Light scattering calculator: coated sphere
Table of contents
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.
The program is intended to be used in the Windows environment. A minimum screen resolution of 1024 x 768 is expected. The program is NOT INTENDED to be run on from a network location. See also other limitations.
In light scattering calculations, the absolute particle diameter, expressed - for example - in Ám, is converted to the relative particle size, i.e. the particle diameter relative to the wavelength of light in the medium surrounding the particle (see relative particle size). Hence, if the "relative particle size" or "relative size" is used in this document without qualification, it means that the size is relative to the wavelength. Otherwise, qualifiers are used, such as "core diameter relative to shell diameter".
In addition to introducing options, version 3.3 significantly modified the preferences system. Neither is compatible with earlier versions of the program. If the version of the calculator that you purchased is different from 3.4, please contact MJC Optical Technology (see Contact info for comments and questions) to obtain an update.
The graphical user interface of the calculator is shown below. Any disabled (grayed) control of the interface implies that the operation associated with that control cannot be performed at the moment. The top-most element of this interface is the menu (see Menu). The main element of this interface, the page control located below the menu bar, contains two tabbed pages: Define parameters and Read parameters from file. Located below the page control, a bar is used to display the program's status messages. Longer messages, including those expecting the user action, are shown in their own dialog boxes.
Options define general features of the user interface, in contrast to preferences, which define what data should be used and which calculations should be performed. Options were introduced in version 3.3 of the MJC Light Scattering Calculator.
Options are saved in a binary file "LscCoatSph.opt" expected to be located in the folder (directory) containing the program. If this file exists on startup, can be successully read, and is consistent with the computer's directory structure, options contained in it are adopted for the user interface. If the file does not exist on startup, it is created by using hard-coded default options. If the options file exists, can be successully read when the program starts, but is not consistent with the computer's directory structure, the default directories are used instead (see also comments on the defective options file).
If the options file is found defective on startup (for example, at least one directory listed in the options file does not exist on the user's computer), the program tries to save it to an archive file "LscCoatSph.op~" and replaces it with the hard-coded default options. If the defective file cannot be archived, the user is asked whether to overwrite the defective options file with the default options.
The current options are used by the program during a calculation session. These options can be modified by either using the Options dialog or saving a preference or a results file to a directory different than that specified in the current options (see directories). The current options are saved:
The hard-coded default options are used to initialize options on program startup if the options file (see the saved options) is not available. To revert to the default options, delete the "LscCoatSph.opt" file in the program directory before starting the program.
Clicking the Options menu item in the program user interface form opens the options dialog:
The options dialog has the following functionality:
This program is designed for use on a single computer (see also system requirements). Hence, the options file stores absolute paths to directories. If you save the options file on one computer and use this file on another computer with different directory structure, an error message (or messages, if more than one directories contained in the options file cannot be found) will appear when starting the program and preferences and/or results files may be saved to the program directory. See also a comment on defective options file
Clicking the Limits menu item displays the acceptable ranges of the calculation parameters. See also Valid ranges of input data. You can Copy all limits to the Windows clipboard and paste it to another Windows program which accepts text input. You can also copy part of the limits text. To do that, select that part with the mouse and click "ctrl-c". When you are ready to paste the selection to another Windows program, click "ctrl-v". You can also Save the limits to a text file.
Clicking the About menu item displays the program info box, including the program version.
Preferences are a set of user-adjustable calculation parameters. On program startup, preferences stored in a binary file in the program directory are used. If that file does not exists, the program uses hard-coded default preferences and saves them to a binary preferences file. Since version 3.3, each of the two calculation modes ("Define parameters" mode [DPM] and "Read parameters from file" mode [RFF], see also the user interface form) has a separate set of preferences.
The preferences system can be used to document the calculations if the results file name is synchronized with the preference file name. Such synchronization is proposed by the program when you save results of calculations.
Preferences for the Define parameters page are stored in a binary file with extension "prd" in a directory specified in the current options (see Directories ...). These preferences are referred to as the DPM preferences (Define ParaMeters). The DPM preferences include the status of each control of the Define parameters page as well as the numerical/textual values of the calculation parameters. If the Wavelength choice is clicked in the Output as a function of ... of the Define parameters tab, both the refractive index file path and the data from the first data line of that file are updated in the preferences set.
