1D Finite Planar Photonic Structure - Transfer Matrix Method

In summary: The reflectivity is 0.0133, the transmittivity is 0.9867In summary, after a lengthy dialogue with my supervisor, the problem was solved by realizing that energy flow was not being conserved due to an implicit assumption. By constructing a function to calculate the time-averaged Poynting vector, the correct values for reflectivity and transmittivity were obtained.
  • #1
aphirst
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Homework Statement


I'm implementing the transfer matrix method (manually) for an EM wave through a 1D layered structure. Basically I'm just considering a plane wave in the positive-x direction, conserving E and H across each material interface, and constructing interface matrices, the appropriate matrix product of which allows you to work out forward/reverse coefficients in each region, and the reflectivity/transmittivity.

I have it set up such that I just make a big list of (complex) refractive indices, and the widths of each sublayer (or the positions of each interface); also specifying the angle of incidence (in the xz plane), and the polarisation (TE or TM, i.e. either E or H respectively entirely in the y-direction).

Whenever the refractive index on the far LHS (i.e. the material that extends to [itex]-\infty[/itex]) is equal to that on the far RHS (i.e. extending to [itex]+\infty[/itex]), reflectivity and transmittivity both work perfectly.

However when they differ, for instance in the case with only one interface; reflectivity behaves perfectly, while transmittivity is no longer bounded under 1.

Homework Equations


E-mail to my supervisor, in which I walk through the cases that do and don't work. Includes plots.
https://dl.dropbox.com/u/3219541/Project/email.pdf [Broken], or the attached email.pdf

A transcript of my working for this method, including some explicit calculations to demonstrate the fact that R behaves, while T does not.
https://dl.dropbox.com/u/3219541/Project/calculations.pdf [Broken], or the attached calculations.pdf

The Attempt at a Solution


I went to see him to discuss the problem yesterday: he told me it was to do with that RHS/LHS difference (rather than an artefact of only performing the method for a single interface, which is what I had thought). He then said something about phase velocities, and about having to scale something according to the ratio of permittivities (or, complex refractive index squared), but wasn't massively clear where that was supposed to come up in the maths. I've read over the relevant sections of some EM textbooks, but I can't see how that's supposed to change what I've worked out.If any of you have the time to read over what I've done, and (if I'm fortunate) point me in the right direction, I'd really appreciate it.
 

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  • #2
After a lengthy dialogue with my supervisor, I solved the problem!

I had been implicitly assuming that since I was conserving the values of E and H at each interface, I was therefore automatically conserving energy flow through the structure. This of course was not the case.

So I constructed a function to calculate the time-averaged Poynting vector, and took the ratio of the energy flow just after the final interface and the energy flow of only the incident light. And what do you know, this gave me exactly what I expected for T.

For instance, here's the output for TM light at the interface between air and glass.
RT_angle_H_1550nm.png
 

1. What is a 1D finite planar photonic structure?

A 1D finite planar photonic structure is a type of optical device that consists of multiple layers of materials with different refractive indices, arranged in a one-dimensional pattern. It is used to manipulate the propagation of light, such as reflecting, transmitting, or filtering specific wavelengths of light.

2. What is the Transfer Matrix Method?

The Transfer Matrix Method (TMM) is a mathematical tool used to analyze the behavior of light as it passes through a layered optical structure. It involves breaking down the structure into multiple layers and calculating the transfer matrix for each layer, which represents the change in the electric and magnetic fields as light passes through the layer. The matrices are then multiplied together to obtain the overall transfer matrix for the entire structure, which can be used to calculate properties such as reflectance and transmittance.

3. How does the Transfer Matrix Method work for 1D finite planar photonic structures?

The TMM works for 1D finite planar photonic structures by calculating the transfer matrix for each layer of the structure, starting from the input layer and moving towards the output layer. The matrices are multiplied together to obtain the overall transfer matrix, which can be used to calculate the reflectance and transmittance of the structure for a given wavelength of light.

4. What are the advantages of using the Transfer Matrix Method for 1D finite planar photonic structures?

The TMM is a versatile and efficient method for analyzing the behavior of light in layered optical structures. It allows for accurate calculations of properties such as reflectance and transmittance, and can be easily adapted for different types of structures. It also provides insights into the effects of varying layer thicknesses and refractive indices on the behavior of light, making it a valuable tool for designing and optimizing photonic devices.

5. What are some applications of 1D finite planar photonic structures and the Transfer Matrix Method?

1D finite planar photonic structures and the TMM have a wide range of applications in optics and photonics. They are commonly used in the design of optical filters, mirrors, and anti-reflection coatings. They also play a crucial role in the development of photonic integrated circuits and devices for telecommunications, sensing, and imaging. Additionally, the TMM is used in research to study the behavior of light in various materials and structures, providing valuable insights for advancing optical technologies.

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