Calculate the transmitted & reflected energy balance through a single-layer etalon cavity.

Model Assumptions:

◼
  • The incident E field is an idealized monochromatic plane wave.
    • ◼
  • The material stack consists of 3 layers,
    {
    n
    1
    ,
    n
    2
    ,
    n
    3
    }, with layer
    n
    2
    forming the etalon cavity.
    • ◼
  • The incident wave is assumed to enter from within layer
    n
    1
    .
    • ◼
  • All media are characterized purely by their refractive indices,
    {
    n
    1
    ,
    n
    2
    ,
    n
    3
    }, which may be either purely real or complex-valued.
    • ◼
  • All dielectric materials are non-absorbing at the input λ.
    • ◼
  • Hence, the range of permissible values for λ is may be implicitly restricted.
    • ◼
  • All boundaries between adjacent media are plane-parallel surfaces.
    • ◼
  • All media are isotropic & homogenous.
    • ◼
  • The responses of all media to the incident field are assumed to be linear wrt the field intensity.

    • Unit conversion functions:

    In[]:=
    NDegtoRad[x_]:=QuantityMagnitude[x]*
    π
    180
    ;NRadtoDeg[x_]:=x*
    180
    π
    °
    ;

    AOI & Cavity Layer Definitions:

    In[]:=
    n:={1.0,1.641,1.50+0I};λ:=633
    nm
    ;numPts:=90;θ:=Range[0,numPts]
    °
    ;​​d:=
    Range[0,numPts]
    10
    microns
    ;

    Intermediate Results:

    In[]:=
    ν:=
    n[[2]]
    n[[1]]
    ,
    n[[3]]
    n[[2]]
    ;
    θ
    crit
    :=ArcSin[ν[[2]]];
    θ
    Brewster
    :=NRadtoDeg[ArcTan[ν[[1]]]];​​ϕ[x_]:=ArcSin
    NDegtoRad[x]
    ν[[1]]
    ,
    ArcSin
    ArcSin
    NDegtoRad[x]
    ν[[1]]
    
    ν[[2]]
    
    NDegtoRad[x]<
    θ
    crit
    -ArcSin
    NDegtoRad[x]
    ν[[1]]
    
    NDegtoRad[x]>=
    θ
    crit
    

    Calculate & Plot the Angle of Incidence (AOI) at The Cavity
    Boundary
    12
    &
    Boundary
    23
    .

    In[]:=
    {ν,θ,NRadtoDeg[
    θ
    crit
    ],
    θ
    Brewster
    ,ϕ[θ]}
    Out[]=
    In[]:=
    Plot{NRadtoDeg[ϕ[x][[1]]],NRadtoDeg[ϕ[x][[2]]]},{x,Min[θ],Max[θ]},PlotPoints->Automatic,PlotRange->Full,PlotStyle->{Green,Black},Exclusions->Automatic,ExclusionsStyle->{Dashed},PlotLegends->{"
    AOR
    12
    ","
    AOR
    23
    "},PlotLabel->"Angle of refraction/reflection at the cavity boundaries (Degree Units)",AxesLabel->{"AOI","AOR"},AxesOrigin->{0,-Max[θ]},Ticks->{Range[QuantityMagnitude[-Max[θ]],QuantityMagnitude[Max[θ]],10]},GridLines->{Range[QuantityMagnitude[-Max[θ]],QuantityMagnitude[Max[θ]],10]},EpilogPointSize[Large],Point
    NRadtoDeg[
    θ
    crit
    ]
    °
    ,
    r
    tm
    [
    θ
    crit
    ][[1]],Inset"
    θ
    crit
    ",
    NRadtoDeg[
    θ
    crit
    ]
    °
    ,Center
    Out[]=
    AOR
    12
    AOR
    23

    Calculate & Plot the Fresnel reflection coefficients at the cavity boundaries :

    In[]:=
    r
    tm
    [x_]:=
    -
    2
    ν[[1]]
    *Cos[x]+
    2
    ν[[1]]
    -
    2
    Sin[x]
    2
    ν[[1]]
    *Cos[x]+
    2
    ν[[1]]
    -
    2
    Sin[x]
    ,
    Re
    -
    2
    ν[[2]]
    *Cos[ϕ[x][[2]]]+
    2
    ν[[2]]
    -
    2
    Sin[ϕ[x][[2]]]
    2
    ν[[2]]
    *Cos[ϕ[x][[2]]]+
    2
    ν[[2]]
    -
    2
    Sin[ϕ[x][[2]]]
    
    NDegtoRad[x]<
    θ
    crit
    Re
    -
    2
    ν[[2]]
    *Cos[ϕ[x][[2]]]+I*
    2
    Sin[ϕ[x][[2]]]
    -
    2
    ν[[2]]
    2
    ν[[2]]
    *Cos[ϕ[x][[2]]]+I*
    2
    Sin[ϕ[x][[2]]]
    -
    2
    ν[[2]]
    
    NDegtoRad[x]>=
    θ
    crit
    ,
    Im
    -
    2
    ν[[2]]
    *Cos[ϕ[x][[2]]]+
    2
    ν[[2]]
    -
    2
    Sin[ϕ[x][[2]]]
    2
    ν[[2]]
    *Cos[ϕ[x][[2]]]+
    2
    ν[[2]]
    -
    2
    Sin[ϕ[x][[2]]]
    
