Mathematical Optics

9.6.3    OPTICAL ACTIVITIES

For convenience, let us change the parameters θ and μ as

$\text{θ}=2\text{α}z,\text{μ}=2\text{β}z,$

(9.118)

and the R(θ) and S(μ) matrices as

$R(\text{α}z)=\left(\begin{array}{ll}\mathrm{c}\mathrm{o}\mathrm{s}(\text{α}z)& -\mathrm{s}\mathrm{i}\mathrm{n}(\text{α}z)\\ \mathrm{s}\mathrm{i}\mathrm{n}(\text{α}z)& \mathrm{c}\mathrm{o}\mathrm{s}(\text{α}z)\end{array}\right),$

(9.119)

and the rotation angle is increased as the beam propagates along the z-direction. This version of optical activity is well known.

In addition, we can consider the squeeze operation

$S(-\text{β}z)=\left(\begin{array}{cc}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}(\text{β}z)& -\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\text{β}z)\\ -\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\text{β}z)& \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}(\text{β}z)\end{array}\right).$

(9.120)

Here, the squeeze parameter increases as the beam moves. The negative sign for β is for convenience.

If this squeeze is followed by the rotation of Equation 9.119, the net effect is

$\left(\begin{array}{ll}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}(\text{β}z)& -\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\text{β}z)\\ -\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\text{β}z)& \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}(\text{β}z)\end{array}\right)\left(\begin{array}{ll}\mathrm{c}\mathrm{o}\mathrm{s}(\text{α}z)& -\mathrm{s}\mathrm{i}\mathrm{n}(\text{α}z)\\ \mathrm{s}\mathrm{i}\mathrm{n}(\text{α}z)& \mathrm{c}\mathrm{o}\mathrm{s}(\text{α}z)\end{array}\right),$

(9.121)

where z is in a macroscopic scale, perhaps measured at the order ...