The default DPM preferences are stored in the "DpmDft.prd" file in the program directory. These preferences may be (1) the hard-coded default preferences or (2) the user-defined default preferences.
Preferences for the Read parameters from file page are stored in a binary file with extension "prr" in a directory specified in the current options (see Directories ...). These preferences are referred to as the RFF preferences (Read parameters From File). The RFF preferences include only the input file path and the status of each checkbox of the Read parameters from file page.
The default RFF preferences are stored in the "RffDft.prr" file in the program directory. These preferences may be (1) the hard-coded default preferences or (2) the user-defined default preferences.
The name of the currently used preferences file is displayed in the Preferences ... group box of each page of the page control (see User interface). Long file names are truncated. Fly the cursor over the file name label to display the full file name in a hint box that appears near the cursor tip. The file names can be displayed in two formats: the file name only or the full file path (including both the directory and the file name), depending on whether the check box Show full file paths in the options dialog is unchecked or checked, respectively. The file name display format can be changed on the fly. If no file name is displayed in the Preferences ... box, the preferences have been changed from the set of preferences most recently loaded or saved.
The following buttons are available in the Preferences group box of each page of the user interface form:
Open this page of the page control (see User interface) by clicking on its tab or via loading a DPM preferences file that opens it automatically and presets the data and output handling parameters (see the description of the preferences box).
Note that editing of numerical data in edit boxes (such as wavelength) is not complete until the edit box looses focus, i.e. the cursor exits the edit box, or the RETURN key is pressed on the keyboard.
abs size check box
Edit x button
Edit diam button
If the output is to be a function of the scattering angle (referred to as theta, i.e. Output as a function of ... Scattering angle is selected) only the start parameter of the size range can be edited. If the output is requested to be a function of the particle size (see Setting the 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 abs size check box. Based on these parameters, the relative size list is generated and displayed (on the calculations' completion) in the output form as follows:
where m = 0 ... countsize range - 1. For example, if the size range contains n values, the countsize range = n. The size range must fulfill this requirement: 0 < xstart <= xmax. Parameter xmax is defined in the Limits. See also the Range editors section.
shell thickness / particle radius edit
shell thickness step count edit
Shell m' and m" edits
Core m' and m" edits
Edit theta button
Range editors for the particle size and scattering angle (see a sample scattering angle editor, named Theta range editor) accept any inputs that are consistent with the abstract range definition (with one exception detailed further in this section). Thus, range definitions may be accepted and displayed in the main form that do not fulfill either the calculation parameters' limits (click the Limits menu item) for the relevant variables, or algorithm requirements. The data are validated only when you click the Run button.
One of the range items is always calculated. This item's Calc radio button is selected. You can force an item to be calculated (i.e. not editable) by clicking the corresponding radio button in the Calc column of the range editor form. You can preview the effects of changing a range item by clicking Apply to apply the change. If satisfied, click Ok to accept it. You can also click Ok right away after any parameter change.
If a calculation setup implies that only the start value of a range can be edited (for example, the start value of the particle size range while the Output as a function of ... is Scattering angle ...), editing that value can lead to inconsistent range definitions in other cases of Output as a function of ... because in such a setup, the range editor does not verify the mathematical correctness of the range. For example, say that the particle size range was once defined as: start = 2, step = 0.1, count = 11, end = 3, and Output as a function of ... was changed to Scattering angle .... This enables the user to change the start value to, say, 3. When the calculation mode is changed to a mode that requires the whole particle size range definition, the modified range: start = 3, step = 0.1, count = 11, end = 3 will be inconsistent. However, this error will only be flagged when the Run button is clicked.
Output as a function of ...
If the results are functions of particle size, the angular patterns (see Results) are displayed for a single scattering angle. Particle size is displayed in the output form as the relative size x [non-dim] or as the absolute size if the abs size check box is un-checked or checked, respectively.
Please see info on a Microsoft ExcelTM spreadsheet (results viewers) pre-formatted to display such results in numerical and publication-ready graphical forms.
Two buttons in Get refractive index spectra ... group box are available for the refractive index spectra import:
... via form
... from file
Once the refractive index file is read in, the refractive index edits (such as shell m') are filled with the data from the first data line of that file. The wavelength edit is filled with the wavelength from that first data line. These edits are disabled in the Wavelength mode.
