Investigating the Properties of Metal-Insulator-Metal (MIM) Waveguides at Optical Frequencies by the Argument Principle Method (APM)

Şükrü Ekin Kocabaş
kocabas@stanford.edu

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notice like this when it starts in an interactive mode:

    <program>  Copyright (C) <year>  <name of author>
    This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
    This is free software, and you are welcome to redistribute it
    under certain conditions; type `show c' for details.

The hypothetical commands `show w' and `show c' should show the appropriate
parts of the General Public License.  Of course, your program's commands
might be different; for a GUI interface, you would use an "about box".

  You should also get your employer (if you work as a programmer) or school,
if any, to sign a "copyright disclaimer" for the program, if necessary.
For more information on this, and how to apply and follow the GNU GPL, see
<http://www.gnu.org/licenses/>.

  The GNU General Public License does not permit incorporating your program
into proprietary programs.  If your program is a subroutine library, you
may consider it more useful to permit linking proprietary applications with
the library.  If this is what you want to do, use the GNU Lesser General
Public License instead of this License.  But first, please read
<http://www.gnu.org/philosophy/why-not-lgpl.html>.

Argument Principle Method Implementation

The block below implements the argument principle method. For more information please look at:

[1]  "A General Transform for Regularizing Planar Open Waveguide Dispersion Relation". Aref Bakhtazad, Habibollah Abiri, R. Ghayour. Journal of Lightwave Technology, vol 15, no 2, Feb 1997, pp 383-390.

[2]  "Multilayer Waveguides: Efficient Numerical Analysis of General Structures". Emmanuel Anemogiannis, Elias N. Glytsis. Journal of Lightwave Technology, vol 10, no 10, Oct 1992, pp 1344-1351.

[3]  "Simple and fast numerical analysis of multilayer waveguide modes". Min-Suk Kwon, Sang-Yung Shin. Optics Communications, vol 233, 2004, pp 119-126.

[4]  "Efficient and accurate numerical analysis of multilayer planar optical waveguides in lossy anisotropic media". Chengkun Chen, Pierre Berini et. al. Optics Express, vol 7, no 8, pp 260-272.

This is the dispersion equation to solve:

x = x; Remove["Global`*"]; 
Off[General::spell]; 
Off[General::spell1];
<<"AuthorTools`"
SetDirectory[NotebookFolder[EvaluationNotebook[]]];
SinglePrint[outputString_]:=
	NotebookWrite[EvaluationNotebook[],Cell[outputString,"Output",TextAlignment->Center,Background->RGBColor[0.8,0.8,0.8] ] ,All];

BeginPackage["APM`"]

SetParameters::"usage" = "";
GetAllRoots::"usage"="";
VisualizeRegion::"usage"="";
PlotMode::"usage"="";
PlotModePub::"usage"="";
PlotModeCont::"usage"="";
PlotModePEC::"usage"="";
PrintErrorEstimate::"usage" = "";

(*
numZeros::"usage" = "";
APM::"usage" = "";
GetSubRegions::"usage" = "";
GetSolutions::"usage" = "";
ModifySearchRegion::"usage" = "";
ErrorEstimate::"usage" = "";
*)


Begin["`Private`"];

(* First type of equation *) 
fRaw1[ki_, km_, d_, δ_] = ϵm*ki*Sinh[ki*2*π*(d/2)] + ϵi*km*Cosh[ki*2*π*(d/2)]; 
fPre1[km_, w_, d_, δ_] = fRaw1[Sqrt[km^2 + w^2*(ϵm-ϵi)], km, d, δ]; 
f1[z_] = fPre1[Exp[z], w, d, δ]; 
df1[z_] = D[f1[z], z]; 
dff1[z_] = Simplify[df1[z]/f1[z]];

(* Second type of equation *)
(* Below the equation is multiplied by ki to knit the riemann surfaces, Otherwise, a discontinuity occurs. *)
fRaw2[ki_, km_, d_, δ_] = ϵm*ki^2*Cosh[ki*2*π*(d/2)] + ϵi*km*ki*Sinh[ki*2*π*(d/2)]; 
fPre2[km_, w_, d_, δ_] = fRaw2[Sqrt[km^2 + w^2*(ϵm-ϵi)], km, d, δ]; 
f2[z_] = fPre2[Exp[z], w, d, δ]; 
df2[z_] = D[f2[z], z]; 
dff2[z_] = Simplify[df2[z]/f2[z]];

(* Functions for error estimation *)
fRaw1dki = D[fRaw1[ki, km, d, δ], ki];
fRaw1dkm = D[fRaw1[ki, km, d, δ], km];
fRaw2dki = D[fRaw2[ki, km, d, δ], ki];
fRaw2dkm = D[fRaw2[ki, km, d, δ], km];
eqnKiKm= ki^2 - km^2 + w^2*(ϵi-ϵm);
eqnKiKmdki = D[eqnKiKm,ki];
eqnKiKmdkm = D[eqnKiKm,km];



