Part 1: Waveguide Basic Concept

ECE 5368/6358                                 han q le - copyrighted
Use solely for students registered for UH ECE 6358/5368 during courses - DO NOT distributed (copyrighted materials)

Segment 2

Load package

TE mode calc

IndexprofSSW[x_,nindex_List,a_]:=If[x≤ -a,nindex[[1]],If[x≤ a,nindex[[2]],nindex[[1]] ] ]  ;
ResolvTE0[nindex_List,a_,λ_,neff_]:= Module[{ϵ1,ϵ2,κ1, k2,k0},
		{ϵ1,ϵ2}=nindex^2 ; k0=2*Pi/λ;
    	κ1=k0*Sqrt[neff^2-ϵ1  ]; k2=k0*Sqrt[ϵ2-neff^2 ] ;
			{ (Cos[a k2] κ1-Sin[a k2] k2) 
,(Cos[a k2] k2+Sin[a k2] κ1) 
,(Cos[a k2] κ1-Sin[a k2] k2) (Cos[a k2] k2+Sin[a k2] κ1)}  ] ;
ResolvTE[nindex_List,a_,λ_,neff_]:= Module[{ϵ1,ϵ2,κ1, k2,k0},
		{ϵ1,ϵ2}=nindex^2 ; k0=2*Pi/λ;
    	κ1=k0*Sqrt[neff^2-ϵ1  ]; k2=k0*Sqrt[ϵ2-neff^2 ] ;
			(Cos[a k2] κ1-Sin[a k2] k2) (Cos[a k2] k2+Sin[a k2] κ1) ] ;

FieldTE[nindex_List,a_,λ_,neff_,x_]:= Module[{ϵ1,ϵ2,κ1, k2,k0,Norm},
		{ϵ1,ϵ2}=nindex^2 ; k0=2*Pi/λ;
    	κ1=k0*Sqrt[Abs[neff^2-ϵ1 ] ]; k2=k0*Sqrt[Abs[ϵ2-neff^2] ] ;
		Norm=(((-cos(4 a k2) κ1+κ1+2 a ((k2^2)+((κ1)^2))))/(2 (k2^2)))+
			(((k2-κ1) (k2+κ1) sin(4 a k2))/(4 (k2^3)))+(1/(κ1)) ;
(1/(Sqrt[Norm])) If[Abs[x]<a,{ cos((a+x)k2)+((sin((a+x) k2) κ1)/k2)          (* this is Ey  *)
,-I ( cos((a+x) k2) κ1-sin((a+x) k2) k2)/k0   (* this is Hz  *)
,- neff ( cos((a+x)k2)+((sin((a+x) k2) κ1)/k2))}   (* this is Hx  *)
	, If [x<0, {(E^(κ1 (x+a) ))                                      (* this is Ey  *) 
,-I κ1 (E^(κ1 (x+a) ))/k0                        (* this is Hz  *)
 ,- neff  (E^(κ1 (x+a) ))}                            (* this is Hx  *)
, ((sin(2 a k2)  ((k2^2)+((κ1)^2)))/(2 k2 κ1))  {(E^(-κ1 (x-a)     ))(* this is Ey  *) 
, I κ1 (E^(-κ1 (x-a) ))/k0  (* this is Hz  *)
,- neff  (E^(-κ1 (x-a) ))}  (* this is Hx  *) ] 
]
] ;

ParmTE[nindex_List,a_,λ_,neff_]:= Module[{ϵ1,ϵ2,κ1, k2,k0,Norm},
		{ϵ1,ϵ2}=nindex^2 ; k0=2*Pi/λ;
    	κ1=k0*Sqrt[Abs[neff^2-ϵ1 ] ]; k2=k0*Sqrt[Abs[ϵ2-neff^2] ] ;
		Norm=(((-cos(4 a k2) κ1+κ1+2 a (k0^2) (ϵ2-ϵ1) ))/(2 (k2^2)))
			+(((k2-κ1) (k2+κ1) sin(4 a k2))/(4 (k2^3)))+(1/(κ1)) ;
{a, neff, κ1, k2,k0,Norm}  ] ;
FieldTE2[parm_List,x_]:= Module[{a, neff, κ1, k2,k0,Norm} ,
{a, neff, κ1, k2,k0,Norm}= parm  ; 
(1/(Sqrt[Norm])) If[Abs[x]<a, 
		{ cos((a+x)k2)+((sin((a+x) k2) κ1)/k2)          (* this is Ey  *)
				,-I ( cos((a+x) k2) κ1-sin((a+x) k2) k2)/k0   (* this is Hz  *)
				,- neff ( cos((a+x)k2)+((sin((a+x) k2) κ1)/k2))}   (* this is Hx  *)
	, If [x<0, {(E^(κ1 (x+a) ))                                        (* this is Ey  *)
					          ,-I κ1 (E^(κ1 (x+a) ))/k0                   (* this is Hz  *)
					            ,- neff  (E^(κ1 (x+a) ))}                     (* this is Hx  *)
				 , ((sin(2 a k2)  ((k2^2)+((κ1)^2)))/(2 k2 κ1))  {(E^(-κ1 (x-a)     ))(* this is Ey  *)
					                                               , I κ1 (E^(-κ1 (x-a) ))/k0  (* this is Hz  *)
						                                ,- neff  (E^(-κ1 (x-a) ))}  (* this is Hx  *)
			 ] ]
					
		      ] ;

