Impulse response v. scattering angle

As explained on the previous page, the MiePlot computer program (freely available from here) can simulate the scattering of short pulses (e.g. duration of 5 fs) of light by homogeneous spherical particles using all of the calculation methods available in MiePlot, such as:

• rigorous methods, such as Mie theory and the Debye series.
• approximate methods, such as ray tracing and ray tracing with interference;

The previous page showed examples of the impulse response calculated using Mie theory for scattering at θ = 150°. Similar calculations using the Debye series can identify the order p of the scattering process where:

• p = 0 implies diffraction or reflection from the exterior of the sphere;
• p = 1 implies transmission through the sphere;
• p = 2 implies one internal reflection;
• p = 3 implies two internal reflections; and so on .....

The diagrams on this page have been calculated for the following conditions:

• Nominal wavelength λ = 650 nm;
• Pulse duration: 5 fs (half-amplitude duration, raised-cosine pulse shape);
• Pulse bandwidth: 564 nm – 767 nm (-3 dB); 404 nm – 1664 nm (-40 dB);
• Sphere radius r = 10 μm;
• Refractive index of sphere: n₁ = 1.33257 + i 1.67E-08 at nominal wavelength λ = 650 nm (N.B. The real part of the refractive index of water is 1.344 at 404 nm and 1.313 at 1664 nm, whilst the imaginary part is negligible for the current purposes.);
• Refractive index of medium: n₀ = 1.

Note that all of the diagrams on this page have been calculated using the assumption that the refractive index of water is constant across the bandwidth of the 5 fs pulse. This assumption is NOT correct (as indicated on the previous page), but it facilitates comparisons with the results of ray tracing calculations which assume a monochromatic source of light at the nominal wavelength λ = 650 nm.

Ray tracing using geometrical optics can identify the ray paths contributing to the scattering –and can even be extended to include the effects of surface waves. It is clear that examining the impulse response at a particular scattering angle θ can give valuable information about the scattering processes at that value of θ. However, much more information is revealed if the impulse response is plotted as a function of θ, as in Fig. 1 below.

Fig. 1   Mie theory calculations of the impulse response of a water droplet of radius r = 10 μm for a 5 fs pulse of red light (nominal wavelength λ = 650 nm) as a function of scattering angle θ.
The refractive index of the sphere n₁ = 1.33257+ i 1.67E-08 and the refractive index of the medium n₀ = 1 are assumed to be constant across the bandwidth of the pulse.
The intensity of the scattered pulses is coded according to the false-color scale shown above the diagram.

Fig. 1 has been calculated using Mie theory, but Debye series calculations have been used to identify the order p of the scattering processes. The maximum intensity occurs for p = 0 at θ = 0° (forward scattering) and t ≈ 67 fs. Note the time reference (t = 0) corresponds to p = 0 reflection from the exterior of the sphere at θ = 180°

Fig. 2   As Fig. 1, but using geometrical optics.

Fig. 2 is similar to Fig.1 except that the calculations are based on geometrical optics. There are significant differences between these two diagrams (mainly due to that fact that Fig.2 excludes diffraction and surface waves), but the overall pattern is similiar. This similarity will be used to explain many of the features seen in Fig. 1.

Comparison of Figs. 1 and 2 shows that the intensity of scattering predicted by geometrical optics can be inaccurate, but it is useful to consider the propagation of rays and their associated time delays. In particular, a ray with impact parameter b results in scattering at angle θ with a time delay t – as defined by the following equations for a sphere of radius r and refractive index n₁ immersed in a medium with refractive index n₀:

t = 2 r / c [ n₀ [1 - √(1 - b²)] + n₁ p cos [arcsin [b n₀ / n₁]] ] where c is the speed of light in a vacuum = 3 x 10⁸ metres per second.

i = arcsin (b)

for p = 0:    θ = 2 i - 180°

for p = 1:    θ = 2 i - 2 arcsin [b n₀ / n₁]

for p > 1:    θ = 2 i - 2 p arcsin [b n₀ / n₁] + (p - 1) 180°

These equations have been used to identify the detailed scattering mechanisms in the following diagrams which show the results of Debye series calculations for individual values of p.

Fig. 3   Impulse response for p = 0 as a function of scattering angle θ	Fig. 4   Impulse response for p = 1 as a function of scattering angle θ

Fig. 3 shows the impulse response calculated using the Debye series for p = 0 (external reflection + diffraction). The curve on the left side of Fig. 1 corresponds to reflection from the exterior of the sphere – as shown by the line marked with the values of the impact parameter b obtained from ray-tracing calculations. In this case, b = 0 corresponds to θ = 180° and t = 0, whilst b = 1 corresponds to θ = 0° and t = 66.7 fs. The p = 0 scattering contributions at t > 66.7 fs seem to be exclusively the result of diffraction, as described in more detail in the papers entitled "Separating diffraction from scattering: the million-dollar challenge" and "Mie scattering in the time domain. Part II. The role of diffraction" (both available here).

Fig. 4 shows the impulse response calculated using the Debye series for p = 1 (transmission through the sphere). Again, the time delays calculated by ray-tracing have been superimposed on the curves: in this case, b = 0 corresponds to θ = 0° and t = 88.8 fs, whilst b = 1 corresponds to θ = 82.75° and t = 125.4 fs. The p = 1 scattering contributions for t > 125.4 fs are due to surface waves – as marked by the dashed lines.