Refractive index file format and sample
Run button initiates the scattering properties calculations. Upon completion of the calculations it is disabled and a display form opens showing the results. If you close the results display form, the View results button will be shown as long as you do not modify the calculation parameters. Click this button to redisplay the results form without repeating the calculations.
Once you modify the calculation parameters, the Run button has to be clicked in order to recalculate and view the results.
Copy button of the display form copies all the results to the Windows clipboard. You can also copy a section of the results by simply selecting that section with the mouse and clicking "ctrl-c". When you are ready to paste the results in a Windows program accepting text, click "ctrl-v".
Save button of the display form saves the results to a space-delimited text file. The file name suggested by the Save As dialog box is that of the DPM preferences file (with extension "txt"). This enables synchronization of the names of the preferences file (see Setting preferences) and the corresponding results file. Please see Directories ... for the location of the results files.
Open this page of the page control (see User interface and the screen shot below) by clicking on its tab or via loading a RFF preferences file that opens it automatically and presets the calculations (see the description of the preferences box).
Once the page is shown, a group box containing the Select input file button is displayed (see also Input data file). If the input data file has not been pre-selected by loading a preferences file, this group contains only the Select input file button. Click it to open the Select input data file dialog. This dialog enables one to select an input data file which contains specification(s) of coated sphere(s) to process. The selected file is evaluated for correctness. If the evaluation is successful, i.e. if:
then, the Select types of results to include in the output: group box will be shown. If the check box for any result type is checked, a group box containg the Run button is shown below. Click that button to process the input data file and calculate scattering properties of coated sphere(s) specified therein.
During calculations, the currently processed data line of the input data file is displayed in the Run button box, next to the Line # label, where # is the number of the data line in the file. This detail is not shown in the screen shot above. The number of the first line of the file is 1. The line count includes all lines (i.e. the text lines and data lines).
Once the calculations are completed, the Run button is disabled, and a results form opens. You can copy all results to the Windows clipboard (click the Copy button) or save the results to a text file (click the Save button). Note that you can also copy selected results to the WindowsTM clipboard. To do that, select the data with the mouse and click ctrl-c. To paste the selection to another text-accepting WindowsTM program, click ctrl-v when the cursor is located in the text-accepting area of that program's window.
If the results form is closed, a View results button is displayed. This makes possible re-viewing of the results without repeating the calculations. This button is not shown if a data error is encountered.
If a data error is encountered in the input file (see Handling of input data errors), the calculations stop and an error message is displayed below the last processed data line. The results form will show the results calculated thus far.
The input data file is a tab-delimited, space-delimited, and/or comma-delimited text file. Multiple delimiters are treated as one delimiter, i.e. "2.0D2.2" is equivalent to "2.0DD2.2" (where D stands for a delimiter). In addition to the data required by the program, the file may contain text lines, such as the user comments, and heading(s), as well as additional data (see User's data) in each or some data lines.
A line is considered a text line if it starts with an item (field) that cannot be converted to a numerical value. A text line may, for example, contain the user's column headings. Note that the program uses its own headings when displaying the results. However, the user's heading can be included in the output file if the Copy text lines to output check box in the 'Read parameters from file' calculation mode is checked.
All text lines are ignored. This enables the user to:
Here, the intention was to separate the two data groups by a group heading which is split into two lines: "_Relative core diameter varies" and "_in the following group". However, given that the last group heading line is used by the program, the resulting group heading is: "_in the following group".
If the Copy text lines to output check box in the Read parameters from file tab is checked, all text lines are copied to the output. The initial block of text lines is copied to the beginning of the output. This block of text lines is shown at the beginning, separated from the results. Lines beginning with an underline ("_") are embedded in the output at the same locations as those in the input file.
A line of the input file is considered a data line if it contains at least 7 items (fields) (if the scattering angle is not defined in the first data line) or 8 items (if the scattering angle is defined) that can be converted each to a numerical value. This is checked when the file is opened and the first data line is displayed. This line is referred to in the Read parameters from file page as the "First data line". The first data line defines the setup of the entire file. Hence, if the first data line contains 7 (or 8) items, all data lines following the first line must have at least 7 (or 8) data items.
