Light scattering calculator: homogeneous sphereMiroslaw JonaszTable of contents
IntroductionThis program calculates the scattering, absorption, and attenuation parameters, as well as the angular scattering patterns of a single homogeneous sphere according to Mie theory. This program is intended to be used in the
Windows environment. A screen resolution of 1024 x 768 is expected. Quick start If needed, modify the parameters displayed
(see "User Interface") and click on the "Run" button to
calculate the scattering properties. The results will be displayed
automatically. Results can be copied to the Windows clipboard
("Copy") and saved to a space-delimited text file ("Save").
User interfaceSetting preferencesPreferences are all user-adjustable program parameters. On program startup you will be requested to open a preferences file (extension=pre). If the file is not opened, the default preferences will be used. To save preferences, simply set the parameters in the main form and click the Preferences | Save menu item (this notation means the Save item of the Preference submenu) or Preferences | Save As.. The preferences system, intended as a convenience, can also be used to document the calculations if the results file name is synchronized with the preference file name. The preferences file name is displayed in the
Preferences file name box. Long fille names are cut to fit. Position the
cursor over the cut name to display it in full in a hint box that appears near
the cursor tip. Save - saves the current preferences to
a file (extension=pre) Miscellaneous menu itemsLimits - displays the limits of the
program parameters. Modifying program input dataEdit x button opens the sphere size
range editor. If the output is requested to be the function of the scattering angle (theta, see Setting output format) only the start parameter of the size range is displayed. If the output is requested to be the function of the particle size (see Setting output format), the remaining particle size range parameters: step, count, and end are displayed (and need to be specified) in units implied by the status of the diam check box. Based on these parameters, the relative particle size list is generated and displayed in the output form as follows
where m = 0 ... size_range_count - 1.
The size range parameters must fulfill this requirement: 0 < start <=
end. See also the Range editors section. diam check box toggles the
absolute/relative particle size definition. wavelength edit Re(n) and -Im(n) edits Edit theta button opens the scattering
angle range editor. Range editorsRange editors for the particle size and scattering angle accept any inputs that are consistent with the abstract range definition. Thus, range definitions may be accepted and displayed in the main form that do not fulfill either the program limits (see Miscellaneous menu items) on the relevant variables or the algorithm requirements. The data are verified for consistency with the light scattering algorithm only when the user clicks on the Run button. The user is advised about problems with the data if any. One of the range parameters is always
calculated. You can force a parameter to be calculated (and not editable) by
clicking on the corresponding radio button in the Calc column of the
range editor form. You can preview the effects of changing a range parameter by
clicking on Apply to apply the change. If satisifed, click Ok to
accept it. You can also click OK right away after any parameter changes.
Setting output format"Output type" group "Output as a function of"
group Runing calculations, displaying, and saving resultsRun button initiates the scattering properties calculations. If the following buttons are disabled, no
results are available. "Messages" group displays
status of the calculations. AssumptionsThe homogeneous sphere is assumed to be located in a plane monochromatic electromagnetic wave whose wavefront has infinite extent. In practical applications this condition requires that:
[Top] Definitions of the input dataThe light scattering properties of a homogenous sphere are completely described by just two parameters: the refractive index of the sphere material relative to that of the surrounding medium and the sphere diameter. Click the Limits item of the menu to view the limits of these parameters. The refractive index is a complex number:
where i 2 = -1. The real part of the refractive index characterizes the velocity of an electromagnetic wave in the sphere material. The imaginary part characterizes the absorption of light power by the sphere material. If the imaginary part is negative, i.e. ImN > 0, then the material absorbs light. The refractive index is a function of the wavelength of the electromagnetic wave. If the absolute particle size mode is selected by clicking on the diam check box, the refractive index is expected, but not checked, to correspond to the wavelength value entered. In this program ReN can assume walues from a range of 0 < ReN < arbitray_value. The ImN range is defined as 0 <= ImN <= arbitrary_value. The sphere size in the algorithm is represented by the relative particle size x [non-dimensional]
where D is the sphere diameter and Lambda is the wavelength of light in the medium surrounding the sphere. The wavelength range is limited to a range that can be viewed by clicking the Limits item of the menu. The program accepts the particle size as x (relative size) if the diam checkbox is not checked. Otherwise, the particle size is expected to be the absolute size (diameter) expressed in micrometers (10-6 m). The scattering angle, theta [deg], is