(* Choose which equation to solve based on solveFunction variable. By default, solveFunction=1. *)
solveFunction = 1;
f[z_] := If[solveFunction == 1, f1[z], f2[z], f1[z]]; 
df[z_] := If[solveFunction == 1, df1[z], df2[z], df1[z]]; 
dff[z_] := If[solveFunction == 1, dff1[z], dff2[z], dff1[z]];

SetParameters[win_, din_, ϵmetal_, ϵdielectric_, functionToSolve_]:=Block[{},
	w = win;
	d = din;
	ϵm = ϵmetal;
	ϵi = ϵdielectric;
	solveFunction = functionToSolve;
];

numZeros[input__] := Module[{zr1, zr2, zi1, zi2, integralOptions, int1, int2, int3, int4, result},
	zr1 = Hold[input][[1]]; zr2 = Hold[input][[2]];
	zi1 = Hold[input][[3]]; zi2 = Hold[input][[4]];
	If[Length[Hold[input]] <= 4, 
		integralOptions = {MaxRecursion -> 300,PrecisionGoal -> Infinity,AccuracyGoal -> 2};
		,
		integralOptions = {MaxRecursion -> 300,PrecisionGoal -> Infinity,AccuracyGoal -> Hold[input][[5]]}; 
	];
	int1 = NIntegrate[dff[x + I*zi1], {x, zr1, zr2}, Evaluate[integralOptions]];
	int2 = I*NIntegrate[dff[zr2 + I*x], {x, zi1, zi2}, Evaluate[integralOptions]]; 
	int3 = NIntegrate[dff[x + I*zi2], {x, zr2, zr1}, Evaluate[integralOptions]];
	int4 = I*NIntegrate[dff[zr1 + I*x], {x, zi2, zi1}, Evaluate[integralOptions]];
	result = (int1 + int2 + int3 + int4)/(2*Pi*I)
]; 

APM[m_, zr1_, zr2_, zi1_, zi2_, wp_] := Module[{integralOptions,int1, int2, int3, in4, result},
	integralOptions = {MaxRecursion -> 100, WorkingPrecision -> wp};
	int1 = NIntegrate[(x + I*zi1)^m*dff[x + I*zi1], {x, zr1, zr2}, Evaluate[integralOptions]];
	int2 = I*NIntegrate[(zr2 + I*x)^m*dff[zr2 + I*x], {x, zi1, zi2}, Evaluate[integralOptions]];
	int3 = NIntegrate[(x + I*zi2)^m*dff[x + I*zi2], {x, zr2, zr1}, Evaluate[integralOptions]];
	int4 = I*NIntegrate[(zr1 + I*x)^m*dff[zr1 + I*x], {x, zi2, zi1}, Evaluate[integralOptions]]; 
	result = (int1 + int2 + int3 + int4)/ (2*Pi*I)
];

GetSubRegions[zr1Raw_, zr2Raw_, zi1Raw_, zi2Raw_] := Module[
	{zr1, zr2, zi1, zi2, numZerosBad = True, numZerosIntegrationProblem, accuracyAim, maxIter, newRegion, n, nBare, out1, out2, out3, out4, out = {}}, 
	Off[NIntegrate::slwcon];
	zr1 = zr1Raw; zr2 = zr2Raw; zi1 = zi1Raw; zi2 = zi2Raw;
	If[zr2 < zr1 || zi2 < zi1, Print["Domain Error!"]; Abort[]; ];
	numZerosBad = True; 
	numZerosIntegrationProblem = False; 
	accuracyAim = 3;
	maxIter = 10; 
	
	While[numZerosBad && maxIter > 0,
		Check[
			nBare = numZeros[zr1, zr2, zi1, zi2, accuracyAim];
			numZerosIntegrationProblem = False;
			,
			numZerosIntegrationProblem = True; 
			newRegion = ModifySearchRegion[zr1, zr2, zi1, zi2];
			Print["Numerical integration error with region ", {zr1, zr2, zi1, zi2},". Expanding to ", newRegion];
			zr1 = newRegion[[1]]; zr2 = newRegion[[2]]; zi1 = newRegion[[3]]; zi2 = newRegion[[4]];
			nBare = numZeros[zr1, zr2, zi1, zi2, accuracyAim];
			,
			NIntegrate::ploss
	    	];
	    	
