Photonic Band Structure Calculation

[mrjeffy321]

I am trying to calculate a photonic crystal’s band structure for a very simply case of a 1-dimentional periodic medium, or rather I am trying to understand the calculations that I am looking at for this situation.
The end result of what I would like to do is get some function that I can graph and use to predict a photonic bandgap if one exists after specifying the needed parameters (lattice constant, index of refraction, …).

I am looking at the explanation / calculations here:
http://ab-initio.mit.edu/photons/tutorial/L1-bloch.pdf (~5 MB)
and here:
http://ab-initio.mit.edu/photons/tutorial/photonic-intro.pdf
But I don’t really understand what is going on or recognize the result when I see it.

I also am looking a derivation (paper copy, no link) which goes into much more detail.
In this derivation, a 1-dimentional PC’s band structure is analyzed by looking at the electric field in one of the layers in the crystal. Bloch’s theorem is applied and we get two eigenvalues (complex conjugates of each other) which look very similar to the one’s found in the above .pdf documents.

I guess what I am not understanding is how to turn these eigenvalues which are found into something more meaningful.

Also, I don’t really understand the significance of first Brillouin zone in all of this. I only have a vague idea of what it is and how to find it.
Does it really apply in the one-dimensional case? Would it not just be a line in 1-D as opposed to a square in a 2D square lattice, or a semi-round soccer ball looking thing in a FCC cubic lattice?

 

[eljose79]

Thank you for your question regarding calculating the band structure of a 1-dimensional photonic crystal. As a scientist with expertise in this area, I am happy to help you understand the calculations and results you are looking at.

Firstly, let me explain the concept of a photonic bandgap. In a photonic crystal, the periodic arrangement of refractive index allows for the creation of a bandgap, which is a range of frequencies where light cannot propagate through the crystal. This is similar to the bandgap in a semiconductor material, where electrons cannot move through the material within a certain energy range. In a photonic crystal, the bandgap is created by the interference of light waves within the crystal structure.

Now, to understand the calculations and results you are looking at, it is important to understand the concept of Bloch's theorem. This theorem states that in a periodic structure, the eigenvalues (or energy levels) of the system are related by a phase factor, known as the Bloch phase. This allows us to simplify the calculations and only consider the behavior of the system within one unit cell, rather than the entire crystal.

In the case of a 1-dimensional photonic crystal, the Brillouin zone is indeed a line, as you mentioned. This line represents the first order diffraction of the crystal structure. This means that any wave with a wave vector (k-vector) within this line will experience constructive interference and can propagate through the crystal, while waves with k-vectors outside of this line will experience destructive interference and will be prohibited from propagating through the crystal.

To turn the eigenvalues you have found into something more meaningful, you can plot them as a function of the wave vector (k-vector) within the Brillouin zone. This will give you the band structure of the crystal, which shows the allowed and forbidden energy states for light within the crystal.

I hope this helps you understand the calculations and results you are looking at. If you have any further questions or would like more clarification, please do not hesitate to ask.

 

[charng]

I would be happy to help you understand the concept of photonic band structure calculation. The calculation of photonic band structure is a fundamental and important aspect in the study of photonic crystals. It allows us to predict the behavior of light in periodic structures and understand the formation of photonic bandgaps, which are crucial for controlling the propagation of light in these materials.

To start, let's discuss the concept of a photonic crystal. A photonic crystal is a periodic structure with a refractive index that varies periodically in space. This periodicity leads to the formation of photonic bandgaps, which are ranges of frequencies where light cannot propagate through the material. These bandgaps are similar to the energy bandgaps in electronic band structures, but instead of electrons, we are dealing with photons.

Now, let's talk about the calculations involved in determining the photonic band structure. The main tool used for this is Bloch's theorem, which states that the wavefunction in a periodic potential can be written as a product of a periodic function and a plane wave. In simple terms, this means that the wavefunction in a photonic crystal can be described as a repeated pattern of a basic unit cell, multiplied by a plane wave with a certain wave vector. This wave vector is related to the frequency of the light and is crucial in determining the band structure.

To calculate the photonic band structure, we need to solve the Maxwell's equations in the periodic medium. This can be done by using the plane wave expansion method, where we express the electric field as a sum of plane waves with different wave vectors. By applying Bloch's theorem, we can reduce the number of equations to solve and obtain the eigenvalues (frequency) and eigenmodes (electric field) of the system.

The first Brillouin zone is a concept that helps us understand the periodicity of the crystal and its effect on the band structure. It is defined as the smallest repeating unit in the reciprocal lattice, which is the Fourier transform of the original crystal lattice. In 1D, the first Brillouin zone is simply a line segment, and it plays a crucial role in determining the band structure and the formation of bandgaps.

I understand that the calculations and concepts involved in photonic band structure can be complex and may require some time to fully grasp. I would suggest consulting more resources and discussing with your peers or a mentor to gain a better understanding. It is also