Wavelength difference is a big deal, I know. It can solely change the whole interaction between the chiral molecule & the light. But I am not sure what's the mechanism by which light of different wavelengths produces different angles of rotation.
So, why and how wavelength matters?
The rotation of plane polarised light by a solution of, say, sucrose depend on the ability of the oscillating magnetic filed of the light to induce an electric dipole moment in the molecule and the ability of the oscillating electric field of the light to induce a magnetic dipole moment.
For these interaction to have any magnitude it is supposed that the electrons in a molecule move in a helical path, or, alternatively that there are two linear dipoles generated by electron motion which are in planes that are at some angle to one another. Although this model is clearly artificial it emphasises the fact that movement of charge must follow somewhat crooked pathways under the influence of the radiation. The theory of this is very complicated but the result from a quantum calculation is relatively straightforwards and is that the molecular rotation M at wavelength $\lambda$ is given by
$$M_\lambda = a\sum_i \frac{\lambda_{0i}^2X}{\lambda^2-\lambda_{0i}^2}$$
where i represents all the electronic states of the molecule and the wavelength $\lambda_{0i} = c/\omega_{0i}$ where $\omega_{0i} = (E_i-E_0)/\hbar$ is the frequency of the $i^{th}$ excited state at energy $E_i$. The parameter $X=\mathrm{Im}(\mu_{m_{0i}}\cdot\mu_{e_{i0}})$ is the value imaginary part of the complex dot product of the induced magnetic and electric dipole moments and a are a set of constants independent of wavelength.
From this formula it is seen that the polarisation rotation depends on the wavelength. (The dependence of rotation angle on wavelength has been called optical rotatory dispersion.) We can also conclude
that experiments should be performed where the molecule has little absorbance but that the rotation will be larger as an absorption band is approached. (in an absorption band elliptically polarised light can be formed)
The strength of any optical transition is not important as the dot product X depends on induced moments and the angle between them. This also means that weak optical transitions can be as important as strong one in rotation the polarisation. If the induced dipoles are perpendicular then the rotation vanishes as the dot product is zero.
That many excited states i can be involved and their effects can cancel to some extent so the value of the rotation is hard to predict.
The polarisation rotation of two mirror image molecules are equal and opposite in size and if a molecule is identical with its mirror image polarisation rotation must be zero and the term X is zero.
When the wavelength $\lambda < \lambda_{0i} $ i.e. the wavelength crosses a transition then the signal changes sign and this is called the Cotton effect.
Light travels at different speeds in different materials. The ratio of the speed at which light travels in a vacuum compared to the speed light travels in a given material is given by the index of refraction ($n$) of the material.
$${{n}=\mathrm{\frac{speed~ of~ light ~in~ vacuum}{speed ~of~ light~ in~ material}}}$$
The index of refraction is a basic property of a material. Further, the index of refraction of a material is wavelength dependent.
The beam of plane polarized light that we use in our polarimeter experiments is created by combining a beam of left circularly polarized light with a beam of right circularly polarized light. Those two beams interfere with each other to produce the plane polarized light we use in the polarimeter experiment. It is important to note that both the right and left circularly polarized beams are chiral as they trace out a right and left handed helix.
If the material in the polarimeter is not chiral, then the angle of the plane polarized light will not be rotated as it passes through the sample. However, if a chiral material is placed in the path of a plane polarized beam, then the left circularly polarized component of the beam will be rotated a different amount than the right circularly polarized component (the 2 beams are chiral and interact differently with the chiral sample) and we observe a rotation of our light beam as it passes through the polarimeter. In other words, the index of refraction of the left circularly polarized light ($n_L$) is different from the index of refraction of the right circularly polarized light ($n_R$) in chiral media.
Since $n$ itself is wavelength dependent, both $n_L$ and $n_R$ are also wavelength dependent and the observed rotation from our sample will change as we change the wavelength of our light beam.
This property of $n_L$ and $n_R$ is the basis of the Cotton effect. If you would like to read more about the Cotton effect and its use in chemistry see this earlier answer.