	    	If[Accuracy[Re[nBare]] > 2,
    			If[Abs[FractionalPart[Re[nBare]]] < 0.1 || Abs[FractionalPart[Re[nBare]]] > 0.9, 
    				numZerosBad = False; 
		    		,
 		   		newRegion = ModifySearchRegion[zr1, zr2, zi1, zi2];
   		 		(*
   		 		Print["Bad integration region with nBare="*nBare*" and region ",{zr1, zr2, zi1, zi2}, 
    				". Expanding to ", newRegion];
		    		*)
 		   		zr1 = newRegion[[1]]; zr2 = newRegion[[2]]; zi1 = newRegion[[3]]; zi2 = newRegion[[4]]; 
   		 	]; 
   		 	, 
   		 	accuracyAim = accuracyAim + 1; 
   		 ];
   		 maxIter = maxIter - 1; 
	]; 
	
	If[Accuracy[Re[nBare]] < 2, 
		Print["Accuracy aim failed: Accuracy of ", Accuracy[Re[nBare]]]; 
	];
	If[Abs[FractionalPart[Re[nBare]]] < 0.1 || Abs[FractionalPart[Re[nBare]]] > 0.9, 
		n = Re[nBare]; 
		n = Round[n];
		If[ !IntegerQ[n], Print["Not an integer: ", n]];
		If[n < 0, Print["Warning: Negative number of zeros! nBare = ", nBare]]; 
		, 
		Print["Fractional part problem. Solution discarded!"];
		n = 0; 
	]; 
	If[n == 1, out = {{zr1, zr2, zi1, zi2}}];
	If[n == 0, out = {}]; 
	If[n < 0, Print["Negative n! Aborting..."]; Abort[]; ]; 
	If[n > 1, 
		out1 = GetSubRegions[zr1, zr1 + (zr2 - zr1)/2,zi1, zi1 + (zi2 - zi1)/2]; 
		out2 = GetSubRegions[zr1 + (zr2 - zr1)/2, zr2, zi1, zi1 + (zi2 - zi1)/2]; 
		out3 = GetSubRegions[zr1 + (zr2 - zr1)/2, zr2, zi1 + (zi2 - zi1)/2, zi2];
		out4 = GetSubRegions[zr1, zr1 + (zr2 - zr1)/2, zi1 + (zi2 - zi1)/2, zi2]; 
		out = Join[out1, out2, out3, out4]; 
		If[Length[out] < n, Print["Something wrong with n=", n, " out=", out]]; 
	];
	out	
];

GetSolutions[regions__] := Module[ {i, out = {}, zr1, zr2, zi1, zi2, root,rootNew}, 
	For[i = 1, i <= Length[regions], 
		zr1 = regions[[i,1]]; zr2 = regions[[i,2]]; zi1 = regions[[i,3]]; zi2 = regions[[i,4]];
		Off[NIntegrate::precw]; 
		root = APM[1, zr1, zr2, zi1, zi2, $MachinePrecision];
		rootNew = root;
		If[ErrorEstimate[root] > 0,
			(* Debug code *)
			(*
			Print["Checking error estimate"];
			*)
			If[zr1 < Re[root] < zr2 && zi1 < Im[root] < zi2,
				rootNew = z /. FindRoot[f[z], {z, root, zr1 + I*zi1, zr2 + I*zi2}]; 
				, 
				rootNew = z /. FindRoot[f[z], {z, 0.5* (zr1 + I*zi1 + zr2 + I*zi2), zr1 + I*zi1, zr2 + I*zi2}]; 
			];
			If[zr1 < Re[rootNew] < zr2 && zi1 < Im[rootNew] < zi2,
				(* Debug code *)
				(*
				Print["OK"];
				*)
				1; (* This is just to prevent Mathematica from complaining...*)
				,
				(*
				Print["Problem!!"];
				*)
				1;
				,
				Print["Something's wrong!"];
				Abort[];
			];

			(*
			Print["FindRoot error estimate=", ErrorEstimate[rootNew]," APM error estimate=", ErrorEstimate[root]]; 
			*)
		];
		If[ErrorEstimate[root] > ErrorEstimate[rootNew], root = rootNew; ];
		(* This is specifically for the lossless case *)
		(* (zi1*zi2)<0 makes sure that the search region includes the real line *)
		If[ErrorEstimate[root] > ErrorEstimate[Re[root]] && (zi1*zi2)<0, root = Re[root]; ];
		(* Debug code *)
		(*
		Print["zr1=",zr1," zr2=",zr2," zi1=",zi1," zi2=",zi2];
		Print["root=",root," errorEstimate=",ErrorEstimate[root]];
		Print["rootNew=",rootNew, " errorEstimate=",ErrorEstimate[rootNew]];
		Print["-----------------------"];
		*)
		out = Join[out, {root}]; 
		i = i + 1; 
	];
	out
]; 

