Example1​​PropagationinthezdirectionofaLaser-Beamwiththespeedoflight:c=
1
ϵ0μ0
Book:
RisingoftheJamesWebbSpaceTelescope​
​anditsFundamentalBlindness
​​ Page15,Equation(19)​​​​
- 
1
2
c
∂
¯
E
×
¯
Η

∂t
+
ε
0
¯
E
∇. 
¯
E
−
ε
0
¯
E
×∇×
¯
E
+
μ
0
¯
Η
∇. 
¯
Η
−
μ
0
¯
Η
×∇×
¯
Η

= 0
(19)​​​​ExampleofaLASER-BEAMwithaGaussianIntensitydivison
-K2
2
x
+
2
y

,combinedwithanarbitrarydivisiong[x,y]inthe(x,y)plane,propagatinginthez-directionwiththespeedoflightc=
1
ϵ0μ0
.
In[]:=
Needs["DifferentialEquations`NDSolveProblems`"]
In[]:=
Needs["DifferentialEquations`NDSolveUtilities`"]
In[]:=
Needs["DifferentialEquations`InterpolatingFunctionAnatomy`"]
In[]:=
ϵ0=.
In[]:=
μ0=.
In[]:=
x=.
In[]:=
y=.
In[]:=
z=.
In[]:=
t=.
In[]:=
Get["VectorAnalysis`"]
General
:VectorAnalysis` is now obsolete. The legacy version being loaded may conflict with current functionality. See the Compatibility Guide for updating information.
In[]:=
InverseFunctions->True
Out[]=
InverseFunctionsTrue
In[]:=
Needs["DifferentialEquations`NDSolveProblems`"]
In[]:=
Needs["DifferentialEquations`NDSolveUtilities`"]
In[]:=
Needs["DifferentialEquations`InterpolatingFunctionAnatomy`"]
In[]:=
SetCoordinates[Cartesian[x,y,z]]
Out[]=
Cartesian[x,y,z]
In[]:=
{Coordinates[Cartesian],CoordinateRanges[Cartesian]}
Out[]=
{{x,y,z},{-∞<x<∞,-∞<y<∞,-∞<z<∞}}
In[]:=
K2=.
In[]:=
f[x,y]=g[x,y]
-K2
2
x
+
2
y

Out[]=
-K2
2
x
+
2
y

g[x,y]
In[]:=
ev={f[x,y]*g[t-(K1/z+1)*z*Sqrt[ϵ0]*Sqrt[μ0]],0,0}
Out[]=

-K2
2
x
+
2
y

gt-1+
K1
z
z
ϵ0
μ0
g[x,y],0,0
In[]:=
mv=(1/Sqrt[μ0])*Sqrt[ϵ0]*​​{0,f[x,y]*g[t-(K1/z+1)*z*Sqrt[ϵ0]*Sqrt[μ0]],0}
Out[]=
0,
-K2
2
x
+
2
y

ϵ0
gt-1+
K1
z
z
ϵ0
μ0
g[x,y]
μ0
,0
Book:
RisingoftheJamesWebbSpaceTelescopeanditsFundamentalBlindnes
:Page15,Equation(19)​​ExampleofaLASER-BEAMwithaGaussianIntensitydivison
-K2
2
x
+
2
y

,combinedwithanarbitrarydivisiong[x,y]inthe(x,y)plane,propagatinginthez-directionwiththespeedoflightc=
1
ϵ0μ0
.​​TheinputfortheElectricFieldIntensity{E(x,y,z,)=ev}andtheMagneticFieldIntensity{H(x,y,z,)=mv}hasbeensubstitutedintheFieldEquationfortheElectromagneticField(BookEquation19,page16).​​​​
- 
1
2
c
∂
¯
E
×
¯
Η

∂t
+
ε
0
¯
E
∇. 
¯
E
−
ε
0
¯
E
×∇×
¯
E
+
μ
0
¯
Η
∇. 
¯
Η
−
μ
0
¯
Η
×∇×
¯
Η

= 0
Equation(19)​​​​Whichcanbewrittenas:​​​​term1=-
1
2
c
∂
¯
E
×
¯
Η

∂t
​​​​term2=
ε
0
¯
E
(∇. 
¯
E
)​​​​term3=−
ε
0
¯
E
×(∇×
¯
E
)​​​​term4=
μ
0
¯
Η
∇. 
¯
Η
​​​​term5=−
μ
0
¯
Η
×∇×
¯
Η
​​​​Equation(19)equals:​​​​term1+term2+term3+term4+term5=0
In[]:=
Div[ev]
Out[]=
-
-K2
2
x
+
2
y

K2xgt-1+
K1
z
z
ϵ0
μ0
g[x,y]
2
x
+
2
y
+
-K2
2
x
+
2
y

gt-1+
K1
z
z
ϵ0
μ0

(1,0)
g
[x,y]
In[]:=
Div[mv]
Out[]=
-
-K2
2
x
+
2
y

K2y
ϵ0
gt-1+
K1
z
z
ϵ0
μ0
g[x,y]
2
x
+
2
y
μ0
+
-K2
2
x
+
2
y

ϵ0
gt-1+
K1
z
z
ϵ0
μ0

(0,1)
g
[x,y]
μ0
In[]:=
FullSimplify[%]
Out[]=
-K2
2
x
+
2
y

ϵ0
gt-(K1+z)
ϵ0
μ0
-K2yg[x,y]+
2
x
+
2
y
(0,1)
g
[x,y]
2
x
+
2
y
μ0
Results for the electromagentic force densities in resp x-direction, y-direction, z-direction: