Fast Zernike fitting of freeform surfaces using the Gauss-Legendre quadrature.

Zernike polynomial orthogonality, an established mathematical principle, is leveraged with the Gauss-Legendre quadrature rule in a rapid novel approach to fitting data over a circular domain. This approach provides significantly faster fitting speeds, in the order of thousands of times, while maintaining comparable error rates achieved with conventional least-square fitting techniques. We demonstrate the technique for fitting mid-spatial-frequencies (MSF) prevalent in small-tool-manufacturing typical of aspheric and freeform optics that are poised to soon permeate a wide range of optical technologies.

Validity of the perturbation model for the propagation of MSF structures in 3D.

Mid-spatial frequency (MSF) structures on optical surfaces degrade system performance and a perturbation model is typically used to simplify the assessment of their effects. In this simple model, MSF phase structures are dragged along the nominal rays of a system to yield estimates of wavefronts in the exit pupil that may be used for further analysis. However, the validity of the perturbation model remains an open area of study. We extend our previous assessment of the validity of this model [K. Liang, Opt. Express 27, 3390-3408 (2019)] that was focused on the analysis of single-frequency MSF structures in two dimensions to now include error estimates for broad-spectra MSF structures in three dimensions.

Rapidly decaying Fourier-like bases.

In many applications it is natural to seek to extract a characteristic scale for a function's variations by reference to a frequency spectrum. Although the moments of a spectrum appear to promise simple options to make such a connection, standard Fourier methods fail to yield finite moments when the function's domain is itself finite. We investigate a family of Fourier-like bases with rapidly decaying spectra that yield well-defined moments for such cases. These bases are derived by considering classes of functions for which a normalised mean square derivative is stationary. They are shown to provide precisely the type of spectrum needed to complete a recent investigation of mid-spatial frequency structure on optical surfaces [K. Liang, Opt. Express 27, 3390-3408 (2019)].

Validity of the perturbation model for the propagation of MSF structure in 2D.

Assessment of the performance degradation caused by the mid-spatial frequency (MSF) structure on optical surfaces often relies on a perturbation method that dovetails with the familiar sequence of models based on geometrical and physical optics. In the case of imaging systems, the perturbative step yields estimates of wavefronts in the exit pupil which are, in turn, used to extract performance measures such as MTF, PSF, and Strehl ratio. To date, the validity of that perturbation appears to be poorly understood. We present methods to estimate the errors of this approach and thereby arrive at a rule of thumb for its accuracy: the error is approximately equal to the RMS of the MSF structure at its source multiplied by the square of the ratio between a particular Fresnel zone size and a characteristic length of the MSF structure.

Strehl ratio as the Fourier transform of a probability density of error differences.

To give useful insight into the impact of mid-spatial frequency structure on optical performance, the Strehl ratio is shown to correspond to the Fourier transform of a simple statistical characterization of the aberration in the exit pupil. This statistical description is found simply by autocorrelating a histogram of the aberration values. In practice, the histogram itself can often be approximated by a convolution of underlying histograms associated with fabrication steps and, together with the final autocorrelation, it follows from the central limit theorem that the Strehl ratio as a function of the scale of the phase error is generally approximated well by a Gaussian.

Orthogonal basis with a conicoid first mode for shape specification of optical surfaces: comment.

Potentially misleading results follow from an error in a recent paper, namely [Opt. Express24, 5448-5462 (2016)], that contains a comparative analysis of schemes for specifying shape. Some corrections are presented for clarification. Additional comments are offered in relation to practical goals in this area of research.

Fitting freeform shapes with orthogonal bases.

Orthogonality is exploited for fitting analytically-specified freeform shapes in terms of orthogonal polynomials. The end result is expressed in terms of FFTs coupled to a simple explicit form of Gaussian quadrature. Its efficiency opens the possibilities for proceeding to arbitrary numbers of polynomial terms. This is shown to create promising options for quantifying and filtering the mid-spatial frequency structure within circular domains from measurements of as-built parts.

Characterizing the shape of freeform optics.

A recently introduced method for characterizing the shape of rotationally symmetric aspheres is generalized here for application to a wide class of freeform optics. New sets of orthogonal polynomials are introduced along with robust and efficient algorithms for computing the surface shape as well as its derivatives of any order. By construction, the associated characterization offers a rough interpretation of shape at a glance and it facilitates a range of estimates of manufacturability.

Manufacturability estimates for optical aspheres.

Some simple measures of the difficulty of a variety of steps in asphere fabrication are defined by reference to fundamental geometric considerations. It is shown that effective approximations can then be exploited when an asphere's shape is characterized by using a particular orthogonal basis. The efficiency of the results allows them to be used not only as quick manufacturability estimates at the production end, but more importantly as part of an efficient design process that can boost the resulting optical systems' cost-effectiveness.

Robust, efficient computational methods for axially symmetric optical aspheres.

Whether in design or the various stages of fabrication and testing, an effective representation of an asphere's shape is critical. Some algorithms are given for implementing tailored polynomials that are ideally suited to these needs. With minimal coding, these results allow a recently introduced orthogonal polynomial basis to be employed to arbitrary orders. Interestingly, these robust and efficient methods are enabled by the introduction of an auxiliary polynomial basis.

Robust and fast computation for the polynomials of optics.

Mathematical methods that are poorly known in the field of optics are adapted and shown to have striking significance. Orthogonal polynomials are common tools in physics and optics, but problems are encountered when they are used to higher orders. Applications to arbitrarily high orders are shown to be enabled by remarkably simple and robust algorithms that are derived from well known recurrence relations. Such methods are demonstrated for a couple of familiar optical applications where, just as in other areas, there is a clear trend to higher orders.

