometric progression. It is useful to know how many terms of the Debye series are needed to replicate the Mie result. For example, if we want the amplitude of the Debye series result to be a fraction k of the amplitude of the Mie result at resonance, the required number of terms pmax is given by:
Applying Eq. 5 for k = 0.99 at each of the resonances shown in Fig. 1 gives the results shown in Table 2. The calculated values of pmax = 268 and pmax = 16,683 for the first two resonances listed in Table 2 are consistent with the estimates from Fig. 6 of pmax ≈ 200 and pmax > 5,000 respectively. Table 2 also indicates that R121n decreases as l increases, resulting in extremely high values of pmax for the two resonances with l = 1. The width δx (FWHM) of the amplitude of the resonance can be calculated by:
Having determined δx, it is simple to determine the Q factor for each resonance using the relationship Q = x/δx. For example, looking at the resonances listed in Table 2, the calculated values of Q vary between 1,650 (for n = 105 and l = 4) and 5.73×1011 (for n = 124 and l = 1).
It is important to recognize that the results shown in Table 2 represent ideal results. For example, the calculations assume that the spherical particle is assumed to be non-absorbing (i.e. the imaginary part of the refractive index is zero). Any absorption within the particle would reduce the value of |R121n|, thus reducing the values of pmax and Q.
As Mie resonances are caused by the combined effects of many terms in the Debye series, it is not obvious how the Debye series can be used to analyze these resonances. In fact, the Debye series can reveal much interesting information about resonances. For example, the animation in Fig. 8 shows that the Debye series contributions vary rapidly in phase in the vicinity of resonances, tending to congregate around 0°. The effects of this behavior can also be seen in Mie results: the nearly-circular loops shown in Fig. 2 can be explained by the Debye series results in Fig. 13.
More importantly, the Debye series expansion in Eq. (4) provides a direct and succinct explanation for these resonances: in particular, resonances for a specified value of n occur when the term |R121n| is positive and real. This criterion ensures that contributions from partial wave n to the scattered field from Debye series terms from p = 1 through p = ∞ are precisely in phase, thus causing the resonance.
The value of |R121n| at the resonance also determines the number of terms pmax in the Debye series that are required to approximate the Mie solution: the above numerical examples show that some broad resonances require only a few hundred terms, whilst very narrow resonances can require billions of terms. Similarly, the widths δx of the resonances and their corresponding p factors can easily be calculated from |R121n|.
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