Fig. 5   Impulse response for p = 2 as a function of scattering angle θ	Fig. 6   Impulse response for p = 3 as a function of scattering angle θ

Fig. 5 shows the intensity of p = 2 scattering (caused by one internal reflection in the sphere) calculated using the Debye series. The primary rainbow corresponds to the zone of maximum intensity occurring at θ = 142° and t ≈ 170 fs. Although this diagram is more complicated than Figs. 3 and 4, it can also be explained by reference to ray-tracing calculations: in this case, b = 0 corresponds to a ray that passes through centre of the sphere and then suffers an internal reflection and is scattered at θ = 180° (backscattering) with a delay t = 177.7 fs. As b increases, the value of θ reduces until it reaches its minimum value at θ = 137.9° (the geometric rainbow angle) when b = 0.8611. Further increases in b cause θ to increase until θ = 165.5° when b = 1. Note that two geometric rays can cause scattering when 137.9° > θ < 165.5°. Interference between these two rays causes the supernumerary arcs on the primary rainbow. Fig.5 shows that the results of the Debye p = 2 calculations agree very well with the time delays from ray-tracing calculations for 0 ≤ b ≤ 1. However, the Debye results also show the effects of surface waves – as indicated by the dashed lines starting from θ = 165.5° and t = 184.1 fs. The V-shaped pattern centred on t = 193 fs is caused by surface waves. Interference between the "short path" surface waves with t < 193 fs and the "long path" surface waves with t > 193 fs results in the glory (as described in the Applied Optics paper "How are glories formed?" available here).

Fig. 6 shows the intensity of p = 3 scattering (caused by two internal reflections in the sphere) calculated using the Debye series. The secondary rainbow corresponds to the zone of maximum intensity occurring at θ ≈ 125° and t ≈ 235 fs. Fig. 6 can also be explained by reference to ray-tracing calculations: in this case, b = 0 corresponds to a ray that passes through centre of the sphere and then suffers two internal reflection and is scattered at θ = 0° (forward scattering) with a delay t = 266.5 fs. As b increases, the value of θ increases until itreaches its maximum value at θ = 129.2° (the geometric rainbow angle) when b = 0.9503. Further increases in b cause θ to decrease until θ = 111.8° when b = 1. The results of the Debye p = 3 calculations coincide with time delays derived from ray-tracing calculations in Fig. 6 for 0 ≤ b ≤ 1, but the Debye results also show the effects of surface waves – as indicated by the dashed lines starting from θ = 111.8° and t = 242.8 fs.

On the previous page, explanations were offered for all of the pulses scattered at θ = 150°, except for a p = 3 pulse at t ≈ 220 fs (marked in Fig. 6 by the + symbol) which is not far from the secondary rainbow at θ ≈ 125° and t ≈ 235 fs. Fig. 6 shows that this pulse coincides with the complex ray, which appears as the diagonal "finger" of intensity in a zone where there are no geometric p = 3 rays and where the Debye calculations show decreasing intensity. Note that, in this zone, the time delay t decreases as θ increases.

Fig. 7   Impulse response for p = 4 as a function of scattering angle θ	Fig. 8   Impulse response for p = 5 as a function of scattering angle θ

Fig. 7 shows the intensity of p = 4 scattering (caused by 3 internal reflections in the sphere) calculated using the Debye series. The tertiary rainbow corresponds to the zone of maximum intensity occurring at θ ≈ 39° and t ≈ 295 fs. In this case, ray-tracing calculations show that b = 0 corresponds to a ray that passes through centre of the sphere and then suffers 3 internal reflections and is scattered at θ = 180° (backscattering) with a delay t = 355.4 fs. As b increases, the value of θ decreases until it reaches θ = 0° when b = 0.7631. Then the value of θ increases with increasing b until it reaches its maximum value at θ = 41.9° (the geometric rainbow angle for p = 4) when b = 0.9738. Further increases in b cause θ to reduce until θ = 29° when b = 1. The results of the Debye p = 4 calculations coincide with the time delays derived from ray-tracing calculations for 0 ≤ b ≤ 1. However, the Debye results also show the effects of surface waves – as indicated by the dashed lines starting from θ = 29° and t = 301.5 fs which form the V-shaped pattern centred on t = 319 fs.

Fig. 8 shows the intensity of p = 5 scattering (caused by 4 internal reflections in the sphere) calculated using the Debye series. The quaternary rainbow corresponds to the zone of maximum intensity occurring at θ ≈ 47° and t ≈ 360 fs. In this case, ray-tracing calculations show that b = 0 corresponds to a ray that passes through centre of the sphere and then suffers 4 internal reflection and is scattered at θ = 0° (forward scattering) with a delay t = 444.2 fs. As b increases, the value of θ increases until it reaches θ = 180° when b = 0.5589. Further increases in b cause θ to decrease until its minimum value at θ = 43.5° (the geometric rainbow angle for p = 5) when b = 0.9837. Further increases in b cause θ to increase until θ = 53.7° when b = 1. The results of the Debye p = 5 calculations coincide with time delays derived from ray-tracing calculations for 0 ≤ b ≤ 1, but the Debye results also show the effects of surface waves – as indicated by the dashed lines starting from θ = 53.7° and t = 360.1 fs which form the V-shaped pattern centred on t = 436 fs.

Page updated on 16 March 2010