TM mode calc

ResolvTM[nindex_List,a_,λ_,neff_]:= Module[{ϵ1,ϵ2,κ1, k2,k0},
		{ϵ1,ϵ2}=nindex^2 ; k0=2*Pi/λ;
    	κ1=k0*Sqrt[Abs[neff^2-ϵ1 ] ];
		 k2=k0*Sqrt[Abs[ϵ2-neff^2] ] ;
			 (cos(a k2) ϵ2 κ1-sin(a k2) k2 ϵ1) (cos(a k2) k2 ϵ1+sin(a k2) ϵ2 κ1)  ] ;

FieldTM[nindex_List,a_,λ_,neff_,x_]:= Module[{ϵ1,ϵ2,κ1, k2,k0,Norm},
		{ϵ1,ϵ2}=nindex^2 ; k0=2*Pi/λ;
    	κ1=k0*Sqrt[Abs[neff^2-ϵ1 ] ]; k2=k0*Sqrt[Abs[ϵ2-neff^2] ] ;
Norm=((((neff/(ϵ2))))^2)((1/(2 (k2^2) ((ϵ1)^2))) ϵ1 ϵ2 κ1-ϵ1 ϵ2 Cos[4 a k2] κ1+2 a ((k2^2) ((ϵ1)^2)+((ϵ2)^2) ((κ1)^2))  
+(1/(4 (k2^3) ((ϵ1)^2)))(k2 ϵ1-ϵ2 κ1) (k2 ϵ1+ϵ2 κ1) Sin[4 a k2])
+((((neff/(ϵ1))))^2)(1/( κ1))  ;
If[Abs[x]≤ a,{ Cos[(a+x)k2]+((Sin[(a+x) k2]  ϵ2  κ1)/(k2 ϵ1)) 
,I  (((Cos[(a+x)k2] ϵ2  κ1)/(ϵ1))-Sin[(a+x) k2]k2)/(ϵ2 k0)
,neff (Cos[(a+x)k2]+((Sin[(a+x) k2] ϵ2  κ1)/(k2 ϵ1)))/ϵ2 }
, If [x<0, {(E^(κ1 (x+a) )),I κ1 (E^(κ1 (x+a) ))/(ϵ1*k0 ) , neff  (E^(κ1 (x+a) ))/ϵ1 }
,{(E^(-κ1 (x-a) )), -I κ1 (E^(-κ1 (x-a) ))/(ϵ1*k0 ) , neff  (E^(-κ1 (x-a) ))/ϵ1   }]]
]

4. Key concept summary of waveguides

4.1 Discussion

Don’t get lost among the trees! see the forest.
Although the above illustration is only for the simplest waveguide, it is very useful to see the essential concepts and key properties of guided waves that are applicable to all waveguides.

1- The purpose of waveguide is ...to guide lightwave! Actually, it is about to confine lightwave to a region as the light is traveling (light always travels) so that the light does not escape and loses energy. Hence, a basic term describing a guided lightwave is (e^(i (β s-ω t))) where s is a length coordinate that doesn’t have to be straight line!. A waveguide path can have any shape:

2- The propation constant β, which is a function of ω, is a key property. The dependence of β[ω] on ω is called the dispersion relation of the waveguide, and we will learn in latter chapters why dispersion is so important.
If there is ONLY one mode, the waveguide is said to be single-mode. If it has more than 1, it is called multi-mode. A waveguide may not have any mode.

3- As it turns out... the details about the EM field profile F[ξ,η] in the other (non-traveling) dimensions {ξ,η} aren’t that important! People do engineer and design the profile F[ξ,η] but only for certain properties for specific applications - but there is NO fundamental guidelines what they should look like. Likewise, there is NO fundamental preference how the confinement is done: total internal reflection of photonic bandgap structure, or in general, any diffractive structure that prevents light from escaping. The mode profile may not be constant throughout the waveguide, which can be by design (adibatically profile-shaping waveguide) or unintentional consequence of the fabrication processes.

4- But the field profile F[ξ,η] does have an important property useful to calculate for waveguide applications: their orthogonality:
                  ∫(Subscript[F, m ])[ξ,η] (SuperStar[(Subscript[F, n ])])[ξ,η]  dξ dη= (Subscript[δ, m,n])  
where m, n are the labels of the modes (assuming all the profiles are normalized for convenience), and the modes are of the same polarization state.
It entails that a general monochromatic wave in the waveguide is a linear sum of these modes:
                      E=((∑_(m=1)^N)(Subscript[a, m])  (Subscript[F, m ])[ξ,η] (e^(i (Subscript[β, m])s)) ) (e^(-i ω t))

Note that the orthogonality of the modes come from using the Green’s theorem:  
(Subscript[E, n])(x,y)  ((Subscript[∇, T]^2) -(((Subscript[β, m])^2)-((ω^2)/(c^2))ϵ(x,y)) (Subscript[E, m])(x,y)=0 
(Subscript[E, m])(x,y)  ((Subscript[∇, T]^2) -(((Subscript[β, n])^2)-((ω^2)/(c^2))ϵ(x,y)) (Subscript[E, n])(x,y)=0 
hence:
∫ ((Subscript[E, n])(x,y)  (Subscript[∇, T]^2)(Subscript[E, m ]) -(Subscript[E, m])(x,y)  (Subscript[∇, T]^2)(Subscript[E, n]))dxdy=(((Subscript[β, m])^2)-((Subscript[β, n])^2))∫ (Subscript[E, n ])(Subscript[E, m ])dxdy

The left hand side of the above equation vanishes for bounded solutions, and hence:  
(((Subscript[β, m])^2)-((Subscript[β, n])^2))∫ (Subscript[E, n ])(Subscript[E, m ])dxdy =0  ⇒ ∫ (Subscript[E, n ])(Subscript[E, m ])dxdy =0
as   (Subscript[β, m])≠(Subscript[β, n]) 
There are 2 common normalization approaches. One is to define a unitless profile for the electric field such that
∫ (Subscript[E, n ])(Subscript[E, m ])dxdy =(Subscript[δ, m,n]) 
Another is to define the profile such that the optical power is normalize:

4.2 Relevance of waveguide properties to applications

1- In practice, no light guide is purely just a conduit for light. Light changes after traveling through a waveguide: power, spectral, temporal and even phase and spatial profile changes that are beyond those that are simple consequence of the propagation.

