Experimental super-resolution withprolate spheroidal functionsKevin Piche, Jeff Z. Salvail, Jonathan Leach and Robert W. Boyd

University of Ottawa

Introduction

Simple figure of a 4-f system.

I Diffraction: Distortion of imagespassing through an optical system.

I Super-resolution: Class of techniquesthat resolve images passed thediffraction or Rayleigh limit [1],i.e. reduce the effects of diffraction.

I 4-f system: Basic diffraction limitedoptical system.

Theory

I Eigenmode: Image that passesthrough an optical system with achange in intensity, but is otherwiseunchanged.

I All images are superpositions ofeigenmodes, P`,p(r, θ).

I Super-resolution: Divide the measuredcoefficients, D`,p by thetransmittance.

I(r, θ) =∞∑

`=−∞

∞∑p=0

I`,pP`,p(r, θ)

=∞∑

`=−∞

∞∑p=0

D`,p

λ`,pP`,p(r, θ)

` and p are the angular and radialindexes, respectively. The eigenvalueλ`,p is the transmitance of therespective eigenmode. i.e. the fractionby which the intensity changes.

Simulation

Images

Mode Coefficients

Original Image

Original Image

Diffracted Image

Diffracted Image Reconstructed Image

Reconstructed Image

l

-2

-1

0

1

2

0 1 2 p

0 1 2 p

0 1 2 p

l

-2

-1

0

1

2

l

-2

-1

0

1

2

Computer simulations of eigenmode super-resolution.

Experiment

SLMCCDcamera

1st order

0th order

Pupil plane

0.5 m 0.5 m 0.5 m 0.5 m

0.5mm

650 nm laser Single mode fiber

Results

Sample Images Coefficients

(b) Diffraction limited images

(c) Reconstructedimages

(a) Originalimages

(i)

(ii)

(iii)

1

0

inten

sity

1

0

inten

sity

1

0

inten

sity

1

0

inten

sity

1

0

inten

sity

1

0

inten

sity

1

0

inten

sity

1

0

inten

sity

1

0

inten

sity

(b) Diffraction limited coeffs.

(c) Reconstructedcoeffs.

(a) Originalcoeffs.

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

00 0 0

0 0 0

0 0 0

p p p

p

p p p

p p

0

3

-3

l 0

3

-3

l 0

3

-3

l

0

3

-3

l

0

3

-3

l

0

3

-3

l 0

3

-3

l

0

3

-3

l 0

3

-3

l

3

3

3 3

3 3

3

3 3

Experiments show that super-resolution is possible!

Conclusions

We have demonstrated that super-resolution is experimentally possible. The next step tothis research is to generalise this to more complicated optical systems and ultimately to

reach the quantum limit to resolution due to quantum fluctuations [2].

References

[1] J. W. S. Rayleigh, Collected Optics Papers of Lord Rayleigh.Optical Society of America, 1994.

[2] V. N. Beskrovny and M. I. Kolobov, “Quantum-statistical analysis of superresolution for optical systemswith circular symmetry,” Phys. Rev. A, vol. 78, p. 043824, Oct 2008.

Quantum Photonics http://www.quantumphotonics.uottawa.ca/ kpich049@uottawa.ca