Light scattering calculator: coated sphere

Miroslaw Jonasz

Table of contents

• Introduction
• Quick start
• User interface
  • Setting preferences
  • Miscellaneous menu items
  • Modifying program input data
  • Range editors
  • Setting output format
  • Running calculations, displaying, and saving results
• Assumptions
• Definitions of the input data
• Output
  • Definitions
  • Types
• Validation
  • Overview
  • Comparison with other published results
  • Homogeneous sphere limits
  • Sample results
• Applications
  • Single particle
  • Suspension
  • Angular scattering pattern
• References
• Contact info for comments and questions
• Disclaimer

Introduction

This program calculates the scattering, absorption, and attenuation parameters, as well as the angular scattering patterns of a single coated sphere according to Aden-Kerker theory.

This program is intended to be used in the Windows environment. A screen resolution of 1024 x 768 is expected.
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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" button) and saved to a space-delimited text file ("Save" button).
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User interface

Setting preferences

Preferences 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.
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Save - saves the current preferences to the currectly active preferences file (extension=pre)
Save As - saves the current preferences to a file named by the user (extension=pre)
Load - loads the preferences from a preferences file (extension=pre)
Default - loads default pereferences

Miscellaneous menu items

Limits - displays the limits of the program parameters.
Help - displays this document that can also be displayed off line with your favourite web browser.
About - displays the program info box.
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Modifying program input data

Edit x button opens the coated sphere (whole) size range editor.
The Edit x caption is shown if the diam check box is unchecked. The sphere size is then expressed by the relative size parameter x (see Definitions of the input data). The button caption is modified to Edit diam if the diam check box is checked. The sphere size is then expressed as the sphere diameter.

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 also 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.
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diam check box toggles the absolute/relative particle size definition.
If this box is checked, the particle size is expressed as the sphere diameter. The wavelength edit box is then displayed and can be modified. Otherwise, the size is expressed as a relative size parameter x. Checking the diam check box does not affect the numerical value of the particle size as displayed in the main form. It is just taken to represent the absolute particle diameter. However, the relative size, x, is now different and is correctly displayed in the results form. A similar procedure in reverse applies when the diam check box is un-checked.

shell thick. rel to x edit
This edit is enables modifying the sphere coating (shell) thickness. The shell thickness is expressed as a fraction of the whole particle x or diameter (depending on the status of the diam check box).

wavelength edit
This edit is displayed only when the diam check box is checked. It enables editing of the wavelength of light in the medium surrounding the sphere. The relative size parameter, used internally by the program, is calculated in this case and depends on the wavelength value.

Core Re(n) and -Im(n) edits
Enable setting the real and imaginary parts of the sphere core refractive index relative to the nonabsorbing medium surrounding the sphere.

Shell Re(n) and -Im(n) edits
Enable setting the real and imaginary parts of the sphere coating (shell) refractive index relative to the nonabsorbing medium surrounding the sphere.

Edit theta button opens the scattering angle range editor.
If the output is selected to be a function of the particle size (see Setting output format), only the start parameter of the theta range is displayed in the main form and can be edited. If the output is requested to be the function of the scattering angle (see Setting output format), the remaing angle range parameters: step [deg], count, and end [deg] are also displayed. The scattering angle list is generated and displayed in the output form according to the prescription used for the size range. The theta range parameters must fulfill this requirement: 0 <= start <= end <= 180 deg.
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Range editors

Range 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.
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Setting output format

"Output type" group
Factors - results are shown as the Mie sums.
Efficiencies - results are shown as efficiencies.
Please the Output section in this document for details.

"Output as a function of" group
Scattering angle - results are displayed for a single particle size as angular scattering patterns.
Particle size: const. shell thick. - results are displayed for a single scattering angle as a function of the particle size with the absolute shell thickness being the same for all sizes.
Particle size: const. ratio shell thick. - results are displayed for a single scattering angle as a function of the particle size with the ratio of the shell thickness to the whole particle size being the same for all sizes.
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Runing calculations, displaying, and saving results

Run button initiates the scattering properties calculations.

If the following buttons are disabled, no results are available.
View opens the results display form.
Copy copies the results to the Windows clipboard.
Save saves the results to a space-delimited text file.

"Messages" group displays status of the calculations.
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Assumptions

The coated sphere is assumed to be located in a plane monochromatic electromagnetic wave whose wavefront has infinite extent. In practical applications this condition requires that:

• 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 focus of a lens is also plane. Therefore, the theory of light scattering by coated sphere 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.
• The medium surrounding the sphere does not absorb light.

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Definitions of the input data

The light scattering properties of a coated sphere are completely described by four parameters: the refractive indices of the sphere core and shell relative to that of the surrounding medium as well as the whole sphere diameter and the shell thickness. 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 (click the Limits item of the menu to view limits).

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 shell thickness is expressed relative to the whole particle size.

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.
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Output

Definitions

This program calculates the light scattering properties of a coated sphere following the Toon and Ackerman (1981) algorithm. A similar algorithm has been recently introduced by Kaiser and Schweiger (1993). These algorithms are based on the theory of light scattering by a coated sphere that was developed by Aden and Kerker (1951) In the notation borrowed from the theory of light scattering by homogeneous sphere, 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.
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Types

Particle size is always displayed in the output form as the relative size x [non-dim] of the whole particle.

