Mie scattering for inhomogeneous medium

In summary, the conversation discusses different methods for using Mie scattering theory to calculate extinction coefficients for a half particle surrounded by two different media. The first method involves treating the particle as two independent particles and using Mie scattering theory to calculate the results for each. The second method involves considering only the medium in the outgoing direction and using modal expansion in spherical harmonics. However, this method may not work for more complex situations. The conversation also raises the question of whether taking the average of extinction coefficients calculated for pure air and glass would yield the correct result. This approach is not a solution to the field boundary value problem and would not work for different media, such as silver. Ultimately, the conversation highlights the challenges and limitations of using Mie scattering
  • #1
Haorong Wu
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Homework Statement
Suppose a particle of Ag with diameter of 100nm is partially embeded in a glass substrate so that half of the particle is surrounded by glass, while the other half is surrounded by air. The refractive index of the glass substrate is 1.5. Use Mie scattering theory to analyze the scattering and absorption spectrums between 300nm and 1000nm.
Relevant Equations
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It is an open question. The professor asked us to find a suitable method to solve it with the help of a computer.

When I learned the Mie scattering, the equations are given for particles in homogenous medium. But now half of the particle is surrounded by glass while the other half by air.

First, I want to treat the particle as two independent particles one of which is surrounded only by glass, while the other by air only. Then I can use Mie scattering theory to calculate independently for them and then sum the results. I believe this could be an approach but it may not give a satisfying result.

Second, I think maybe I only need consider the medium in the outgoing direction. I note that the coefficients for extinction, scattering, and absorption are related to ##a_n## and ##b_n##. And ##a_n## and ##b_n## are related to the scattered fields $$ \mathbf {E} _{s}=\sum _{n=1}^{\infty }E_{n}\left(ia_{n}\mathbf {N} _{e1n}^{(3)}(k,\mathbf {r} )-b_{n}\mathbf {M} _{o1n}^{(3)}(k,\mathbf {r} )\right).$$ Besides, the incident light will excite the atoms based on its frequency ##\omega## which is not related to the refractive index of the medium. Also the scattering directions are randomly distributed, so I can calculate the scattering coefficients for pure air and for pure glass, and then add them up and divide them by 2. The same method will be applied to the extinction coefficients.

Is there any more reasonable method?
 
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  • #2
Mie theory relies on separation of variables in spherical coordinates of Maxwell's equations. The half space of dielectric breaks the rotational symmetry the problem so this separation of variables doesn't apply directly. I would look at using spherical modes in each half space. The angular part of these modes in each half space (vector spherical harmonics) are the same. The only difference are the radial functions which carry the only dependence on wave number(the wavelength changes between media). What this leaves you with are one dimensional mode matching problems for each of the radial functions for each angular mode. These should be all independent.

P.S. I've never attempted this solution so I'm spitballing here.

P.S.P.S One must consider a modal expansion in spherical harmonics of the incident fields. This should be doable for your case because reflection from a dielectric interface like this has a known solution in terms of plane waves in each half space. I would start with this observation.
 
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  • #3
Thanks, @Paul Colby . I have tried the easiest situation.
3-1.jpg
This is the only configuration that I can solve, because, as you mentioned, the symmetry is crucial for Mie theory, and this configuration keeps most of symmetries.

From the equation of extinction cross section, $$W_{ext}=\frac 1 2 \Re \int_0^{2 \pi} \int_0^{\pi} (E_{i\phi} H_{s\theta}^{*} -E_{i\theta} H_{s\phi}^{*} -E_{s\theta} H_{i\phi}^{*} +E_{s\phi} H_{i\theta}^{*})r^2 \sin \theta d\theta d \phi ,$$ where ## E_{i \theta} = \frac {\cos \phi} {\rho} \sum_{n=1}^{\infty} E_n(\psi_n \pi_n-i \psi_n^{'} \tau_n ), ## ##H_{i\theta} = \frac k {\omega \mu} \tan \phi E_{i\theta}##, etc.

We note that all variables in ##Es## and ##Hs##, such as ##\psi_n, \pi_n, \tau_n , \xi_n## do not depend on ##\phi##. So after integrating ##\theta## we have $$ W_{ext}\propto\Re \int d\phi (\alpha \sin^2 \phi - \beta \cos^2 \phi)$$ where ##\alpha## and ##\beta## are some constants. If the particle is surrounded by homogeneous medium, the integration will be carried over ##0## to ##2\pi##, yielding ##W_{ext} \propto \Re \pi (\alpha -\beta)##. However, if the system is configured as in the above picture, then the integration will be split into two part, one from ##0## to ##\pi## with the refractive index ##n_1##, the other from ##\pi## to ## 2\pi## with the refractive index ##n_2##, giving ##W_{ext} \propto \Re \frac 1 2 [\pi (\alpha_1 -\beta_1)+\pi (\alpha_2 -\beta_2)]##.

Since the extinction coefficient is proportional to its cross section, we could first calculate the extinction coefficients when the particle is surrounded by pure air and pure glass, respectively, i.e., $$ Q_{ext}^{air},~~~ Q_{ext}^{glass},$$ and then take their average $$Q_{ext}=\frac 1 2 ( Q_{ext}^{air}+ Q_{ext}^{glass}).$$

If we consider other more general situations, the integration over ##\theta## would play an important role and it would be quite difficult to figure out its analytic solution. Therefore, we will not attempt to solve them.
 
  • #4
Haorong Wu said:
and then take their average
But this isn't a solution of the field boundary value problem. Replace the refractive index of the half space with that of Ag. Do you still think this average yields the correct answer? Clearly it doesn't.
 
  • #5
@Paul Colby . Thanks, I made a mistake. I forgot that the change of the medium will affect all the fields inside the particle. I will re-think your post. Well, to be honest, my professor did not taught us those details. He just gave us the conclusion. It is a good time that I will go through the theory step by step.
 
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  • #6
The problem as stated isn't an easy boundary value problem and it's not clear to me that a series solution with known coefficients is attainable. At best you might obtain a series of equations for the coefficients which need to be solved. These equations could well be too complex to solve with pencil and paper. Just setting them up is a significant task in itself. I guess it depends on how badly one want's the answer. If your professor doesn't know the solution and would like it, I have very reasonable rates. :rolleyes:

[Edit] Ah, I missed the light being along the plane separating half spaces. This could well be a special easy case that is solvable. My comments assumed arbitrary incidence angles to the plane. They still apply.
 
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Related to Mie scattering for inhomogeneous medium

1. What is Mie scattering for inhomogeneous medium?

Mie scattering is a phenomenon that occurs when light interacts with small particles in a medium that is not uniform in composition. In this case, the particles have a size comparable to the wavelength of light, causing the light to scatter in all directions.

2. How is Mie scattering different from Rayleigh scattering?

Mie scattering is different from Rayleigh scattering in that it takes into account the size of the particles, while Rayleigh scattering assumes that the particles are much smaller than the wavelength of light. This means that Mie scattering is more accurate for larger particles, while Rayleigh scattering is more accurate for smaller particles.

3. What factors affect Mie scattering for inhomogeneous medium?

The factors that affect Mie scattering for inhomogeneous medium include the size and shape of the particles, the refractive index of the particles and the surrounding medium, and the wavelength of the incident light. These factors can all influence the scattering pattern and intensity.

4. How is Mie scattering used in scientific research?

Mie scattering is used in a variety of scientific research fields, including atmospheric science, remote sensing, and biomedical imaging. It can provide information about the size, shape, and composition of particles in a medium, which is useful for understanding the properties and behavior of various substances.

5. What are some applications of Mie scattering for inhomogeneous medium?

Mie scattering has many practical applications, such as in weather forecasting, air pollution monitoring, and medical diagnostics. It is also used in industries such as agriculture, food processing, and pharmaceuticals to study the properties of particles in various materials.

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