Exact diffraction calculation from fields specified over arbitrary curved surfaces

Abstract

Calculation of the scalar diffraction field over the entire space from a given field over a surface is an important problem in computer generated holography. A straightforward approach to compute the diffraction field from field samples given on a surface is to superpose the emanated fields from each such sample. In this approach, possible mutual interactions between the fields at these samples are omitted and the calculated field may be significantly in error. In the proposed diffraction calculation algorithm, mutual interactions are taken into consideration, and thus the exact diffraction field can be calculated. The algorithm is based on posing the problem as the inverse of a problem whose formulation is straightforward. The problem is then solved by a signal decomposition approach. The computational cost of the proposed method is high, but it yields the exact scalar diffraction field over the entire space from the data on a surface.

Introduction

When the input field is specified over a planar surface, the calculation of monochromatic scalar optical diffraction can be accomplished in a straightforward manner by plane wave decomposition or the Rayleigh–Sommerfeld diffraction integral, or by other methods derived from these. Integration over the planar surface allows computation of the exact diffraction field over the entire space. However, if the input field is specified over a curved surface, rather than a planar surface, straightforward integration over the curved surface may not provide the exact field over the entire space. Calculation of the exact diffraction field from a curved surface requires greater care and is the subject of this work [16].

Diffraction field calculation by direct integration over the surface on which the input field is specified, is essentially a weighted superposition of the free-space diffraction kernel. However, direct integration gives the exact field only when the integrated surface field value remains unaltered by the propagation from other surface elements. If we simply ignore such mutual interactions, the calculated field will be different from the actual field. The method we set forth is based on the following observation. No such interactions exists when the input field is specified over a plane; therefore it is straightforward to express the field on an arbitrary curved surface (and indeed any region of the entire space) as a weighted superposition integral of the free-space diffraction kernel over a planar surface. In the problem we wish to solve, the field is known over a curved surface and we wish to obtain the field over a planar surface (which would then also enable us to calculate it over the entire space). Since it is not straightforward to express the field on the planar surface in terms of the field on the curved surface, we express the field on the curved surface in terms of that on the planar surface, and solve an inverse problem to obtain the field on the planar surface. The inverse problem arising from this exact formulation can be solved by employing several methods and standard algorithms, each with their pros and cons. In this paper, we propose a signal decomposition algorithm for this purpose.

Our interest in diffraction calculations from curved surfaces stems from our work on computer generated holography (CGH) and three-dimensional imaging and television [1], [2], [3], [4], [11], [17], [20], [21], [22], [23]. Since the diffraction field from an arbitrarily shaped object is the field that we desire to recreate at the display end, its accurate calculation is of utmost importance.

In both computer graphics and CGH, objects are commonly modeled as a set of sample points distributed over space [8], [9], [14], [15]. It is assumed that the characteristics of the continuous object can be sufficiently represented by these sample points. A straightforward approach to compute the diffraction field created by an object is to superpose the fields created by each sample point of the object; doing so amounts to treating each sample point as a light source. We will refer to diffraction field calculation approaches based on superposition of the fields at each sample point of the object as “source model” approaches. In these approaches, it is assumed that the value of each source is independent of the field at other points. Then, the independently computed fields from these points are superposed. The calculated field will be the same as the actual field only if the points truly act as sources (i.e., if the values of these sources are not perturbed by the superposed field generated by the other sources). However, usually there are complicating interactions. Consequently, the field calculated using the source model will not be exact or may even be significantly in error. Diffraction field calculations based on the source model have the advantage of having reasonable computational complexities, but they are not necessarily exact except when all the sample points are given over a planar surface.

With the term mutual interaction we refer to the fact that the field at a given input point is not independent of the field at the other input points; in other words, it is not possible to specify them independently and arbitrarily.

Ignoring the mutual interactions and straightforwardly superposing the specified input field values will not give exact results. Instead, a simultaneous calculation of the diffraction field due to the given input points is necessary. We will refer to approaches based on such simultaneous calculation of the diffraction field as “field model” approaches. The diffraction field computation method presented in this paper is based on such an approach and uses a decomposition of the field specified over an orientable manifold onto a function set obtained from the intersection of the propagating plane waves by the manifold.

The algorithm we propose can be used for both two-dimensional (2D) and three-dimensional (3D) spaces. For simplicity we will first discuss the 2D case. In the 3D case, numerical issues due to larger data sets arise. Nevertheless, as a proof of concept the extension of the proposed algorithm to the 3D case is also presented.