Design of pupil filter for extended depth of focus and lateral superresolution in Optical Coherence Tomography

Evgenia Bousi*, Stelios Timotheou, and Costas PitrisKIOS Research Center for Intelligent System and Networks, Dept. of Electrical and Computer

Engineering, University of Cyprus, 75 Kallipoleos St, Nicosia, Cyprus, 1678 ABSTRACT

A seven-zone pupil filter which can extend the depth of focus (DOF) and provide lateral super resolution in Optical Coherence Tomography is designed. Both amplitude and phase of the transmitted light beam are modified in each zone. The scalar diffraction theory is used to optimize the zone parameters. A broadband source spectrum and a Gaussian beam profile has been taken into account for the design of this filter. Several filters have been reported to extend the depth of focus (DOF) and improve the resolution but they take the assumption that a monochromatic and uniform light beam is used. With the proposed filter the DOF in OCT can be extended by 14 times while the lateral resolution can be improved by a factor of 1.47 and maintained constant within this DOF.

Keywords: Optical Coherence Tomography, Pupil filter, Extended depth of focus, Lateral resolution improvement

1. INTRODUCTIONIn Optical Coherence Tomography (OCT), the lateral resolution is fundamentally limited by the focusing of the

delivery optics. A high lateral resolution and a wide depth of focus (DOF) are mutually exclusive which imposes significant limitations. Although a higher effective numerical aperture (NA) enhances the lateral resolution, it limits the DOF which is inversely proportional to the square of the effective numerical aperture (NA) of the optical system 1.

Adaptive optics 1, axicon lens 2, dynamic focusing or focus tracking 3-4 is used for maintaining high lateral resolution over a large imaging depth. However, each technique requires special hardware in the system design and may limit the scanning speed and its application in real-time. A different approach for achieving uniform lateral resolution is the use of inverse scattering 5-6 algorithms. Interferometric Synthetic Aperture Microscopy (ISAM) can achieve depth independent resolution throughout a volume where the focus is fixed at one depth 7.

In this work we propose spatially modulating the pupil function of the focusing optics by a seven-zone binary pupil filter. To achieve superresolution along extended DOF many designs of pupil filters have been presented, including amplitude filters, pure phase filters, and complex filters. Amplitude filters are not energy efficient since a certain part of the pupil is obstructed. The design of phase only pupil filters8-9 pays more attention in recent years due to its better transmittance which make it more practical. Four-zone binary phase pupil filters were used for superresolution along extended depth of focus in a Swept Source Optical Coherence Microscopy system10. A gain in DOF from 8.9-11.5 and a superresolution factor of 0.7 has been reported by these kind of filters. However higher sidelobes are always observed in phase only filters. Complex pupil filters were introduced to improve the superresolving power of an optical imaging system11 assuming a uniform monochromatic optical wave. In this abstract the performance of a seven zone pupil filter especially designed for optical coherence tomography is demonstrated. A Gaussian beam profile and a broadband source spectrum are taken into account to simulate the light beam in the sample arm of a typical OCT system. An increased depth of focus by 14 times and an improved lateral resolution by a factor of 1.47 is demonstrated by this filter.