defined as the angle between the direction of the incident wave (beam axis) and
the observation direction. The angular pattern of light scattered by a sphere
is axially symmerical about the direction of the incident light. Thus, theta is
the only directional parameter required. OutputDefinitionsThis program calculates the light scattering properties of the homogeneous sphere following the Mie theory (Mie, 1908) with algorithms introduced by Deirmendijan (1969), Kattawar and Plass (1967), and Wiscombe (1980). A modern explanation of Mie theory is given, for example, by Bohren and Huffman (1983). In reference to Mie theory in its modern notation, the nondimensional versions of these properties are defined as follows:
and
where a[n] and b[n] are functions of the particle size and refractive index, and pi[n] and tau[n] are functions of the scattering angle. These functions are defined, for example, by Bohren and Huffman (1983). S1 and S2 are complex functions. Thus, their (real) magnitudes |S1|2 and |S2|2 are of interest in comparing with results of an experiment and are displayed. The S11 that is also displayed, is a real function of the scattering angle and is defined as follows:
This latter quantity is the element [1][1] of
the Mueller matrix, which completely describes the amplitude and polarization
of light scattered by a particle. See, for example, Bohren and Huffman (1983), for an in-depth discussion of
Mueller matrices and polarization effects. TypesParticle size is always displayed in the output form as the relative size x [non-dim]. Factors Efficiencies
and
Multiply the efficiencies by the geometric
cross section of the sphere (PI * D 2/ 4) in order to obtain
the attenuation, scattering, and absorption cross sections, as well as the
S1, S2, and S11 differential cross section for scattering
of light polarized perpendicular to the scattering plane, parallel to the
scattering plane, and unpolarized respectively. The scattering plane contains
the incident beam and observation directions. ValidationOverviewThe Mie-theory algorithms involve recurrence-based calculations of functions related to Bessel functions. Such calculations are known to be unstable if traversed upwards, i.e. in the direction of the increasing summation index of the Mie series. Specifically, such instabilities may occur while calculating the A[n] function [in the notation of Deirmendijan (1969)] in the upper range of the summation index n. With an increase in the numerical precision from single floating point precision to double and long double precision the onset of such instabilities in the upward recurrence algorithms is merely postponed for particles typical of many natural and industrial processes. One way to counteract such instabilities is to use downward recurrence for the Bessel-derived functions of a complex argument, specifically A[n]. Calculations for the refractive index ReN < 1 (for example, an air bubble in water) specifically require downward recurrence or the Lentz (1976) algorithm that permits to calculate each value of A[n] independently. Downward recurrence in double precision is adopted in the present program for the calculations of the A[n] function. The "initial" value of that function at the maximum value of the summation index, n, is calculated by using the Lentz (1976) method, with the iteration precision set to 10-12. This method gives, for example, A[100] = 0.34728219013 - i 1.0813043393 in 25 iterations for N = 1.28 - i 1.37 and x = 62 in good agreement with a Lentz's (1976) sample case as far as it can be assessed from his Fig.1. The user is nevertheless advised to exert caution when using this program in the extremities of the nominally allowed input data ranges. As a guidance the following rules should be observed:
These relationships must hold to within the
relevant numerical precision, here generally on the order of 10-12.
On request a diagnostic version of this program is available. That version
writes values of key functions of the algorithm (vs. the summation index) to a
text file that can be imported to a spreadsheet for examination. A simple
spreadsheet viewer for these functions is also available. Comparison with other published resultsResults of this program have been tested against the known behavior of the relevant functions and also compared with published results. Excellent correspondence has been obtained between this program results and those reported by Grehan and Gouesbet (1979). This correspondence attests to the capability of the downward recursion combined with the Lentz (1976) method for obtaining the initial value of the A[n]. The Grehan and Gousebet (1979) SUPERMIDI program is reported to use the Lentz method altogether. Table 1. Comparison between the S1-function values obtained with the present program and those obtained by Grehan and Gouesbet (1979) with their SUPERMIDI program for N = 1.33 - i * 0 and various values of x. At scattering angles of 0 and 180 deg, S1 = S2.
For N = 0.1 and x = 9999 we
obrain S1(0 deg) = 2.5088e15 , and S1(180 deg) = 1.55674e8 Table 2. Comparison between the S1-function values obtained with the present program and those obtained by Grehan and Gouesbet (1979) with their SUPERMIDI program for x = 8 and ReN = 1.33 with various values of ImN.
Sample angular patterns of non-absorbing (ImN = 0) and absorbing (ImN = 1000) water droplet in air obtained with this program are shown in Fig. 1.
