The scattering of light from a spherical object has been dealt with by many investigators since the time Gustav Mie worked out the general theory. The formal solution by Mie assumed that the object is composed of a homogeneous, isotropic and optically linear material irradiated by an infinitely extending plane wave. The formal solution of Mie can be extended to include any laser beam mode. The following is a general solution to the boundary value problem in vector spherical harmonics by separating the incident, scattered and internal electromagnetic fields and satisfying the Helmholtz equation:

(1.1)

where is a scalar field in spherical coordinates . The electric and magnetic fields are then expanded in terms of the even and odd multipole vector fields. The fields are expressed as a function of the scalar field, , by

(1.2)

(1.3)

The general solution can be expressed as a sum over the vector spherical harmonics. The scattered, internal and incident electric fields respectively are

(1.4)

(1.5)

(1.6)

where a, b, c, and d are expansion coefficients and the o and e notation refers to odd and even multipoles. Similar expression exist for the magnetic fields Hs, Ht, and Hi.

The vector spherical harmonic functions, when is the radius vector, are defined as the following:

(1.7)

(1.8)

where (1.9)

and are the spherical Bessel functions and of order n. At this point the subscripts can be labeled as the following: = the mode order and n = the mode number. The mode order actually gives the number of maxima in the radial dependence. The mode number gives the number of maxima in the angular dependence between 0 and 180 degrees. The mode number also specifies the order of the Bessel or Hankel function that constitute the radial part of the MDR. The expansion coefficients are the Mie coefficients which remain the same for any incident laser beam mode:

(1.10)

(1.11)

(1.12)

(1.13)

where the variable x is called the size parameter and is equal to 2pa/, a being the radius of the sphere and m is the refractive index of the scattering medium. The primes in the scattering coefficients denote differentiation with respect to the arguments. Kim and Lee7 have worked out the theory for light scattering of a dielectric sphere in a Gaussian beam and their results are similar except that the Gaussian amplitude gives extra expansion coefficients.

The characteristic modes are called morphology dependent resonances or MDR's since they depend on the shape, size and the characteristics of the scattering medium. These resonance locations can be determined by calculating the poles of the expansion coefficients given by equations (1.10) and (1.12), since the denominators of are the same as those of, , respectively. We get for the conditions,

for TM modes, (1.14)

for TE modes (1.15)

from which the resonance x values or size parameters can be determined. The quality factor or Q of these resonances is defined as the following,15

(1.16)

or

(1.17)

where x is the full width at half maximum (FWHM) of the resonance peak centered at x. The period is to be interpreted in terms of the resonant frequency of the cavity (period =). In a system where losses, such as absorption, must be taken into consideration, an effective quality factor is then,

(1.18)

The resonance locations can be experimentally observed in optical levitation experiments where the spherical particle changes in size by means of evaporation. The signal the experimentalist measures by collecting the light scattered from the drop and collected by a detector is called the scattering intensity, F, which is a function of the angles q and f.

(1.19)

where the scattering functions, , and are the nonzero components of the scattering matrix and contain information on the amplitude and phase of the scattered light. The scattered intensity,, is a dimensionless quantity and normalized with respect to the incident energy flux. These scattering functions have been calculated13 and are found to be of the following form:

(1.20a)

(1.20b)

where and

A parameter can be defined which describes the internal electric field intensity due to the incident laser field. This function is called the source function S,

(1.21)

This parameter has been calculated theoretically by several investigators to determine the internal field's contribution to processes depending on the electric field. The computations show that for droplets transparent to the incident radiation, the electric field intensity is higher at the shadow side of the droplet. For droplets that absorb at the incident radiation wavelength, the highest intensity is at the front face of the droplet. When the droplet size parameter coincides with a resonance, or an x that satisfies equation (1.14) or (1.15), the field is strongest near the perimeter of the droplet. The field intensities have been calculated to be on the order of 10 to higher than the incident beam intensity off resonance and up to on a resonance. These resulting field intensities can then be high enough to initiate stimulated processes such as stimulated Raman emission, stimulated fluorescence emission, stimulated Brilluoin emission and other multi-photon processes. For the most part, these processes have a spontaneous component from which the stimulated intensity grows.

Return to Single Particle Laser Spectroscopy Page