## Light scattering calculator: homogeneous sphere- Introduction
- Quick start
- User interface
- Assumptions
- Input data: Definitions
- Output
- Validation
- Sample applications
- References
- Contact info for comments and questions
- Disclaimer
## IntroductionThis program calculates the total (angle-integrated) attenuation, scattering, and absorption parameters, as well as the scattering matrix (Mueller matrix) of a single homogeneous sphere or a size distribution of spheres, according to Mie theory. The sphere material may absorb light. However, the medium surrounding the sphere is assumed to be nonabsorbing. This program is intended to be used in the Windows environment. A screen resolution of 1024 x 768 or higher is expected. ## Quick start If needed, modify the parameters displayed in the program main form (see User interface) and click the ## User interface## Setting preferencesPreferences are all user-adjustable program parameters. On program startup default preferences will be used. To save preferences, simply set the parameters in the program's main form and click the The preferences are saved to a binary file. That file name is displayed in the When a preferences file is loaded, the values of the adjustable parameters of the program are used to update the main form. If you subsequently change any of these parameters, the "mod" indicator is shown in the
## Miscellaneous menu items
## Modifying program input data
If the output is requested to be the function of the scattering angle (
where
## Range editorsRange editors for the particle size and scattering angle accept any inputs that are consistent with an abstract range definition. Thus, range definitions may be accepted in a range editor form and displayed in the main
form that do not fulfill either the program limits (see Miscellaneous menu items) of 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 One of the range parameters is always calculated. You can force a parameter to be calculated (which prevents it from being edited) by clicking on the corresponding radio button in the ## Setting the output format
Please also see the Output section in this document for details.
## Integrating scattering properties over a particle size rangeIf the particle size scale is absolute (the The integration results are displayed as the attenuation, scattering, and absorption coefficients (the total scattering properties) and the scattering (Mueller) matrix (the differential scattering properties). You can easily convert the The following PSDs are available: - power-law size distribution defined as
*D*^{ -slope} **log-normal size distribution**defined as 1/*D*exp[-(ln*D*- ln*D*_{gm})^{2}) / ( 2 * Var( ln*D*) )], where ln*D*_{gm}is the geometric mean diameter of the dispersion of particles, Var( ln*D*) is the variance of the natural logarithm of the particle diameter in that dispersion.
The scale factor of the PSD (with a unit of 1 / µm As much as the single-particle calculations performed by this program can be considered numerically "exact", the integration results should be viewed as approximate. Such integration is performed numerically by using the trapezoidal rule. Its result depends on the particle diameter range step chosen by invoking the [Top] ## Runing calculations, displaying, and saving results
If these latter three buttons are disabled (grayed), no results are yet available for the current set of values of the program parameters. Note that the results can be conveniently displayed in pre-formatted, publication-ready graphs by using optional viewers for angle-dependent and particle size-dependent patterns (
## AssumptionsThe homogeneous sphere is assumed to be located in a uniform nonabsorbing and nonscattering medium illuminated by a plane monochromatic electromagnetic wave whose wavefront has infinite extent. In practical applications, this assumption translates to the following conditions: - The light beam illuminating the particle has a plane wave front in the vicinity of the particle. Usually such a beam is "collimated" i.e. its divergence is limited. However, the wavefront in the very focus of a lens is also plane. Therefore, the Mie theory applies to situations where the particle(s) is(are) illuminated by a focused beam - this can greatly increase the incident power and thus increase the power of light scattered by the particle.
- The diameter of the beam is large as compared with the particle.
- The particle is located near the beam axis to avoid edge effects.
[Top] ## Input data: Definitions The light scattering properties of a homogenous sphere are completely described by just two parameters: the The refractive index is a complex number:
where diam check box, the refractive index is expected, but not checked, to correspond to the current value of the wavelength.In this program, m < arbitray_value. The _{r}m range is defined as 0 ≤ _{i}m ≤ arbitrary_value._{i}The sphere size in the algorithm is represented by the relative particle size
where The program accepts the particle size as The scattering angle, ## Output## DefinitionsThis program calculates the light scattering properties of the homogeneous sphere following the Mie theory (Mie G 1908) with algorithms introduced by Deirmendijan D 1969, Kattawar GW and Plass 1967, and Wiscombe WJ 1980. A modern explanation of Mie theory is given, for example, by Bohren CF and Huffman 1983. In reference to Mie theory in its modern notation, the nondimensional versions of these properties are defined as follows:
where
where The program also calculates the nonzero elements of the scattering matrix, also referred to as the Mueller matrix, which for the sphere are
The scattering matrix completely describes the amplitude and polarization of light scattered by a particle. See, for example, Bohren CF and Huffman 1983, for an in-depth discussion of Mueller matrices and polarization effects. The element ## Types Particle size is displayed in the output form as the relative size
and the scattering matrix elements. Multiply the efficiencies by the geometric cross section of the sphere ( ## Validation## Small particle limitThe MJC light scattering calculator for homogeneous spheres can handle the relative particle size, The results agree with the small particle limit approximation of the Mie theory to within ~0.1% for x = ~5e-7. The agreement gets better (~0.001%) as the particle size increases. As the particle size increases further, the agreement with the small particle approximation worsens because the approximation is simply inadequate for the large particle sizes. Here are samples of the calculator performance at the small particle size limit for: - a sphere with a refractive index
*m*= 0.098 – i 3.086 (gold in water at a wavelength of 530 nm in vacuum)
where Q_sca is the scattering efficiency (*see*Efficiencies), Q_abs is the absorption efficiency, and M11 is the first element of the scattering (Mueller) matrix at a scattering angle of 0 deg. For the small particles (i.e. particles much smaller than the wavelength of light) Q_abs ~ x^{-1}, Q_sca ~ x^{-4}, and M11 ~ x^{-6}. As it is evident from the above graph, the absorption of light is the dominant mechanism of attenuation of light by the sample sphere in the small particle size range. - a sphere with
*m*= 1.33 – i 0 (a water sphere in air)
Note that Q_abs = 0 in this latter case, because the refractive index of the sphere material is real, i.e. the sphere material does not absorb light.
