NEAR-FIELD OPTICS

Remove["Global`*"]

Highlights

Abstract

Introduction

Off[General::"spell", General::"spell1"]; Needs["Graphics`ParametricPlot3D`"]Needs["Graphics`Arrow`"]; \(Needs["Graphics`PlotField`"]; 
Needs["Algebra`ReIm`"]; 

    Needs["Calculus`VectorAnalysis`"]; 
\(tera = 10\^12; 
    micro = 1\/10\^6; \)\)giga = (10^9); nano = (1/(10^9)); mega = (10^6); pico = (1/(10^12)); kilo = (10^3); femto = (1/(10^15)); milli = (1/(10.^3)); atto = (1/(10^18)); NSub
       = {c -> 2.99792458 (10^8), e -> (1.6021773/(10^19)), 
        h -> (6.6260755/(10.^34)), hbar -> (1.05457/(10.^34)), 
        K -> (1.380658/(10^23)), me -> (9.109389/(10^31)), 
        mu0 -> ((4. π)/(10^7)), eps0 -> (8.85419/(10^12))}; ESi = 0.179 tera; rhoSi = 2330; nuSi = 0.28; ESiO2 = 0.06 tera; rhoSiO2 = 2200; nuSiO2 = 0.3; ESi3N4 = 0.15 tera; rhoSi3N4 = 3100; nuSi3N4 = 0.24; EAu = 0.08 tera; rhoAu = 19300; nuAu = 0.42; ENi = 0.207 tera; rhoNi = 8900; nuNi = 0.41; EW = 0.345 tera; rhoW = 19300; nuW = 0.28; EPt = 0.17 tera; rhoPt = 21400; nuPt = 0.39; EIr = 0.52 tera; rhoIr = 22500; nuIr = 0.3; EHOPG = 0.2 tera; rhoHOPG = 2250; nuHOPG = 0.3; EPol = 0.001 tera; rhoPol = 1000; nuPol = 0.3; xx = 0.8; EPtIr = xx 0.8 EPt + (1 - xx) EIr; rhoPtIr = xx rhoPt + (1 - xx) rhoIr; nuPtIr = 0.3; \(opt = {PlotRange -> All, AxesOrigin -> {0, 0}, 
        Frame -> True, GridLines -> Automatic, 
        DefaultFont -> {"Times-Roman", 8}, 
        PlotStyle 
          -> {Dashing[{.0001}], Dashing[{.0500}], Dashing[{.0200}], 
            Dashing[{.0001}]}}; 

    opt2 = {PlotRange -> All, AxesOrigin -> {0, 0}, 
        DefaultFont -> {"Times-Roman", 8}, 
        FrameLabel -> {"x", "y"}, Frame -> True, 
        GridLines -> Automatic}; 

num = 45; 


    optBW = {PlotPoints -> num, PlotRange -> All, 
        Axes -> True, DefaultFont -> {"Times-Roman", 8}, 
        Axes -> True, Frame -> True, Mesh -> False}; 

    \(optColor
       = {PlotPoints -> num, PlotRange -> All, Axes -> True, 
        DefaultFont -> {"Times-Roman", 8}, Axes -> True, 
        Frame -> True, Mesh -> False, ColorFunction -> Hue}; 
    \)\)

Operating Conditions

lambda = 0.5 micro; a = (lambda/10);

case
       = {k -> ((2 π)/lambda), 
        w -> ((2 π c)/lambda) /. NSub, 
        m0 -> -((2 (a^3))/(3 π))};

Clear[atn]; atn[x_, y_] := If[x == 0 && y == 0, 0, ArcTan[x, y]]

opt1 = optBW;

Far-Field Solution

Vector Potential

Ad = 
      (1/(Rr^2)) 
        (m0 (1 - I k Rr) Exp[I (k Rr - w t)] 
            Sin[Ttheta] {0, 0, 1});

Ad0 = 
      ((m0 Sin[Ttheta] (Sqrt[1 + (((k Rr))^2)]) {0, 0, 1})/(Rr^2)); \(psi = k\ Rr + ArcTan[1, \(-k\)\ Rr] - w\ t; 
\(Ad = Ad0\ Exp[I\ psi]; 
    \)\)

Electric Field

Ed = -(∂_t) (Ad/c);

Ed0 = -I Simplify[Ed Exp[-I psi]] /. case /. NSub;

Clear[Edx]; Edx[x_, y_, z_] := 
      Sin[Pphi] Ed0[[3]] /. 
            Ttheta -> ArcCos[(y/(Sqrt[(x^2) + (y^2) + (z^2)]))] /. 
          Rr -> (Sqrt[(x^2) + (y^2) + (z^2)]) /. Pphi -> atn[x, z];

Clear[Edz]; Edz[x_, y_, z_] := 
      Cos[Pphi] Ed0[[3]] /. 
            Ttheta -> ArcCos[(y/(Sqrt[(x^2) + (y^2) + (z^2)]))] /. 
          Rr -> (Sqrt[(x^2) + (y^2) + (z^2)]) /. Pphi -> atn[x, z];

