experimental super-resolution with · experimental super-resolution with prolate spheroidal...
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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/ [email protected]