{"id":290561,"date":"2023-06-27T13:18:01","date_gmt":"2023-06-27T13:18:01","guid":{"rendered":"\/forum\/forums\/reply\/290561\/"},"modified":"2023-06-27T13:18:01","modified_gmt":"2023-06-27T13:18:01","slug":"290561","status":"publish","type":"reply","link":"https:\/\/innovationspace.ansys.com\/forum\/forums\/reply\/290561\/","title":{"rendered":"Reply To: Unexpected S matrix from EME propagation"},"content":{"rendered":"<p>&lt;p&gt;&lt;span style=&#8221;font-size: 12pt;&#8221;&gt;I have tried to implement everything in Lumerical Script.&nbsp;&lt;\/span&gt;&lt;\/p&gt;&lt;p&gt;&lt;span style=&#8221;font-size: 12pt;&#8221;&gt;My structure looks like this:&nbsp;&lt;\/span&gt;&lt;\/p&gt;&lt;p&gt;<img loading=\"lazy\" decoding=\"async\" src=\"\/forum\/wp-content\/uploads\/sites\/2\/2023\/06\/27-06-2023-1687870846-mceclip0.png\" width=\"450\" height=\"262\" \/>&lt;\/p&gt;&lt;p&gt;&lt;span style=&#8221;font-size: 12pt;&#8221;&gt;The periodic group is set only in cells 2 and 3. To my understanding, the <em>internal S-matrix<\/em> is the S matrix from before cell 2 to after cell 3. This is based on my understanding from the Ansys video.&nbsp;&lt;\/span&gt;&lt;\/p&gt;&lt;p&gt;&lt;span style=&#8221;font-size: 12pt;&#8221;&gt; <img loading=\"lazy\" decoding=\"async\" src=\"\/forum\/wp-content\/uploads\/sites\/2\/2023\/06\/27-06-2023-1687871027-mceclip1.png\" width=\"546\" height=\"455\" \/>&lt;\/span&gt;&lt;\/p&gt;&lt;p&gt;&lt;span style=&#8221;font-size: 12pt;&#8221;&gt;My Lumerical script does the following:&nbsp;&lt;\/span&gt;&lt;\/p&gt;<\/p>\n<ol>\n<li>&lt;span style=&#8221;font-size: 12pt;&#8221;&gt;Set the periods to 1 and run Eme Propagate. &lt;\/span&gt;\n<ul>\n<li>&lt;span style=&#8221;font-size: 12pt;&#8221;&gt;The simulated structure is [1, (2,3)^1, 4].&lt;\/span&gt;<\/li>\n<li>&lt;span style=&#8221;font-size: 12pt;&#8221;&gt;Extract the&nbsp;<em>Internal User Matrix<\/em>. As far as I understood, this represent the S matrix of my unit cell.&lt;\/span&gt;<\/li>\n<\/ul>\n<\/li>\n<li>&lt;span style=&#8221;font-size: 12pt;&#8221;&gt;Set the periods to 2 and run EME Propagate&lt;\/span&gt;\n<ul>\n<li>&lt;span style=&#8221;font-size: 12pt;&#8221;&gt;The simulated structure is [1, (2,3)^2, 4].&lt;\/span&gt;<\/li>\n<li>&lt;span style=&#8221;font-size: 12pt;&#8221;&gt;Extract the <em>Internal User Matrix.<\/em> At this point, this should represent the structure of two cascaded unit cells.&lt;\/span&gt;<\/li>\n<\/ul>\n<\/li>\n<li>&lt;span style=&#8221;font-size: 12pt;&#8221;&gt;Calculate the S matrix of the two cascaded unit cells analytically.&lt;\/span&gt;\n<ul>\n<li>&lt;span style=&#8221;font-size: 12pt;&#8221;&gt;This has been implemented with a Lumerical script function and it is a known result.&lt;br&gt;&lt;\/span&gt;<\/li>\n<li>&lt;span style=&#8221;font-size: 12pt;&#8221;&gt;The validity of the function has been tested by calculating the S matrix of two cascaded networks whose result was known.&lt;\/span&gt;<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p>&lt;p&gt;&lt;span style=&#8221;font-size: 12pt;&#8221;&gt;Again, by comparing the S-matrix of point 2 and 3, I find quite a difference in the results.&lt;br&gt;I am pretty confident that the analytical function is correct, thus I believe there is either a mistake in the way that I interpret the S matrix or the EME algorithm is doing something beyond just cascading S matrices which I cannot find on the documentation.&lt;br&gt;I would like to have some highlights on that.&nbsp;&lt;br&gt;&lt;br&gt;&lt;\/span&gt;&lt;\/p&gt;&lt;p&gt;&lt;span style=&#8221;font-size: 12pt;&#8221;&gt;Thank you&lt;br&gt;&lt;br&gt;&lt;br&gt;&lt;\/span&gt;&lt;\/p&gt;&lt;p&gt;&nbsp;&lt;\/p&gt;&lt;p&gt;&lt;span style=&#8221;font-size: 12pt;&#8221;&gt;Below the script that I used:&nbsp;&lt;br&gt;&lt;\/span&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;# This function cascades two S matrices&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;function cascade_S(S){&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp; rows = size(S);&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp;&nbsp;&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp; rows = rows(1);&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp;&nbsp;&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp; #Extract blocks&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp; S11 = S(1:rows\/2, 1:rows\/2);&lt;\/span&gt;&lt;\/p&gt;&lt;p&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp; S12 = S(1:rows\/2, rows\/2+1:rows);&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp;&nbsp;&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp; S21 = S(rows\/2+1:rows, 1:rows\/2);&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp;&nbsp;&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp; S22 = S(rows\/2+1:rows, rows\/2+1:rows);&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp;&nbsp;&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp; # Identity&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp; I = matrix(rows\/2, rows\/2);&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp; for (i = 1:rows\/2){I(i,i) = 1;}&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp;&nbsp;&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp; # Cascade step &nbsp; &nbsp;&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp; S11f = S11 + mult(S12, mult(S11, mult(inv(I-mult(S22, S11)), S21)));&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp; S12f = mult(S12 , mult(inv(I-mult(S11,S22)), S12));&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp; S21f = mult(S21 , mult(inv(I-mult(S22, S11)), S21));&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp; S22f = S22 + mult(S21 , mult(S22, mult(inv(I-mult(S11,S22)), S12)));&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp;&nbsp;&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp; # Final S&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp; cascade = matrix(rows, rows);&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp;&nbsp;&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp; cascade(1:rows\/2, 1:rows\/2) = S11f;&lt;\/span&gt;&lt;\/p&gt;&lt;p&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp; cascade(1:rows\/2, rows\/2+1:rows) = S12f;&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp;&nbsp;&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp; cascade(rows\/2+1:rows, 1:rows\/2) = S21f;&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp;&nbsp;&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp; cascade(rows\/2+1:rows, rows\/2+1:rows) = S22f;&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp;&nbsp;&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp; return cascade;&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp;&nbsp;&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;}&lt;\/span&gt;&lt;\/p&gt;&lt;p&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;### Just to test that cascade works&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;#x = [ &nbsp;-0.0011 &#8211; 0.0605i, &nbsp;-0.2268 + 0.1546i, &nbsp;-0.0967 &#8211; 0.0686i, &nbsp; 0.0477 + 0.0444i;&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; #-0.2265 + 0.1545i, &nbsp;-0.0693 &#8211; 0.0377i, &nbsp; 0.0419 + 0.0414i, &nbsp;-0.0998 &#8211; 0.0465i;&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; #-0.0961 &#8211; 0.0692i, &nbsp; 0.0420 + 0.0415i, &nbsp;-0.0499 &#8211; 0.0681i, &nbsp;-0.2246 + 0.1676i;&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp;#0.0478 + 0.0445i, &nbsp;-0.0996 &#8211; 0.0468i, &nbsp;-0.2244 + 0.1673i, &nbsp;-0.0546 &#8211; 0.0231i];&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp;&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;#known_casca_x = &nbsp; &nbsp;[0.0030 &#8211; 0.0585i , -0.2312 + 0.1531i , &nbsp;0.0060 + 0.0165i , -0.0053 &#8211; 0.0136i;&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; #-0.2309 + 0.1530i, &nbsp;-0.0664 &#8211; 0.0368i, &nbsp;-0.0042 &#8211; 0.0124i , &nbsp;0.0090 + 0.0124i;&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; &nbsp;#0.0058 + 0.0166i, &nbsp;-0.0041 &#8211; 0.0124i, &nbsp;-0.0463 &#8211; 0.0669i, &nbsp;-0.2291 + 0.1665i;&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;&nbsp; #-0.0052 &#8211; 0.0137i, &nbsp; 0.0089 + 0.0124i, &nbsp;-0.2290 + 0.1662i, &nbsp;-0.0511 &#8211; 0.0223i];&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;#print(x);&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;#Sf=cascade_S(x);&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;#print(Sf &#8211; known_casca_x);&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;#print(max(abs(Sf &#8211; known_casca_x)));&lt;\/span&gt;&lt;\/p&gt;&lt;p&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;### &#8212;&#8212;&#8212;&#8212;&#8212;&#8211; Actual code &#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&lt;\/span&gt;&lt;\/p&gt;&lt;p&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;#&#8212;&#8212;&#8212;&#8212;&#8212; Point 1&lt;\/span&gt;&lt;\/p&gt;&lt;p&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;setemeanalysis(&#8220;override periodicity&#8221;, 1);&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;setemeanalysis(&#8220;periods&#8221;, [1]);&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;emepropagate();&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;S_user_1cell = getresult(&#8220;EME&#8221;, &#8220;user s matrix&#8221;);&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;S_internal_1cell = getresult(&#8220;EME&#8221;, &#8220;internal s matrix&#8221;);&lt;\/span&gt;&lt;\/p&gt;&lt;p&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;rows = size(S_internal_1cell);&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;rows = rows(1);&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;idx = linspace(0,rows, rows);&lt;\/span&gt;&lt;\/p&gt;&lt;p&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;# Plot internal S&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;#image(idx, idx, abs(S_internal_1cell));&lt;\/span&gt;&lt;\/p&gt;&lt;p&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;# &#8212;&#8212;&#8212;&#8212;&#8212; Point 2&lt;\/span&gt;&lt;\/p&gt;&lt;p&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;# Set number of Cell on 2&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;setemeanalysis(&#8220;periods&#8221;, [2]);&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;emepropagate();&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;S_user_2cell = getresult(&#8220;EME&#8221;, &#8220;user s matrix&#8221;);&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;S_internal_2cell = getresult(&#8220;EME&#8221;, &#8220;internal s matrix&#8221;);&lt;\/span&gt;&lt;\/p&gt;&lt;p&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;# Plot internal S&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;#image(idx, idx, abs(S_internal_2cell));&lt;\/span&gt;&lt;\/p&gt;&lt;p&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;# &#8212;&#8212;&#8212;&#8212;&#8212;- Point 3&lt;\/span&gt;&lt;\/p&gt;&lt;p&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;# Cascade the single unit S and plot&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;S_cascade = cascade_S(S_internal_1cell);&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;#image(idx, idx, abs(S_cascade));&lt;\/span&gt;&lt;\/p&gt;&lt;p&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;# Plot difference between matrix in point 2 and point 3&lt;\/span&gt;&lt;br&gt;&lt;span style=&#8221;color: rgb(45, 194, 107);&#8221;&gt;image(idx, idx, abs(S_cascade &#8211; S_internal_2cell));&lt;\/span&gt;&lt;\/p&gt;<\/p>\n","protected":false},"template":"","class_list":["post-290561","reply","type-reply","status-publish","hentry"],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 4.9.10 - aioseo.com -->\n\t<meta name=\"description\" content=\"I have tried to implement everything in Lumerical Script. 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My structure looks like this: The periodic group is set only in cells 2 and 3. To my understanding, the internal S-matrix is the S matrix from before cell 2 to after cell 3. 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My structure looks like this: The periodic group is set only in cells 2 and 3. To my understanding, the internal S-matrix is the S matrix from before cell 2 to after cell 3. 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