Fig. 1. The S11 functions of the scattering angle for a sphere with x = 70, and ReN = 1.33 for two values of the imarginary part of the refractive index, ImN. These patterns are created by this program at a 0.1 deg increment in theta in good agrement with the results published byGrehan and Gousebet (1979). [Top] Results of this program have also been compared with numerical results reported by Dave (1969). For example, this program generates Qabs = 0.790966, vs. 0.7910 obtained by Dave, for x = 50*PI and N = 1.342 - i1. Results obtained with this program also compare
well with a range of other results available in graphical form. In some cases
discrepancies have been noted. One source of such discrepancies, aside from the
recurrence direction, is the way by which the rounding errors accumulate during
calculations. This way dependes on the form of a particular implementation of
the Mie algorithm. ApplicationsIntegrated scattering, absorption, and attenuationSingle particleIn practical applications one frequently needs to calculate the total power F [W] (flux) of light attenuated, scattered, or absorbed by a sphere. This can be accomplished with the use of the attenuation and scattering cross sections as follows:
where Ein[W m-2] is the irradiance [power per unit area] of the light beam. Note that in physics, the power per unit area is traditionally refered to as the intensity. The cross sections [m2] are defined as follows:
where "xxx" stands for
"Abs", "Att", or "Sca" and k = 2 * PI /
Lambda [m-1] is the wave number. SuspensionFor M [m-3] identical particles per unit volume of the suspension we have
where V [m3] is the illuminated volume of the suspension, A [m2] is the cross section area of the light beam, and z[m] is the thickness of the volume V measured along the beam axis. The power transmitted through a suspension volume of thickness z is reduced by Fatt.. The effect of the suspension interfaces (for example, water/air) is not accounted for. In the limit of the small changes, the reduction, dF, in the transmitted power due to attenuation by a suspension slab of thickness dz, can be expressed as follows:
where F is the power incident at the entrance face of the slab. Again, the effect of reflection at the slab faces is not taken into account. The solution of this equation is
where M * AttnCrossSec is the attenuation coefficient, usually denoted by c [m-1]. One can calculate in a similar manner the power scattered in all directions by the volume, V, of the suspension. If the suspension consists of particles with
various diameters and refractive indices, the particle concentration M
is replaced by the size-refractive index distribution of the particles and the
total powers become integrals of the respective cross sections weighed by that
distribution. Such integration must at present be provided off line, for
example by using a spreadsheet. Angular scattering patternSimilarly we can calculate the intensity (power per unit solid angle, units of W sr-1) of light scattered by a single sphere or suspension of spheres at a specific direction (theta, phi) relative to that of the incident wave:
where
The dimension of the AngCrossSec is m2sr-1. Note that in physics the intensity is defined as the power per unit area, a reverse of the convention traditionally used in the light scattering studies. The azimuthal angle phi is measured in a plane perpendicular to the wave direction (beam axis). The scattering pattern of a sphere is independent of the azimuthal angle. Thus, we have:
If the acceptance solid angle of a detector, Wdet [sr], about the direction (theta, phi) is known, the power of the scattered light received by that detector can be calculated as follows:
One may also be interested in obtaining the power scattered into an annular region between the angles of theta1 and theta2. This power equals:
where
INTt1t2 denotes integration over t
in a range of t1 to t2. Forward and backward scattering ratiosAlthough the light scattering calculator does not provide these values directly, they can be evaluated by first integrating the scattering pattern S11 off-line within the scattering angle ranges of 0 to 90° (forward scattering) and/or 90-180° (backward scattering). Then the ratios can be formed as follows:
for forward scattering, and
where Qsca f is
the forward scattering efficiency, and Qsca b is
the backward scattering efficiency, and INT denotes an integral. The
integration should be carried out with theta expressed in radians. Determination of the refractive index of spherical particlesThe scattering efficiency, Qsca, of a particle of given size is a function of its refractive index. This functional relationship can be read in reverse, i.e. with the refractive index being a function of the scattering efficiency. An example of such a relationship is shown in Fig. 2 for nonabsorbing particles in water and in Fig. 3 for nonabsorbing particles in air. This relationship forms the basis of the determination of the refractive index of particles from values of their scattering parameters. With monodisperse homogeneous particles one can simply derive the scattering efficiency from measurements of light scattering by the particles and, by using a curve of the type shown in the following two figures, one could determine the refractive index of such particles given that the particle size is known. Note that the relationship between Qsca and N may become multivalued for some particle sizes (Fig. 3). This may be counteracted by adjusting the wavelength of light that changes the x-parameter (the relative size) of the particles and/or by using the angular light scattering pattern at optimized angles (for example, Tycko et al. 1985) instead of the integrated light scattering properties such as Qsca .
