Light scattering calculator: homogeneous sphere

Miroslaw Jonasz

Table of contents

• Introduction
• Quick start
• User interface
  • Setting preferences
  • Miscellaneous menu items
  • Modifying program input data
  • Range editors
  • Setting the output format
  • Integrating scattering properties over a particle size range
  • Running calculations, displaying, and saving results
• Assumptions
• Input data: Definitions
• Output
  • Definitions
  • Types
• Validation
  • Overview
  • Comparison with other published results
• Sample applications
  • Total scattering, absorption, and attenuation
  • Angular scattering pattern
  • Determination of the refractive index of spherical particles
  • Determination of the particle size
• References
• Contact info for comments and questions
• Disclaimer

Introduction

This 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.
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Quick start

If needed, modify the parameters displayed in the program main form (see User interface) and click the Run button to initiate calculations. The results will be displayed automatically. These results can be copied to the Windows clipboard (click Copy) and saved to a space-delimited text file (Save). You can also select a portion of the output text, copy it to the clipboard with CTRL-C, and paste it with CTRL-V to a Windows program that accepts text from the Windows clipboard.
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User interface

Setting preferences

Preferences 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 Preferences | Save menu item (this notation means the Save item of the Preferences 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 are saved to a binary file. That file name is displayed in the Preferences file name box. Long file names are cut to fit into that box. Position the cursor over the abbreviated name to display it in full in a hint box that appears near the cursor tip. In order to simplify data archiving, the name of the results file proposed in the Save homogeneous sphere results file save dialog is the same as that of the current preferences file name. Hence, you should avoid including the "Pref" string or a similar string in the preferences file name.

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 Preferences file name ... section until you save the modified parameters to the same or another preferences file.
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Save - saves the current preferences to a 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. The primary purpose of the limits is to prevent invalid numerical operations (for example, division by 0). However, even with the program adjustable parameters all within the limits, the calculation results may look strange, for example, all scattering properties reported may be identical to zero when integrating the scattering properties over particle size. This may prevent the proper completion of the calculations, namely the normalization of all elements of the scattering matrix other than M₁₁ by that element. Strange-looking results or parameter values may certainly be also a result of the programming error. Hence, we would greatly appreciate your letting us know about it at info@mjcopticaltech.com.
Help - displays this document
About - displays the program info box.
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Modifying program input data

Edit x button opens the sphere 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 Input data: Definitions). 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 and the wavelength of light (in the medium surrounding the sphere) must also be specified.

If the output is requested to be the function of the scattering angle (theta, see Setting the output format), then 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 the output format), the remaining particle size range parameters: step, count, and end are displayed (and can be edited) in units implied by the status of the diam check box: if that box is unchecked, then the units are "nondimensional", otherwise these are the units of the wavelength [μm]. Based on these parameters, the relative particle size list is generated as follows

where n = 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. The relative size parameter is used internally by the program as the base size scale. Note that checking the diam check box converts the numerical values of the relative particle size range parameters, as displayed in the main form, to values representing the abolute size scale. A reverse procedure applies when the diam check box is un-checked. Also note that the correspondence between the two particle size scales hold only for the wavelength that is specified at the time of the conversion. When the wavelength is changed, the correspondence between the two particle size scales applies at this new wavelength.

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.

mr and mi edits
Enable setting of the real and imaginary parts, respectively, of the sphere refractive index relative to that of 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 the output format), the remaining scattering angle range parameters: step [deg], count, and end [deg] are also displayed and can be edited. The scattering angle list is generated and displayed in the output form according 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 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 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 (which prevents it from being edited) 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 the Apply button to apply the change. If satisfied, click Ok to accept it. You can also click Ok right away after any parameter changes.
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Setting the output format

"Output integral properties as"
Factors - total scattering properties are shown as the Mie sums (see Equations 5 and 6).
Cross sections - total scattering properties are shown as cross sections (this option is available only when the diam check box is checked)
Efficiencies - total scattering properties are shown as efficiencies.

Please also see the Output section in this document for details.

"Output as a function of"
Scattering angle - results are displayed for a single particle size as angular scattering patterns. If this radio button is selected and the particle size scale is absolute (the diam check box is checked) then an option to integrate the scattering properties over the particle size is displayed (the integrate over particle size checkbox). See Integrating scattering properties over a particle size range.
Particle size - results are displayed for a single scattering angle as a function of the particle size. [Top]

Integrating scattering properties over a particle size range

If the particle size scale is absolute (the diam checkbox is checked) and the output is set to be a function of the scattering angle (the Scattering angle in Output as a function of is selected) then you can integrate the scattering properties, weighed by the particle size distribution (PSD), over a particle diameter range as defined in the main form.

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 M₁₁ element of the scattering matrix into the volume scattering function of the dispersion of spherical particles characterized by the particle size distribution type and parameters chosen, by multiplying M₁₁ by a factor of 1 / K ², where K = 2 π / λ, where λ is the wavelength of light in the medium surrounding the particle.

The following PSDs are available:

• power-law size distribution defined as D ^-slope
• log-normal size distribution defined as 1/D exp[-(lnD - lnD_gm)² ) / ( 2 * Var( lnD) )], where lnD_gm is the geometric mean diameter of the dispersion of particles, Var( lnD) 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 / µm³), which represents the particle concentration is assumed to be unity in both cases. You can easily convert the results to represent another particle concentration by multiplying them by the desired numerical value of the scale factor. Note that the particle diameter, D, in the above formulas for the particle size distribution is really a nondimensional diameter. It is made so by dividing it by the diameter of 1 µm. Hence, its numerical value remains identical to that of the absolute particle diameter.

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 Edit diam button (note that this button caption is shown only when the diam checkbox is checked). Hence, you may want to "experiment" with the step size to find the minimum step size below which the integration results do not change to within the desired numerical accuracy.

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Runing calculations, displaying, and saving results

Run button initiates the calculations of the scattering properties.

View opens the results display form.
Copy copies the results to the Windows clipboard.
Save saves the results to a space-delimited text file. In order to simplify data archiving, the name of the results file proposed in the Save homogeneous sphere results file save dialog is the same as that of the current preferences file name. Hence, you should avoid including the "Pref" string or a similar string in the preferences file name.

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 (see the Graphical data viewers) in "Light scattering calculator: homogeneous sphere".

Messages displays status of the calculations.
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Assumptions

The 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.

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Input data: Definitions

The 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 and other adjustable 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 (in the sign convention adopted here), i.e. m_i > 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 current value of the wavelength.

In this program, m_r can assume walues from a range of 0 < m_r < arbitray_value. The m_i range is defined as 0 ≤ m_i ≤ arbitrary_value.

The sphere size in the algorithm is represented by the relative particle size x [nondimensional]

where D is the sphere diameter and λ is the wavelength of light in the medium surrounding the sphere.

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, θ [deg], referred to as theta in the program output, 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 symmetrical 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 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: