the 90 degree phase shift between the x and y components that we have decomposed each circular polarization into. As usual when dealing with phasor notation, it is understood that such quantities are to be multiplied by and then the actual electric field at any instant is given by the real part of that product. Substituting these expressions for and into the equation for we obtain: The last equation shows that the resulting vector has the x and y components in phase and oriented exactly in the direction, as we had intended, justifying the representation of any linearly polarized state at angle as the superposition of right and left circularly polarized components with a relative phase difference of . Now let us assume transmission through an optically active material which induces an additional phase difference between the right and left circularly polarized waves of . Let us call the result of passing the original wave linearly polarized at angle through this medium. This will apply additional phase factors of and to the right and left circularly polarized components of : Using similar math as above we find: thus describing a wave linearly polarized at angle , thus rotated by relative to the incoming wave: We defined above the difference in the refractive indices for right and left circularly polarized waves of . Considering propagation through a length L in such a material, there will be an additional phase difference induced between them of (as we used above) given by: , where is the wavelength of the light (in vacuum). This will cause a rotation of the linear axis of polarization by as we have shown. In general, the refractive index depends on wavelength (see dispersion) and the differential refractive index will also be wavelength dependent. The resulting variation in rotation with the wavelength of the light is called optical rotatory dispersion (ORD). ORD spectra and circular dichroism spectra are related through the Kramers–Kronig relations. Complete knowledge of one spectrum allows the calculation of the other. So we find that the degree of rotation depends on the color of the light (the yellow sodium D line near 589 nm wavelength is commonly used for measurements), and is directly proportional to the path length through the substance and the amount of circular birefringence of the material which, for a solution, may be computed from the substance's specific rotation and its concentration in