    NDegtoRad[x]<
    θ
    crit
    Im
    -
    2
    ν[[2]]
    *Cos[ϕ[x][[2]]]+I*
    2
    Sin[ϕ[x][[2]]]
    -
    2
    ν[[2]]
    2
    ν[[2]]
    *Cos[ϕ[x][[2]]]+I*
    2
    Sin[ϕ[x][[2]]]
    -
    2
    ν[[2]]
    
    NDegtoRad[x]>=
    θ
    crit
    ;​​
    r
    te
    [x_]:=
    Cos[x]-
    2
    ν[[1]]
    -
    2
    Sin[x]
    Cos[x]+
    2
    ν[[1]]
    -
    2
    Sin[x]
    ,
    Re
    Cos[ϕ[x][[2]]]-
    2
    ν[[2]]
    -
    2
    Sin[ϕ[x][[2]]]
    Cos[ϕ[x][[2]]]+
    2
    ν[[2]]
    -
    2
    Sin[ϕ[x][[2]]]
    
    NDegtoRad[x]<
    θ
    crit
    Re
    Cos[ϕ[x][[2]]]-I*
    2
    Sin[ϕ[x][[2]]]
    -
    2
    ν[[2]]
    Cos[ϕ[x][[2]]]+I*
    2
    Sin[ϕ[x][[2]]]
    -
    2
    ν[[2]]
    
    NDegtoRad[x]>=
    θ
    crit
    ,
    Im
    Cos[ϕ[x][[2]]]-
    2
    ν[[2]]
    -
    2
    Sin[ϕ[x][[2]]]
    Cos[ϕ[x][[2]]]+
    2
    ν[[2]]
    -
    2
    Sin[ϕ[x][[2]]]
    
    NDegtoRad[x]<
    θ
    crit
    Im
    Cos[ϕ[x][[2]]]-I*
    2
    Sin[ϕ[x][[2]]]
    -
    2
    ν[[2]]
    Cos[ϕ[x][[2]]]+I*
    2
    Sin[ϕ[x][[2]]]
    -
    2
    ν[[2]]
    
    NDegtoRad[x]>=
    θ
    crit
    ;
    In[]:=
    Plot{
    r
    te
    [NDegtoRad[x]][[1]],
    r
    tm
    [NDegtoRad[x]][[1]]},{x,Min[θ],Max[θ]},PlotPoints->Automatic,PlotRange->Full,PlotStyle->{Red,Blue},Exclusions->Automatic,ExclusionsStyle->{Dashed},PlotLegends->{"
    r
    te12
    ","
    r
    tm12
    "},PlotLabel->"Fresnel Reflection Coefficient, r, vs AOI at
    Boundary
    12
    ",AxesLabel->{"AOI","r"},Ticks->{Range[QuantityMagnitude[Min[θ]],QuantityMagnitude[Max[θ]],10],Range[-1,1,0.1]},GridLines->{Range[QuantityMagnitude[Min[θ]],QuantityMagnitude[Max[θ]],10],Range[-1,1,0.1]},EpilogPointSize[Large],Point
    θ
    Brewster
    °
    ,
    r
    tm
    [NDegtoRad[
    θ
    Brewster
    ]][[1]],Inset"
    θ
    Brewster
    ",
    θ
    Brewster
    °
    ,Center
    Out[]=
    r
    te12
    r
    tm12
    In[]:=
    Plot{
    r
    te
    [NDegtoRad[x]][[2]],
    r
    tm
    [NDegtoRad[x]][[2]],
    r
    te
    [NDegtoRad[x]][[3]],
    r
    tm
    [NDegtoRad[x]][[3]]},{x,Min[θ],Max[θ]},PlotPoints->Automatic,PlotRange->Full,PlotStyle->{Red,Blue,{Red,Dashed,Blue,Dashed}},Exclusions->Automatic,ExclusionsStyle->{Dashed},PlotLegends->{"Re[​
    r
    te23
    ​]","Re[​
    r
    tm23
    ​]","Im[​
    r
    te23
    ​]","Im[​
    r
    tm23
    ​]"},PlotLabel->"Fresnel Reflection Coefficient, r, vs AOI at
    Boundary
    23
    ",AxesLabel->{"AOI","r"},Ticks->{Range[QuantityMagnitude[Min[θ]],QuantityMagnitude[Max[θ]],10],Range[-1.0,1.0,0.1]},GridLines->{Range[QuantityMagnitude[Min[θ]],QuantityMagnitude[Max[θ]],10],Range[-1.0,1.0,0.1]},AxesOrigin->{0,-Max[θ]},EpilogPointSize[Large],Point
    NRadtoDeg[
    θ
    crit
    ]
    °
    ,
    r
    te
    [
    θ
    crit
    ][[2]],Inset"
    θ
    Crit
    ",
    NRadtoDeg[
    θ
    crit
    ]
    °
    ,Center
    Out[]=
    Re[​
    r
    te23
    ​]
    Re[​
    r
    tm23
    ​]
    Im[​
    r
    te23
    ​]
    Im[​
    r
    tm23
    ​]
    In[]:=
    r
    tm
    [θ]
    Out[]=
    {
    ⋯1⋯
    }
    Full expression not available
    (
    original memory size:
    487 kB)