## Large particle limitFor a large sphere, the Mie series have to be summed over a large number of terms. These terms are calculated by using 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 One way to counteract such instabilities is to use downward recurrence for the Bessel-derived functions of a complex argument, specifically 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 WJ 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 m = 1.28 - i 1.37 and x = 62 in good agreement with a Lentz's 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: *F_sca*must equal*F_att*if*m*= 0, i.e. if the particle material does not absorb light_{i}*F_sca*must be less than*F_att*if*m*> 0._{i}
These relationships must hold to within the relevant numerical precision, here generally on the order of 10 ## Comparison with published results Results 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 G and Gouesbet 1979. This correspondence attests to the capability of the downward recursion combined with the Lentz WJ 1976 method for obtaining the initial value of the
x = 9999 we obrain S1(0 deg) = 2.5088e15, and S_{1}(180 deg) = 1.55674e8.[Top]
m = 1000) water droplet in air obtained with this program are shown in Fig. 1._{i}
[Top] Results of this program have also been compared with numerical results reported by Dave JV 1969. For example, this program generates 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 manner by which the rounding errors accumulate during calculations. This dependes on the form of a particular implementation of the Mie algorithm. Table 3 contains results for the attenuation efficiency and scattering efficiency ( b functions by using the _{n}A-function approach (as defined by Deirmendijan D 1969). The MIECPP calculates the a and _{n}b functions by using ratios of the successive orders of the Ricatti-Bessel functions._{n}
Table 4 contains results for the attenuation efficiency obtained with MJC Light Scattering Calculator for Homogeneous Sphere (HS) as compared to those obtained by Penndorf RB 1957.
Table 5 contains results for the attenuation efficiency and absorption efficiency obtained with MJC Light Scattering Calculator for Homogeneous Sphere (HS) as compared to those obtained and quoted by Fenn RW and Oser 1965, who developed one of the early coated-sphere scattering programs. The results listed by Fenn and Oser refer to the limiting case of the vanishing shell. Fenn and Oser do not state algorithm details.
Table 6 contains results for the |
## Sample applications## Total scattering, absorption, and attenuation## Single particleIn practical applications one frequently needs to calculate the total power
where ^{-2}] is the irradiance [power per unit area] of the light beam. Note that the program reports the cross sections in µm^{2}. Hence, either these values needs to be converted to m^{2}, or the irradiance should be converted to W µm^{-2}. The cross sections are defined in Equations 14 through 16.[Top] ## SuspensionFor
where The solution of this equation is
where the product By substituting If the suspension consists of particles with various diameters and refractive indices, the particle concentration ## Angular scattering pattern Similarly we can calculate the intensity (power per unit solid angle; units of W sr
The product If the acceptance solid angle of a detector, theta is known, the power of the scattered light received by that detector can be calculated as follows:
[Top] ## Determination of the refractive index of spherical particlesThe scattering efficiency, Note that the relationship between N may become multivalued for some particle sizes Fig. 3. This could 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 DH et al 1985) instead of the total (i.e., angle-integrated) light scattering properties such as Q_sca.
## 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. ## ReferencesArnold S., Neuman M., Pluchino A. B. 1984. Molecular spectroscopy of a single aerosol particle. Ashkin A., Dziedzic J. M. 1981. Observation of optical resonances of dielectric spheres by light scattering. Bohren C. F., Huffman D. 1983. Dave J. V. 1969. Scattering of electromagnetic radiation by a large, absorbing sphere. Deirmendijan D. 1969. Du Hong 2004. Mie-scattering calculation. Fenn R. W., Oser H. 1965. Scattering properties of concentric soot-water spheres for visible and infrared light. Grehan G., Gouesbet G. 1979. Mie theory calculations: new progress, with emphasis on particle sizing. Heintzenberg J., Charlson R. J. 1996. Design and application of the integrating nephelometer: a review. Jones A. R. 1983. Calculation of the ratios of complex Riccati-Bessel functions for Mie scattering. Kattawar G. W., Plass G. N. 1967. Electromagnetic scattering from absorbing spheres. Lentz W. J. 1976. Generating Bessel functions in Mie scattering calculations using continued fractions. 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). Penndorf R. B. 1957. New tables of total Mie scattering coefficients for spherical particles of real refractive indexes (1.33 < n < 1.50). 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. 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. Wiscombe W. J. 1996. Mie scattering calculations: Advances in technique and fast, vector-speed computer codes. NCAR Tech. Note. NCAR/TN-140+STR, orig. publ. June 1979, NCAR, Atmos. Analysis Prediction Div., Boulder, CO, 64 pp. Wiscombe W. J. 1980. Improved Mie scattering algorithms. [Top] ## 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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