Fig. 2. The effect of the refractive index (relative to the medium) of a nonabsorbing spherical particle on the scattering efficieny, Qsca of a sphere in water. The values were obtained with the MJC light scattering calculator. The sphere x parameter labels the curves. Note that the Qsca axis is logarithmic. The x range shown corresponds to a diameter range of 0.1 to 1.1 µm at a wavelength of 0.633 µm (HeNe laser light).
Fig. 3. The effect of the refractive index (relative to the medium) of a nonabsorbing spherical particle on the scattering efficieny, Qsca, of a sphere in air. The values were obtained with the MJC light scattering calculator. The sphere x parameter labels the curves. Note that the Qsca axis is logarithmic. The x range shown corresponds to a diameter range of 0.1 to 1.1 µm at a wavelength of 0.633 µm (HeNe laser light). The relationship between the refractive index and Qsca may become multivalued (top curves) when looked upon from the point of view of the Qsca. This may preclude the determination of the refractive index from the value of the scattering efficiency for particles with size and refractive index in this and other similar ranges of these parameters unless the wavelength of light can be changed to change the relative particle size, x. The following discussion assumes that experimental data on the scattering properties of the particles, specifically, the scattering coefficient, b [1/m] are available. Such coefficient can on one hand be readily measured with an integrating nephelometer (for example, Heintzenberg and Charlson, 1996) and on the other hand, derived from the LSC-provided data as will be shortly demonstrated. The refractive index and the particle size can also be simultaneously and very accurately derived from the measurements of light scattering by individual spherical particles at specific angles. An example of such experiment is given by Ashkin and Dziedzic (1981). That experiment opened a possibility to study molecular spectroscopy of minute samples of liquids as present in single aerosol particles (for example, Arnold et al. 1984). The light scattering calculator can provide direct calculation support for such experiments. Consider first a simple case when the particles are monodisperse. With aerosols, quasi-monodisperse particles are obtained from a polydispersed particle population, for example, at the output of a differential mobility analyzer. If this output can be coupled directly to the nephelometer input and the particle concentration can be measured independently at the output of the nephelometer, then such an arrangement simplifies the determination of the refractive index of the particles and enables the measurement of the particle size/refractive-index distribution. In this case, the incremental scattering coefficient, db, can be calculated as follows:
where db can be interpreted as a contribution of particles with sizes between D and D + dD [m] to the scattering coefficient of the particles, Qsca [dimensionless] is the scattering efficiency that can be calculated by the the LSC, A [m2] is the particle cross section area, m [particles/m4] is the particle size distribution (PSD), and Lambda [m] is the wavelength of light. The PSD m(D) referred to here is the differential PSD, i.e. the number concentration of the particles per unit volume and unit particle size interval, hence the exponent of 4 in the PSD unit. With this definition, the product m(D) dD is the number concentration of particles with sizes in an interval [D, D + dD]. Given that the particle size and wavelength are known, one parameter remains unknown in that equation: the refractive index, N, of the particles. It is a free parameter in this model and its value can be assessed from the knowledge of the scattering coefficient and the PSD data as sketched at the beginning of this section. Thus, the experiment as described above would enable the determination of the refractive index of the particles. With independent knowledge of how this parameter is affected by the composition/structure of the particles one should be able to characterize this aspect of the aerosol/suspension by performing such experiments. If the particle population is polydisperse, the scattering coefficient can be calculated with data provided by the light scattering calculator (LSC) and with independent information on the particle size distribution. Please note that the LSC does not calculate the scattering coefficient directly because that coefficient depends on the size and refractive-index distribution of the particles. Instead, the LSC provides optical properties for a single particle of specified size, D, and refractive index, N. In a simple case when all particles have the same refractive index (and thus composition), the scattering coefficient, b, [1/m] is expressed as follows
where INT() stands for an intgeral with the limits specified in the parentheses. The integration limits, Dmin and Dmax are determined by the contribution of the particles to the scattering coefficient. With realistic size distributions, the lower limit is set by the scattering efficiency of the particles that rapidly decrasese with the particle size. The upper limit is set by the particle concentration decreasing rapidly with size. Some reasonably simple additional calculations would need to be performed on the LSC-provided data, namely the integration over the particle size range as prescribed by the above formula. Please note several constraints. To start with, this discussion applies accurately only to spherical particles as the LSC calculates the optical properties of such particles. The discussion of how closely the relevant optical properties of irregular particles are represented by those of spheres is beyond the scope of this note. Further, the sensitivity of this method, and in fact the possibility of meaningful determination of the refractive index given experimental errors, depends on the particle size range relative to the wavelength of light, the PSD, and also on the refractive index value itself. If a reasonable compromise cannot be reached for the scattering coefficient, it may help to use the ratio of the backscattering coefficient to the scattering coefficient (for example, Twardowski et al. 2001). The backscattering coefficient refers to the backward hemisphere (scattering angles from 90° to 180°) as opposed to the scattering coefficient that refers to the whole sphere of the directions relative to that of the incident light beam (scattering angles from 0° to 180°). Finally, there may be instrumental limitations.
