Design of Computer-Generated Hologram With Ring Focus for Nonmechanical Corneal Trephination With Er:YAG Laser in Penetrating Keratoplasty

Abstract

Purpose: To calculate a beam-shaping optical element for homogeneous intensity distribution within a focal ring to be used in nonmechanical trephination with the Er:YAG laser in penetrating keratoplasty instead of a spot guiding device. Methods: The phase distribution behind a holographic optical element (HOE) kΨ( u) can be described by the addition of the hologram phase Φ H ( u) to the beam phase Φ E( u ): k Ψ( u ) = Φ H( u ) + Φ E( u ), k = 2 π/ λ , where u denotes the coordinates inside the hologram aperture, k an integer, and λ the laser wavelength. To avoid discontinuous wavefronts leading to speckle noise, a smooth phase function is necessary. After transforming the hologram aperture coordinates into the focal plane x in a focal distance f, Ψ can be retrieved from the slope equation: ▿ Ψ( u ) = x( u ) − u /f . Results: Creating a ring focus can be reduced to an essentially one-dimensional problem by separation of variables due to the symmetry condition. We calculated a computer-generated eight-level phase-only HOE with 4096 × 4096 pixels from a Gaussian-distributed 2.94 Er:YAG laser spot with a beam diameter of 10 mm and a focal distance of 100 mm. Thereby, a ring focus with an inner/outer radius of 7/8 mm can be created. To avoid Poisson's spot, the symmetry of the problem was broken by circular modulation of the phase leading to a spiral-like structure. The calculated efficiency of the HOE relating the energy within the ring to the total energy was 91%. Conclusion: With an HOE it is possible to redistribute the energy along the desired focal ring. The HOE design can be adapted to the intensity distribution of the impinging laser beam with its characteristic aperture shape. A circular homogeneous corneal trephination depth is possible, because the energy fluctuation from pulse to pulse does not locally affect the ablation process. A ring focus for the Er:YAG laser has the potential to render superfluous a manual beam control via micromanipulator and to allow a more rapid and more regular corneal trephination along aperture masks.