Circularly Polarized Wave by Two
Linearly Polarized Waves
2003/01/06 Takuichi Hirano (Tokyo Institute of
Technology)
Fig. 1 Coordinates
The condition that the sum of two linearly polarized
waves 
, traveling toward 
direction as
shown in Fig. 1, becomes circularly polarized wave is proved here,
where 
is the
wavenumber.
The coordinates are shown in Fig. 1. In 
coordinates, 
and 
axes are crossing
with angle 
and and 
axes coincide
with 
and 
axes,
respectively. 
,
,
and 
indicate unit
vectors of ,
, and axes,
respectively. and are expressed by 
and 
as follows.
(1)
Consider the sum
of two linearly polarized waves whose axes of polarization are and , respectively, as shown in Fig. 1.
(2)
where 
and 
are complex
numbers (Phasor representation) as follows
(3)
where 
,
,
and 
are real numbers.
By substituting
Eq.(3) into Eq.(2) and using Eq.(1)
The circular
polarization condition in 
plane is applied
here.
where + sign (upper)
in right hand side means the right hand circularly polarized wave (RHCP) while –
sign (lower) means the left hand one (LHCP) when the wave propagates toward +z
direction.
By Eq.(4)
(6)
Substituting this
equation into Eq. (5) yields
where 
is an arbitrary
integer number.
By substituting
this into Eq.(6)
The following two
conditions are obtained for circular polarization
where + sign
(upper) in right hand side means the right hand circularly polarized wave
(RHCP) while – sign (lower) means the left hand one (LHCP) when the wave
propagates toward +z direction.