ModifySearchRegion[zr1_, zr2_, zi1_, zi2_] := Module[ {scale, delR, delI, cenR, cenI, out},
	scale = 1.05; 
	delR = zr2 - zr1; 
	delI = zi2 - zi1; 
	cenR = (zr2 + zr1)/2;
	cenI = (zi2 + zi1)/2; 
	delR = delR*scale; 
	delI = delI*scale;
	out = {cenR - delR/2, cenR + delR/2, cenI - delI/2, cenI + delI/2}
];

ErrorEstimate[soln_] := Module[ 
	{kFound, kiFound, kmFound, f1, f2, df1ki, df1km, df2ki, df2km, f1Found, f2Found, errorSolution, deltaki, deltakm}
	,
	kmFound = Exp[soln]; 
	kiFound = Sqrt[kmFound^2 - w^2*(ϵi - ϵm)]; 
	kFound = Sqrt[kmFound^2 + w^2*ϵm];
	If[solveFunction == 1,
		df1ki=fRaw1dki /. {ki -> kiFound, km -> kmFound};
		df1km=fRaw1dkm /. {ki -> kiFound, km -> kmFound};
		df2ki=eqnKiKmdki /. {ki -> kiFound, km -> kmFound};
		df2km=eqnKiKmdkm /. {ki -> kiFound, km -> kmFound};
		f1Found=fRaw1[ki, km, d, δ] /. {ki -> kiFound, km -> kmFound};
		f2Found=eqnKiKm /. {ki -> kiFound, km -> kmFound};
		,
		df1ki=fRaw2dki /. {ki -> kiFound, km -> kmFound};
		df1km=fRaw2dkm /. {ki -> kiFound, km -> kmFound};
		df2ki=eqnKiKmdki /. {ki -> kiFound, km -> kmFound};
		df2km=eqnKiKmdkm /. {ki -> kiFound, km -> kmFound};
		f1Found=fRaw2[ki, km, d, δ] /. {ki -> kiFound, km -> kmFound};
		f2Found=eqnKiKm /. {ki -> kiFound, km -> kmFound};				
	]; 

	Off[RowReduce::"luc"];
	errorSolution = NSolve[{f1Found == df1ki*Δki + df1km*Δkm, f2Found == df2ki*Δki + df2km*Δkm}, {Δki, Δkm}];
	On[RowReduce::"luc"];
	
	If[Abs[kiFound] > 10^(-6),
		deltaki = Abs[(Δki/kiFound)*100 /. errorSolution]; 
		,
		deltaki = Abs[Δki*100 /. errorSolution]; 
	];
	If[Abs[kmFound] > 10^(-6),
		deltakm = Abs[(Δkm/kmFound)*100 /. errorSolution];
		,
		deltakm = Abs[Δkm*100 /. errorSolution]; 
	]; 
	Max[deltaki, deltakm]
];

PrintErrorEstimate[soln_] := Print[ErrorEstimate[soln]];

GetAllRoots[zrMin_, zrMax_, ziMin_, ziMax_]:=Block[{regions,solutionList={}},
	regions=GetSubRegions[zrMin, zrMax, ziMin, ziMax];
	solutionList= GetSolutions[regions];
	SetPrecision[solutionList, $MachinePrecision]	
];

VisualizeRegion[zrmin_, zrmax_, zimin_, zimax_,solutionList_]:=Block[{z,zr,zi,epi,c0,zrminPlot,zrmaxPlot},
	z=Log[zr]+I*zi;
	zrminPlot =  Exp[zrmin];
	zrmaxPlot = Exp[zrmax];
	epi={RGBColor[1,1,1]};
	epi = Append[epi, 
			Table[Text[Style[m,Bold,14],{Exp[Re[solutionList[[m]]]], Im[solutionList[[m]]]},
					Background -> Opacity[0.6,RGBColor[0,0,0]]
				  ],					
					{m,1,Length[solutionList]}]
		  ];
	Off[General::ovfl];
	c0=ContourPlot[Log[1+Abs[f[z]]], {zr,zrminPlot,zrmaxPlot},{zi,zimin,zimax}, 
		Frame->True,
		Mesh->None, 
		FrameTicks->{Automatic, Table[i*π/4,{i,-4, 4}]}, 
		FrameLabel->{Abs[Subscript["κ","m"]]/Subscript["k","p"],Arg[Subscript["κ","m"]]}, 
		PlotPoints->50,
		ColorFunction->(Opacity[1.0,RGBColor[Random[],Random[],Random[]]]&),
		LabelStyle -> {FontFamily->"Helvetica",FontSize -> 10},
		Contours -> Table[10^x, {x, -3, 1, 0.1}], 
		ColorFunctionScaling -> False, 
		ImageSize -> 72*4,
		Exclusions -> None,
		Epilog-> epi]	
];

