Equation 47 from Lock's 1993 paper (reproduced above) calculates the amplitude S1 of the scattered light for perpendicular polarisation as a function of the scattering angle θ and the tilt φ of the scattering plane. Most of the results on this page assume the conditions specified in Lock's 1993 paper which considered scattering from a sphere of radius 43.3 μm and refractive index n = 1.33, refractive index of medium = 1, λ = 0.5145 μm, φ = 90° with the incident Gaussian beam defined by x0 = 0, y0 = -40 μm, z0 = 0 and ω0 = 20 μm. A diagram showing the coordinate system adopted by Lock is available elsewhere on this web site.
Evaluation of the above equation involves three separate terms, each of which involves summation of terms involving l and/or m. The limit of lmax is well known from algorithms for Mie scattering and is typically defined as:
lmax = x + 4.05 (x)1/3 + 2
where x = 2 π r / λ for a scattering sphere of radius r and light of wavelength λ.
For the conditions assumed in Lock's 1993 paper, x ≅ 528.7 and the resulting value of lmax = 563. Although the summations are well-defined in terms of l, the situation regarding m is less clear. The second and third terms of equation 47 involve summations from m = 1 to m = ∞. In practice, numerical algorithms need to truncate such infinite sums at some value of m, such as at mmax. In his 1993 paper, Lock suggested that mmax = 5 was sufficient for the condition being investigated. However, in his 1995 paper, equations 57 & 58 were introduced to determine mmax for arbitrary conditions.
The previous page illustrated the importance of using the correct value of mmax. For the conditions assumed in Lock's 1993 paper, Mie calculations of sufficient accuracy could be achieved with mmax = 10, whereas Debye calculations required mmax = 20 (i.e. in agreement with the value given by equations 57 & 58 in the 1995 paper). As doubling the value of mmax approximately doubles the computation time, it is obviously worth avoiding use of unnecessarily high values of mmax
This page addresses the general issue of setting appropriate values of lmax and mmax by examining the contributions of the various terms in Lock's equation 47 as a function of l and m.
Fig. 1 False-colour map showing the relative amplitude of the contributions due to specific values of l & m in Lock's equation 47
Mie calculations for scattering of light at θ = 120° (perpendicular polarisation only)
Fig. 1 shows the amplitudes of the various contributions to S1 in in the 3 terms of Lock's equation 47 as a function of l and m. The false-colour scale at the top right is logarithmic - so that, for example, yellow-green zones correspond to values of l and m which contribute amplitudes of about 10-6 relative to the maximum value of 9.265 (which occurs in term 3 for l = 504 and m = 1).
The very rapid decline in the amplitudes of the contributions above l = 550 confirms the validity of the above equation which gives lmax = 563. However, Fig. 1 shows that mmax is poorly defined in that the amplitudes of the contribution decline gradually with m. Fig. 1 suggests that mmax ≅ 20 would be appropriate for a lower limit of 10-14 or mmax ≅ 12 for a limit of 10-6.
Although Fig. 1 shows the results for θ = 120°, similar patterns appear at other values of θ. Computation time could be reduced by performing only those calculations for values of l and m inside the boundary defined by the yellow dashed lines.
The previous page indicated that mmax = 10 gave satisfactory results on Mie calculations and that little benefit in terms of accuracy was achieved by using higher values of mmax. Nevertheless, the Debye series calculations required mmax = 20 (much more than for the Mie calculations). It is therefore instructive to produce a similar false-colour diagram for the Debye series calculations - shown below as Fig. 2.
Fig. 2 False colour map showing the relative amplitude of the contributions due to specific values of l & m in Lock's equation 47
Debye p = 3 calculations for scattering of light at θ = 120° (perpendicular polarisation only)
Although Fig. 2 shows the results for scattering angle θ = 120° (the most sensitive value of θ for Debye p = 3 calculations according to Fig. 7b on the previous page), it seems to indicate that the Debye series calculations are less demanding in terms of mmax than the Mie calculations shown in Fig. 1. This finding contradicts the results given on the previous page, thus indicating that the type of diagram shown in Figs. 1 and 2 can give misleading results.
