content_image/344291.jpg content_image/344292.jpg content_image/344290.jpg content_image/344289.jpg content_image/344288.jpg content_image/344287.jpg
fields, they are mutually orthogonal; besides, the magnetic field is transverse, ( h ( 1 $^{)}$k ) = 0, while the electric field, generally, is not: ( e ( 1 $^{)}$k ) ̸ = 0; it is transverse for the special case of propagation along the external fields k$_{⊥}$ = 0. It is seen again that a massless excitation is possible in mode 1 as long as it propagates along the external field, otherwise the fields (32) vanish if k 2 = 0, unless k$_{⊥}$ = 0, when the square root turns to unity. 

The electric and magnetic fields carried by the vectors ( ˜ Fk )$_{µ}$ and ( Fk )$_{µ}$, respectively, are 

<img src='content_image/344291.jpg'>

<img src='content_image/344292.jpg'>

The electric fields carried by the vectors c + µ and c − $_{µ}$, respectively, are 

<img src='content_image/344290.jpg'>

<img src='content_image/344289.jpg'>

while their magnetic fields are 

<img src='content_image/344288.jpg'>

The orthogonality of the electric fields ( e $^{+}$e $^{−}$) = ( e $^{(1)}$e $^{−}$) = 0 , seen in eqs. (32), (35), (36), originates from the fact that, in the special frame, the time-component of one of their mutually orthogonal ( c $^{+}$c − = 0), and 4-transversal ( c $^{±}$k = 0) vector-potentials c ± µ disappears: c − 0 = 0. The 3-vector e − is directed along the axis (call it axis 1 ), orthogonal to the plane, where the external fields and the propagation vector k lie (the plane (3,2) ). The vector e + lies in that plane. It makes the universal angle α = arctan( k$_{3}$k$_{⊥}$/ ( k 2 0 − k 2 $_{3}$)) with the direction of the external fields 3 . Also the magnetic field h − lies in the plane spanned by the external fields and the propagation direction, while h + is orthogonal to this plane. 

The electric fields in the eigenmodes 2 and 3, e (2 , $^{3)}$, lie both in the common plane spanned by the vectors e $^{±}$: 

<img src='content_image/344287.jpg'>

