# Impulse response of a sphere

Mie theory offers a rigorous solution to the problem of scattering of light by a homogeneous spherical particle; but it reveals no information about the scattering processes. Fortunately, greater understanding of the scattering mechanisms can be obtained by using:

Although ray tracing cannot match the accuracy of Mie theory and the Debye series, it is nevertheless useful because it offers a simple explanation for many optical effects such as rainbows. It is also a valuable starting point for investigations of scattering mechanisms, as in Fig. 1.

Fig. 1   Geometric rays that contribute to scattering at θ = 150°
assuming that the refractive index of sphere n1 = 1.33257 and the refractive index of medium n0 = 1

The sphere in Fig. 1 is illuminated by a beam of light - which can be considered as an infinite set of parallel rays arriving from the left of the diagram. Fig. 1 shows selected rays that result in scattering angle θ = 150° for p ≤ 5 (i.e. for rays suffering 4 or fewer internal reflections). Each of the rays is associated with an impact parameter b – where b = 0 indicates a "central" ray aimed at the centre of the sphere, whilst b = ± 1 indicates "edge" rays that are tangential to the top (or bottom) of the sphere. Although the individual rays in Fig. 1 all leave the sphere at θ = 150°, they take very different paths - and consequently have varying path lengths. Table 1 shows the calculated time delay t for each ray (measured between the dashed reference lines "before" and "after" the sphere) using the following equations:

t = 2 r / c [ n0 [1 - √(1 - b2)] + n1 p cos [arcsin [b n0 / n1]] ]
where:
b is the impact parameter
r is the radius of the sphere
n1 is the refractive index of the sphere
n0 is the refractive index of the medium
c is the speed of light in a vacuum = 3 x 108 metres per second.
 Ray p Impact parameter b Time delay t fs A 0 -0.2588 2.3 B 2 0.5351 173.1 C 2 0.9862 175.1 D 4 0.1307 354.2 E 5 0.6475 404.1 F 5 -0.4686 423.6 G 6 0.9198 426.2 H 6 -0.7876 455.6 I 6 0.0747 532.4 J 7 0.4249 595.7 K 7 0.3056 608.5 L 8 0.6562 634.9 M 8 0.5608 656.2

Table 1  Propagation parameters for rays A - F shown in Fig. 1 (together with rays G - M) all of which result in θ = 150°
assuming a sphere of radius r = 10μm, refractive index of sphere n1 = 1.33257 and refractive index of medium n0 = 1.
N.B. The time delays are quoted in femtoseconds (1 fs = 10-15 seconds).

The time delays listed in Table 1 suggest that it might be possible to identify the contributions made by different rays by transmitting a short pulse of light towards the sphere and by measuring the time of arrival of the various pulses as received by a detector located at θ = 150°. In practice, it would be difficult to perform such an experiment, but the MiePlot computer program allows you to simulate this process, using the techniques outlined by Bech & Leder in the following references:

• H. Bech and A. Leder, “Particle sizing by ultrashort laser pulses - numerical simulation,” Optik 115, 205–217 (2004);
• H. Bech and A. Leder, “Particle sizing by time-resolved Mie calculations - A numerical study,” Optik 117, 40–47 (2006).

The MiePlot program first performs a Fast Fourier Transform (FFT) on the pulse shape in the time domain to determine the spectrum of the pulse, which is then multiplied by the results of scattering calculations for a range of scattering angles θ at a number of discrete wavelengths across the bandwidth of the pulse. The results for a given value of θ as a function of wavelength are then subjected to another FFT so as to produce the time domain impulse response for that value of θ – as shown in Fig. 2 below:

Fig. 2   Impulse response of a water droplet of radius r = 10 μm for a 5 fs pulse of red light (nominal wavelength λ = 650 nm) at scattering angle θ = 150°.
The top and middle graphs show the results of Mie calculations, whereas the bottom graph shows the separate terms of the Debye series.
The top graph takes account of dispersion (i.e. due to the varying refractive index n1 of the sphere across the bandwidth of the pulse).
The middle and bottom graphs assume that the refractive index of the sphere n1 = 1.33257 + i 1.67E-08 and that this does not change with wavelength.

