Geometric Optics with Lie Groups

Donald Barnharth, from Optical Software introduced me with
another very interesting application of Lie groups in geometric optics.

This should not be a surprise because geometric optics as well as classical mechanics can be expressed in terms of a variational principle. In classical mechanics we have the minimal action and in geometric optics we have Fermat's principle, which states that the trajectory of light rays minimize the optical path length defined as

where n is the refraction index, ds is the differential arc length and C is the path. The whole machinery of classical mechanics can be translated to geometric optics with the corresponding Hamiltonian formulation as the phase space formulation of geometric optics.

The key idea of Hamiltonian mechanics is that the trajectory of the particles can be seen as active continuous symplectic transformations. For example, one has the following time-evolution operator



where H is the Hamiltonian (independent of time), with the Lie operator defined in terms of the Poisson brackets as



and


The same formalism can be applied to geometric optics with some particular adjustments. The time-evolution operator is replaced by the transformation that propagates the optical phase-state in the space. For practical applications this transformation is factorized in a perturbative-like product expansion that reminds the Fer expansion as



where, the first factor represents the para-axial approximation while the rest represent the corresponding higher order corrections.

References

1. V. Lakshminarayanan, Ajoy Ghatak, and K. Thyagarajan, Lagrangian Optics
   
2. Alex J. Dragt, A Lie connection between Hamiltonian and
   Lagrangian optics
   
3. Kurt B. Wolf, Geometric Optics on Phase Space, Springer, 2004
   
   

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