2- Some changes are just the effect of the propagation and properties of the waveguide (e. g. optical fiber). Some changes can be by design (intentional) for certain applications (modulators and photonic integrated curcuit to process optical signals).

3- In the latter cases, the intentional or designed changes can be passive or active. Passive is simply about changing the light by the waveguide without providing any physical stimulus, energy or power to the waveguide. Active is when some physical, energy or power is applied to the waveguide to effect the desired changes.

Bottom line: waveguides are simply devices that we use to spatially confine light for transportation and processing, because spatially freely propagating and diffracting light is generally more difficult to control and process for applications.

5. Important parameters of guided modes

Although we study a particular waveguide geometry above, the slab waveguide, several important concepts are applicable to any waveguide, and can be illustrated with the slab waveguide.

5.1 Power and energy flow

5.1.1 Poynting vector - TE mode

Intensity inside a waveguide is obtained by evaluating the Poynting vector. The time-averaged Poynting vector is:  
                                            ⟨(Overscript[S, →])⟩=(c/(8 π))Re[ (Overscript[E, ->]) ×(Overscript[(SuperStar[H]), ->])]                         (5.1.1)

Consider the example of slab waveguide.
For TE mode:
                                  (Overscript[E, ->]) =  E[x,z] (OverHat[y ])(e^(-i ω t ))                                                  (5.1.2)
                                (Overscript[H, ->])=-(i/(μ (Subscript[k, 0]))) ∇×(Overscript[E, ->])  = (({{(OverHat[x]), (OverHat[y]), (OverHat[z])}, {(∂_x), (∂_y), (∂_z)}, {0, (Subscript[E, y]), 0}})) 
=-(i/(μ (Subscript[k, 0])))(({{(OverHat[x]), (OverHat[y]), (OverHat[z])}}))(({{-(∂_z)(Subscript[E, y])[x,z]}, {0}, {(∂_x)(Subscript[E, y])[x,z]}}))(e^(-i ω t))                    (5.1.3)
                   ⟨(Overscript[S, →])⟩=(c/(8 π))Re[ (Overscript[E, ->]) ×(Overscript[(SuperStar[H]), ->])] 
           =(c/(8 π)) (1/(μ (Subscript[k, 0])))Re|({{(OverHat[x]), (OverHat[y]), (OverHat[z])}, {0, (Subscript[E, y]), 0}, {-i(∂_z)(SuperStar[(Subscript[E, y])])[x,z], 0, i (∂_x)(SuperStar[(Subscript[E, y])])[x,z]}})|       (5.1.4)

Det[(({{(OverHat[x]), (OverHat[y]), (OverHat[z])}, {0, (Subscript[Ef, y])[x,z], 0}, {-I(∂_z)(SuperStar[(Subscript[Ef, y])])[x,z], 0, I (∂_x)(SuperStar[(Subscript[Ef, y])])[x,z]}}))]

⟨(Overscript[S, →])⟩=c/(8 π)Re[ (Overscript[E, ->]) ×(Overscript[(SuperStar[H]), ->])]
        =(c/(8 π μ (Subscript[k, 0])))Re[ (OverHat[x]) i  (Subscript[E, y])(x,z)  ((∂ (SuperStar[(Subscript[E, y])]))/(∂ x)) +(OverHat[z]) i (Subscript[E, y])(x,z) ((∂ (SuperStar[(Subscript[E, y])]))/(∂ z))]         (5.1.5)
Is it possible for a net energy to flow along x direction? Intuitively, no. The energy can bounce back and forth from within the waveguide, but the net roundtrip (or the total space average) must be zero. 
If (Subscript[E, y])(x,z) is real, then so is ((∂ (SuperStar[(Subscript[E, y])]))/(∂ x))  and Re[ (OverHat[x]) i  (Subscript[E, y])(x,z) ((∂ (SuperStar[(Subscript[E, y])]))/(∂ x))]= 0: Obviously the intensity along x direction = 0. But the principle must be true much more generally, not for just this case of TE mode and real (Subscript[E, y])(x,z).
The intensity of the z-traveling component is given by:
                       |⟨(Overscript[S, →])⟩. (OverHat[z])| =(c/(8 π μ (Subscript[k, 0]))) |Re[ i (Subscript[E, y])(x,z) ((∂ (SuperStar[(Subscript[E, y])]))/(∂ z))]|                            (5.1.6)
and the power is given by:  
            P=(∫_(-∞)^∞)|⟨(Overscript[S, →])⟩. (OverHat[z])|dx =(c/(8 π μ (Subscript[k, 0]))) (∫_(-∞)^∞)|Re[ i (Subscript[E, y])(x,z) ((∂ (SuperStar[(Subscript[E, y])]))/(∂ z))]|dx        (5.1.7)
For mode m:   
                                   ((∂ (SuperStar[(Subscript[E, y])]))/(∂ z))= -i (Subscript[β, m]) (e^(-i (Subscript[β, m]) z)) (SuperStar[(Subscript[E, m])])[x]                                      (5.1.8)
The intensity is:
                            |⟨(Overscript[(Subscript[S, m]), →])⟩ . (OverHat[z])| =(c/(8 π μ (Subscript[k, 0])))(Subscript[β, m]) (|(Subscript[E, m])[x]|)^2 
                  =((c (Subscript[n, eff;m]))/(8 π μ))  (|(Subscript[E, m])[x]|)^2                                 (5.1.9)
                                       (Subscript[P, m])= ((c (Subscript[n, eff;m]))/(8 π μ))(∫_(-∞)^∞)(((|(Subscript[E, m])[x]|))^2)dx                               (5.1.10)
These expressions are very similar to planewave, with the difference being the effective index, which is mode-dependent.
This is the reason why normalization with respect to power is slightly different from normalization with respect to wavefunction:
For just wavefunction:             (∫_(-∞)^∞)(|(Subscript[E, m])[x]|)^2 = 1.                            (5.1.11a)
For power:                         (Subscript[n, eff;m])(∫_(-∞)^∞)(|(Subscript[E, m])[x]|)^2dx = 1                    (5.1.11b)
(all other constants are dropped for convenient).
Either way is fine as long as one remembers what to use in each calculation.

