Scattering of Gaussian beams by spherical particles

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.

• 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

MiePlot uses Lock's definitions for the beam coordinates (x₀, y₀ and z₀) measured relative to the centre of the scattering sphere - as illustrated below.

Fig. 1   Coordinate system for a Gaussian beam relative to the centre of the scattering sphere
Reproduced by kind permission of J. A. Lock ©

The above diagram shows the general case in which x₀ ≠ 0 and y₀ ≠ 0. The beam propagates in the z direction, so that an "on axis" beam (specified by x₀ = 0 and y₀ = 0) would pass through the centre of the sphere. The waist (minimum width) of the focussed Gaussian beam is located at (x₀, y₀, z₀) where the half width of the beam is defined by ω₀.

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   Polar plot of Mie theory calculations for scattering of light of λ = 0.5145 μm, from a sphere of radius 43.3 μm and refractive index n = 1.33, refractive index of medium = 1 with an incident Gaussian beam defined by x₀ = 0, y₀ = -40 μm, z₀ = 0 and ω₀ = 20 μm.

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 ω₀ = 20 μm is less than the sphere's radius. As it may be hard to interpret the polar plot in Fig. 2, the data has been replotted in Fig. 3 below using a logarithmic scale for intensity and a linear scale for scattering angle θ.

Fig. 3   Mie theory calculations for scattering of light of λ = 0.5145 μm, from a sphere of radius 43.3 μm and refractive index n = 1.33, refractive index of medium = 1 with an incident Gaussian beam defined by x₀ = 0, y₀ = -40 μm, z₀ = 0 and ω₀ = 20 μm

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 p indicates that the light that has suffered p - 1 internal reflections. For example, the primary rainbow is due to p = 2 contributions (corresponding to one internal reflection), whilst the secondary rainbow is due to p = 3 contributions (corresponding to two internal reflections). Fig. 4 below shows the results of Debye series calculations for individual values of p.

Fig. 4   Debye series calculations for scattering of light of λ = 0.5145 μm, from a sphere of radius 43.3 μm and refractive index n = 1.33, refractive index of medium = 1 with an incident Gaussian beam defined by x₀ = 0, y₀ = -40 μm, z₀ = 0 and ω₀ = 20 μm (perpendicular polarisation only)

Fig. 4 indicates that the primary rainbow (p = 2) resulting from incident light with a beam centre y₀ = -40 μm appears around θ = 138°, whilst the secondary rainbow (p = 3) appears around θ = -128°. &nsbp; Note that, if the beam centre was changed to y₀ = 40 μm, the primary rainbow (p = 2) would appear at around θ = -138°, whilst the the secondary rainbow (p = 3) would appear at around θ = 128°.

As the beam centre y₀ = -40 μm corresponds to an impact parameter b = y₀ / r = -40 / 43.3 ≅ -0.9238, this beam is well-positioned to generate rainbows: geometric optics indicates that, for n = 1.33, the impact parameter corresponding to the primary rainbow angle at θ = 137.5° is b ≅ -0.8624 whilst that for the secondary rainbow angle near θ = -129.9° is b ≅ 0.9507. Higher order rainbows require impact parameters b very close to 1 (e.g the p = 12 rainbow corresponds to b = 0.9973 which is very close to an edge ray). Thus Fig. 4 shows many rainbows of various orders - some of which cannot be seen when the sphere is illuminated by plane waves. For exanple, the p = 6 rainbow shown around θ = 130° is normally swamped by the stronger p = 3 rainbow.

Fig. 5   Comparison of Mie theory and Debye series (p = 0 to 12) calculations for scattering of light of λ = 0.5145 μm, from a sphere of radius 43.3 μm and refractive index n = 1.33, refractive index of medium = 1 with an incident Gaussian beam defined by x₀ = 0, y₀ = -40 μm, z₀ = 0 and ω₀ = 20 μm (perpendicular polarisation only)

The blue curve in Fig. 5 shows that the vector sum of the Debye series calculations for p = 0 through p = 12 is essentially equivalent to the Mie solution.

