Phase space methods in optical design and simulation

The concept of phase space

Phase space methods are well known and extensively used in classical mechanics and quantum mechanics, where the position and the velocity of a particle define its location in phase space.

In optics, the phase space location indicates the position and angle of a single ray within an optical system. Thus classical ray-tracing corresponds to phase space trajectories:

Fig.1: Ray-tracing of a single ray through an optical system and the corresponding trajectories in phase space.

Phase space transformations

In phase space representation an optical system acts as transformation of the input lightdistribution onto the light distribution at the image plane. Therefore an analysis of these transformation properties provides a complete picture of the optical functionality. Thus phase space provides a different perspective onto optical systems as compared to standard ray-tracing picture.

Fig. 2: Simple phase space transformations: a) initial distribution, b) free paraxial propagation along a distance z along the optical axis, c) propagation through a thin lens of focal length f, d) free space propagation over a distance z for non-paraxial angles,  e) propagation from the front focal plane of a lens of focal length f to the back focal plane of the lens, f) free space propagation in combination with a reflection of a plane mirror parallel to the optical axis.

Phase space in illumination design

Especially for illumination design problems, where the general transport of radiance is important, phase space provides an interesting access towards illumination design. For example light mixing devices, such as mixing rods, can be quite confusing if viewed from a ray-tracing perspective. In contrast a phase space representation can often easily resolve the optical functionality:        

Fig. 3: Ray-tracing picture (top) and Phase space transformation of a mixing rod (bottom): a) Initial distribution; b) Propagated distribution in the absence of the rod side-walls; c) Final distribution resembling the back-folding effect of the rod.   

Recent Publications

Rausch D., Herkommer A.M. “Phase space approach to the use of integrator rods and optical arrays in illumination systems”, Advanced Optical Technologies, Vol. 1, page 69-78, (2012).

Herkommer, A. M., Rausch D. “Phase space optics – an engineering tool for illumination design”, Proc. SPIE 8429, 84290C (2012).