5.1.2 For TM mode

⟨(Overscript[S, →])⟩=(c/(8 π)) (1/(ϵ  (Subscript[k, 0]))) Re[-i (OverHat[x]) (SuperStar[(Subscript[H, y])]) (Subscript[H, y]^((1,0)))(x,z) -i (OverHat[z]) (SuperStar[(Subscript[H, y])]) (Subscript[H, y]^((0,1)))(x,z)]       (5.1.12)
The intensity is given by:  
                                        |⟨(Overscript[S, →])⟩.(OverHat[z])| =(c/(8 π ϵ  (Subscript[k, 0]))) |Re[ -i (SuperStar[(Subscript[H, y])])(x,z) ((∂ (Subscript[H, y]))/(∂ z))]|            (5.1.13)
Now we need to convert to Ex:
                                  (Subscript[E, x]) =- (i/(ϵ  (Subscript[k, 0])))(∂_z)(Subscript[H, y])[x,z] .                           (5.1.14)
 For a given mode m: 
                                     (Subscript[E, m;x]) =- (i/(ϵ  (Subscript[k, 0])))i (Subscript[β, m]) (Subscript[H, y])[x,z] 
            =((Subscript[β, m])/(ϵ  (Subscript[k, 0]))) (Subscript[H, y])[x,z]  = ((Subscript[n, eff;m])/ϵ) (Subscript[H, y])[x,z]          (5.1.15)
Thus:                                    (Subscript[H, y])[x,z]= (ϵ/(Subscript[n, eff;m]))(Subscript[E, m;x ]).                               (5.1.16)
Then: 
                                    |⟨(Overscript[S, →])⟩.(OverHat[z])| =(c/(8 π )) (ϵ/(Subscript[n, eff;m]))|Re[ (SuperStar[(Subscript[E, m;x])]) (Subscript[E, m;x])]|
            = (c/(8 π )) (ϵ/(Subscript[n, eff;m]))(((|(Subscript[E, m;x])|))^2)               (5.1.17)
Indeed that the intensity for TM mode is proportional to the transverse component of the electric field, as expected. 
The power of the mth mode is given by:  
                       (Subscript[P, m])=(∫_(-∞)^∞)|⟨(Overscript[(Subscript[S, m]), →])⟩.(OverHat[z])|dx = (c/(8 π (Subscript[n, eff;m ])))(∫_(-∞)^∞)ϵ[x](((|(Subscript[E, m;x])|))^2)dx       (5.1.18)
Notice that (∫_(-∞)^∞)ϵ[x](((|(Subscript[E, m;x])|))^2)dx can be written as:  
                         (∫_(-∞)^∞)ϵ[x](((|(Subscript[E, m;x])|))^2)dx =(((∫_(-∞)^∞)ϵ[x](((|(Subscript[E, m;x])|))^2)dx)/((∫_(-∞)^∞)(((|(Subscript[E, m;x])|))^2)dx))(∫_(-∞)^∞)(((|(Subscript[E, m;x])|))^2)dx 
                                          = ⟨ϵ⟩(∫_(-∞)^∞)(((|(Subscript[E, m;x])|))^2)dx     (5.1.19)
where  ⟨ϵ⟩ is defined as the spatial average of ϵ. Thus:
                                                    (Subscript[P, m])=(c/(8 π  ))((⟨ϵ⟩)/(Subscript[n, eff;m]))(∫_(-∞)^∞)(((|(Subscript[E, m;x])|))^2)dx .               (5.1.20)
Notice the suttle difference in the power expression of TE and TM mode. The ratio ((⟨ϵ⟩)/(Subscript[n, eff;m]))can be considered as a modal average index.

