Needs["VectorAnalysis`"]

Extract ϵ and μ matrices from LeLmLocal notebook and rename x-> Xx since we will use package VectorAnalysis to compute curl and div.

LeMat ={{0,Cos[x],-λ Sin[x]},{-Cos[x],λ Sin[x] Sin[x+ϕ],Cos[x] Sin[x+ϕ]},{λ Sin[x],Cos[x+ϕ] Sin[x],λ Cos[x] Cos[x+ϕ]}} /. x-> Xx;

LmMat={{0,Cos[x+ϕ],-λ Sin[x+ϕ]},{-Cos[x+ϕ],λ Sin[x] Sin[x+ϕ],Cos[x+ϕ] Sin[x]},{λ Sin[x+ϕ],Cos[x] Sin[x+ϕ],λ Cos[x] Cos[x+ϕ]}} /. x-> Xx;

LeMat//MatrixForm

LmMat//MatrixForm

Doublecheck that LeMat coincides with expressions in paper

(* pricipal part of permittivity ϵ *)
Simplify[(LeMat + Transpose[LeMat])/2]//MatrixForm
(* skewon part of permittivity ϵ *)
Simplify[(LeMat - Transpose[LeMat])/2]//MatrixForm
(* pricipal part of permeability μ *)
Simplify[(LmMat + Transpose[LmMat])/2]//MatrixForm
(* skewon part of permeability μ *)
Simplify[(LmMat - Transpose[LmMat])/2]//MatrixForm

Note: the above computations show that both permittivity ϵ and permeability μ have a skewon component.

Define fields as in Example 3.16

Efield= Cos[t]{0,λ Sin[ Xx], Cos[ Xx]};
Hfield= -Sin[t]{0,λ Sin[ Xx+ϕ], Cos[Xx+ϕ]};

Dfield = Simplify[Simplify[LeMat.Efield]/.λ^2 -> 1]
Bfield = Simplify[Simplify[LmMat.Hfield]/.λ^2 -> 1]

Check that permittivity Le and permeability Lm are invertible

Simplify[Simplify[Det[LeMat]/. λ^2 -> 1]/. λ^2-> 1]
Simplify[Simplify[Det[LmMat]/. λ^2 -> 1]/. λ^2-> 1]

Check that Maxwell's equations hold for Efield, Hfield, Dfield and Bfield

Simplify[Curl[Efield]==-D[Bfield,t]]
Simplify[Curl[Hfield]== D[Dfield,t]]
Div[Dfield]
Div[Bfield]

Compute energy of solution

Simplify[Efield.Dfield + Hfield.Bfield]

Compute the Poynting  vector of solution:

Simplify[Cross[Efield, Hfield]]