{"id":218533,"date":"2022-07-14T20:57:23","date_gmt":"2022-07-14T20:57:23","guid":{"rendered":"\/forum\/forums\/topic\/incorporating-radial-birefringence-breaking-cylindrical-symmetry-of-the-modes\/"},"modified":"2022-07-14T20:57:23","modified_gmt":"2022-07-14T20:57:23","slug":"incorporating-radial-birefringence-breaking-cylindrical-symmetry-of-the-modes","status":"closed","type":"topic","link":"https:\/\/innovationspace.ansys.com\/forum\/forums\/topic\/incorporating-radial-birefringence-breaking-cylindrical-symmetry-of-the-modes\/","title":{"rendered":"Incorporating radial birefringence breaking cylindrical symmetry of the modes?"},"content":{"rendered":"

Hello, Learning Forum!<\/p>\n

I am working on a research project in which I’ve been investigating light transmission, especially light in guided modes, through different micron-scale structures. I’ve been using Lumerical’s FDTD software to build the models and run simulations, and I’ve run into some unexpected mode results that I’ve described in detail below. If I should be posting this question somewhere else, or contacting customer support directly, please let me know!<\/p>\n

—<\/p>\n

I have changed a cylindrically-symmetric model to include radial birefringence in one of the materials, and now when I calculate the modes I am observing odd asymmetric behavior. I am wondering what could have caused this (my model is still symmetric as far as I can tell) and if there’s anything I can do in the software to see symmetric modes for this model — or if it is some intrinsic asymmetry (which I think is unlikely).<\/p>\n

 <\/p>\n

My basic structure is a single solid cylinder being enclosed by a hollow cylinder (outer radii 0.6 microns and 1 micron respectively). The outer layer (the hollow cylinder) has a higher refractive index than the inner cylinder, and I am investigating how light is guided through this structure.<\/p>\n

<\/p>\n

In previous simulations, I specified these structures to have constant refractive index (outer layer 1.44, inner layer 1.38, environment 1.00), and found that the modes reflected this symmetry — see especially mode 3 and mode 6 in the picture below.<\/p>\n

<\/p>\n

Now, I am specifying the outer structure to have positive birefringence along radial optic axes. I have followed the steps to encode an anisotropic material ( https:\/\/optics.ansys.com\/hc\/en-us\/articles\/360034394694-Creating-anisotropic-optical-materials-in-FDTD-and-MODE ) in the software. <\/p>\n

The diagonalized permittivity matrix (being in cylindrical coordinates) looks like the matrix below. Note n_{e} is the refractive index for light travelling radially, n_{o} is the refractive index for light travelling in the phi or z directions. <\/p>\n

epsilon = [ n_{e}^2, 0, 0; 0, n_{o}^2, 0; 0, 0, n_{o}^2 ]<\/p>\n

Since this anisotropic material is diagonal in cylindrical coordinates, I have encoded this in the material using the material database (which uses Cartesian coordinates, but I later convert this to cylindrical coordinates). Diagonal anisotropy can be specified in a material by specifying n_{x}, n_{y}, n_{z} as seen in the form n = [n_{x}, 0, 0; 0, n_{y}, 0; 0, 0, n_{z} ]. I have included a screenshot of this as well. I am currently using the values: n_{x} = 1.8 (which specifies this for the radial direction), n_{y} = 1.4 (phi direction), and n_{z} = 1.4 (z direction). The settings I have input, and the index monitor profiles, are shown in the picture below. Again, everything looks like it should here.<\/p>\n

<\/p>\n

Since I am using a cylindrical coordinate system here, I have made use of the matrix transform grid attribute ( https:\/\/optics.ansys.com\/hc\/en-us\/articles\/360034915173-Matrix-Transformation-Simulation-object ) in the software. This essentially specifies new unit vectors in a given area of the structure. I have generated cylindrical unit vectors at each point in the hollow cylinder material (which is centered around the origin and has well-known inner and outer radii). <\/p>\n

