Superresolving Phase Filters J. McOrist, M. Sharma, C. Sheppard

Introduction A lens brings light to a focus Geometric optics the focus is a point Physical optics the focus is a distribution of light known as a point spread function We can control the point spread function by changing the light at the aperture

Basic Imaging System Back Focal Plane Front Focal Plane

Focal Distributions The point spread function has two components: - Transverse - Axial Central peak is the central lobe, and the secondary peaks are the side lobes. Resolving power is related to the size of the central lobe

What is Superresolution? Superresolution in general, is reducing the size of the central lobe below the classical Raleigh limit Normally achieved by placing a filter in the back focal plane of the lens While resolution is improved, the effectiveness is limited by: - the size of the side lobes (M) - Strehl Ratio - central lobe intensity (S)

Superresolving PSF

Problems and Motivation Amplitude filters have two main problems:  Central lobe intensity  Fabrication of the filters Little theoretical work in phase filters, in particular axial behaviour Phase modulation is now possible with Diffractive Optics and Spatial Light Modulators

This is the first type of mask we examined Consists of two concentric zones Sales and Morris first examined this type of Mask in the Axial Direction Toraldo Phase Masks Zone masks are very simple, both to produce and to analyse mathematically  Phase change of  0 No phase change

Theoretical Considerations In the Fresnel Approximation we can describe the axial amplitude as 1 For a filter with two zones of equal area we get an intensity distribution 1. C.J.R. Sheppard, Z.S. Hegedus, J. Opt Soc. Am. A 5 (1988) 643.

Theoretical Considerations Due to its simple form we can easily determine the properties of the pupil filter We determined values for the Strehl Ratio (S), Spot Size, and axial position. We can also model the point spread function for values of  0

PSF of Two zone Filter The PSF of two-zone mask as the phase varies from 0 to Pi

Axial Behaviour of a Two-Zone The Strehl Ratio of a Two-Zone Element

Conclusions - Two Zone Filter Experiences a displaced focal spot from the focal plane Large increase in sidelobes Superresolution characteristics aren’t desirable Semi agreement with Sales and Morris 1 1. Sales., T.R.M., Morris.,G.M., Optics Comm. 156 (1998) 227

Higher Dimensional Filters If we increase N, the number of zones we find there are solutions for Superresolution We examined a three-zone filter, and a five- zone filter. We also generalised to a N-zone filter

Binary N-Zone Filters Consists of N concentric annuli called zones We only consider equal area annuli, and zones of equal phase difference, normally Pi. Indeed in the case of Pi, we get an expression for the axial point spread function

Three-zone Filter PSF Centered at Focal Spot Centered at the Focal Plane Plots of the PSF at centered at different positions. The dashed line is the diffraction limit.

Five-zone Filter Centered at Focal Spot Centered at the Focal Plane Plots of the PSF at centered at different positions. The dashed line is the diffraction limit.

Conclusions Three and Five zone filters exhibit similar behaviour: -Sidelobes displaced from the central spot -Focal Spot displacement increases Spot size is about half the diffraction limited case – Amplitude filters S = 0

Generalisation to N-Zone Filter We showed following common properties are exhibited for N-Zone Filters when N is odd: - Sidelobes are increasingly displaced in proportion to 2N - Central Lobe displaced in proportion to N - No loss in Strehl Ratio - No increase in Spot Size

Applications Large scope for applications of filters -Confocal Microscopy - Scanning resolution and control depth of scanning -Optical Data Storage -Optical Lithography -Astronomy Production is now much more possible than in the past 10 years

Summary – The Future Superresolution is the ability to resolve past the classical limit Pupil plane filters provide a way to do this – in particular phase only filters Superresolution appears to improve as the number of annuli is increased Possible to control the position of the focal spot?