The graphs in Fig. 2 shows MiePlot's calculated impulse response for scattering angle θ = 150° using Mie theory 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 at -3 dB points; but the bandwidth of the pulse has been truncated at -40 dB points (404 nm – 1664 nm);
• 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.

The effects of dispersion can be seen by comparing the top and middle graphs in Fig. 2. Each of these graphs has a pulse matching the pulse marked "p = 0" in the bottom graph corresponding to ray A in Fig. 1 caused by reflection from the exterior of the sphere. Note that these p = 0 pulses have essentially identical shapes in top two graphs, thus indicating that dispersion has little or no effect on these pulses. But, for p > 0, the pulses in the top graph are broadened and slightly shifted in time relative to the middle graph. Note, for example, that the p = 4 pulses have very different shapes and, furthermore, show maximum intensity at t = 359.1 fs in the upper graph and t = 354.2 fs in the middle graph.

Although the middle and bottom graphs incorrectly assume that the refractive index of the sphere n1 is independent of wavelength, it is nevertheless useful to see that the pulses on the bottom graph of Fig. 2 coincide with timings from Table 1, as indicated by the dotted vertical lines marked A to H. The very close agreement between the values of t given by the various independent methods of calculation (i.e. Mie, Debye and ray-tracing) gives considerable confidence in the results.

However, it is also important to recognize that some of the scattered pulses shown for θ = 150° in Fig. 2 were not predicted by the ray tracing exercise in Fig. 1 – for example, there is an extra p = 2 pulse at t ≈ 210 fs, a p = 3 pulse at t ≈ 220 fs and a p = 5 pulse at t ≈ 416 fs. What causes these “non-geometrical” pulses?

The fact that the p = 2 pulse at t ≈ 210 fs and the p = 5 pulse at t ≈ 416 fs are both dominated by parallel polarization suggests the involvement of surface waves – which are typically generated by rays with impact parameter b = 1 or b = -1.

Fig. 3   A p = 2 ray path involving surface waves travelling 44.5° anti-clockwise along the circumference of the sphere resulting in scattering angle θ = 150°.

Fig. 3 shows an incident ray arriving from the left of the diagram with impact parameter b = -1. This ray is refracted into the sphere at point A and then suffers an internal reflection at point B. Ray tracing suggests that this ray would travel to point C where it would leave the sphere, being refracted so that it travels along the tangent at C (shown as a black line with an arrow), thus experiencing a total anti-clockwise deviation of 165.5°. However, such rays can also generate surface waves which travel along the circumference of the sphere. In this particular case, surface waves travelling from C to D through an arc of 44.5° would result in anti-clockwise deviation of 165.5° + 44.5° = 210° which is equivalent to the required value of θ = 150°. Assuming that the surface waves propagate at a speed determined by the refractive index n0 of the medium (rather than by the refractive index n1 of the sphere), the calculated time delay is t = 210 fs for the ray path shown in Fig. 3, which agrees well with the bottom graph of Fig. 2.

Fig. 4   A p = 5 ray path involving surface waves travelling 96.3° clockwise along the circumference of the sphere resulting in scattering angle θ = 150°.

A similar explanation is shown in Fig. 4 for the p = 5 pulse at t ≈ 416 fs. The ray path in Fig. 4 is more complicated than in Fig. 3 because it involves 4 internal reflections (at points B, C, D and E) before exiting the sphere at point F. According to geometrical optics, the ray should follow the tangent at F (shown as a black line with an arrow) and thus experience a total deviation of 413.7°;. However, surface waves travelling from F to G (an arc length of 96.3°) would result in clockwise deviation of 413.7° + 96.3° = 510° corresponding to the required value of θ = 150°. The calculated time delay for the ray path shown in Fig. 4 is t = 416.3 fs, which is consistent with the bottom graph of Fig. 2.

Surface waves seem to provide a good explanation for the "non-geometrical" pulses at t ≈ 210 fs and t ≈ 416 fs, but the cause of the p = 3 pulse at t ≈ 220 fs needs further investigation – as shown on the next page.

Page updated on 15 May 2011