(* This function plots the transverse magnetic field of the TM mode that is found. It uses the simplified expressions derived.*)
PlotMode[solution_]:= Block[ {xmin,xmax,ymin,ymax,Km,Ki,R,Hy,TitleParam,HyImag,pReal,pImag},
	
	xmin=-2*d;
	xmax=3*d;
	ymin=0.0;
	ymax=0.0;
	Km = Exp[solution];
	Ki = Sqrt[Km^2 + w^2*(ϵm-ϵi)];
	R := -(Km/ϵm) / (Ki/ϵi);
	Hy = {};
	If[solveFunction == 1 ,
		If[ Abs[R/Tanh[Ki*2*π*(d/2)] - 1] > 10^(-3), Ki = -Ki]; (*Ki might need a negative sign*)
		(*If R still doesn't satisfy the required relationship, throw an error. Note that R is defined by := *)
		If[ Abs[R/Tanh[Ki*2*π*(d/2)] - 1] > 10^(-3), Throw[badSolution,{R, Tanh[Ki*2*π*(d/2)]}]];
		(* Here is the definition of Hy *)
		Hy = Join[Hy,Table[{x,Exp[Km*2*π*x]},{x,xmin,0,d/100}]];
		Hy = Join[Hy,Table[{x,Cosh[Ki*2*π*(x-d/2)]/Cosh[Ki*2*π*(d/2)]},{x,0,d,d/100}]];
		Hy = Join[Hy,Table[{x,Exp[-Km*2*π*(x-d)]},{x,d,xmax,d/100}]];	
	];
	
	If[solveFunction == 2,
		If[ Abs[R/Coth[Ki*2*π*(d/2)] - 1] > 10^(-3), Ki = -Ki]; (*Ki might need a negative sign*)
		(*If R still doesn't satisfy the required relationship, throw an error. Note that R is defined by := *)
		If[ Abs[R/Coth[Ki*2*π*(d/2)] - 1] > 10^(-3), Throw[badSolution,{R, Coth[Ki*2*π*(d/2)]}]];
		(* Here is the definition of Hy *)
		Hy = Join[Hy,Table[{x,Exp[Km*2*π*x]},{x,xmin,0,d/100}]];
		Hy = Join[Hy,Table[{x,-Sinh[Ki*2*π*(x-d/2)]/Sinh[Ki*2*π*(d/2)]},{x,0,d,d/100}]];
		Hy = Join[Hy,Table[{x,-Exp[-Km*2*π*(x-d)]},{x,d,xmax,d/100}]];	
	];
	
	(* Hy[[All,2]] = Abs[ Hy[[All,2]] ]; *)
	HyImag = Hy;
	HyImag[[All,2]] = Im[ HyImag[[All,2]] ];
	ymin= Min[Join[ Re[Hy[[All,2]]],Im[Hy[[All,2]]]  ]];
	ymax=Max[Join[ Re[Hy[[All,2]]],Im[Hy[[All,2]]]  ]];
	TitleParam = "ω="<>ToString[w] <>" d=" <>ToString[d]<>" ϵm="<>ToString[ϵm]<>" ϵi=" <>ToString[ϵi];
	
	pReal=ListPlot[Re[Hy],PlotRange->{{xmin,xmax},{ymin,ymax}}, Joined->True,
		PlotStyle->{Thickness[0.005]}, ImageSize -> 72*4,
		TextStyle -> {FontFamily->"Helvetica",FontSize -> 10},Axes -> False, 
		Frame->True, FrameLabel->{"x"/Subscript["λ","p"],None,TitleParam,None},
		FrameTicks->Automatic, 
		Prolog -> {{Opacity[0.5,RGBColor[0,(160/255)^0.5,(198/255)^0.5]], Rectangle[{xmin*0.99, ymin*0.99}, {0, ymax*0.99}]}, 	
				{Opacity[0.5,RGBColor[0,(160/255)^0.5,(198/255)^0.5]], Rectangle[{d*0.99, ymin*0.99}, {xmax*0.99, ymax*0.99}]}},
		DisplayFunction->Identity
		];
	
	pImag=ListPlot[Re[HyImag],
	PlotRange->{{xmin,xmax},{ymin,ymax}}, Joined->True,
	PlotStyle->{Thickness[0.005],Dashing[{0.01}]}, ImageSize -> 72*4, 	
	TextStyle -> {FontFamily->"Helvetica",FontSize -> 10},Axes -> False, 
	Frame->True,FrameLabel->{"x"/Subscript["λ","p"],None,TitleParam,None},
	FrameTicks->Automatic, DisplayFunction->Identity];
	Show[{pReal,pImag}]
];

