Multiple-prism dispersion theory

The generalized mathematical description of multiple-prism dispersion, as a function of the angle of incidence, prism geometry, prism refractive index, and number of prisms, was introduced as a design tool for multiple-prism grating laser oscillators by Duarte and Piper, Also, higher order phase derivatives have been derived using a Newtonian iterative approach. This extension of the theory enables the evaluation of the Nth higher derivative via an elegant mathematical framework. Applications include further refinements in the design of prism pulse compressors and nonlinear optics. Single-prism dispersion For a single generalized prism (), the generalized multiple-prism dispersion equation simplifies to \frac{\partial\phi_{2,1}}{\partial\lambda} = \frac{\sin\psi_{2,1}}{\cos\phi_{2,1}} \frac{\partial n_1}{\partial\lambda} + \frac{\cos\psi_{2,1}}{\cos\phi_{2,1}} \tan\psi_{1,1} \frac{\partial n_1}{\partial\lambda} If the single prism is a right-angled prism with the beam exiting normal to the output face, that is \phi_{2,m} equal to zero, this equation reduces to \frac{\partial\phi_{2,1}}{\partial\lambda} = \tan\psi_{1,1} \frac{\partial n_1}{\partial\lambda} to provide tuning in a dye laser. ==Intracavity dispersion and laser linewidth==

The first application of this theory was to evaluate the laser linewidth in multiple-prism grating laser oscillators. For the special case of zero dispersion from the multiple-prism beam expander, the single-pass laser linewidth is given by : \Delta\lambda \approx \Delta \theta \left(M {\partial\theta\over\partial\lambda}\right)^{-1} where M is the beam magnification provided by the beam expander that multiplies the angular dispersion provided by the diffraction grating. In practice, M can be as high as 100-200. When the dispersion of the multiple-prism expander is not equal to zero, then the single-pass linewidth is given by : \Delta\lambda \approx \Delta \theta \left(M {\partial\theta\over\partial\lambda} + {\partial\phi_{2,m}\over\partial\lambda} \right)^{-1} where the first differential refers to the angular dispersion from the grating and the second differential refers to the overall dispersion from the multiple-prism beam expander (given in the section above). ==Further applications==

In 1987 the multiple-prism angular dispersion theory was extended to provide explicit second order equations directly applicable to the design of prismatic pulse compressors. The generalized multiple-prism dispersion theory is applicable to: • Amici prisms • laser microscopy, • narrow-linewidth tunable laser design, • prismatic beam expanders ==See also==