Clear[Edy]; Edy[x_, y_, z_] := 0;

Electric Vector Field in the $x-z$ Plane

Fig16x1 = 
      PlotVectorField[
        Evaluate[{Edx[x a, a, z a], -Edz[x a, a, z a]}], {x, -2, 
          2, .5}, {z, -2, 2, .5}, 
        AxesLabel -> {"x in a", "z in a"}, 
        DefaultFont -> {"Times-Roman", 8}, 
        AxesOrigin -> {-2.24, 0}, PlotRange -> All, 
        Axes -> True, Frame -> True];

Electric Vector Field in the $x-y$ Plane

Fig16x2 = 
      PlotVectorField[
        Evaluate[{Edx[x a, y a, a], Edy[x a, y a, a]}], {x, -2, 
          2, .5}, {y, -2, 2, .5}, PlotRange -> All, 
        Axes -> True, AxesOrigin -> {-2.18, 0}, 
        AxesLabel -> {"x in a", "y in a"}, 
        DefaultFont -> {"Times-Roman", 8}, Frame -> True];

Electric Field in the $x-y$ Plane

Fig16x3 = 
      DensityPlot[
        Evaluate[Abs[Edx[x a, a y, a]]^2], {x, -2, 2}, {y, -2, 2}, 
        AxesLabel -> {"x in a", "y in a"}, Evaluate[opt1]];

Fig16x4 = 
      DensityPlot[
        Evaluate[Abs[Edz[x a, y a, a]]^2], {x, -2, 2}, {y, -2, 2}, 
        AxesLabel -> {"x in a", "y in a"}, Evaluate[opt1]];

Magnetic Field

SetCoordinates[Spherical]; Hd = Curl[Ad];

Hd01 = Simplify[Hd Exp[-I psi]];

Hd0 = 
      {Hd01[[1]], 
            √(
                (((I ComplexExpand[Hd01][[2, 1
                          ]]))^2) + 
                  ComplexExpand[Hd01][[2, 2
                      ]]^2), 0} /. case /. NSub;

Poynting Vector and Intensity

Sd = ((c Ed0\[Cross]Hd0)/(4 π)) /. NSub;

Id = (Sqrt[Sd.Sd]);

Fig16x5 = 
      SphericalPlot3D[
        Evaluate[Id /. Rr -> a 200], {Ttheta, (π/100), π}, {
          Pphi, (π/2), -(π/2)}, Axes -> True, 
        Boxed -> True, 
        PlotRange -> {{0, 5}, {-5, 5}, {-5, 5}}, 
        ViewVertical -> {1, 0, 0}, 
        DefaultFont -> {"Times-Roman", 8}, 
        AxesLabel -> {"z", "x", "y"}];

Clear[Idc]; Idc[x_, y_, z_] := 
      Id /. Ttheta -> ArcTan[y, (Sqrt[(x^2) + (z^2)])] /. 
          Rr -> (Sqrt[(x^2) + (z^2) + (y^2)]) /. Pphi -> atn[z, x];

Intensity in the $x-z$ Plane

Fig16x6 = 
      DensityPlot[
        Evaluate[Idc[x a, a, z a]], {x, -2, 2.01}, {z, -2, 2.01}, 
        AxesLabel -> {"x in a", "z in a"}, Evaluate[opt1]];

Intensity in the $x-y$ Plane

Fig16x7 = 
      DensityPlot[
        Evaluate[Idc[x a, y a, a]], {x, -2, 2.01}, {y, -2, 2.01}, 
        AxesLabel -> {"x in a", "y in a"}, Evaluate[opt1]];

Near-Field Solution

a = 50 nano;

sol = 
      Solve[{(x^2) + (y^2) == (a^2) (1 - (u^2)) (1 + (v^2)), 
          z == a u v}, {u, v}];

\(uu = sol[[4, 1, 2]]; 

    vv = sol[[4, 2, 2]]; \)

Electric Field

clear[Eax]; Eax[x_, y_, z_] = 
      k z - (1/π) 
              (2 k a u 
                  (1 + v ArcTan[v] + (1/(3 ((u^2) + (v^2)))) + 
                      (((x^2) - (y^2))/(3 (a^2) ((u^2) + (v^2)) 
                          (((1 + (v^2)))^2))))) /. {u -> uu, 
            v -> vv} /. case;

Clear[Eay]; Eay[x_, y_, z_] = 
      -((4 k x y u)/(3 π a ((u^2) + (v^2)) 
                  (((1 + (v^2)))^2))) /. {u -> uu, v -> vv}
             /. case;

Clear[Eaz]; Eaz[x_, y_, z_] = 
      ((4 k x v)/(3 π ((u^2) + (v^2)) (1 + (v^2)))) /. {
            u -> uu, v -> vv} /. case;

Clear[Ea]; Ea[x_, y_, z_] = {Eax[x, y, z], Eay[x, y, z], Eaz[x, y, z]};