The nephelometer may use polychromatic or quasi-monochromatic light, in which
case the scattering coefficient is an average over the wavelength band specific
for the nephelometer. The integrating nephelometer measures only the major part
of the scattering and backscattering coefficients. Such instrumental factors
may need to be accounted for depending on the accuracy required. Determination of the particle sizeArguments similar to those given in the
preceding section can be given regarding the determination of the particle
size, given the knowledge of the particle refractive index. Particle shape discrimination: sphere vs. nonsphereIf the integrating nephelometer allows the
measurement of the forward and backward scattering coefficients of the
particles, and if the refractive index of the particles and particle size are
known beforehand, one may attempt to determine whether the particles are
spherical or not. This determination would exploit the difference between the
angular pattern of light scattering by a sphere and that of nonspherical
particles of the "same size" and refractive index. Again, the
discussion of such differences is beyond the scope of this note. However, by
merely comparing the ratio of the backward-to-forward or backward-to-total
scattering coefficients measured with the nephelometer to that calculated with
the help of the light scattering calculator (LSC), you may be able to determine
whether the particles are spherical or not. Some reasonably simple additional
calculations would need to be performed on the LSC-provided data, namely the
integration of the angular pattern calculated by the LSC within the forward and
backward hemisphere (scattering angle ranges of 0 to 90° and 90° to
180°) to yield the forward and backward scattering coefficients
respectively. ReferencesArnold S., Neuman M., Pluchino A. B. 1984. Molecular spectroscopy of a single aerosol particle. Opt. Lett. 9: 4-6 Ashkin A., Dziedzic J. M. 1981. Observation of optical resonances of dielectric spheres by light scattering. Appl. Opt. 20: 1803-1814. Bohren C. F., Huffman D. 1983. Absorption and scattering of light by small particles. Wiley, New York. Dave J. V. 1969. Scattering of electromagnetic radiation by a large, absorbing sphere. IBM J. Res. Developm. 13: 302-313. Deirmendijan D. 1969. Electromagnetic scattering on spherical polydispersions. Elsevier, Amsterdam Grehan G., Gouesbet G. 1979. Mie theory calculations: new progress, with emphasis on particle sizing. Appl. Opt. 18: 3489-3493. Heintzenberg J., Charlson R. J. 1996. Design and application of the integrating nephelometer: a review. J. Atmos. Oceanic Technol. 13: 987-1000. Kattawar G. W., Plass G. N. 1967. Electromagnetic scattering from absorbing spheres. Appl. Opt. 6: 1377-1382. Lentz W. J. 1976. Generating Bessel functions in Mie scattering calculations using continued fractions. App. Opt. 15: 668-671. Mie G. 1908. A contribution to the optics of turbid media special colloidal metal solutions (in German: Beitrage zur Optik trüber Medien speziell kolloidaler Metallösungen). Ann. Phys. 25: 377-445. Twardowski M. S., Boss E., MacDonald J. B., Pegau W. S., Barnard A. H., Zaneveld J. R. V. 2001. A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle compositions in Case I and Case II waters. J. Geophys. Res.C 106: 14,129-14,142. Tycko D. H., Metz M. H., Epstein E. A., Grinbaum A. 1985. Flow-cytometric light scattering measurements of red blood cell volume and hemoglobin concentration. Appl. Opt. 24:1355-1365. Wiscombe W. J. 1980. Improved Mie scattering
algorithms. Appl. Opt. 19: 1505-1509. Contact info for comments and questionsPlease direct your comments and questions regarding this software, as well as questions on other MJC Optical Technology software and services to: Dr. Miroslaw Jonasz DisclaimerThe information contained in this document is believed to be accurate. However, neither the author nor MJC Optical Technology guarantee the accuracy nor completeness of this information and neither the author nor MJC Optical Technology assumes responsibility for any omissions, and errors, or for damages which may result from using or misusing this information. |
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