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3. OPTIMIZATION METHODTo obtain useful annular pupil filters, the optimal combination of parameters (e.g. number of concentric annular regions, phase shift, transmittance, radii of the annular regions) needs to be determined so that some design specifications such as the pass-band ripples and side-lobes of the transverse and axial intensity distributions, the Strehl ratio, and the transverse and axial intensity gain factors, are simultaneously met. Classical (uni-objective) optimization requires that these design specifications are imposed as constraints with the aim of achieving a desired threshold, while a weighted combination of one or more specifications can be optimized by incorporating them into the problem objective. The current state of practice in the literature is to derive such designs either using direct search methods such as genetic algorithms, or indirect search methods (e.g. conjugate gradient and quasi-newton algorithms) that attempt to solve approximated versions of the considered problem. Nevertheless, these algorithms cannot guarantee optimally of the derived solutions and often locally optimal solutions that are significantly worse than the global optimal solution. Our approach significantly diverges from the current state-of-practice by formulating an approximated version of the original problem as a Mixed Integer Linear Program (MILP) which can be solved using standard MILP solvers to provide solutions very close to the globally optimal solution. The level of approximation provides a trade-off between the level of accuracy and the solution difficulty of the problem, as finer approximation implies better solution quality but larger problem size and hence higher execution times. The key idea behind our approach is to consider a large number of annular regions with small and constant radius difference and determine the characteristics and clustering of these regions (into groups of consecutive regions with identical characteristics) that solve the considered optimization problem. Each of these identical region groups is essentially one annular region, while the total number of regions can be restricted through an appropriate constraint imposed to limit the number of allowed clusters. For example, if 100 regions are considered with radii 0-0.01, 0.01-0.02, …, 0.99-1, respectively, and a clustering of regions 0.05-0.06 – 0.22-0.23 is derived, in essence this means that an annular region with radii 0.05-0.23 is derived with phase shift and transmittance corresponding to those of the group. The key advantage of this approach is that all nonlinear operations (mostly associated with the integration of Eqs. (1) and (3)) can be pre-computed through discretization of the radial and axial coordinates. The resulting problem is a non-convex separable quadratic optimization problem with binary constraints which can be transformed into a MILP optimization problem through piecewise linear approximation of the nonlinear terms, and solved using standard MILP solvers. Another important advantage of the developed methodology is that it allows the formulation and solution of several variations of the considered problem in terms of the objective function and constraints (minimization of the ripple in the pass-band of the intensity distribution, maximization of the transverse or axial intensity gain factors) as well as in terms of the design parameters considered (continuous ([0-1]), binary ({0,1}), or constant (1) transmittance).

4. RERULTSThe filter is optimized for maximum depth of focus and lateral resolution simultaneously, with the constraints that the ripple should be minimized and the Strehl Ratio should be maximized. The parameters of the filter after optimization are: ri=[0.22,0.5,0.68,0.69,0.9,0.98,1], Ti=[0.9631,0.5856,0.8687,0.1597,1,0.2497,0.9467], φi=[π,0,π,0,0,0,0]. The Strehl Ratio is 0.11, which means that the intensity at the focus with the filter is reduced to 11% of the intensity at the focus without the filter. The transmittance and phase coefficients are selected to be 0-1, and 0/π. The simulations are performed for a low numerical aperture objective of NA=0.07, and a Gaussian source spectrum of 110nm spectral range. The central wavelength of light source is 1300nm. Fig. 3 shows the axial and transverse intensity distribution of the system without filter (red line) and with filter (black line). From the figures we can see that with the proposed filter an extended DOF by a factor of 14 can be achieved with the optimized filter, while the transverse resolution is improved by a factor of 1.47 (the DOF is the full width at 0.9 maximum). Also from Fig. 4 we can see that a constant spot size is maintained within the extended DOF. Fig.5 shows the 3D intensity distribution. As a result of the extended DOF the transverse intensity suffers from increased sidelobes. The level of sidelobes for the proposed design (0.18) is significantly better than that reported in 10 (0.31). Those sidelobes can be compressed by using signal processing

techniques such as deconvolution14. Also from Fig. 2(b) we can see that the axial intensity has a ripple of less than 0.03. Pure binary (transmittance 0/1 and phase 0/π) filters can also be designed with our optimization technique but in the cost of lower Strehl Ratio and increased ripple.

Figure. 3. (a) Transverse intensity distribution in the focal region: (red line) conventional system and (black line) system with filter. (b) Axial intensity (red line) conventional system and (black line) system with filter. (R and Z axis in m)

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5. CONCLUSIONIn this summary, a seven zone pupil filter is designed for application in OCT systems. A broadband source spectrum and a Gaussian beam profile has been taken into account for the design of this filter. Several filters have been reported to extend the DOF and improve the resolution but they take the assumption that a monochromatic and uniform light beam is used. With the proposed filter the DOF in OCT can be extended by 14 times while the lateral resolution can be improved by a factor of 1.47 and maintained constant within this DOF.

ACKNOWLEDGMENTS This work was funded by the Research Promotion Foundation of Cyprus under the Framework Program for Research, Technological Development, and Innovation 2009-2010 (Project Identifier: ΤΕΧΝΟΛΟΓΙΑ/ΜΗΧΑΝ/0311(ΒΙΕ)/10).

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