Electric Field in the $x-y$ Plane

Fig16x8 = 
      Table[DensityPlot[
          Evaluate[Abs[Eax[x a, y a, (((10^jj) a)/1000)]^2]], {x, -2, 
            2}, {y, -2, 2}, 
          AxesLabel -> {"x in a", "y in a"}, 
          PlotLabel -> jj, Evaluate[opt1], 
          DisplayFunction -> Identity], {jj, 1, 3, 1}];

Show[
      GraphicsArray[{Fig16x8[[1]], 
          Fig16x8[[2]], 
          Fig16x8[[3]]}]];

tbl1 = 
      TableForm[{N[Abs[Eax[0, 0, (a/100)]]^2], N[Abs[Eax[0, 0, (a/10)]]^2], 
          N[Abs[Eax[0, 0, a (1/1)]]^2]}, 
        TableHeadings -> {{"a/100", "a/10", "a"}}]

Fig16x9 = 
      Table[DensityPlot[
          Evaluate[Abs[Eay[x a, y a, (((10^jj) a)/1000)]^2]], {x, -2, 
            2}, {y, -2, 2}, 
          AxesLabel -> {"x in a", "y in a"}, 
          PlotLabel -> jj, Evaluate[opt1], 
          DisplayFunction -> Identity], {jj, 1, 3, 1}];

Show[
      GraphicsArray[{Fig16x9[[1]], 
          Fig16x9[[2]], 
          Fig16x9[[3]]}]];

Fig16x10 = 
      Table[DensityPlot[
          Evaluate[Abs[Eaz[x a, y a, (((10^jj) a)/1000)]^2]], {x, -2, 
            2}, {y, -2, 2}, 
          AxesLabel -> {"x in a", "y in a"}, 
          PlotLabel -> jj, Evaluate[opt1], 
          DisplayFunction -> Identity], {jj, 1, 3, 1}];

Show[
      GraphicsArray[{Fig16x10[[1]], 
          Fig16x10[[2]], 
          Fig16x10[[3]]}]];

Magnetic Field

Clear[Hax]Hax[x_, y_, z_] = 
      -((4 x y v)/(π (a^2) ((u^2) + (v^2)) 
                  (((1 + (v^2)))^2))) /. {u -> uu, v -> vv}
             /. case;

Clear[Hay]; Hay[x_, y_, z_] = 
      1 - ((2 (ArcTan[v] + (v/((u^2) + (v^2)))))/π) + 
            ((2 ((x^2) - (y^2)) v)/(π (a^2) ((u^2) + (v^2)) 
                (((1 + (v^2)))^2))) /. {u -> uu, v -> vv} /. 
        case;

Clear[Haz]; Haz[x_, y_, z_] = 
      -((4 a y u)/(π (a^2) ((u^2) + (v^2)) 
                  (1 + (v^2)))) /. {u -> uu, v -> vv} /. 
        case;

Clear[Ha]; Ha[x_, y_, z_] = {Hax[x, y, z], Hay[x, y, z], Haz[x, y, z]};

Poynting Vector and Intensity

SetCoordinates[Cartesian];

Clear[Sa]; Sa[x_, y_, z_] := 
      ((c CrossProduct[Ea[x, y, z], Ha[x, y, z]])/(4 π)) /. NSub

Clear[Ia]; Ia[x_, y_, z_] := N[(Sqrt[Sa[x, y, z].Sa[x, y, z]])]

Fig16x11 = 
      Table[ListDensityPlot[
          Table[Ia[x a, y a, (((10^jj) a)/1000)], {x, -2, 2, .5}, {y, 
              -2, 2, .5}], 
          AxesLabel -> {"x in a", "y in a"}, 
          PlotLabel -> jj, Evaluate[Take[opt1, -6]], 
          DisplayFunction -> Identity], {jj, 1, 3, 1}];

Show[
      GraphicsArray[{Fig16x11[[1]], 
          Fig16x11[[2]], 
          Fig16x11[[3]]}]];

tbl2 = 
      TableForm[{N[micro Ia[0, 0, (a/100)]], N[micro Ia[0, 0, (a/10)]], 
          N[micro Ia[0, 0, a (1/1)]]}, 
        TableHeadings -> {{"a/100", "a/10", "a"}}]

Discussion of the Models

Electric Field

Fig16x12 = 
      Plot[Evaluate[{Abs[Edx[a x, (a/1000), a]], 
            Abs[Eax[a x, (a/1000), a]]}], {x, -2, 2}, 
        FrameLabel -> {"x in a", "Ex in a.u"}, Evaluate[opt]];

Intensity

Fig16x13 = 
      Plot[Evaluate[{micro Idc[a x, (a/1000), a], 
            micro Ia[a x, (a/1000), a]}], {x, -2, 2.1}, 
        FrameLabel -> {"x in a", "Intensity in a.u"}, 
        Evaluate[opt]];

Fig16x11