(* This function plots the transverse magnetic field of the TM mode that is found. It uses the simplified expressions derived.*)
(* This is the one used for publication purposes *)
PlotModePub[solution_]:= Block[ {xmin,xmax,ymin,ymax,Km,Ki,R,Hy,TitleParam,HyImag,pReal,pImag},
	
	xmin=-2*d;
	xmax=3*d;
	ymin=0.0;
	ymax=0.0;
	Km = Exp[solution];
	Ki = Sqrt[Km^2 + w^2*(ϵm-ϵi)];
	R := -(Km/ϵm) / (Ki/ϵi);
	Hy = {};
	If[solveFunction == 1 ,
		If[ Abs[R/Tanh[Ki*2*π*(d/2)] - 1] > 10^(-3), Ki = -Ki]; (*Ki might need a negative sign*)
		(*If R still doesn't satisfy the required relationship, throw an error. Note that R is defined by := *)
		If[ Abs[R/Tanh[Ki*2*π*(d/2)] - 1] > 10^(-3), Throw[badSolution,{R, Tanh[Ki*2*π*(d/2)]}]];
		(* Here is the definition of Hy *)
		Hy = Join[Hy,Table[{x,Exp[Km*2*π*x]},{x,xmin,0,d/100}]];
		Hy = Join[Hy,Table[{x,Cosh[Ki*2*π*(x-d/2)]/Cosh[Ki*2*π*(d/2)]},{x,0,d,d/100}]];
		Hy = Join[Hy,Table[{x,Exp[-Km*2*π*(x-d)]},{x,d,xmax,d/100}]];	
	];
	
	If[solveFunction == 2,
		If[ Abs[R/Coth[Ki*2*π*(d/2)] - 1] > 10^(-3), Ki = -Ki]; (*Ki might need a negative sign*)
		(*If R still doesn't satisfy the required relationship, throw an error. Note that R is defined by := *)
		If[ Abs[R/Coth[Ki*2*π*(d/2)] - 1] > 10^(-3), Throw[badSolution,{R, Coth[Ki*2*π*(d/2)]}]];
		(* Here is the definition of Hy *)
		Hy = Join[Hy,Table[{x,Exp[Km*2*π*x]},{x,xmin,0,d/100}]];
		Hy = Join[Hy,Table[{x,-Sinh[Ki*2*π*(x-d/2)]/Sinh[Ki*2*π*(d/2)]},{x,0,d,d/100}]];
		Hy = Join[Hy,Table[{x,-Exp[-Km*2*π*(x-d)]},{x,d,xmax,d/100}]];	
	];
	
	(* Hy[[All,2]] = Abs[ Hy[[All,2]] ]; *)
	HyImag = Hy;
	HyImag[[All,2]] = Im[ HyImag[[All,2]] ];
	ymin= Min[Join[ Re[Hy[[All,2]]],Im[Hy[[All,2]]]  ]];
	ymax=Max[Join[ Re[Hy[[All,2]]],Im[Hy[[All,2]]]  ]];
	TitleParam = "ω="<>ToString[w] <>" d=" <>ToString[d]<>" ϵm="<>ToString[ϵm]<>" ϵi=" <>ToString[ϵi];
	
	pReal=ListPlot[Re[Hy],PlotRange->{{xmin,xmax},{ymin,ymax}}, Joined->True,
		PlotStyle->{Black,Thickness[0.005]}, ImageSize -> 72*3.5,
		BaseStyle -> {FontSize -> 8, FontFamily -> "Myriad"},Axes -> False, 
		Frame->False, 
		Prolog -> {{Opacity[0.5,RGBColor[0,(160/255)^0.5,(198/255)^0.5]], Rectangle[{xmin*1.01, ymin*1.01}, {0, ymax*1.01}]}, 	
				{Opacity[0.5,RGBColor[0,(160/255)^0.5,(198/255)^0.5]], Rectangle[{d*1.01, ymin*1.01}, {xmax*1.01, ymax*1.01}]}},
		DisplayFunction->Identity
		];
	
	pImag=ListPlot[Re[HyImag],
		PlotRange->{{xmin,xmax},{ymin,ymax}}, Joined->True,
		PlotStyle->{Black,Thickness[0.005],Dashing[{0.01}]}, ImageSize -> 72*3.5, 	
		BaseStyle -> {FontSize -> 8, FontFamily -> "Myriad"},Axes -> False, 
		Frame->False,
		FrameTicks->Automatic, 
		DisplayFunction->Identity];
	Show[{pReal,pImag}]
];

(* This function plots the transverse magnetic field of the TM mode that is found. It uses the simplified expressions derived.*)
(* This is the continuous spectrum *)

