Passive phase adjustment via Kramers-Kronig (K-K) phase in coherent beam combining is reviewed and assessed. While recent experiments and modeling show that K-K does not improve the average output power when it is the only passive phasing mechanism in a resonator, K-K does improve the combining efficiency in the presence of other effects such as wavelength tuning, Kerr nonlinearity, and regenerative phase. For a two-laser array, an improvement in output power can be achieved by K-K alone if the constant of proportionality α between the gain and the K-K phase is very large (e.g., α = 100 rad/Neper), but this is unlikely to be practical in available materials, and does not appear to scale with array size.
Advances in 3D fabrication have afforded new freedoms in the design of custom optical elements. We explore a design space where the refractive index may vary freely with position in a material by considering applications in coherent beam shaping. First, we will describe experimental results where we have used femtosecond laser writing to fabricate custom GRIN waveguides in a planar geometry. We show a 2D device that converts a Gaussian beam to a flat top in one transverse direction, and is a single-mode waveguide in the other. Second, we describe a numerical method for designing 3D refractive index profiles. As an example, we will use the 3D method to design a device that transforms a circular beam with a Gaussian intensity profile into a square beam with a flat top.
Design methods are illustrated to produce beam shaping elements in which the refractive index is a continuous function of position. An index profile yielding a desired gradual transformation of the field can be computed in two ways. A ray theory approach yields a solution consistent with the eikonal equation, while diffraction effects can be incorporated into the index profile by using a split step representation of the medium and performing a series of phase retrieval calculations. The methods are demonstrated in an example of mode conversion and coherent laser beam combining, where a near-unity conversion efficiency can theoretically be achieved.
The refractive index profile of one-dimensional gradient-index (GRIN) samples can be measured using the incident and exit beam angles of multiple beams passing through the sample at different positions along the index gradient. Beginning from a region of known refractive index, the collective angular deflection measurement of multiple beams is bootstrapped to compute the index profile of the entire sample. An alternative method using an approximate beam displacement model and a corrective algorithm is also presented. The two techniques are used to measure the index profile of a thick GRIN sample, and experimental results show good agreement with a maximum discrepancy of 1.5×10 −3 in the calculated index. An index accuracy of 5×10 −4 is predicted for the bootstrap method employing typical micron-level spatial measurements.