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Scattering of Gaussian beams by a spherical particle

Numerical evaluation


MiePlotís calculations for scattering of Gaussian beams by spherical particles are based on the algorithms in the following papers: Having written the computer subroutines on the basis of Lock's algorithms, the first test was naturally to try to replicate the graphs shown in Lock's 1993 papers, such as those reproduced in Figs. 1a and 1b below:

Fig. 1a Lock's results for y0 = -40 μm
Reproduced by kind permission of J. A. Lock ©
Fig. 1b Lock's results for y0 = 40 μm
Reproduced by kind permission of J. A. Lock ©
Fig. 2a Results from MiePlot for y0 = -40 μm
Fig. 2b Results from MiePlot for y0 = 40 μm
Fig. 1 & 2    Mie theory calculations for scattering of light from a sphere of radius 43.3 μm and refractive index n = 1.33, refractive index of medium = 1, λ = 0.5145 μm with a Gaussian beam: x0 = 0, y0 = -40 μm (Figs. 1a and 2a) or y0 = 40 μm (Figs. 1b and 2b), z0 = 0, ω0 = 20 μm (perpendicular polarisation only).

Figs. 1 and 2 compare the Mie calculations from Lock's 1993 paper with the equivalent graphs produced by MiePlot. Although the graphs are superficially similar, there are some subtle differences: for example, around θ = 115° for y0 = -40 μm and around θ = 160° for y0 = 40 μm. However, the differences are much bigger with the Debye series calculations - as illustrated in Figs. 3 and 4 below.

Fig. 3a Lock's results for y0 = -40 μm
Reproduced by kind permission of J. A. Lock ©
Fig. 3b Lock's results for y0 = 40 μm
Reproduced by kind permission of J. A. Lock ©
Fig. 4a Results from MiePlot for y0 = -40 μm
Fig. 4b Results from MiePlot for y0 = 40 μm
Fig. 3 & 4    Debye series calculations for scattering of light from a sphere of radius 43.3 μm and refractive index n = 1.33, refractive index of medium = 1, λ = 0.5145 μm with a Gaussian beam: x0 = 0, y0 = -40 μm (Figs. 3a and 4a) or y0 = 40 μm (Figs. 3b and 4b), z0 = 0, ω0 = 20 μm (perpendicular polarisation only).


Comparison of Figs. 3a and 4a (for y0 = -40 μm) shows that Lock's results are effectively identical to MiePlot's results for p = 1, p = 2, p = 6 and p = 10. Despite this close agreement, MiePlot's results show much stronger scattering for p = 5, p = 9 and p = 11, whilst showing weaker scattering for p = 0 (for θ > 20°) and p = 3 (for θ ≅ 125°).

Similarly, comparison of Figs. 3b and 4b (for y0 = 40 μm) show that Lock's results are effectively identical to the MiePlot results for p = 0, p = 3, p = 4, p = 7 and p = 11. However, MiePlot's results show much stronger scattering for p = 8 and p = 12, whilst showing weaker scattering for p = 1 (for θ > 10°), p = 2 (for θ < 165°) and p = 12.

These inconsistencies were initially very puzzling - especially as there is close agreement between some of the curves (such as p = 0) over certain ranges of θ but major differences at other values of θ. Despite careful checking of the MiePlot subroutines for Gaussian beam scattering, no reason for these discrepancies was obvious.

In the absence of any independent source of results of Gaussian beam scattering, it was unclear how to proceed. Fortunately, Tak Shun Chan and Professor W.K. Lee of the Chinese University of Hong Kong had developed a Mathematica program for Gaussian beam scattering which was also based on Lock's algorithms - and they generously provided me with the results of their Debye series calculations as shown in Fig. 5 below.
Fig. 5a Results for y0 = -40 μm
Reproduced by kind permission of T. S. Chan & W. K. Lee ©
Fig. 5b Results for y0 = 40 μm
Reproduced by kind permission of T. S. Chan & W. K. Lee ©

Fig. 5    Results from a Mathematica program for Debye series calculations for scattering of light from a sphere of radius 43.3 μm and refractive index n = 1.333, refractive index of medium = 1, λ = 0.5145 μm, Gaussian beam: x0 = 0, y0 = -40 μm (Fig. 5a) or y0 = 40 μm (Fig. 5b), z0 = 0, ω0 = 20 μm, angular resolution = 0.0225° (perpendicular polarisation only) .

Comparison of Figs. 3 and 5 indicates that the Mathematica program gives results which are essentially identical to those published by Lock in 1993. This suggested that MiePlot's results are incorrect - which would not be surprising given the complexity of the Gaussian beam algorithms!

However, detailed examination of Lock's results revealed some puzzling behaviour. For example, Lock's results shown in Fig. 3a for p = 2 scattering with y0 = -40 μm show a primary rainbow near θ = 139° with maximum intensity of about 105, whereas Fig. 3b for y0 = 40 μm shows a maximum (also near θ = 139°) of about 102. Apart from this difference in intensity by a factor of about 103, these p = 2 curves are very similar in shape.

Beam centre y0
(μm)
y
(μm)
Abs[y - y0]
(μm)
Intensity relative
to on-beam intensity
-40
-40
0
1
-40
  -37.34
   2.7
        0.9653
 40
  -37.34
77.3
    1.03 x 10-13

Table 1  Intensity of Gaussian beams with ω0 = 20 μm related to primary rainbow

Geometrical optics suggests that p = 2 scattering causes the primary rainbow at θ = 137.5° corresponding to incident rays with an impact parameter b ≅ -0.8624 (equivalent to to y ≅ -37.3 μm for a scattering sphere of radius 43.3 μm). Table 1 examines the intensity characteristics of the relevant Gaussian beams: it shows that a beam of half-width ω0 = 20 μm and a beam centre of y0 = -40 μm would generate a strong primary rainbow (because the intensity of such a Gaussian beam at y = -37.3 μm would be about 0.965 of its on-beam intensity). On the other hand, a beam centred on y0 = 40 μm would generate an extremely weak primary rainbow at θ = 137.5° (because the intensity of such a Gaussian beam at y = -37.3 μm would be about 10-13 of its on-beam intensity). In other words, the primary rainbow caused by a Gaussian beam with y0 = 40 μm would have an intensity of about 10-13 relative to that caused by a beam with y0 = -40 μm, instead of a relative intensity of 10-3 suggested by Lock's results.

Applying this type of analysis to p = 3 scattering indicates further anomalies in Lock's results. Lock's results in Fig. 3b for p = 3 scattering with y0 = 40 μm show a secondary rainbow near θ = 129° with maximum intensity of about 104, whereas those for y0 = -40 μm show a maximum (also near θ = 129°) of about 10. These p = 3 curves are very similar in shape, apart from a difference in intensity by a factor of about 103.

Beam centre y0
(μm)
y
(μm)
Abs[y - y0]
(μm)
Intensity relative
to on-beam intensity
   40
40
0
1
   40
     41.16
     1.16
         0.9932
 -40
     41.16
   81.16
    4.95 x 10-15

Table 2  Intensity of Gaussian beams with ω0 = 20 μm related to the secondary rainbow

In this case, geometrical optics suggests that the secondary rainbow at θ = 129.9° is caused by incident rays with an impact parameter b ≅ 0.9507 corresponding to y ≅ 41.16 μm for a scattering sphere of radius 43.3 μm. Table 2 indicates that a beam centre of y0 = 40 μm would generate a strong secondary rainbow (because the intensity of such a Gaussian beam at y = 41.16 μm would be about 0.993 of its on-beam intensity). On the other hand, a beam centred at y0 = -40 μm would generate an extremely weak primary rainbow (because the intensity of such a Gaussian beam at y = 37.3 μm would be about 5 x 10-15 of its on-beam intensity). Hence, the secondary rainbow caused by a Gaussian beam with y0 = -40 μm should have an intensity of about 5 x 10-15 relative to that caused by a beam with y0 = 40 μm, instead of a relative intensity of 10-3 suggested by Lock's results.

As the above analysis is based on geometrical optics, it might be considered rather simplistic - especially as geometrical optics fails near the rainbow angle. Nevertheless, Fig. 6 below shows good agreement between MiePlot's results for p = 3 scattering using the Debye series and using a modified version of Young's method. Note that Fig. 6 contradicts Lock's results (shown in Fig. 3a and 3b above) which indicate a intensity difference of only about 10-3 between the secondary rainbows at θ ≅ 129° caused by beams with y0 = -40 μm and y0 = 40 μm. Fig. 6 shows that geometric optics calculations can provide independent verification of the Debye series calculations: the close agreement between these independent methods is very reassuring.