5.2 Intensity plot

In the lab, if we look at a waveguide facet, what to we see? We see the intensity.

indexprof={3.4,3.5}; a=2.5 ; λ=1.5 ;
segment={3.499,3.495,3.48,3.47,3.45,3.42,3.405}; neff={} ;
For[ i=1, i≤ 6, i++,
	AppendTo[neff, 
	 x /. FindRoot[ResolvTE[indexprof,a,λ,x]==0,{x,segment[[i+1]],segment[[i]]}]
                            ] ]

indexprof={3.4,3.5}; a=2.5 ; λ=1.5 ;
segment={3.499,3.495,3.48,3.47,3.45,3.42,3.405}; neff={} ;
For[ i=1, i≤ 6, i++,
	AppendTo[neff, 
	 x /. FindRoot[ResolvTE[indexprof,a,λ,x]==0,{x,segment[[i+1]],segment[[i]]}]
                            ] ] ;
alldp={};
For[i=1,i<Length[neff]+1,i++,
	AppendTo[alldp, DensityPlot[neff[[i]]*Abs[FieldTE[indexprof,a,λ,neff[[i]],x][[1]]]^2,{y,0,20},{x,-4,4}
	 ,PlotPoints->{2,50}, Mesh->False, AspectRatio->0.25, ColorFunction->GrayLevel
	,ImageSize->{400,120}] ];

             ] ;
alldp

Intensity plot
,,,
,,}

We recall earlier that for some TM mode, if the index difference is large, the E field leakage is also quite large. We can see how it is in this example. Here, we have a thin 1 μm semiconductor film surrounding by air:

nindex={1,3.5}; a=0.5 ; λ=1.5 ;
Plot[ResolvTM[nindex,a,λ,x],{x,nindex[[1]],nindex[[2]]}]

segment2={3.49,3.4,3,2.5,1.5,1.01}; neffTM={} ;
For[ i=1, i<Length[segment2], i++,
	AppendTo[neffTM, 
	 x /. FindRoot[ResolvTM[nindex,a,λ,x]==0,{x,segment2[[i+1]],segment2[[i]]}]
                            ] ] ;
neffTM

This is the normal (perpendicular) E field, (Subscript[E, x]):

allEfTM={};
For[i=1,i<Length[neffTM]+1,i++,
	    AppendTo[  allEfTM, Plot[FieldTM[nindex,a,λ,neffTM[[i]],x][[3]],{x,-1.5,1.5}
, PlotRange->All, PlotStyle->{Hue[(i-1)/(0.1+Length[neffTM]),1,0.4]}
,Filling-> {1-> {Axis, Hue[(i-1)/(0.1+Length[neffTM]),1,1,0.35]}} 
	 ] ]
             ] ;
allEfTM

allHyTM={};
For[i=1,i<Length[neffTM]+1,i++,
	    AppendTo[  allHyTM, Plot[FieldTM[nindex,a,λ,neffTM[[i]],x][[1]],{x,-1.5,1.5}
, PlotRange->All, PlotStyle->{Hue[(i-1)/(0.1+Length[neffTM]),1,0.4]}
,Filling-> {1-> {Axis, Hue[(i-1)/(0.1+Length[neffTM]),1,1,0.35]}} 
	 ] ]
             ] ;
allHyTM

alldpTM={};
Eps[x_]:= If[Abs[x]<=a, nindex[[2]]^2,nindex[[1]]^2]; 
For[i=1,i<Length[neffTM]+1,i++,
	    AppendTo[   alldpTM,   DensityPlot[(Eps[x]/neffTM[[i]])*Abs[FieldTM[nindex,a,λ,neffTM[[i]],x][[3]]]^2,{y,0,20},{x,-1.5,1.5}
	 ,PlotPoints->{2,50}, Mesh->False, ColorFunction->GrayLevel,AspectRatio->0.25
	,ImageSize->{400,120}] ]
             ] ;
alldpTM

Note the highest mode: the light is barely confined, and it hugs the cladding mostly.

5.3 Confinement factor and mode size

An important concept (for all waveguide, not just slab): confinement factor: How much of the electric field is really in the core region? The fraction of the field in the core is called the confinement factor. (in complex structure, we need to define what "core" region is the region of our interest) 
                                (((∫_core^□)(((|E[x,y]|))^2)dxdy  )/((∫_all^□)(((|E[x,y]|))^2)dxdy))                       (5.3.1)
Note: the confinement factor is a convenient definition and NOT a fundamental quantity like propagation constant, power, or field profile. 
Why is it relevant? because in many waveguides, the core region often contains the structure of interest, such as quantum wells for a laser or modulator. Knowing “how much” of the mode is there allows us to determine the strength of the interaction between guided modes and the structure of interest. It is a “design” parameter to allow tailoring, optimization certain property of the waveguide device.

