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 FocalPlane

Focal Distributions

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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

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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 as1

1

02exp2 dttPvU iut

For a filter with two zones of equal area we get an intensity distribution

22

22 cossin uucuI

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

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1Strehl Ratio of Two - zone element

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 Morris1

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

2

22

cos

cossin

Nu

u

N

ucuI

Three-zone Filter PSF

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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

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Centered at Focal Spot Centered at the Focal Plane

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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?