IMPORTANT: If the first data line contains 8 items, the 8th item is interpreted as the scattering angle. Hence, if the first data line of your data file contains more than 7 numerical fields but you intend not to define the scattering angle (i.e. the 8th field), you must separate the first 7 fields from the remaining numerical fields by a text field (see User's data).
The input data file can contain empty lines, although this may be confusing for the human reader and is not recommended. If you would like to separate groups of data lines visually for the human reader, please use group headings.
Each of the first 7 (or 8, if the scattering angle is defined) data items of a data line must be convertible to a numerical value. These items must always be arranged in the following order (counting from the beginning of the line; the space denotes one of the approved delimiters: space, tab, or comma):
Wave DiaS DiaC MReS MImS MReC MImC ScaAng User's text or data here ...
All data items intended for use by this program must fall into their respective validity ranges. These ranges can be viewed by clicking the Limits menu item. The acceptable ranges of coated sphere diameters are expressed in the relative particle size scale when the limits are displayed.
An error is reported and the program stops processing the input data file if:
Here, the shell diameter (DiaS) is the variable data item, other items are kept constant.
Results of the calculations are written to a display form as a sequence of output lines. Each output line begins with the copy of the relevant input data line, except the scattering angle (if defined). The following output items may be appended next:
The angle-dependent output items can only be appended to an output text line if the scattering angle data item is present in the first data line of the input data file. The output format is illustrated in sample output file.
For definitions of the these items, please refer to the Results section.
Once the output items are selected (see Setting the output format), the Run button is displayed (see Read parameters from file). Calculations are initiated by clicking that button. The program displays (in the box containing the Run button) each data input line it processes.
If the Copy text lines to output check box is checked (see Read parameters from file), text lines in the input data file are copied to the results. See details in the Text lines section. If the Copy user data to output check box is checked, the user data (the part of a data line following the items required by the program) are copied to the output. See User's data for more detail.
When calculations are completed without a data error, the output display form is shown and the results can be viewed, copied as text, or saved to a text file.
"Rff" (RFF file name prefix) +This file name will be suggested as the results file name (extension "txt") when the results are chosen to be saved.If a data error occurs, calculations stop and an error message is displayed under the last read-in input line. The results obtained so far are shown in a separate output form. All items in the RFF page, except the Select input file button, are disabled (but remain visible). In particular, the Run button is disabled. It can only be enabled again by clicking the Select input file button (see Read parameters from file).
The form displaying the results allows one to copy the (whole) results to the Windows clipboard (for transfer to a text-accepting Windows program) or save it to a space-delimited text file. Please see Directories ... for the location of the results files.
Sample results obtained for the sample input data file are shown below (numerical values are truncated to 4 decimals for this webpage only; the program outputs results with a precision of 6 decimals). The wavelength is displayed even if it is set to 0 in the input file. This is done to preserve the structure of the block of fields of the input data.
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 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 menu item to view the acceptable ranges of these parameters.
The refractive index is a complex number:
where i 2 = -1. The real and imaginary parts of the complex refractive index may also be referred to in this document and in some output forms as "ReM" and "ImM" or "MRe" and "MIm", respectively, and the refractive index as a whole can be referred to as "RI".
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, m" is positive, i.e. m" > 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 the abs size check box, the refractive index is expected to correspond to the wavelength value entered.
In this program, m' can assume values from a range of 0 < m' < m'max. The m" range is defined as 0 <= m" <= m"max, where the upper limits are arbitrarily chosen. The actual range definitions can be viewed by clicking the Limits menu item.
Spectral data for the complex refractive indices of the core and shell of the particle can be loaded from a text file if the option Output as a function of ... Wavelength is selected.
where D is the sphere diameter and λ 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 menu item.
The program accepts the particle size as x (relative size) if the abs size checkbox is not checked. Otherwise, the particle size is expected to be the absolute size (diameter) expressed in micrometers (1 Ám = 10-6 m).
The shell thickness is expressed relative to the whole particle radius, i.e. the relative shell thickness = shell thickness [length unit] / particle radius [length unit].
The scattering angle, θ [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, θ is the only directional parameter required.