Another concept is the mode size: how big is the mode. Again, the mode profile function such as E[x,y] is a fundamentally defined quantity, but its “size” is only a convenient definition, the same way as we say how large the spot size of a Gaussian beam is. A common definition is based on power and intensity. 
Recalling that the intensity of mode m:
                            |⟨(Overscript[(Subscript[S, m]), →])⟩ . (OverHat[z])| =(c/(8 π μ (Subscript[k, 0])))(Subscript[β, m]) (|(Subscript[E, m])[x]|)^2 
                  =((c (Subscript[n, eff;m]))/(8 π μ))  (|(Subscript[E, m])[x]|)^2                          (5.1.9)
We can define spatially averaged intensity as:
                               (Subscript[I, m;ave])=(( ∫((|(Subscript[E, m])[x,y]|)^2) |⟨(Overscript[(Subscript[S, m]), →])⟩ . (OverHat[z])| dxdy  )/(∫((|(Subscript[E, m])[x,y]|)^2)dxdy))                                    
                                 =((c (Subscript[n, eff;m]))/(8 π)) (( ∫((|(Subscript[E, m])[x,y]|)^4)  dxdy  )/(∫((|(Subscript[E, m])[x,y]|)^2)dxdy))                                        (5.3.2)
In other words, it is the spatially averaged value of ((|E[x,y]|)^2) by the mode profile ((|E[x,y]|)^2) itself. 
We can write (5.3.2) alternatively:
                             (Subscript[I, m;ave])=((c (Subscript[n, eff;m]))/(8 π)) ∫((|(Subscript[E, m])[x,y]|)^2)dxdy   (( ∫((|(Subscript[E, m])[x,y]|)^4)  dxdy  )/(((∫((|(Subscript[E, m])[x,y]|)^2)dxdy))^2))    
Or:                        (Subscript[I, m;ave])=(Subscript[P, m])/  ( (( (((∫((|(Subscript[E, m])[x,y]|)^2)dxdy))^2) )/(∫((|(Subscript[E, m])[x,y]|)^4)  dxdy )))                     (5.3.3)     
where (Subscript[P, m]) is the mode power defined in (5.1.10). We recognize that we can define:
                             (Subscript[A, m])≡(( (((∫((|(Subscript[E, m])[x,y]|)^2)dxdy))^2) )/(∫((|(Subscript[E, m])[x,y]|)^4)  dxdy ))                                 (5.3.4)
as the “effective” mode area (or mode size) of mode m. This definition is of course by no mean unique. But it is conveniently used when dealing with power and intensity of the mode, and we can write:
                            (Subscript[P, m]) =(Subscript[I, m;ave]) (Subscript[A, m])                                            (5.3.5)

Exercise: use the definition (5.3.4) to obtain the effective area of a Gaussian beam at its beam waist (prove that it is π (w^2) as expected).

Exercise: Find the effective mode size (1 dimensional only) for all the modes of the slab waveguide in the example of 3.4

6 Introduction to multi-mode propagation

6.1 Concept

If we launch an optical beam into a multimode waveguide, what do we get out at the other end? Again we apply the principle of linear superposition:
                                                     Φ[x]=(∑_(m=1)^N)(Subscript[a, m])(Subscript[F, m])[x]                          (6.1.1)
We can represent the input signal as a finite-rank vector: 
Φ=(({{(Subscript[a, 1])}, {(Subscript[a, 2])}, {.}, {(Subscript[a, N])}})). The full expression of the electric field is: 
                    (Overscript[E, →])=(∑_(m=1)^N)(Subscript[a, m])(Subscript[(Overscript[E, →]), m])[x](e^(i ((Subscript[β, m])z-ω t)))  =(e^(-i ω t))(∑_(m=1)^N)(Subscript[a, m])(Subscript[(Overscript[E, →]), m])[x](e^(i (Subscript[β, m])z))    (6.1.2)
We notice one very important feature: we cannot separate the x and z dependent component of the net field (Overscript[E, →]) anymore. Thus: while the z-dependent and the x-dependent component can be separated for every eigenmode of the waveguide, they can not be so for any arbitrary wave.
The reason is modal dispersion as discussed above: each mode has a different propagation constant (Subscript[β, m]).

What does this entail? Remember how when you add waves together, not only amplitude but also phase is very important? Let's consider the input signal at z=0: 
                                        (Overscript[E, →])[x,z=0]=(e^(-i ω t))(∑_(m=1)^N)(Subscript[a, m])(Subscript[(Overscript[E, →]), m])[x] .              (6.1.3)
At z=L, we have:
                                       (Overscript[E, →])[x,z=L]=(e^(-i ω t))(∑_(m=1)^N)(Subscript[a, m])(e^(i (Subscript[β, m])L))(Subscript[(Overscript[E, →]), m])[x] 
                       =(e^(-i ω t))(∑_(m=1)^N)(Subscript[a, m])'(Subscript[(Overscript[E, →]), m])[x]       (6.1.4)
where                                                (Subscript[a, m])'=(Subscript[a, m])(e^(i (Subscript[β, m])L)).
Thus, the profile (Overscript[E, →])[x,z=L] can be very different from that at z=0.
This very important aspect is the foundation for waveguide circuit: modulate and control of light path through waveguide.

6.2 Illustration example

6.2.1 Example set up

Consider a slab waveguide with 2 modes:

nindex={1.4,1.5}; a=1.25 ; λ=1.5 ;
Plot[ResolvTE[nindex,a,λ,x],{x,nindex[[1]],nindex[[2]]}]

segment={1.49,1.46,1.42}; neff={} ;
For[ i=1, i<Length[segment], i++,
	AppendTo[neff, 
	 x /. FindRoot[ResolvTE[nindex,a,λ,x]==0,{x,segment[[i+1]],segment[[i]]}]
                            ] ]

neff

Let's look at the waveguide mode

Plot[{FieldTE[nindex,a,λ,neff[[1]],x][[1]]
		   ,FieldTE[nindex,a,λ,neff[[2]],x][[1]]},{x,-6,6}
	, PlotRange->{-1.,1}, PlotStyle->{{Hue[0.2,1,0.8]},{Hue[0.8,1,0.8]}}]

At z=0, let's have a wave that consists of half and half of the low mode and the high mode:

w=0.5;
Plot[{w*FieldTE[nindex,a,λ,neff[[1]],x][[1]]
		   +(1-w)*FieldTE[nindex,a,λ,neff[[2]],x][[1]]},{x,-4,4}
	, PlotRange->{-1.,1}, PlotStyle->{{Hue[0.2,1,0.8]},{Hue[0.8,1,0.8]}}]

