Susana Ríos

Determination of phase mode components in terms of local wave-front slopes: an analytical approach.

An analytical formulation that relates the modal expansion coefficients of a given wave front to its local transverse phase derivatives is proposed. The modal coefficients are calculated as a weighted integral over the wave-front slopes. The weighting functions for each mode are the components of a two-dimensional vector whose divergence equals the corresponding mode function. This approach is useful for analytical phase reconstruction from the input data provided by shearing interferometers or Hartmann-Shack wave-front sensors. Numerical results for a simulated experiment in terms of a set of Zernike polynomials are given.

Performance analysis of curvature sensors: optimum positioning of the measurement planes

Curvature sensors are used to measure wave-front aberrations in a number of different applications ranging from adaptive optics to optical testing. In practice, their performance is limited not only by the quality of the detector used for irradiance measurements but also by the separation between measurement planes used for the calculation of the axial derivative of intensity. This work resolves the problem of determining the separation between intensity measurement planes thus optimizing the variance in experimental measurements. To do this, the variance of the local curvature of the phase will be analyzed as a function of the noise level in the measurements and the separation between planes. Moreover, error bounds will be established for experimental measurements.

Minimum-variance phase reconstruction from Hartmann sensors with circular subpupils

Analytical expressions for the minimum-variance phase reconstructor for Hartmann sensors with circular subpupils are presented in this work. The approach assumes a wavefront expansion in terms of orthogonal Zernike polynomials, and takes advantage of the a priori knowlegde of the phase correlation function associated to phase distortions produced by atmospheric turbulence with Kolmogorov statistics.

Hartmann sensing with Albrecht grids

Abstract The modal coefficients of a wavefront can be estimated by suitably weighted integrals of the wavefront slopes. When the basis functions chosen to expand the wavefront are orthogonal (e.g. Zernike polynomials), the reconstruction problem becomes orthogonal itself, so that each coefficient can be estimated independently from the others. Modal cross-coupling can in this way be avoided. The obtained coefficients correspond to a direct least-squares fit between the real and estimated wavefronts. The evaluation of the modal integrals from the finite data set of measurements supplied by Shack-Hartmann sensors or shearing interferometers requires the use of efficient numerical integration methods. In this paper it is shown that Albrecht cubatures are a good candidate to perform this task. The wavefront slopes have to be measured at the nodal points of the chosen cubature, which in general are located in an unevenly spaced grid. Numerical results show that the method of orthogonal reconstruction with Albrecht cubatures allow to recover the modal coefficients with better accuracy than other usual approaches, including the standard non-orthogonal least-squares fit between the gradients of the orginal and reconstructed wavefronts.

Modal phase estimation from wavefront curvature sensing

Abstract A model to perform a maximum likelihood fit of the modal coefficients of a given wavefront to the data provide by curvature sensing devices is proposed in this work. The fit is directly done from the modal expansion of the wavefront, and not from its first- and second-order derivatives, so that the orthogonality properties of the basis functions are preserved in the final result. The least-squares estimate of each coefficient can be obtained as a sum of weighted integrals over the wavefront Laplacian inside a domain and over its outwards normal derivative along the domain edge. Analytical solutions for the weighting functions are given for a modal wavefront expansion in terms of Zernike polynomials.

Integral evaluation of the modal phase coefficients in curvature sensing: Albrecht’s cubatures

The modal coefficients describing a given wave front as a linear combination of a set of orthogonal functions can be determined independently of each other by a suitable integration of the signal provided by a curvature sensor. This fact allows us to avoid the use of matrix inversion routines and overcomes some problems related to the modeling error arising in conventional modal fits. Several procedures for evaluating these integrals from a discrete set of intensity measurements are compared. The results show that, for a given number of sampling points, the combination of an Albrecht’s cubature in the inner pupil region with a composite trapezoidal integration of the edge signal can give more accurate results with smaller noise propagators than other methods also analyzed.

Modal wavefront projectors of minimum error norm

Sets of auxiliary vector functions may be derived which enable the modal coefficients of a wavefront expressed in terms of a given basis to be directly projected as weighted integrals of the wavefront slopes. We derive the necessary and sufficient condition for these functions to have minimum error norm and show that for the specific case of a basis set comprising the Zernike circular polynomials, they are precisely the Gavrielides functions.

Role of boundary measurements in curvature sensing

The role of boundary measurements in curvature sensing is analyzed in basis of the method of weighting functions. It is shown that depending on the orthogonal basis chosen to expand the wavefront some modes can be exactly restored from the curvature data only, without the boundary information. This fact implies that curvature sensing technique can be used without contour measurements to evaluate some given modes. Therefore, if the contribution to the total phase distortion of the non-estimated modes is relatively small compared to the estimated ones, the method can be successfully applied avoiding edge measurements.

Coupling of single-mode fibers to single-mode GRIN waveguides by butt-joining

We present an analytical expression for the mode field radius of the fundamental mode of a planar surface GRIN waveguide for the optimum coupling of radiation from a given single-mode fiber with a Gaussian mode field distribution. It is shown that the fundamental mode of a single-mode ion-exchanged waveguide can be represented by a Hermite-Gaussian profile. The method presented here allows the determination of suitable waveguide parameters for optimum coupling. Theoretical predictions are in good agreement with the experimentally measured efficiency when other losses are accurately accounted for.

Variational solution for modal wave-front projection functions of minimum-error norm.

Common wave-front sensors such as the Hartmann or curvature sensor provide measurements of the local gradient or Laplacian of the wave front. The expression of wave fronts in terms of a set of orthogonal basis functions thus generally leads to a linear wave-front-estimation problem in which modal cross coupling occurs. Auxiliary vector functions may be derived that effectively restore the orthogonality of the problem and enable the modes of a wave front to be independently and directly projected from slope measurements. By using variational methods, we derive the necessary and sufficient condition for these auxiliary vector functions to have minimum-error norm. For the specific case of a slope-based sensor and a basis set comprising the Zernike circular polynomials, these functions are precisely the Gavrielides functions.