Shape specification for axially symmetric optical surfaces.

Advances in fabrication and testing are allowing aspheric optics to have greater impact through their increased prevalence and complexity. The most widely used characterization of surface shape is numerically deficient, however. Furthermore, with regard to tolerancing and to constraints for manufacturability, this representation is poorly suited for design purposes. Effective alternatives are therefore presented for working with rotationally symmetric surfaces that are either (i) strongly aspheric or (ii) constrained in terms of the slope in the departure from a best-fit sphere.

Using rays better. IV. Theory for refraction and reflection.

A new ray-based method is extended to include the modeling of optical interfaces. The essential idea is that the wave field and its derivatives are always expressed as a superposition of ray contributions of flexible width. Interfaces can be analyzed in this way by introducing a family of surfaces that smoothly connects them. Even though the ray-to-wave link may appear to be obscured at caustics, the standard Fresnel coefficients (for plane waves at flat interfaces between homogeneous media) are shown to be universally applicable on a ray-by-ray basis. Thus, in the interaction at the interface, the surface's curvature and any gradients in the refractive indices influence only the higher asymptotic corrections. Further, this method finally gives access to such corrections.

Using rays better. III. Error estimates and illustrative applications in smooth media.

A new method for computing ray-based approximations to optical wave fields is demonstrated through simple examples involving wave propagation in free space and in a gradient-index waveguide. The analytic solutions that exist for these cases make it easy to compare the new estimates with exact results. A particularly simple RMS error estimate is developed here, and corrections to the basic field estimate are also discussed and tested. A key step for any ray-based method is the choice of a family of rays to be associated with the initial wave field. We show that, for maximal accuracy, not only must the initial field be considered in choosing the rays, but so too must the medium that is to carry the wave.

Using rays better. I. Theory for smoothly varying media.

We present a method for computing ray-based approximations to optical fields that not only offers unprecedented accuracy but is also accompanied by accessible error estimates. The basic elements of propagation through smooth media, refraction and reflection at interfaces, and diffraction by obstacles give the foundations for the new framework, and the first of these is treated here. The key in each case is that the wave field and any relevant derivatives are expressed consistently as a superposition of delocalized ray contributions. In this way, the mysteries surrounding the sometimes perplexing tenaciousness of ray-based estimates are clearly resolved. Further, an essential degree of freedom in this approach offers an attractive resolution of part of the apparent conflict of particle/wave duality.

Using rays better. II. Ray families to match prescribed wave fields.

A key step in any ray-based method for propagating waves is the choice of a family of rays to be associated with the initial wave field. We develop some basic prescriptions for constructing initial ray families to match two particular types of waves. Various Gaussian and Bessel beams are separately given special treatment because of their general interest. These ideas are directly useful for a newly developed method for ray-based wave modeling. The new method expresses the wave as a superposition of ray contributions that is independent of the width of the field element associated with each ray. This insensitivity is investigated here even when the elemental width varies from ray to ray. The results increase the applicability of the new wave-modeling scheme.

Phase-space distributions for high-frequency fields.

The Wigner distribution function and various windowed Fourier transforms are examples of phase-space distributions that are used, among other things, to formalize the link between ray and wave optics. It is well known that, in the limit of high frequencies, these distributions become localized for simple wave fields and therefore that the localization can be used to define the associated ray families. This localized form is characterized here for both the Wigner distribution function and a Gaussian windowed Fourier transform. Aside from the greater understanding of the distributions themselves, these results promise a clearer intuition of phase-space-based methods for optical modeling. In particular, regardless of the context, the geometric construction that is presented for estimating the Wigner distribution function gives a valuable appreciation of its highly structured and sometimes surprising form.

Uncertainty products for nonparaxial wave fields.

Although maximal localization is a basic notion in the consideration of phase-space representations of fields, it has not yet been pursued for general wave fields. We develop measures of spatial and directional spreads for nonparaxial waves in free space. These measures are invariant under translation and rotation and are shown to reduce to the conventional ones when applied to paraxial fields. The associated uncertainty relation sets limits to joint localization in coordinate and frequency space. This relation provides a basis for the definition of a joint localization measure that is analogous to the beam propagation factor (i.e., M2) of paraxial optics. The results are first developed for two-dimensional fields and then generalized to three dimensions.

Fiber-diameter measurement by occlusion of a gaussian beam.

The amount of light occluded by a fiber as it passes through alaser beam can be used as the basis for fiber-diametermeasurement. This technique is analyzed with a two-dimensionalrigorous model. The occlusion seen for dielectric fibers as afunction of their diameter is highly oscillatory owing to interferencebetween the light transmitted by the fiber and the rest of thediffracted field. Scalar diffraction theory is shown to be adequatein modeling this effect. The oscillation sets a limit to theaccuracy of simple diameter measurement systems and is confirmedexperimentally for glass fibers. However, wool fibers are found tobe better treated as an absorbing material. The effect of beampolarization is investigated and found to be negligible for dielectricfibers but significant for metal fibers of small diameter.

Diffraction effects in the radiometry of coherent beams.

High-accuracy radiometry requires an optical beam in which all the light is contained within the radius of the smallest detector to be calibrated. We analyze a common configuration of the optical components used to prepare such a beam and show that diffraction rings are formed in the far field although the irradiance is zero along the limiting aperture's edge. The beam profile is calculated and used to find the radius of the smallest detector that can be calibrated with this beam.