My Research and Inventions

### History and Future

Transverse Magneto-Optical effect

## How to describe mathematically  elliptical polarization of light

 Fig.1 Conventional longitudinal ellipticity of light

The following describes conventional polarization of light, which light may have when it propagates through a bulk material.

Figure 1 shows conventional longitudinal ellipticity of light. The polarization is rotating in xy-plane, which is perpendicular to light propagation direction. The y-component of the polarization legs behind x-polarization. Therefore, the polarization can be describe as

In cases of a=1 and a=-1, the polarization is left and right circular, respectively. In cases of a=0 and a=infinity, the polarization is linear. In all other cases, the polarization is elliptical.

## When light may have transverse-elliptical polarization

 Fig.2 Transverse ellipticity of light

Only light, which polarization is transverse elliptical, may experience the transverse MO effect. Since an electromagnetic wave is transverse and transverse elliptical polarization clearly has a component along propagation direction, it seems that such a polarization is forbidden. However, there is one special case when such a polarization is possible. It is the case when light has an evanescent component along the direction perpendicular to the wave propagation direction. Light may have the evanescent component when it propagates in vicinity of an interface between two different materials.

For example, if the wave propagates along z-direction, but it has evanescent component along x-direction. In this case the x-component of the wave vector has only an imaginary part and the z-component has only a real part. So the wave is described as

In the case of polarization in the xz-plane, the condition for the wave to be transverse (wave vector is perpendicular to wave polarization) is

or

The ratio (4) is similar to the ratio (1). If the ratio (1) describes conventional longitudinal ellipticity of light, the ratio (4) describes the transverse ellipticity.

In the case

the polarization is circular and the wave will experience strongest transverse MO effect. It is because in the case when the polarization is circular and the axis of polarization rotation is along the magnetic field, light will interact only with electrons of one spin direction and therefore experiences the strongest transverse MO effect.

## Transverse Ellipticity of a Transverse Electromagnetic Wave (Contradiction??)

From Fig.2 it is very clear, that wave, which polarization is transverse elliptical, has polarization component along propagation direction.

#### How it is possible? Electromagnetic wave is transverse and should not have longitudinal component of polarization!!

It is not completely true. The wave (2) is transverse, but it has a polarization component along the propagation direction. The seeming contradiction in the above-mentioned properties is explained as follows. As required for a transverse wave, the wave polarization is perpendicular to the direction of the wave vector , but the  direction of wave vector is not the propagation direction for this wave. For an electromagnetic wave the propagation direction should be determined by a direction of electromagnetic energy flow, which is defined by time-averaged Poynting vector

and for the wave (2) it is along z-direction, along which the wave has a polarization component.

Fig.3 Rotation of polarization of wave is similar to rotation of wheels of train in case of transverse ellipticity and to rotation of airplane propeller in case of conventional longitudinal ellipticity

## Transverse ellipticity in the case of light propagation in metals

It is interesting to known how large the transverse ellipticity in known cases when the transverse MO is experimentally observed.

The transverse ellipticity is calculated as

Figure 4 shows calculated transverse ellipticity in cases of a waveguide mode propagating in Fe/AlGaAs, Co/AlGaAs, Ni/AlGaAs waveguides (buffer Al0.5Ga0.5As(300 nm)/core Al0.3Ga0.7As(1200 nm)/ clad Al0.5Ga0.5As)  and in cases of a surface plasmon propagating along Fe/air, Co/air, Ni/air interfaces. In all cases the traverse ellipticity is significant. The ellipticity is close to circular for the waveguiding mode. That is the reason why the experimentally observed transverse MO effect in waveguides is substantial.

In both cases of the hybrid waveguide modes and the surface plasmons, the transverse ellipticity is substantial.

 Fig.4 Transverse ellipticity of waveguide mode propagating in metal/AlGaAs waveguide and surface plasmon propagating at metal/air interface. 100% corresponds to a circular polarization. The upper and lower insets show the optical field distribution of a waveguide mode and a surface plasmon, respectively.