Programmable Phase Optics
The Generalised Phase Contrast method
The imaging and visualisation of optical phase, such as wavefront disturbances or aberrations is a challenging yet often vital requirement in optics. A number of techniques can be applied in fields ranging from optical component testing through to wavefront sensing whenever a qualitative or quantitative analysis of an optical phase disturbance is required. In general, a phase disturbance can not be directly viewed and a method must therefore be sought to extract information about the wavefront from an indirect measurement. An example of this is the generation of fringe patterns in an interferometer, which gives information about the flatness of an optical component without requiring a physical measurement of the component surface. In this article, we describe a powerful phase-contrast technique that we have developed for the visualisation of phase disturbances, outlining the considerable improvements this method offers over previous analyses and discussing some potential applications.
A number of interferometric phase visualisation techniques can be classed as common path interferometry where the signal and reference beam travel along the same optical axis and interfere at the output of the optical system. Put simply, this means that we perturb a portion of the wavefront we wish to measure to create a reference wavefront and interference between the unperturbed wavefront information and this synthetically generated reference allows the visualisation of the phase information in the original wavefront. A schematic representation of the operation of a common path interferometer (CPI) is shown in Figure 1. Possibly, the most widely known implementation of a CPI is the Zernike phase contrast method. They exist however, in many different forms such as the point diffraction, dark central ground filtering, and field absorption methods. Although special cases such as the Zernike method have been previously treated, a comprehensive approach to the analysis of the generic CPI is lacking. Our recent work has thus concentrated on the establishment of a rigorous analytical framework to describe the operation, design and optimisation of this class of interferometers. The approach we use is based on a generalisation of the Zernike technique. The Generalised Phase Contrast (GPC) method is not limited by the operational constraints of the Zernike technique and by careful choice of the parameters for the Fourier filter to match the phase disturbance, it is possible to convert the phase information into a high contrast intensity distribution with a minimal loss of photons.
Based on the theoretical framework of the GPC method we can design CPI systems for a range of applications and achieve optimal performance in terms of fringe accuracy, visibility and peak irradiance. The GPC approach is an extension of the Zernike method and in the following sections, we therefore use this as the starting point in our explanation of the requirements for a generalised description. In the remainder of this article, we give an overview of some potential applications that exploit the GPC method. These include phase-visualisation, wavefront sensing, wavefront generation, programmable optical tweezers and optical encryption systems.
Zernike phase contrast
The Zernike phase contrast technique allows the visualisation of phase perturbations by the use of a Fourier plane phase shifting filter. The Dutch physicist Fritz Zernike received the Nobel Prize in 1953 for inventing this method, which led to a break-through in medicine and biology by making essentially transparent cell or bacteria samples clearly visible under a microscope. Its successful operation, however, requires that the spatial phase distribution, f(x,y), at the input is limited to a "small-scale" phase approximation where the largest phase is typically taken to be significantly less than p/3. If the phase distribution at the input is thus restricted, then a Taylor expansion to first order is sufficient for the mathematical treatment so that the input wavefront can be written as,
The light corresponding to the two terms in this "small-scale" phase approximation can be separated spatially by use of a single lens, where the phase distribution is located in the front focal plane and the corresponding spatial Fourier transformation is generated in the back focal plane of the lens. In this first order approximation, the constant term represents the amplitude of on-axis light focused by the lens in the back focal plane and the second spatially varying term represents the off-axis light. Zernike realised that a small phase shifting quarter wave plate acting on the focused light makes it possible to obtain an approximately linear visualisation of small phase structures by generating interference between the two phase quadrature terms in Eq. (1):
Generalised phase contrast
In the general case, we do not wish to be restricted to a limited phase range and we therefore cannot make the first order approximation of the Zernike technique. Higher order terms in the expansion need to be taken into account, so the expansion takes the form:
In this expression however, the contribution of spatially varying terms cannot be separated from the supposedly focussed light represented by the first term in this Taylor series expansion. In fact, all of these spatially varying terms can contribute to the strength of the on-axis focused light. For a significant modulation in the input phase, this contribution from the spatially varying terms can result in a significant modulation of the focal spot amplitude in the back focal plane of the lens. These terms can thus result in either constructive or destructive interference with the on-axis light, the net result of which will be an attenuation of the focused light amplitude which can only have a maximum value for a perfectly plane wave at the input.
Thus for phase objects or wavefronts breaking the Zernike approximation we must find an alternative mathematical approach to that of the Taylor expansion given in Eq. (3). We have chosen a Fourier analysis as a more suitable technique for completely separating the on-axis and higher spatial frequency components. This gives the following form for exp(if(x,y)), where (x,y) e W:
In this Fourier decomposition, the first term is a complex valued constant linked to the on-axis focused light from a phase disturbance defined within the spatial region. The second term describes light scattered by spatially varying structures in the phase object. Comparing Eq. (3) and Eq. (4) it is apparent that the first term of Eq. (3) is a poor approximation to that of Eq. (4) when operating beyond the Zernike small-scale phase regime.
point in the Generalised Phase Contrast method is the identification of the
operating regime where a match can be achieved between the strength of the
focused light (first term of Eq. 4) and the Fourier filter parameters (see
Fig. 1). If the system is applied to wavefront sensing or the visualisation
of unknown phase objects the GPC method specifies the filter parameters for
achieving optimal performance in extracting and displaying the phase
information carried by the incoming wavefront. In the case where we have
control over the incoming wavefront or phase modulation the method provides
extra means of optimisation by encoding the phase distribution itself in
addition to the filter parameters. This approach is particularly useful when
the filter parameters have a restricted dynamic range or are fixed. The
rigourous derivation of the equations for choosing these parameters can be
found in ref. 
and the references therein.