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.