Fig. 6    MiePlot results for y0 = -40 μm for Debye series calculations (red curves) and a modified version of Young's method of ray tracing calculations (blue curves) for p = 3 scattering of light from a sphere of radius 43.3 μm and refractive index n = 1.33, refractive index of medium = 1, λ = 0.5145 μm, Gaussian beam: x0 = 0, z0 = 0, ω0 = 20 μm (perpendicular polarisation only).

Although the result shown in Fig. 6 suggested that the MiePlot results were correct, it was very difficult to determine the reasons for the anomalous results obtained by Lock and by Chan & Lee. All of the calculations were based on Lock's 1993 paper, except that MiePlot had incorporated various techniques proposed in Lock's 1995 paper entitled Improved Gaussian beam-scattering algorithm. A lot of effort was spent comparing the results of various algorithms within Lock's two papers, but eventually it was realised that MiePlot was using equations 57 and 58 in the 1995 paper to determine the value of mmax whilst the other two programs had adopted mmax = 5 as proposed in section 3 of Lock's 1993 paper. In practice, MiePlot used mmax = 20 for Lock's conditions. This simple difference makes a dramatic difference to the accuracy of the Debye series calculations - as can be seen in Fig. 7 below.

Fig. 7a Debye series results for p = 0
Fig. 7b Debye series results for p = 3

Fig. 7    Effect of varying mmax on Debye series calculations for p = 0 (Fig. 7a) and p = 3 (Fig. 7b) scattering of light from a sphere of radius 43.3 μm and refractive index n = 1.33, refractive index of medium = 1, λ = 0.5145 μm, Gaussian beam: x0 = 0, y0 = -40 μm, z0 = 0, ω0 = 20 μm (perpendicular polarisation only)

Fig. 7 indicates that, for the conditions selected by Lock, the value of mmax is crucially important at θ ≅ 70° for Debye series p = 0 calculations and at θ ≅ 130° for Debye series p = 3 calculations. Note that Fig.7a includes some green curves for mmax = 20 which are almost entirely hidden by the blue curves for mmax = 15, thus indicating that mmax = 15 would probably be adequate for these conditions. On the other hand, Fig.7b shows that mmax = 20 is needed for p = 3 calculations (whereas mmax = 25 gives identical results to mmax = 20). Although increasing the value of mmax gives more accurate results, this necessarily increases the computation time (to a first approximation, doubling the value of mmax will double the computation time). As the Gaussian beam algorithms are already very time-consuming (especially for large spheres and narrow beams), we need to find the optimum balance between accuracy and computation time. The next page explores this topic in more detail.

Care must be taken to avoid computer modelling of unrealistic situations - for example, there is no point in making calculations for extremely narrow beams: Even if the above conditions are satisfied, it must be emphasised that the algorithms for scattering of Gaussian beams are only approximations. Fig. 8 below shows that the Debye p = 3 calculations give results comparable to those for ray tracing for ω0 = 30 μm and ω0 = 20 μm - but not for ω0 = 15 μm.


Fig. 8    Effect of varying half-width ω0 of the Gaussian beam on Debye series calculations (coloured lines) and on a modified version of Young's method of ray tracing calculations (black curves) for p = 3 scattering of light from a sphere of radius 43.3 μm and refractive index n = 1.33, refractive index of medium = 1, λ = 0.5145 μm, Gaussian beam: x0 = 0, y0 = -40 μm, z0 = 0 (perpendicular polarisation only)


Note that, unlike the problems shown in Figs. 7 a and 7b, the problem highlighted in Fig. 8 cannot be rectified by increasing the value of mmax. The reason for the errors shown in Fig. 8 may be due to the beam shape coefficients not giving an accurate representation of the required Gaussian beam – as shown in Figs. 1 and 2 of the following paper: To avoid such problems, it is obviously sensible to to compare MiePlot's results for Mie and Debye calculations with the results of Young (mod) calculations.



Page updated on 24 April 2012
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