## The correlation confocal microscope

Optics Express, Vol. 18, Issue 10, pp. 9765-9779 (2010)

http://dx.doi.org/10.1364/OE.18.009765

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### Abstract

A new type of confocal microscope is described which makes use of intensity correlations between spatially correlated beams of light. It is shown that this apparatus leads to significantly improved transverse resolution.

© 2010 Optical Society of America

## 1. Introduction

## 2. The Standard Confocal Microscope.

32. R.H. Webb, “Confocal Optical Microscopy,” Rep. Prog. Phys. **59**, 427–471 (1996). [CrossRef]

*same*lens, with reflection rather than transmission occurring at the sample. (In this paper we will for simplicity always draw the transmission case, but most of the considerations will apply equally to the reflection case.) This lens serves as the objective; it has focal length

*f*and radius

*a*, and serves to focus the light going in and out of the sample. The sample is represented at point

**y**by a function

*t*(

**y**); depending on the setup,

*t*(

**y**) will represent either the transmittance or reflectance of the sample. At the first lens, the distances are chosen so that the imaging condition

*P*in the sample. Any stray light not focused to this point is blocked by the pinhole, thus providing the first improvement in contrast between between

*P*and neighboring points. The distances at the second lens also satisfy the imaging condition, so the second lens performs the inverse of the operation carried out by the first one, mapping the diffraction disk in the sample back to a point at the detection plane. The pinhole in this plane blocks any light not coming from the immediate vicinity of

*P*, thus providing further contrast. Together, the two pinholes serve to pass light from a small in-focus region in the sample and to block light from out-of-focus regions. The in-focus point is then scanned over the sample. The end result is a significant improvement in contrast over the widefield microscope.

*h*(

**y**) and transverse point-spread function (PSF) of the microscope. Let

*h*(

_{i}*ξ*,

*y*) (

*i*= 1,2) be the impulse response functions for the first and second lenses individually (including the free space propagation before and after the lens). Up to multiplication by overall constants, these are of the form

*p*̃(

**q**) is the Fourier transform of the aperture function

*p*(

**x**

^{′}) of the lens.

**q**and

*k*respectively denote the transverse and longitudinal momenta of the incoming photon. We assume that

*q*<<

*k*. From now on, we will also assume a circular, abberation-free lens of radius

*a*, in which case:

*q*is the magnitude of

**q**and

*J*

_{1}is the Bessel function of first order. Applying a pinhole at one end, we may also define

**y**is

*t*(

**y**) =

*δ*(

**y**), the impulse response becomes the coherent spread function

*z*

^{′}

_{i}=

*z*for

_{i}*i*= 1,2), use of Eq. (3) gives us ([32

32. R.H. Webb, “Confocal Optical Microscopy,” Rep. Prog. Phys. **59**, 427–471 (1996). [CrossRef]

*p*̃ is a sharply-peaked function, the higher power in the confocal result leads to a further increase of sharpness and a resulting improvement in resolution.

## 3. Correlations versus Confocality.

**q**entering the pinhole, the spectrum leaving the pin-hole is approximately flat; all transverse momentum correlations are lost. Thus it seems that we must choose between keeping either the correlations or the source pinhole, but not both.

- One method, often used in the standard confocal microscope, is to have the beam hit the lens parallel to the axis and focus to a point one focal length
*f*away [Fig. 2(a)]. We can then introduce correlations by arranging for pairs of narrow, well-localized beams of light to strike the lens at equal distances from the axis. - A second method is to use pairs of photons produced by spontaneous parametric down-conversion at correlated angles, so that if they are traced backward they both seem to emanate from the same point [Fig. 2(b)]; that point would be then be analogous to the pinhole. The crystal would have to be far enough from the lens for the cross-section of the pump beam to appear to be pointlike.
- A third method is to have a very narrow beam reflecting from a fixed point on a rotating mirror, then pass through a beam splitter. The two beams then have anticorrelated directions, and both trace back to the same illuminated point on the mirror. The mirror, as it rotates, fills the entire lens aperture with light over time.