PlotModeCont[Km_]:= Block[ {xmin,xmax,ymin,ymax,Ki,Hy,TitleParam,HyImag,pReal,pImag},
	
	xmin=-2*d;
	xmax=3*d;
	ymin=0.0;
	ymax=0.0;
	Ki = Sqrt[Km^2 + w^2*(ϵm-ϵi)];
	Hy = {};
	If[solveFunction == 1 ,
		(* Here is the definition of Hy *)
		Hy = Join[Hy,Table[{x,Cosh[Km*2*π*(-x)]+(Ki/ϵi) / (Km/ϵm) * Tanh[Ki*2*π*(d/2)]*Sinh[Km*2*π*(-x)]},{x,xmin,0,d/100}]];
		Hy = Join[Hy,Table[{x,Cosh[Ki*2*π*(x-d/2)]/Cosh[Ki*2*π*(d/2)]},{x,0,d,d/100}]];
		Hy = Join[Hy,Table[{x,Cosh[Km*2*π*(x-d)]+(Ki/ϵi)/(Km/ϵm) * Tanh[Ki*2*π*(d/2)]*Sinh[Km*2*π*(x-d)]},{x,d,xmax,d/100}]];	
	];
	If[solveFunction == 2,
		Throw[notImplemented];
	];
	
	(* Hy[[All,2]] = Abs[ Hy[[All,2]] ]; *)
	HyImag = Hy;
	HyImag[[All,2]] = Im[ HyImag[[All,2]] ];
	ymin= Min[Join[ Re[Hy[[All,2]]],Im[Hy[[All,2]]]  ]];
	ymax=Max[Join[ Re[Hy[[All,2]]],Im[Hy[[All,2]]]  ]];
	Hy[[All,2]] = Hy[[All,2]] / Max[Abs[{ymin,ymax}]];
	HyImag[[All,2]] = HyImag[[All,2]] / Max[Abs[{ymin,ymax}]];
	ymin = ymin / Max[Abs[{ymin,ymax}]];
	ymax = ymax / Max[Abs[{ymin,ymax}]];
	
	TitleParam = "ω="<>ToString[w] <>" d=" <>ToString[d]<>" ϵm="<>ToString[ϵm]<>" ϵi=" <>ToString[ϵi];
	
	pReal=ListPlot[Re[Hy],PlotRange->{{xmin,xmax},{ymin,ymax}}, Joined->True,
		PlotStyle->{Black, Thickness[0.005]}, ImageSize -> 72*3.5,
		Axes -> False, 
		Frame->False, 
		FrameTicks->Automatic, 
		Prolog -> {{Opacity[0.5,RGBColor[0,(160/255)^0.5,(198/255)^0.5]], Rectangle[{xmin*1.01, ymin*1.01}, {0, ymax*1.01}]}, 	
				{Opacity[0.5,RGBColor[0,(160/255)^0.5,(198/255)^0.5]], Rectangle[{d*1.01, ymin*1.01}, {xmax*1.01, ymax*1.01}]}},
		DisplayFunction->Identity
		];
	
	pImag=ListPlot[Re[HyImag],
	PlotRange->{{xmin,xmax},{ymin,ymax}}, Joined->True,
	PlotStyle->{Black,Thickness[0.005],Dashing[{0.01}]}, ImageSize -> 72*3.5, 	
	Axes -> False, 
	Frame->False,
	FrameTicks->Automatic, DisplayFunction->Identity];
	Show[{pReal,pImag}]
];

PlotModePEC[solution_]:= Block[ {xmin,xmax,ymin,ymax,Hy,n,TitleParam,p},
	
	xmin=-2*d;
	xmax=3*d;
	ymin=0.0;
	ymax=0.0;
	Hy = {};
	If[solveFunction == 1 ,
		n = 2*(solution -1);
		(* Here is the definition of Hy *)
		Hy = Join[Hy,Table[{x,(-1)^(solution+1)*Cos[n*π*(x-d/2)/d]},{x,0,d,d/100}]];
	];
	
	If[solveFunction == 2,
		n =2*(solution -1)+1;
		(* Here is the definition of Hy *)
		Hy = Join[Hy,Table[{x,Cos[n*π*(x-d/2)/d]},{x,0,d,d/100}]];
	];
	
	ymin= -1.05;
	ymax= 1.05;
	TitleParam = "PEC Mode for ω="<>ToString[w] <>" d=" <>ToString[d]<>" ϵi=" <>ToString[ϵi] <> " n="<>ToString[n];
	
	p=ListPlot[Hy,PlotRange->{{xmin,xmax},{ymin,ymax}}, PlotJoined->True,PlotStyle->{Thickness[0.005]}, ImageSize -> 72*7, 	TextStyle -> {FontFamily->"Helvetica",FontSize -> 10},Axes -> False, Frame->True,FrameLabel->{"x"/Subscript["λ","p"],None,TitleParam,None},FrameTicks->Automatic, Prolog -> {{RGBColor[0,(160/255)^0.5,(198/255)^0.5], Rectangle[{xmin*0.99, ymin*0.99}, {0, ymax*0.99}]}, {RGBColor[0,(160/255)^0.5,(198/255)^0.5], Rectangle[{d*0.99, ymin*0.99}, {xmax*0.99, ymax*0.99}]}},DisplayFunction->Identity];
	Show[p,DisplayFunction->$DisplayFunction]
];

