Efficient and robust recurrence relations for the Zernike circle polynomials and their derivatives in Cartesian coordinates

Advanced Optical Analysis of Divergence Between the Foci of the Neodymium-Doped Yttrium Aluminum Garnet (Nd:YAG) and Aiming Beam Lasers

Purpose

To evaluate the divergence between the neodymium-doped yttrium aluminum garnet (Nd:YAG) surgical laser and the aiming diode laser beams foci.

Design

Optical analysis and measurements were performed using a Volk Goldmann 3-mirror lens with a Nidek YC-1800 Nd:YAG laser apparatus.

Subjects

None.

Methods

We used the Zemax OpticStudio program for the model of Nd:YAG treatment in a human eye. Additionally, theoretical calculations were performed.

Main Outcome Measures

The divergence between the Nd:YAG laser focus and the intersection of the 2 aiming beams inside the eye.

Results

Focal points of the 2 laser beams converge 8 mm behind the cornea. Posterior to this point, the intersection of the diode laser aiming beams lies in front of the focal point of the Nd:YAG treatment laser, with distance between the 2 foci progressively increasing up to 305 microns at 24 mm behind the cornea.

Conclusions

We report the degree of divergence between the 2 lasers' focal points due to the difference in refraction between the corresponding wavelengths. These results have high practical relevance, as they provide a starting point for increasing the accuracy of Nd:YAG laser treatment, particularly when applied to the posterior segment, thereby minimizing the risk of complications. Current Nd:YAG laser devices have the built-in ability to modify the focal point of the aiming beam along the z-axis, thus providing possibility for an immediate application of our findings in clinical practice.

Financial Disclosures

The authors have no proprietary or commercial interest in any materials discussed in this article.

Measuring spatio-temporal couplings using modal spatio-spectral wavefront retrieval

Knowledge of spatio-temporal couplings such as pulse-front tilt or curvature is important to determine the focused intensity of high-power lasers. Common techniques to diagnose these couplings are either qualitative or require hundreds of measurements. Here we present both a new algorithm for retrieving spatio-temporal couplings, as well as novel experimental implementations. Our method is based on the expression of the spatio-spectral phase in terms of a Zernike-Taylor basis, allowing us to directly quantify the coefficients for common spatio-temporal couplings. We take advantage of this method to perform quantitative measurements using a simple experimental setup, consisting of different bandpass filters in front of a Shack-Hartmann wavefront sensor. This fast acquisition of laser couplings using narrowband filters, abbreviated FALCON, is easy and cheap to implement in existing facilities. To this end, we present a measurement of spatio-temporal couplings at the ATLAS-3000 petawatt laser using our technique.

A Tip–Tilt and Piston Detection Approach for Segmented Telescopes

Accurate segmented mirror wavefront sensing and control is essential for next-generation large aperture telescope system design. In this paper, a direct tip–tilt and piston error detection technique based on model-based phase retrieval with multiple defocused images is proposed for segmented mirror wavefront sensing. In our technique, the tip–tilt and piston error are represented by a basis consisting of three basic plane functions with respect to the x, y, and z axis so that they can be parameterized by the coefficients of these bases; the coefficients then are solved by a non-linear optimization method with the defocus multi-images. Simulation results show that the proposed technique is capable of measuring high dynamic range wavefront error reaching 7λ, while resulting in high detection accuracy. The algorithm is demonstrated as robust to noise by introducing phase parameterization. In comparison, the proposed tip–tilt and piston error detection approach is much easier to implement than many existing methods, which usually introduce extra sensors and devices, as it is a technique based on multiple images. These characteristics make it promising for the application of wavefront sensing and control in next-generation large aperture telescopes.

Integral-based parallel algorithm for the fast generation of the Zernike polynomials

The integral representation of the Zernike radial functions is well approximated by applying the Riemann sums with a surprisingly rapid convergence. The errors of the Riemann sums are found to averagely be not exceed 3 ×10^-14, 3.3×10^-14, and 1.8×10^-13 for the radial order up to 30, 50, and 100, respectively. Moreover, a parallel algorithm based on the Riemann sums is proposed to directly generate a set of radial functions. With the aid of the graphics processing units (GPUs), the algorithm shows an acceleration ratio up to 200-fold over the traditional CPU computation. The fast generation for a set of Zernike radial polynomials is expected to be valuable in further applications, such as the aberration analysis and the pattern recognition.