6.2.2 Mixed mode propagation

What does this wave look like as it propagates along the waveguide?
At z= 0: 
          (Overscript[E, →])[x,z=0]=(1/2)(e^(-i ω t))((Subscript[(Overscript[E, →]), 1])[x]+(Subscript[(Overscript[E, →]), 2])[x] )                          (6.2.1)
          (Overscript[E, →])[x,z=L]=(1/2)(e^(-i ω t))((e^(i (Subscript[β, 1])L)) (Subscript[(Overscript[E, →]), 1])[x]+(e^(i (Subscript[β, 2])L)) (Subscript[(Overscript[E, →]), 2])[x] )        (6.2.2)
                          =(1/2)(e^(-i ω t))(e^(i (Subscript[β, 1])L))( (Subscript[(Overscript[E, →]), 1])[x]+(e^(i ((Subscript[β, 2])-(Subscript[β, 1]))L)) (Subscript[(Overscript[E, →]), 2])[x] )   (6.2.3)
What happens at z=L when (e^(i ((Subscript[β, 2])-(Subscript[β, 1]))L)) = (e^(i π)) ? 
         (Overscript[E, →])[x,z=L]=(1/2)(e^(-i ω t))(e^(i (Subscript[β, 1])L))( (Subscript[(Overscript[E, →]), 1])[x]- (Subscript[(Overscript[E, →]), 2])[x] )                 (6.2.4)
Let's compare the field profile at z=0 and z=L, dropping the oscilling term (e^(-i ω t))(e^(i (Subscript[β, 1])L)) :

w=0.5;
Plot[{w*FieldTE[nindex,a,λ,neff[[1]],x][[1]]
		   +(1-w)*FieldTE[nindex,a,λ,neff[[2]],x][[1]]
	     , w*FieldTE[nindex,a,λ,neff[[1]],x][[1]]
		   -(1-w)*FieldTE[nindex,a,λ,neff[[2]],x][[1]]},{x,-4,4}
	, PlotRange->{-1.,1}, PlotStyle->{{Hue[0.2,1,0.8]},{Hue[0.8,1,0.8]}}]

Remarkable: this tells us that the profile changes quite a bit as the wave propagates: it is not always a constant profile wave in a waveguide! What if it keeps on going?
         (Overscript[E, →])[x,z=2L]=(1/2)(e^(-i ω t))(e^(i 2 (Subscript[β, 1])L))( (Subscript[(Overscript[E, →]), 1])[x]+(e^(i 2 ((Subscript[β, 2])-(Subscript[β, 1]))L)) (Subscript[(Overscript[E, →]), 2])[x] )            (6.2.5)
and (e^(i ((Subscript[β, 2])-(Subscript[β, 1]))L)) = (e^(i π)) : (Overscript[E, →])[x,z=2L]=(1/2)(e^(-i ω t))(e^(i 2 (Subscript[β, 1])L))( (Subscript[(Overscript[E, →]), 1])[x]+ (Subscript[(Overscript[E, →]), 2])[x] ): we are back to square one again.
We can see all this by looking at its intensity. From the above expression of the Poynting vector: 
              |⟨(Overscript[S, →])⟩ . (OverHat[z])| = (c/(8 π μ (Subscript[k, 0])))|Re[ i (Subscript[E, y])(x,z) ((∂ (SuperStar[(Subscript[E, y])]))/(∂ z))]| 
             (Subscript[E, y])(x,z)= w (Subscript[E, y;1])(x)(e^(i (Subscript[β, 1])z)) +(1-w) (Subscript[E, y;2])(x)(e^(i (Subscript[β, 2])z ))
             ((∂ (SuperStar[(Subscript[E, y])]))/(∂ z))=-i (Subscript[β, 1]) w (Subscript[E, y;1])(x)(e^(-i (Subscript[β, 1])z)) -i (Subscript[β, 2])(1-w) (Subscript[E, y;2])(x)(e^(-i (Subscript[β, 2])z ))
             i (Subscript[E, y])(x,z) ((∂ (SuperStar[(Subscript[E, y])]))/(∂ z))=(w (Subscript[E, y;1])(x)(e^(i (Subscript[β, 1])z)) +(1-w) (Subscript[E, y;2])(x)(e^(i (Subscript[β, 2])z)) )×
		         ((Subscript[β, 1]) w (Subscript[E, y;1])(x)(e^(-i (Subscript[β, 1])z)) + (Subscript[β, 2])(1-w) (Subscript[E, y;2])(x)(e^(-i (Subscript[β, 2])z)) )    (6.2.6)

w=.;

Expand[(w (Subscript[Ef, y;1])(x)(E^(I (Subscript[β, 1])z)) +u (Subscript[Ef, y;2])(x)(E^(I (Subscript[β, 2])z)) )
		                             ((Subscript[β, 1]) w (Subscript[Ef, y;1])(x)(E^(-I (Subscript[β, 1])z)) + (Subscript[β, 2])u (Subscript[Ef, y;2])(x)(E^(-I (Subscript[β, 2])z)) )]