Factors
The following quantities can be displayed, (copied and/or saved): ScaFactor (scattering factor) AttFactor (attenuation factor), and AbsFactor (absorption factor), as well as | S1(theta) |², | S1(theta) |², and S11(theta) where theta is the scattering angle.

Efficiencies
The following quantities can be displayed (copied and/or saved):

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.
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Validation

Overview

The algorithms of the coated sphere theory involve, as it is also the case with the homogeneous sphere theory, 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 relevant series. 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 or to use the Lentz (1976) algorithm that permits to calculate each of the values of the relevant functions independently. Downward recurrence in double precision is adopted in the present program. The "initial" values of the functions at the maximum value of the summation index, n, are calculated by using the Lentz (1976) method, with the iteration precision set to 10^-12.

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:

• ScaFactor must equal AttFactor if Im(N) = 0 for both the core and shell
• ScaFactor must be less than AttFactor if -Im(N) > 0. for either or both the core and shell.

These relationships must hold to within the relevant numerical precision, here generally on the order of 10^-12.
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Comparison with other 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 Kaiser and Schweiger (1993), Bohren and Huffman (1983), and Fenn and Oser (1965). This correspondence attests to the capability of the downward recursion combined with the Lentz (1976) method for obtaining the initial values.

Homogeneous sphere limits

In addition, the results were checked against the results of the homogeneous sphere calculator, also available from MJC Optical Technology, for three cases:

• vanishing shell
• vanishing core
• vanishing refractive index difference

The vanishing shell case was tested by using a composite water-carbon sphere with the carbon shell thickness decreasing to 0. The whole particle x = 70 was assumed, with the water refractive index N = 1.33 - i0, and the carbon refractive index N = 2 - i1. As expected, the efficiencies of the composite particle tend to the the limiting values for the homogeneous water sphere. These limiting efficiencies, obtained with both the homogeneous and coated sphere MJC Optical Technology calculators, assume values of Q_att = Q_sca = 2.02147 and Q_abs = 0.

The vanishing core case was tested by using a composite carbon-water sphere with the carbon core diameter decreasing to 0. The whole particle size was again x = 70, with the refractive index values the same as before. The limiting values of the efficiencies for the homogeneous sphere obtained with the coated sphere calculator for the shell thickness of 0.9999999 (maximum allowed) are Q_att = 2.02147, Q_sca = 2.02147, and Q_abs = 8.88178e-16, i.e. essentially 0 at the numerical precision involved.

The coated sphere calculator was also tested by vanishing the refractive index difference between the core and the shell in which case, the results should and do converge to the homogeneous particle case. For example, the efficiencies of a composite particle (water core, water shell) with x = 70, and a 50%-thick shell yield Q_att = 2.02147, Q_sca = 2.02147, and Q_abs = 4.4e-16, i.e. essentially 0 at the numerical precision involved. With the homogeneous sphere calculator we obtain Q_att = Q_sca = 2.02147, and Q_abs = 0.00000.

Sample results

Fig.3 shows angular pattern results of sample calculations for a water droplet with a whole x = 70, and a 1%-thick coating of carbon [core N = 1.33 - i0 (water), shell N = 2 - i1 (carbon, see Janzen, 1979)]. These results suggest a substantial effect of a highly absorbing thin shell around a non-absorbing core on the light scattering by the composite particle at the large angles.

Fig. 1. The S11 functions of the scattering angle for a homogeneous (uncoated) water sphere and a composite water-carbon sphere. In both cases the whole particle x = 70. The coated sphere shell thickness is 1%. The refractive index of water N = 1.33 - i0. The refractive index of the carbon coating N = 2 - i1.

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Applications

Single particle

In 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 E_in[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.
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Suspension

For 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 F_att.. 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.
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Angular scattering pattern

Similarly 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, W_det [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 INT_t1^t2 denotes integration over t in a range of t1 to t2.
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References

Aden A. L., Kerker M. 1951. Scattering of electromagnetic waves from two concentric spheres. J. Appl. Phys. 22: 1242-1246.

Bohren C. F., Huffman D. 1983. Absorption and scattering of light by small particles. Wiley, New York.

Fenn R. W., Oser H. 1965. Scattering properties of concentric soot-water spheres for visible and infrared light. Appl. Opt. 4: 1504-1509.

Janzen J. 1979. The refractive index of colloidal carbon. J. Colloid Interface Sci. 69: 436-447.

Kaiser T., Schweiger G. 1993. Stable algorithm for computation of Mie coefficients for scattered and transmitted fields of a coated sphere. Computers in Physics 7: 682-686.

Lentz W. J. 1976. Generating Bessel functions in Mie scattering calculations using continued fractions. Appl. Opt. 15: 668-671.

Toon O. B., Ackerman T. P. 1981. Algorithms for the calculation of scattering by stratified sphere. Appl. Opt. 20: 3657-3660.
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Contact info for comments and questions

Please direct your comments and questions regarding this software, as well as questions on other MJC Optical Technology software and services to:

Dr. Miroslaw Jonasz
MJC Optical Technology
217 Cadillac Street
Beaconsfield QC H9W 2W7
Canada

Internet: www.mjcopticaltech.com
e-mail: m.jonasz@mjcopticaltech.com

tel +1 514 695 4132
fax +1 514 695 3315
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Disclaimer

The 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.

Last modified: . Copyright 2000 MJC Optical Technology. All rights reserved.