Superresolving Phase Filters

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Superresolving Phase Filters

• J. McOrist, M. Sharma, C. Sheppard

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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

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Basic Imaging System
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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

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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)

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Superresolving PSF
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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

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Toraldo Phase Masks

• Zone masks are very simple, both to produce and
  to analyse mathematically

• 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

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Theoretical Considerations

• In the Fresnel Approximation we can describe the
  axial amplitude as1

• 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.
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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

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PSF of Two zone Filter

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

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

• Experiences a displaced focal spot from the focal
  plane
• Large increase in sidelobes
• Superresolution characteristics arent desirable
• Semi agreement with Sales and Morris1

1. Sales., T.R.M., Morris.,G.M., Optics Comm. 156
(1998) 227
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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

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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

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

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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

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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

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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?