Laser-assisted in situ keratomileusis (LASIK) for vision correction involves using a 193 nm excimer laser to sculpt the cornea. Most excimer lasers have elliptical-shaped output that is roughly top hat-shaped in one direction and Gaussian in the other, with non-uniformity of 25% or more. With the appropriate application of diffractive optics, however, such a beam takes on a smooth Gaussian profile ideal for raster scanning across the eye. Varying the dwell time enables precise control of the ablation process, and accurate and fast correction.
A typical pattern-generation system of the type used for LASIK includes a computer-generated diffractive or refractive free-form optic that creates a Gaussian angular distribution (see figure). A condenser lens then converts this angular distribution into a spatial distribution at the focal plane of the lens. The final spatial output is actually the convolution of the raw lasers divergence with the angular distribution of the diffractive optic. When this distribution is at least five times larger than the source divergence, the final spatial distribution is dominated by the DOE design, but when the source and diffractive optical element (DOE) divergence are similar, a smoothing and broadening of the source divergence becomes evident.
Interestingly, excimer lasers tend to diverge differently in the two orthogonal directions. For instance, a typical laser used for refractive surgery may have a divergence of 3 × 1 mrad. This can lead to a slight asymmetry in the final beam shape that can easily be corrected by the scanning process. All of these properties need to be considered when choosing your laser source, diffractive design, and lens focal length.
For example, if you want to ensure a high-quality Gaussian beam shape that is 1 mm in diameter, you would choose a lens with a focal length less than 66 mm—given a laser beam divergence of 3 mrad. For this case, the divergence of the diffractive optic would be 15 mrad. Once you have determined the optical profile needed, the next step is to create the DOE able to produce this pattern. This procedure is essentially the formation of a computer-generated hologram (CGH) that diffracts light to form the desired illumination pattern.
The desired numeric aperture (NA) typically describes the angular extent of a DOE output pattern. The NA of a pattern is the sine of the half-angle over which the light will be spread (sin θ = NA). From this initial specification, we can determine some important properties of the DOE. The ideal DOE is aperiodic; the pattern of the DOE does not repeat itself for a relatively large transverse distance. However, the element does contain the correct range of spatial frequencies to produce the desired diffraction pattern within a finite length, or unit cell size. This unit cell can be determined using the grating equation:
dsinθ = mλ
where d is the period of the grating, λ is the wavelength of light used, m is the desired diffraction order from the grating, and θ is the angle at which the desired order is deflected.
To create a Gaussian DOE with an NA of 0.015 for use at λ = 193 nm, we create a circular target grid with a radius of, say, m = 50 orders. From our grating equation we determine that the unit cell in this DOE is 643 µm (m = 50, sin θ = 0.015.) This dimension determines the angular separation between diffraction orders, and that the source map can be defined by specifying the relative intensity (typically 0–255 intensity levels) of the light diffracted into each diffraction order. When the DOE is designed, it will diffract most of the incident light into the target diffraction orders. By selecting a reasonably high value for m, we ensure that the angular separation between orders is small relative to the source divergence of the excimer laser. This diffractive optic, which would have a minimum period of about 13 μm (m = 1, sin θ = 0.015), is easily manufactured with today’s grayscale or multiphase-level lithography.
At deep ultraviolet (DUV) wavelengths, we have until recently been limited to fused silica materials that tend to degrade with continued exposure to the high-peak-power nanosecond pulses. While this has required periodic replacement of the microoptic, and thus added to operating costs, a non-traditional etching technique is becoming readily available to fabricate diffractive and refractive microoptics in CaF2, a material that lasts much longer in the DUV.1 Using stepper and e-beam lithography, it is now possible to create diffractive profiles with >90% efficiency and flexible illumination patterns. The significant increase in material lifetime allows for both reduced operating costs and increased pulse energies of advanced excimer lasers.—MH and TL
REFERENCE
1. U. Wielsch, K. Kanzler, and T. Lindsey, Laser Focus World, 47, 5, 89–93 (May 2011).