Fig. 6   Comparison of Debye series and Young mod. (p = 2) calculations for scattering of light of λ = 0.5145 μm, from a sphere of radius 43.3 μm and refractive index n = 1.33, refractive index of medium = 1 with an incident Gaussian beam defined by x₀ = 0, y₀ = -40 μm, z₀ = 0 and ω₀ = 20 μm

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 p = 2 calculations based on geometric optics and the Debye series (at least for scattering angles θ > 139°).

Fig. 7   Mie theory calculations for scattering of light of λ = 0.5145 μm from a sphere of radius 1000 μm and refractive index n = 1.336, refractive index of medium = 1 with an incident Gaussian beam defined by x₀ = 0, y₀ = -950 μm, z₀ = 0 and ω₀ = 200 μm.

Fig. 8   Mie theory calculations for scattering of light of λ = 0.5145 μm from a sphere of radius 1000 μm and refractive index n = 1.336, refractive index of medium = 1 with an incident Gaussian beam defined by x₀ = 0, y₀ = -950 μm, z₀ = 0 and ω₀ = 200 μm (perpendicular polarisation only)

Figs. 7 and 8 show another example of Mie calculations for scattering of a Gaussian beam: in this case, the beam width ω₀ = 200 μm is about 20% of the 1000 μm radius of the sphere (whereas the previous calculations in Figs. 4 and 5 assumed a beam width of about 46% of the radius). The narrow width of the beam makes it possible to observe higher order rainbows.

Fig. 9   Comparison of Mie theory and Young mod. calculations (for p = 0 through p = 12) for scattering of light of λ = 0.5145 μm from a sphere of radius 1000 μm and refractive index n = 1.336, refractive index of medium = 1 with an incident Gaussian beam defined by x₀ = 0, y₀ = -950 μm, z₀ = 0 and ω₀ = 200 μm (perpendicular polarisation only)

Given the success of ray tracing techniques shown in Fig. 6, it is interesting to compare the vector sum of ray tracing calculations for p = 0 through p = 12 with the results of Mie calculations as shown in Fig. 9. Although there are some significant differences between the red and blue curves in Fig. 9, the very close agreement (especially in areas where p = 0 and p = 1 contributions are dominant) shows that ray tracing can act as a powerful independent technique to check the validity of Mie calculations. It is also important to note that the ray tracing calculations are extremely quick - the blue curve in Fig. 9 took about 5 seconds to compute, compared with several hours for the red curve!

Fig. 10   Debye p = 2 calculations for scattering of light of λ = 0.65 μm from a sphere of radius 10 μm and refractive index n = 1.33257, refractive index of medium = 1 with 2 incident Gaussian beams defined by x₀ = 0, y₀ = -10 μm (red curves) and y₀ = 10 μm (blue curves), z₀ = 0 and ω₀ = 2 μm

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 p = 2 calculations for two Gaussian beams - one with y₀ = -10 μm (red curves) and one with y₀ = 10 μm (blue curves). These beam parameters have been chosen because the beam centres hit the edges of the sphere and thus generate surface waves. The peak around θ ≅ 142° corresponds to the primary rainbow caused by the Gaussian beam with y₀ = -10 μm, whilst the peak around θ ≅ 218° (= 360° - 142°) corresponds to the primary rainbow caused by the Gaussian beam with y₀ = 10 μm. Note that the primary rainbow is dominated by perpendicular polarisation, whereas the scattered light at θ ≅ 180° is due to surface waves which are dominated by parallel polarisation.

Fig. 11   Vector sum of Debye p = 2 calculations for scattering of light of λ = 0.65 μm from a sphere of radius 10 μm and refractive index n = 1.33257, refractive index of medium = 1 with 2 incident Gaussian beams defined by x₀ = 0, y₀ = -10 μm and y₀ = 10 μm, z₀ = 0 and ω₀ = 2 μm