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Most of the calculation methods in MiePlot are concerned with scattering of plane waves from spherical particles. However, it is also interesting to investigate the case of illumination by a narrow beam of light, especially when the beam width is smaller than the particle. The results of such calculations are relevant to laboratory experiments in which a single drop of water is illuminated by a very narrow beam of laser light – thus isolating high-order rainbows as described in various papers such as:

- J.A. Lock,
*Theory of the observations made of high-order rainbows from a single water droplet*

Applied Optics, Vol. 26, No. 24, p. 5291, December 1987 Free download - P. H. Ng, M. Y. Tse and W. K. Lee,
*Observation of high-order rainbows formed by a pendant drop*

J. Opt. Soc. Am. B, Vol. 15, No.11, p. 2782, November 1998

Although Mie theory and the Debye series provide rigorous solutions for plane-wave scattering, there is no rigorous solution for scattering of narrow beams. Following the fundamental work of Gouesbet, Gréhan, Barton and others, it was noted that the mathematical solutions were simpler for the special case of beams with Gaussian intensity profiles. Even so, the results are still approximations – and the algorithms for Mie theory and Debye series scattering calculations are much more complicated (and time-consuming) than those for plane-wave scattering.

MiePlot's calculations for Gaussian beams are based on the algorithms in the following papers:

- J. A. Lock,
*Contribution of high-order rainbows to the scattering of a Gaussian laser beam by a spherical particle*

JOSA A Vol. 10, No. 4, p. 693, April 1993 Free download - J. A. Lock,
*Improved Gaussian beam-scattering algorithm*

Applied Optics, Vol. 34, No. 3, p. 559, January 1995 Free download

Reproduced by kind permission of J. A. Lock ©

The above diagram shows the general case in which x

It is important to note that Lock defined the beam to be polarised in the x-direction (as opposed to the usual practice of defining polarisation as being perpendicular or parallel to the scattering plane).

Following Lock's definition, the intensity of the parallel-polarised component of the scattered light is zero when the scattering plane is parallel to the y-z plane (i.e. tilt φ = 90° relative to the x-plane). Similarly, the perpendicularly-polarised component of the scattered light is zero when the scattering plane is parallel to the x-z plane (i.e. tilt φ = 0°). MiePlot allows users to choose whether to adopt Lock's definition of polarisation or to adopt the more usual definition (noting the latter option is restricted to φ = 90°). All of the results shown on this page assume the latter definition.

When a spherical particle is illuminated by plane waves (i.e. illumination due to a small light source at a large distance), the scattering pattern is symmetrical around scattering angles θ = 0° and θ = 180°,. However, illumination of a spherical droplet by a narrow beam of light generally produces asymmetrical scattering as a function of θ. It is therefore necessary to make Gaussian beam calculations for the full 360° range of scattering angles from θ = -180° to θ = 180° (or θ = 0° to θ = 360°) as shown in Fig.2 below.

Fig.2 shows that the scattering pattern of a Gaussian beam by a sphere of radius 43.3 μm radius is definitely not symmmetrical when the beam is offset from the centre of the sphere by 40 μm and the beam's half-width ω

Fig. 3 again emphasises the asymmetrical nature of the scattering pattern - the scattering patterns for positive scattering angles (i.e. 0° < θ < 180°) are completely different to those for negative scattering angles (i.e. -180° < θ < 0°). At first sight, it is difficult to explain these scattering patterns. Fortunately, Debye series calculations can be used to separate the scattering contributions that have suffered a specific numbers of internal reflections in the sphere: order

Fig. 4 indicates that the primary rainbow (

As the beam centre y

The blue curve in Fig. 5 shows that the vector sum of the Debye series calculations for

Geometrical optics is widely considered to be very inferior to rigorous techniques such as Mie and Debye calculations. However, a modified version of Young's method (involving ray tracing with interference between rays) can produce some remarkably accurate results for scattering of plane waves as shown elsewhere on this web site. Fig. 6 above shows that such techniques are also valid for scattering of Gaussian beams - note the very close agreement between

Figs. 7 and 8 show another example of Mie calculations for scattering of a Gaussian beam: in this case, the beam width ω

Given the success of ray tracing techniques shown in Fig. 6, it is interesting to compare the vector sum of ray tracing calculations for

Gaussian beam calculations can be used to explore the scattering mechanisms resulting in the atmospheric glory. More information on the formation of glories is available elsewhere on this site, but Fig. 10 above shows the results of Debye

If the sphere is simultaneously illuminated by two Gaussian beams with y

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