Fig. 3 Mie calculations
To address this issue, another type of false-colour map has been developed - as shown in Fig. 3 above. Note that this map covers the entire range of scattering angles from θ = -180° to θ = 180° (rather than the single value of θ used in Figs. 1 and 2). In this case, the inner sums involving l in terms 2 and 3 of Lock's equation 47 have been computed for a particular value of m. Superficially, this may not appear to be significantly different to the method used to generate Figs. 1 & 2. However, the summation of all of the terms involving l is actually a vector sum - if all of the individual contributions are roughly in phase for a particular value of m, this vector sum would be much larger than if the individual contributions had random phases.
Fig. 3 shows that the value of mmax for Mie calculations varies with θ and indicates that an amplitude limit of 10-6 would require mmax ≅ 10.
Fig. 4 Debye p = 0 calculations
Fig. 4 is particularly interesting because, unlike the other diagrams shown above, the amplitudes of the contributions do not display a monotonic decrease with increasing values of m. For example, the isolated red areas in term 3 indicate unusually high levels at, for example, m = 12 and θ ≅ 90°. These isolated red areas explain the anomalous results shown in Fig. 7a on the previous page for Debye p = 0 at θ ≅ 70°.
Fig. 5 Debye p = 1 calculations
Fig. 5 shows equivalent results for Debye p = 1 calculations. Note that, in this particular case, negative values of θ require much higher values of m than for positive values of θ.
Fig. 6 Debye p = 2 calculations
Fig. 6 shows equivalent results for Debye p = 2 calculations. In this particular case, the primary rainbow angle at θ ≅ -139° requires the greatest values of m.
Fig. 7 Debye p = 3 calculations
Fig. 7 shows equivalent results for Debye p = 3 calculations - which indicate some isolated areas of high amplitude contributions, such as m = 16 and θ ≅ 130° thus explaining the anomalous results shown in Fig. 7b on the previous page for Debye p = 3 at θ ≅ 120°.
MiePlot's results have been examined for two conditions:
scattering from a sphere of radius 43.3 μm and refractive index n = 1.33, refractive index of medium = 1, λ = 0.5145 μm, φ = 90° with the incident Gaussian beam defined by x0 = 0, y0 = -40 μm, z0 = 0 and ω0 = 20 μm;
scattering from a sphere of radius 1000 μm and refractive index n = 1.336, refractive index of medium = 1, λ = 0.5145 μm, φ = 90° with the incident Gaussian beam defined by x0 = 0, y0 = -950 μm, z0 = 0 and ω0 = 100 μm.
For condition 1, equations 57 and 58 in Lock's 1995 paper suggest mmax = 20 . However, the previous page indicated that, for this condition, mmax = 10 is sufficient for Mie calculations, but mmax = 20 is needed for Debye series calculations. These results suggest that Lock's equations overestimate the value of mmax by a factor of 2 for Mie calculations.
For condition 2, equations 57 and 58 in Lock's 1995 paper suggest mmax = 90 . Fig. 8 below indicates that this value of mmax is appropriate for Mie calculations for this condition - and emphasises that it is unsafe to reduce the computed value of mmax by the factor of 2 applicable for condition 1.
Fig. 8 Comparision of Mie calculations for mmax = 45 and mmax = 90 for condition 2: scattering from a sphere of radius 1000 μm and refractive index n = 1.336, refractive index of medium = 1, λ = 0.5145 μm, φ = 90° with the incident Gaussian beam defined by x0 = 0, y0 = -950 μm, z0 = 0 and ω0 = 100 μm.
The above results indicate that the ampltudes of the contributions for a given value of θ do not always decline monotonically with increasing value of m and, consequently, great care must be taken in setting the value of mmax.