End[];

EndPackage[]

Single Frequency Test

Module[{},
	w=1.0;
	delta=0.05;
	(*
	wforϵm = w;
	ϵm = 1- 1/(wforϵm^2 - I*wforϵm*delta);
	*)
	ϵm = -143.49668392243762-I*9.517339564114454; (* for 1550.0 nm *)
	ϵm = Re[ϵm];
	ϵi = (1.00)^2;
	d = 0.25;
	solveFunction = 1; 
	assym = Pi/1000; 
	ZRmin = -5;
	ZRmax = Log[41];  (* value for d=0.25 *)
		
	ZImin = -4*Pi/4- 2*assym; 
	ZImax = 4*Pi/4 + 1*assym; 


	SetParameters[w, d, ϵm, ϵi, solveFunction];
	solutionList = GetAllRoots[ZRmin, ZRmax, ZImin, ZImax];
	kmList = Exp[solutionList];
	
	(* Radiation condition, km should have a positive real part *)
	kmList = Select[kmList, Re[#1]>0 & ];
	
	kiList = Sqrt[kmList^2 + w^2*(ϵm - ϵi)]; 
	kList = Sqrt[kmList^2 + w^2*ϵm]; 
	(* Modify the kList for radiation condition *)
	For[i=1,i<=Length[kList],i=i+1,
		If[ Im[kList[[i]]] > 0, kList[[i]] = - kList[[i]] ];
	];
	
	Remove[i];
	sortedIndex = Ordering[kList, All, Abs[Im[#1]] < Abs[Im[#2]] & ]; 
	kList = kList[[sortedIndex]]; 
	kmList = kmList[[sortedIndex]]; 
	kiList = kiList[[sortedIndex]];
	solutionList = Log[kmList]; 
	
	Print["For w=", w, " d=", d, " ϵm=", ϵm, " ϵi=", ϵi, " the ", If[solveFunction==1,"even", "odd"]," bound modes found are:"]; 
	Print[Floor[Sqrt[ϵi+Abs[ϵm]]*d]+1, " simple roots"];
	Print[TableForm[Transpose[{kList, kiList, kmList, solutionList}], 
	  TableHeadings -> {Automatic, {"k","ki", "km", "Log[km/kp]"}}]];
	p = VisualizeRegion[ZRmin, ZRmax, ZImin, ZImax, solutionList]
]

power[kz_]:=Block[{Ki,Km,outMetal,outInsulator},
Ki = (Sqrt[(kz^2)-ϵi]);
Km = (Sqrt[(kz^2)-ϵm]);
(* out=(kz/(ϵi))*((If[Re[Ki]!= 0,((Sinh[2π d Re[Ki]])/(Re[Ki])),2 π d]+If[Im[Ki]!= 0,((Sin[2π d Im[Ki]])/(Im[Ki])),2 π d])/(Cosh[2π d Re[Ki]]+Cos[2π d Im[Ki]]))+(kz/(ϵm))(1/(Re[Km])); *)
outMetal=(kz/(ϵm))(1/(Re[Km]));
outInsulator = (kz/(ϵi))*((If[Re[Ki]!= 0,((Sinh[2π d Re[Ki]])/(Re[Ki])),2 π d]+If[Im[Ki]!= 0,((Sin[2π d Im[Ki]])/(Im[Ki])),2 π d])/(Cosh[2π d Re[Ki]]+Cos[2π d Im[Ki]]));
{Re[outMetal],Re[outInsulator],Re[outInsulator+outMetal]}
]

powerList ={};
For[ii=1,ii<=Length[kList],ii++,powerList = Append[powerList,power[kList[[ii]]]]];

Print["For w=",w," d=",d," ϵm=",ϵm," ϵi=",ϵi," the ",If[solveFunction==1,"even","odd"]," bound modes found are:"];
Print[TableForm[Transpose[{kList, kiList, kmList, powerList[[All,1]],powerList[[All,2]],powerList[[All,3]]}], 
	  TableHeadings -> {Automatic, {"k","ki", "km", "Power Metal","Power Insulator","Power Total"}}]];