i (Subscript[E, y])(x,z) ((∂ (SuperStar[(Subscript[E, y])]))/(∂ z))=(w^2) (Subscript[β, 1]) (((Subscript[Ef, 1])(x))^2)+(u^2) (Subscript[β, 2]) (((Subscript[Ef, 2])(x))^2)
			       +((e^(i z ( (Subscript[β, 2])- (Subscript[β, 1])))) (Subscript[β, 1]) +(e^(-i z ( (Subscript[β, 2])- (Subscript[β, 1]))))  (Subscript[β, 2])) u w (Subscript[Ef, 2])(x) (Subscript[Ef, 1])(x)    (6.2.7)
There is a real term: (w^2) (Subscript[β, 1]) (((Subscript[Ef, 1])(x))^2)+(u^2) (Subscript[β, 2]) (((Subscript[Ef, 2])(x))^2) which is the individual sum of the intensity of each component. But more importantly, there is the interference term:
                      ((e^(i z ( (Subscript[β, 2])- (Subscript[β, 1])))) (Subscript[β, 1]) +(e^(-i z ( (Subscript[β, 2])- (Subscript[β, 1]))))  (Subscript[β, 2])) u w (Subscript[Ef, 2])(x) (Subscript[Ef, 1])(x).                  (6.2.8)
What is the effect of this interference term? Let's plot it out:

w=0.5; β=((2π)/λ)neff;
Plot3D[Re[(w*FieldTE[nindex,a,λ,neff[[1]],x][[1]](E^(I β[[1]]z))
		   +(1-w)*FieldTE[nindex,a,λ,neff[[2]],x][[1]](E^(I β[[2]]z)))*
		(w*β[[1]]*FieldTE[nindex,a,λ,neff[[1]],x][[1]](E^(-I β[[1]]z))
		   +(1-w)*β[[2]]*FieldTE[nindex,a,λ,neff[[2]],x][[1]](E^(-I β[[2]]z)))
				 ]
		,{z,0,30},{x,-2.5,2.5}, PlotPoints->{120,40}
	, PlotRange->All, Mesh->False, BoxRatios->{30,5,5},
ViewPoint->{0,0,5}]

Notice what happens: the wave seems to move from right side of the waveguide to left and back to right again and so on... : this is the behavior of a ray that bounces back and forth between the two boundary surface. Where is the shift maximum?

L=(π/(Abs[(β[[2]]-β[[1]])]))

What is a Δβ coupler? why is it called Δβ?

We have a waveguide with this property: cladding index= 1.4 ; thickness 2.5 um, the core index can be tuned with an external electric field from 1.45 to 1.55. A beam with 1.5 um wavelength is coupled to one side of the waveguide (mode1 + mode2), how does the external electric field can change its propagation property?

Show[Plot3D[ResolvTE[{1.4,ncore},a,λ,x],{x,1.4,ncore-0.01},{ncore,1.45,1.55}, Mesh-> False , PlotPoints->{50,10}, PlotRange-> {{1.4,1.55},All}]
,Plot3D[0,{x,1.4,1.55},{ncore,1.45,1.55},  PlotPoints->{4,4}]]

At core=1.45

nindex={1.4,1.45}; a=1.25 ; λ=1.5 ;
Plot[ResolvTE[nindex,a,λ,x],{x,nindex[[1]],nindex[[2]]}]

segment={1.44,1.43,1.401}; neff={} ;
For[ i=1, i<Length[segment], i++,
	AppendTo[neff, 
	 x /. FindRoot[ResolvTE[nindex,a,λ,x]==0,{x,segment[[i+1]],segment[[i]]}]
                            ] ]

neff

w=0.5; β=((2π)/λ)neff;
plot1=Plot3D[Re[(w*FieldTE[nindex,a,λ,neff[[1]],x][[1]](E^(I β[[1]]z))
		   +(1-w)*FieldTE[nindex,a,λ,neff[[2]],x][[1]](E^(I β[[2]]z)))*
		(w*β[[1]]*FieldTE[nindex,a,λ,neff[[1]],x][[1]](E^(-I β[[1]]z))
		   +(1-w)*β[[2]]*FieldTE[nindex,a,λ,neff[[2]],x][[1]](E^(-I β[[2]]z)))
				 ]
		,{z,0,30},{x,-2.5,2.5}, PlotPoints->{120,40}
	, PlotRange->All, Mesh->False, BoxRatios->{30,5,5},
ViewPoint->{0,0,5}]

At core=1.45

nindex={1.4,1.55}; a=1.25 ; λ=1.5 ;
Plot[ResolvTE[nindex,a,λ,x],{x,nindex[[1]],nindex[[2]]}]

segment={1.54,1.5,1.46}; neff={} ;
For[ i=1, i<Length[segment], i++,
	AppendTo[neff, 
	 x /. FindRoot[ResolvTE[nindex,a,λ,x]==0,{x,segment[[i+1]],segment[[i]]}]
                            ] ]

neff

w=0.5; β=((2π)/λ)neff;
plot2=Plot3D[Re[(w*FieldTE[nindex,a,λ,neff[[1]],x][[1]](E^(I β[[1]]z))
		   +(1-w)*FieldTE[nindex,a,λ,neff[[2]],x][[1]](E^(I β[[2]]z)))*
		(w*β[[1]]*FieldTE[nindex,a,λ,neff[[1]],x][[1]](E^(-I β[[1]]z))
		   +(1-w)*β[[2]]*FieldTE[nindex,a,λ,neff[[2]],x][[1]](E^(-I β[[2]]z)))
				 ]
		,{z,0,30},{x,-2.5,2.5}, PlotPoints->{120,40}
	, PlotRange->All, Mesh->False, BoxRatios->{30,5,5},
ViewPoint->{0,0,5}]

Now we can compare:

With an external electric field that “tunes” the index of the core, we see that the beam can be switched from one side of the waveguide to the other.

6.3 Implication of multi-mode propation on spatial profile

See APP on waveguide - mixed mode propagation.