Valid ranges of input data are shown by clicking the Limits menu item. Please note that the values for the particle size and the refractive indices of the shell refer to the individual parameters taken in isolation. In light scattering calculations, stricter limits are imposed by interplay between shell diameter and complex refractive index. This is because the product of these two latter parameters is one of the key parameters used by the scattering calculation function of the program.
This way of presenting the validity ranges enables the user to explore the whole individual range of a parameter, provided that the parameter linked to it is set to a "safe" value. For example, one can explore nearly the entire range of the shell diameter, provided that the shell refractive index magnitude assures that the product of these two parameters is not smaller than the minimum value. Thus, we can have a shell relative diameter of 5e-6 when the shell refractive index magnitude evaluates to a value greater than unity, say 1.1 (as in 1.1 - i 0). However, we cannot (for example) set simultaneously the shell diameter to 5e-6 and the shell refractive index to 0.01 - i 0.
The magnitude of the minimum of the product, z, of the shell diameter and of the refractive index magnitude is related to the fact that the program does not use approximations but an "exact" algorithm for calculation of the scattering properties. This algorithm involves taking differences between functions of z. When z evaluates to a sufficiently small value, differences between very large numbers are calculated. Given a limited number of decimal spaces in the computer representations of these numbers, these differences may evaluate to 0 when they should have a very small non-zero value. This may cause the algorithm to loose accuracy, as shown in the following figure, and even fail when z is allowed to decrease further.
Given the interplay between the particle sizes and refractive indices of a coates sphere, the user of this program is advised to exert caution when running this program in the extremities of the nominally allowed ranges of the input parameters. This applies especially to the cases where several input parameters are nearing their maxima allowed in isolation. For example:
As a guidance, the following rules should be observed:
This program calculates light scattering properties of a coated sphere following the Toon and Ackerman 1981 algorithm. A similar algorithm has been more 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:
where the "Atn", "Sca" and "Abs" prefixes mean attenuation, scattering, and absorption, respectively. We also have:
where a[n] and b[n] are functions of the particle sizes and refractive indices, and π[n] and τ[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| and |S2| are of interest in comparing results of calculations and an experiment. The squares of these magnitudes are displayed. M11, that is also displayed, is a real function of the scattering angle and is defined as follows:
This latter quantity is the element  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.
Angular patterns S1, S2, and M11 represent 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.
Multiply these efficiencies by the geometric cross section of the sphere (π D 2/ 4), where D is the whole sphere diameter, in order to obtain the attenuation, scattering, and absorption cross sections, respectively. This is done by the program when the abs size check box is checked.
Note that one can also define efficiencies for the scattering angle-dependent quantities:
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. Series of these functions are then summed to yield the desired results. Such calculations are known to be unstable if traversed upwards, i.e. in the direction of the increasing summation index. 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 this 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.
Results of this program have been tested against the known behavior of the relevant functions and also compared with published results ofg light scattering calculations. Excellent correspondence has been obtained between this program results and those reported, for example, by Kaiser and Schweiger 1993, Bohren and Huffman 1983. This correspondence attests to the capability of the downward recursion combined with the Lentz 1976 method for obtaining the initial values. Comparisons with other published results are detailed in the following text.
Table 1 contains results for the attenuation efficiency and scattering efficiency obtained with a program developed by Liu Lei et al 2007 and the MJC Light Scattering Calculator for Coated Sphere (CSS). The results obtained with these programs agree to within the numerical precision shown.
Table 2 compares results for the absorption efficiency obtained by Fenn RW and Oser 1965 and those obtained with the MJC Light Scattering Calculator for Coated Sphere (CSS). The absorption efficiency values tend to diverge as the particle size increases. When the particle size increases, the calculations involve an increasingly larger number of terms, suggesting that convergence problems in the program used by Fenn RW and Oser might be the source of this divergence. Similar trend can be observed when comparing results shown in Pilat MJ 1967 (his Fig. 2, middle panel) with those of the CSS program: the two sets of results diverge substantially for the relative particle size greater than ~1. Oscillations that appear beyond the first maxima of the efficiencies in the Pilat results do not seem to be consistent with significant absorption of light in a thick shell of the particle because absorption typically dampens such oscillations. Independent calculations (D. Zelmanovic, personal communication) confirm the CSS result. As shown in Table 1, results obtained with CSS and another program independently developed by Liu Lei et al 2007 for significantly greater particle sizes agree to 7 decimals.
