ABSTRACT (original article): The optical Venn paradox is observed on light crossing three successive linear polarizers whose axes are oriented along x and y and at 45° between x and y. This observation has not yet been understood within the usual formalism of classical electromagnetism. Here, we develop a classical explanation where the theory is based on the use of potentials. CITATION (original article): G. M. Stéphan (2025), Classical theory of the optical Venn paradox, AIP Advances 15 (1): 015119. https://doi.org/10.1063/5.0223729

1. Use of the covariant tensor
a
ki

In[]:=
Unprotect["`*"];ClearAll["`*"];
Gradient tensor
In[]:=
aki={{a11,a12,a13,a14},{a21,a22,a23,a24},{a31,a32,a33,a34},{a41,a42,a43,a44}};
Antisymmetric part : the field tensor
In[]:=
fki=1/2(aki-Transpose[aki]);
Symmetric part : the matter tensor
In[]:=
ski=1/2(aki+Transpose[aki]);
Covariant components of the gradient of the 4-potential: in the first line are the derivatives with respect to time, the second with respect to x...
In[]:=
a11=ϕt;a12=Axt;a13=Ayt;a14=Azt;​​a21=ϕx;a22=Axx;a23=Ayx;a24=Azx;​​a31=ϕy;a32=Axy;a33=Ayy;a34=Azy;​​a41=ϕz;a42=Axz;a43=Ayz;a44=Azz;
Fields polarized along X and Y are defined by a12 - a21 and a13 - a31
In[]:=
Print["ski = ",MatrixForm[ski]];Print["fki = ",MatrixForm[fki]];
ski =
ϕt
Axt+ϕx
2
Ayt+ϕy
2
Azt+ϕz
2
Axt+ϕx
2
Axx
Axy+Ayx
2
Axz+Azx
2
Ayt+ϕy
2
Axy+Ayx
2
Ayy
Ayz+Azy
2
Azt+ϕz
2
Axz+Azx
2
Ayz+Azy
2
Azz
fki =
0
Axt-ϕx
2
Ayt-ϕy
2
Azt-ϕz
2
1
2
(-Axt+ϕx)
0
1
2
(-Axy+Ayx)
1
2
(-Axz+Azx)
1
2
(-Ayt+ϕy)
Axy-Ayx
2
0
1
2
(-Ayz+Azy)
1
2
(-Azt+ϕz)
Axz-Azx
2
Ayz-Azy
2
0
Filtering effect : Polarizers suppress the fields X and Y)
In[]:=
a12=a21;a13=a31;(*orAxt=ϕx;Ayt=ϕy;*)
This is the filtered field before the third polarizer
In[]:=
Print["filtered fki = ",MatrixForm[fki]];
filtered fki =
0
0
0
Azt-ϕz
2
0
0
1
2
(-Axy+Ayx)
1
2
(-Axz+Azx)
0
Axy-Ayx
2
0
1
2
(-Ayz+Azy)
1
2
(-Azt+ϕz)
Axz-Azx
2
Ayz-Azy
2
0
Jacobian matrix of the third polarizer
In[]:=
R=({​​{1,0,0,0},​​{0,Cθ,Sθ,0},​​{0,-Sθ,Cθ,0},​​{0,0,0,1}​​});
Filtered tensor expressed in the reference frame of the third polarizer. (Click on the output matrices for full size)
In[]:=
Cθ=​​Cos[θ];Sθ=Sin[θ];​​akiθ=​​R.aki.​​Transpose[​​R];​​Print["akiθ = ",MatrixForm[akiθ]];
In[]:=
fkiθ=Simplify[1/2(akiθ-Transpose[akiθ])];​​Print["Field tensor fkiθ = ",MatrixForm[fkiθ]];

Conclusion

There is no transverse field, it remains a longitudinal component if Azt different ϕz (depends on the source).

2. Use of the mixed tensor
i
a
k

In[]:=
Unprotect["`*"];ClearAll["`*"];​​aki={{a11,a12,a13,a14},{a21,a22,a23,a24},{a31,a32,​​a33,a34},{a41,a42,a43,a44}};
Covariant components
In[]:=
a11 = ϕt ; a12 = Axt; a13 = Ayt; a14 = Azt;​​a21 = ϕx ; a22 = Axx; a23 = Ayx; a24 = Azx;​​a31 = ϕy ; a32 = Axy; a33 = Ayy; a34 = Azy;​​a41 = ϕz ; a42 = Axz; a43 = Ayz; a44 = Azz;
Metric Tensor
In[]:=
g={{1,0,0,0},{0,-1,0,0},{0,0,-1,0},{0,0,​​0,-1}};
Mixed Tensor
In[]:=
akupi=aki.g;
In[]:=
skupi=1/2(akupi+Transpose[akupi]);fkupi=​​1/2(akupi-Transpose[akupi]);​​Print["skupi = ",​​MatrixForm[skupi]];Print["Mixed Field Tensor fkupi = ",​​MatrixForm[fkupi]];
skupi =
ϕt
1
2
(-Axt+ϕx)
1
2
(-Ayt+ϕy)
1
2
(-Azt+ϕz)
1
2
(-Axt+ϕx)
-Axx
1
2
(-Axy-Ayx)
1
2
(-Axz-Azx)
1
2
(-Ayt+ϕy)
1
2
(-Axy-Ayx)
-Ayy
1
2
(-Ayz-Azy)
1
2
(-Azt+ϕz)
1
2
(-Axz-Azx)
1
2
(-Ayz-Azy)
-Azz
Mixed Field Tensor fkupi =
0
1
2
(-Axt-ϕx)
1
2
(-Ayt-ϕy)
1
2
(-Azt-ϕz)
Axt+ϕx
2
0
Axy-Ayx
2
Axz-Azx
2
Ayt+ϕy
2
1
2
(-Axy+Ayx)
0
Ayz-Azy
2
Azt+ϕz
2
1
2
(-Axz+Azx)
1
2
(-Ayz+Azy)
0
Field filtering : same operation as before in program 1. This is justified by the fact that the time and the spatial components have not the same variance in a Lorentz transformation .
In[]:=
Axt=ϕx;Ayt=ϕy;
Print["Filtered fkupi = ",MatrixForm[fkupi]];
In[]:=
R=({​​{1,0,0,0},​​{0,Cθ,Sθ,0},​​{0,-Sθ,Cθ,0},​​{0,0,0,1}​​});​​Cθ=Cos[θ];Sθ=Sin[θ];
Transformation in the reference frame of the third polarizer. (Click on the output matrices for full size)
In[]:=
akupiθ=​​Simplify[R.akupi.Inverse[R]];​​Print["akupiθ = ",MatrixForm[akupiθ]];
In[]:=
fkupiθ=​​Simplify[1/2(akupiθ-Transpose[akupiθ])];​​Print["fkupiθ = ",MatrixForm[fkupiθ]];

Conclusion

Transverse fields are proportional to the spatial derivatives of the scalar potential. it remains a longitudinal component if Azt different -ϕz

General Conclusion

Further study is needed on the different types of emitters and detectors.

CITE THIS NOTEBOOK

Classical theory of the optical Venn paradox​
by GuyMichel Stephan​
Wolfram Community, STAFF PICKS, March 19, 2025
​https://community.wolfram.com/groups/-/m/t/3420540