In this example we will show how to convert spatially varying stress or strain into a spatially varying refractive index profile, using the (n,k) Material import object. In this example, we assume a simple analytic form for the stress but it could be modified easily to load a spatially varying strain profile from a different simulation.
Background
When introducing the spatially varying refractive index due to stress or strain in Lumerical’s FDTD or MODE, it is important to distinguish situations which will result in diagonal anisotropy from non-diagonal anisotropy. Diagonal anisotropy, which is the subject of this example, can be solved easily using an nk import material, available in both FDTD and MODE.
| Note: If the stress or strain results in a permittivity tensor with off-diagonal elements, then it will be necessary to diagonalize that permittivity tensor and use a matrix transformation to introduce the effect of the strain. The setup of the matrix transform grid attribute is demonstrated in Matrix transformation. |
In general, the change in permittivity due to a given strain is given by
$$ \Delta\left(\frac{1}{n^{2}}\right)_{i}=\Delta\left(\frac{1}{\varepsilon_{r}}\right)_{i}=p_{i j} \varepsilon_{j} $$
where n is the refractive index, εr is the relative permittivity, p is the photoelastic tensor and ε is the strain. We use the contracted tensor notation (1-xx, 2-yy, 3-zz, 4-zy, 5-zx, 6-xy) [1]. Generally, in isotropic media it is possible to simplify the above expression. Furthermore, in some circumstances, it is useful to relate the change in refractive index to the strain rather than the stress, which can be done using the stress-strain relations. For example, in fused silica the change in refractive index in the i-th direction can be related to the strain by
$$ \Delta n_{i}=n_{i}-n_{0}=-B_{2} \sigma_{i}-B_{1}\left(\sigma_{j}+\sigma_{k}\right) $$
where the indices (i,j,k) are cyclical permutations of (x,y,z) and the values of B1 = 4.22e-6 (MPa)-1 and B2 = 0.65e-6 (MPa)-1 [2].
For this example, we assume that we have a fused silica grating that has been bent over a radius, R, much larger than the pitch, a. The dominant elongation component is therefore along the x-axis, and we can neglect the other transverse components to the strain. We assume, therefore, that the stress has the form
$$ \sigma_{y}=\sigma_{z}=C_{12} \frac{y}{R} $$
$$ \sigma_{x}=C_{11} \frac{y}{R} $$
where y is the distance from the bending center, C11 and C12 are components of the elasticity tensor that relate σ (stress) and ε (strain) by Hooke’s law (σ=C*ε). This is similar to the effect of bending a fiber, as shown in [3] but can also be applied to a bent grating. The components are given by
$$ C_{11}=\frac{(1-n) E}{(1+n)(1-2 n)}$$
$$C_{12}=\frac{n E}{(1+n)(1-2 n)} $$
where n = 0.164 is the Poisson number and E = 76 GPa being the Young’s modulus for fused silica [2].
Simulation setup / Workflow
The simulation file stress_strain.fsp contains a grating made from the bent glass which has been etched with a 50% duty cycle. The structure group “strained glass” contains a script that constructs the spatially varying refractive index based on the above formulas and constants for glass. It allows the user to enable or disable the bending, and set the radius of curvature, R. For this example, we use R = 100 microns.
The script file stress_strain_analysis.lsf runs two simulations: once with no bending using the nominal index of the glass (here 1.444). The second simulation enables the bending. We visualize the refractive index of the glass, which can be seen clearly by setting the colourbar min and max to 1.44 and 1.444 respectively.
Results
Finally, the script compares the reflected order efficiencies for the 2 cases, as shown below. We can see that the order efficiencies for the 0th and 1st order efficiencies can be substantially modified by the bending.
Related publications
- D. Lockwood, L. Pavesi, "Silicon Photonics II: Components and Integration", Springer, 2011
- W. Primak and D. Post, "Photoelastic constants of vitreous silica and its elastic coefficient of refractive index", J. Appl. Phys. 30, 779 –788 (1959).
- C. Shulze et al, "Mode resolved bend loss in few-mode optical fibers", Opt Express, 2013.