I have set up a grid of 801 by 801 points, to span the two-dimensional area which is 4 microns along both x and y axes (the cross-section is a square). For points outside the outer radius of the hollow cylinder (the background environment) and for points inside the inner radius of the hollow cylinder (the solid cylinder in the center), I specify the identity matrix as the matrix transform U since there is no coordinate transformation needed here. For points within the outer and inner boundaries of the hollow cylinder, I implement a cylindrical coordinate matrix transformation. I want the software to use the refractive index information for the x direction (n_{x}) as the refractive index for the radial direction; similarly to use n_{y} for the \\phi direction and n_{z} for the z direction (this last one is unchanged). So the matrix U (that ensures a permittivity matrix \\epsilon is diagonal as seen on this page: https:\/\/optics.ansys.com\/hc\/en-us\/articles\/360034394694-Creating-anisotropic-optical-materials-in-FDTD-and-MODE ) at a given point (x, y, z) is:<\/p>\n

U = [ cos(phi), – sin(phi), 0; sin(phi), cos(phi), 0;  0, 0, 1 ] where phi = atan2(y, x)<\/p>\n

So I have calculated (in MATLAB) a U matrix at each point in the 801 by 801 grid I have set up (which, for points within the hollow cylinder cross-sectional area, implements a cylindrical coordinate transformation). (Since position on the z-axis doesn’t affect the matrix transform whatsoever, I duplicate this data so it can be applied at an 801×801 grid at z = 0 microns and an 801×801 grid at z = 4 microns and the software knows it is constant with respect to z.) <\/p>\n

I am confident that this is working as I intended it to, according to the visual aids of the matrix transformation grid attribute that, when the grid attribute is added to the model, indicate the specified unit vectors. On the image below, you can see that within the light yellow area (the hollow cylinder) each blue arrow (usually indicating the x-direction) points radially outward and each green arrow (usually indicating the y-direction) points in the phi direction. (I have changed the colors of the structures so these matrix transformation arrows are more visible.)<\/p>\n

 <\/p>\n

So to my knowledge, everything about the model is working properly and there is well-established cylindrical symmetry about the structure (which I’ve placed on the origin). However, when I calculate the modes for this new model (with the radial birefringence included), I am finding breaking of symmetry that I did not find in the old model. There seem to be modes that arbitrarily choose spots within the high-refractive-index hollow cylinder to have higher intensity than others. Some extreme examples include a single smeared spot of high intensity on one side of the hollow cylinder, with low intensity on the other side. I have not been able to find a mode like mode 3 or mode 6 in the mode picture above. Examples of the modes I am finding with this new model are shown in the picture below.<\/p>\n

<\/p>\n

So, might there be anything in the software that is causing this? Have I done something wrong in implementing my model into the software? Does anyone have any ideas as to why the modes are non-symmetric when I am implementing the radial birefringence? Many thanks in advance!<\/p>\n","protected":false},"template":"","class_list":["post-218533","topic","type-topic","status-closed","hentry","topic-tag-anisotropic","topic-tag-cylinders","topic-tag-cylindrical-coordinates","topic-tag-dielectric-waveguide","topic-tag-fdtd","topic-tag-lumerical","topic-tag-modes","topic-tag-symmetry","topic-tag-waveguides"],"aioseo_notices":[],"acf":[],"custom_fields":[{"0":{"_bbp_subscription":["256430","2592"],"_bbp_author_ip":["204.2.243.124"]," _bbp_last_reply_id":["0"]," _bbp_likes_count":["0"],"_btv_view_count":["3152"],"_bbp_topic_status":["unanswered"],"_bbp_status":["publish"],"_bbp_topic_id":["218533"],"_bbp_forum_id":["27833"],"_bbp_engagement":["2592","256430"],"_bbp_voice_count":["2"],"_bbp_reply_count":["4"],"_bbp_last_reply_id":["232916"],"_bbp_last_active_id":["232916"],"_bbp_last_active_time":["2022-08-13 18:01:08"]},"test":"emily-fredeucalgary-ca"}],"_links":{"self":[{"href":"https:\/\/innovationspace.ansys.com\/forum\/wp-json\/wp\/v2\/topics\/218533","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/innovationspace.ansys.com\/forum\/wp-json\/wp\/v2\/topics"}],"about":[{"href":"https:\/\/innovationspace.ansys.com\/forum\/wp-json\/wp\/v2\/types\/topic"}],"version-history":[{"count":0,"href":"https:\/\/innovationspace.ansys.com\/forum\/wp-json\/wp\/v2\/topics\/218533\/revisions"}],"wp:attachment":[{"href":"https:\/\/innovationspace.ansys.com\/forum\/wp-json\/wp\/v2\/media?parent=218533"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}