# How to Calculate Transmission Spetrum

EDIT: Now that I have thought more I suppose this problem is a classical problem and should therefore be moved to that forum. Sorry for the mistake. Can this be moved?

Hi

I am working on a simulation trying to model light traveling through a periodic grating with subwavelength apertures. I am launching a Gaussian pulse at this structure and I would like to find which frequencies have the highest transmission. I have setup diagnostics that measure the poynting vector verse time of the incident and transmitted wave. Once I collect the data I use fast fourier transform in matlab and then get the power spectrum from the incident and transmitted wave.

From my introductory waves course (I'm an undergrad) I remember that the transmission of a wave (lets just talk about a wave on a string) through a boundary is the
square of the transmitted amplitude/square of the incident amplitude.

So to find the transmission spectrum for my problem should I do something like what I would do with a wave on a string? I could divide the power spectrum of the transmitted wave by the power spectrum of the incident wave?

I do not necessarily need a fully answer. It would be sufficient to just let me know if I am on the right or wrong track. Also can anyone recommend a book, website, or anything that would cover this material and is written at the undergrad/grad level? Thank you in advance for your help

Note: I posted here because this forum cover computational physics but if there is a better place for this please let me know.

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## Answers and Replies

What physical model are you simulating? Also, it would be helpful to see even a crude sketch of the geometry (angles of beams with respect to the grating, orientation of the holes, etc.). I am worried that you are simulating structures comparable to the wavelength of the light, and it is not clear whether or not you are accounting for evanescent wave effects.

My first thought on how to approach this problem is to simulate the interaction of the light with the grating using the finite-difference time-domain (FDTD) method (discretization of Maxwell's equations in space and time, which will account for evanescent wave effects), then Fourier transform the electric field at the output to obtain the transmission spectrum (as you've been doing). Some codes for simulating photonic crystals may be applicable to this situation as well.

I just want to start off by saying thank you for your interest and help. I will try to clarify the best I can but sometimes I have a hard time explaining what I am doing in words.

What physical model are you simulating?
what exactly do you mean by model?

I attached a diagram of the geometry. The grating is periodic in the vertical direction and the light is horizontal. So the light is normal to the surface. I should have been more specific since I only want to look at waves being launched at the grating with magnetic component pointing out of the page in the diagram.

Using FDTD method is what many of the papers I have read have done. I am trying to reproduce some of the results from S. Astilean, P. Lalanne, M. Palamaru, Optics Communications, Volume 175, Issue 4-6, p. 265-273. (I would have attached it but it was too large). They used what they called "the enhanced version of the rigorous coupled-wave analysis" which I believe is just a certain type of FDTD.

Instead of using FDTD I would like to model that same behavior that they did using particle-in-cell (PIC) code. I am using OOPIC Pro, a 2D PIC code. Since my grating is periodic in one direction I can use periodic boundary conditions to reduce my problem to a 1D problem. I have modeled silver in OOPIC by creating stationary positively charged ions and electrons in the same region with the proper number density. I think that assuming I have a sufficiently small enough grid I should have the evanescent waves accounted for.

Does that all make sense?

My current problem is in how to calculate transmission efficiencies if I know the incident and transmitted poynting vector over a period of time.

Thank you again

#### Attachments

• diagram.doc
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Sorry it's taken me a little while to get back to you. First, I just wanted to let you know that rigorous coupled-wave analysis (RCWA) is a different simulation method than finite-difference time-domain (FDTD).

FDTD samples the electric and magnetic field at evenly-spaced points in space and time. If the sampling interval in space is $$\Delta x > 0$$, then FDTD approximates spatial derivatives like $$\frac{df}{dx}$$ by $$\frac{df}{dx} \approx \frac{f(x + \Delta x) - f(x)}{\Delta x}$$. Note that if we take the limit of this approximation as $$\Delta x \to 0$$, we have the definition of the exact derivative. This approximation is extended to partial derivatives with respect to space and time to evaluate Maxwell's equations.

RCWA utilizes the Bloch theorem (known as the Floquet theorem in the one-dimensional case) that the eigenfunctions of the wave equation for a periodic potential (in this case, a grating) have the form of a plane-wave $$\exp(i \vec{k} \cdot \vec{r})$$ multiplied by a function that has the periodicity of the potential. Thus the electric and magnetic field can be represented by a Fourier series where the fundamental frequency corresponds to the periodicity of the grating. RCWA solves for the field by finding a modal (Fourier) decomposition for the field in the grating region and in the incident and transmitted regions, then finding the amplitudes of these modes that satisfy Maxwell's equations and electromagnetic boundary conditions. The mathematics involved in this process include finding matrix eigenvalues and eigenvectors (which correspond to modes), and solving a linear system of equations (to find the mode amplitudes). If you're confused by all this, don't worry--a year ago I was probably as confused or more confused than you are now. The upshot is that the mathematics of RCWA and FDTD are very different.

I must confess that I know very little about particle-in-cell methods, and in particular I have no idea how particle-in-cell methods are used to simulate electromagnetic fields, so I'm afraid I can't help too much with that at this time.