Sun Wenbo et al 2004 provide a set of values of the attenuation efficiency, Qc, scattering efficiency, Qb, and absorption efficiency, Qa, of a coated sphere with a relative size of 10 and 12 for the core and shell, respectively, and with the complex refractive indices of 1.2 - i 0.1 and 1.1 - i 0.05 for the core and shell, respectively (relative to the refractive index of the surrounding, non-absorbing medium). These values:
match very well those obtained with the MJC Light Scattering Calculator for Coated Sphere:
In addition, the results were checked against the results of the homogeneous sphere calculator, also available from MJC Optical Technology, for three cases:
The vanishing shell case was tested by using a composite water-carbon sphere in air with the carbon shell thickness decreasing to 0. The whole particle x = 70 was assumed, with the water refractive index m = 1.33 - i0, and the carbon refractive index m = 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 for the shell thickness of 1e-7 with the coated sphere MJC Optical Technology calculator, assume values of Qatt = 2.02147, Qsca = 2.02141 and Qabs = 0.0000566. The homogenous sphere MJC Optical Technology calculator yields Qatt = Qsca = 2.02141 and Qabs = 0.
The vanishing core case was tested by using a composite carbon-water sphere in air 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 0.999999 are Qatt = 2.12599, Qsca = 1.30296, and Qabs = 0.823029.
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 Qatt = 2.02147, Qsca = 2.02147, and Qabs = 3.71e-16 (the latter value is essentially 0 at the numerical precision involved). With the homogeneous sphere calculator we obtain Qatt = Qsca = 2.02141 and Qabs =~ 0.
Kattawar and Hood 1976 (cited by Toon and Ackerman 1981) note that calculations of light scattering by a coated sphere should be substitued by calculations of light scattering by a homogenous sphere when the contribution of core of the coated sphere to the scattering properties becomes insignificant. This suggestions stems from the fact that the core-related terms contributing to the coefficient of the series representing the scattering properties involve differences of very small numbers. Given limited numerical precision, these differences may carry significant errors.
The following figure shows angular patterns (M11, denoted by S11 in the vertical axis label (see the Results section for a definition of M11) for a water droplet with a whole x = 70, without and with a 1%-thick coating of carbon [core m = 1.33 - i 0 (water), shell m = 2 - i 1 (carbon, see Janzen 1979)]. These results suggest a substantial effect of a highly absorbing thin shell around a non-absorbing core on light scattering by this composite particle at most scattering angles.
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, scattering, and absorption cross sections, Catn, Csca, and Cabs, respectively, as follows:
where Ein[W m-2] is the irradiance of the light beam. Note that in physics, a quantity with a dimension of power per unit area is traditionally refered to as the intensity.
The cross sections [m2] are defined as follows:
where "xxx" means either "Abs", "Atn", or "Sca" and k = 2 π / λ [m-1] is the wave number.
The cross sections can also be defined by using the efficiencies as follows:
where "xxx" means either "abs", "att", or "sca", and D is the whole particle diameter.
For N [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 Fatn. The effect of the suspension layer interfaces (for example, water/air) is not accounted for here. The product N Catn equals the attenuation coefficient, usually denoted by c [m-1].
In the limit of the small changes, a 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
One can calculate in a similar manner the power scattered in all directions by a volume, V, of the suspension.
If the suspension consists of particles with various diameters and refractive indices, the particle concentration N is replaced by the size-refractive index distribution of the particles and the total power of light attenuated, scattered, or absorbed by the suspensions is expressed as a double integral of the respective cross section weighed by that distribution. Such integration must at present be provided off line, for example by using a spreadsheet.
Similarly one can calculate the radiometric intensity (power per unit solid angle; W sr-1) of light scattered by a single sphere or suspension of spheres at a specific direction (θ, φ) relative to that of the incident wave. The azimuthal angle φ is measured in a plane perpendicular to the wave direction (beam axis). Given the axial symmetry of light scattering by a coated sphere (about the direction of the incident light beam), the scattering direction can be defined by using only the scattering angle, θ. Hence:
The dimension of Csca(θ) is m2sr-1.
Given the acceptance solid angle of a detector, Ωdet [sr], the power of 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 θ1 and θ2. This power equals:
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