Gaussian-to-flattop laser beam conversion via a two-lens refractive system

Rong Zhou

Supervisor: Prof. Hill

I.Introduction

Although applications of laser have led to technology revolution both in industry and in scientific experiments, in many circumstances one is confronted with problem that the irradiance profile is undesirable. For instance, laser emitted from a high-Q optical cavity in TEM00 mode or from a single-mode optical fiber possesses a Gaussian profile and has a non-uniform intensity distribution while researchers and engineers often need to generate an uniform illumination over an area. Problems of this kind necessitate the development of beam reshaping techniques, which are already widely applied in the context of fields such as semiconductor lithography, material processing and medical treatment. A brief summary of laser beam reshaping techniques is contained in "Laser beam reshaping" ,Optics &Photon News(April 2003). The design I have adopted belongs to "Field Mappers" mentioned in the article.

The goal of my work in the past six weeks was to design a refractive optical system for converting a Gaussian laser beam into one with flattop intensity profile and uniform phase profile (so that the beam can propagate without being severely spread). Telescopic refractive optical systems with good performances in beam-shaping are already commercially available, yet such design is still useful for convenience in specific tasks and for economic consideration.

II. Description of Approach

The design of the system followed exactly the method mentioned in Hoffnagle et al, with layout shown in Fig 1(adapted from Hoffnagle et al). The scheme consists of two uniaxial aspheric lenses that are rotational symmetric, each with one flat and one curved surface. Hence the task is reduced to calculating the sag curves of lenses.

                                                                          Fig 1. Ray paths through a refractive reshaping system

Since the refractive method is adopted, it is legitimate to model incoming beam as a bunch of geometrical rays differing from each other in intensity and phase, as shown in Fig 1. In our laboratory, the experimental configuration can be adjusted so that the left surface of the first lens is right at the waist of the laser beam. Hence here all incoming rays are assumed to be parallel and of uniform phase, simplifying the treatment

The function of the two uniaxial lenses are to map incoming geometrical rays into outgoing parallel rays in a controlled manner, described by functional relation R=h(r) . In the formula R is the distance between an incoming ray to the axis and r is that between outgoing ray and the axis.

The goal of reshaping the beam can be achieved, provided the following two conditions are satisfied.
(1) Function h is implemented by the lenses so that amount of energy enclosed by r in Gaussian profile is equal to the amount of energy enclosed by R=h(r) in a desired output profile.
(2) The optical path lengths of all rays are equal between input plane and output plane, in order to achieve an uniform phase profile in the output plane.

These two conditions, together with Snell's law, fixed the sag curves z(r) and W(R). Generally a set of two differential equations have to be solved to obtain these curves. Surprisingly, in our simplified situation they can be analytically expressed in terms of function h.

                                            

III. Results

As an illustration of this approach, calculation is done with Gaussian profile as input and Fermi-Dirac profile as output. Sag curves are drawn below with comparisons to those of spherical lenses.
　

   

The code for computation, written in Mathematica, can be found here(.nb). It must be mentioned that the results are not guaranteed to be correct since they haven't been verified in ray-tracing analysis.