## 4. The Correlation Confocal Microscope.

**y**

_{1}and

**y**

_{2}of the sample functions

*t*

_{1}and

*t*

_{2}will be partially linked by the spatial correlation that we will impose; we will find that this leads to an improvement of the lateral or transverse resolution.

*c*and vector field

**b**(

**y**). We assume that

**b**(

**y**) is a radial vector of constant magnitude pointing out from the axis, so that the

*ϕ*(

_{i}**y**) depend only on radial distance

*y*= ∣

**y**∣ and are linear in

*y*. The necessary phase shifts may be produced by a graded index material; for a material. 1

*mm*to 1

*mm*thick, the gradient of index versus radial distance required is of the order of

34. P.F. Carcia, R.H. French, M.H. Reilly, M.F. Lemon, and D.J. Jones, “Optical superlattices-a strategy for designing phase-shift masks for photolithographys at 248 and 193 nm: Application to AIN/CrN”, Appl. Phys. Lett **70**, 2371–2372 (1997)
[CrossRef]

*j*= 1) and lower (

*j*= 2) branch just before the second beam splitter are then given by

*BS*

_{1}, then by using Eqs. (12)–(13 ) we see that the state incident on the second beam splitter is proportional to

*BS*

_{2}. Taking the inner product of the resulting expression with the state having two photons leaving

*BS*

_{2}through port 3, we have the two-photon amplitude in branch 3:

*ϕ*

_{1}=

*ϕ*,

*ϕ*

_{2}= -

*ϕ*, and assumed that the two objects or samples are identical (

*t*

_{1}=

*t*

_{2}=

*t*). The amplitude may be put into the form (up to an overall multiplicative constant):

*ϕ*(

**y**

^{′}) =

*ϕ*(

**y**

^{″}) = 0. So if we arrange for

*ϕ*(

**y**) to drop from

**y**

^{′}and

**y**

^{″}that fall far from the axis, near the edge of the Airy disk (i.e. at the first zero of the Airy function). This is the key to the resolution enhancement. (In the notation introduced earlier, this means

*BS*

_{2}into arm 3, the final beam splitter

*BS*

_{3}simply routes them (50% of the time) to two different detectors, so that a coincidence count may be measured. The result is that, up to overall constants, the coincidence rate will be:

*A*

_{3}is given by Eq. (18). Note that the coincidence rate does not actually give the image of ∣

*t*(

**y**)∣

^{2}, as would normally be obtained from a microscope, but rather it gives the image of ∣

*t*(

**y**)∣

^{4}. So the square root of the coincidence count must be taken before comparison with the images from other microscopes.

*t*(

**y**) to be a delta function. Keeping in mind the square root mentioned above, this then gives us:

*y*is the magnitude of

**y**. We see that the factor sin

^{2}(

*ϕ*(

*y*)) is responsible for the improvement of resolution compared to the standard confocal microscope. This factor suppresses the counting of photon pairs with values of

**y**

^{′}and

*y*

^{″}that differ significantly from each other; this, when combined with the factors of

*p*̃, reduces the contribution to the coincidence rate of points in the outer part of the Airy disk.

*Note that all of this comes about because the factors involving the trigonometric functions in Eq. (18) provide a coupling between the integration variables*

**y**

^{′}and

**y**

^{″}. Without this coupling, which amounts to a spatial correlation between the photons detected from the two branches, Eq. (18) factors into two independent integrals, each of which looks like the amplitude of a standard confocal microscope.

*Thus, without the correlation, nothing is gained that could not be obtained from simply using a standard confocal microscope and squaring the output intensity*.

36. K.W.C. Chan, M.N.O. Sullivan, and R.W. Boyd, “Optimization of thermal ghost imaging: high-order correlations vs. background subtraction”, Optics Express **18**, 5562–5573 (2010). [CrossRef] [PubMed]

17. J.D. Franson, “Nonlocal Cancellation of Dispersion,” Phys. Rev. A **45**3126–3132 (1992). [CrossRef] [PubMed]

20. C. Bonato, A.V. Sergienko, B.E.A. Saleh, S. Bonora, and P. Villoresi, “Even-Order Aberration Cancellation in Quantum Interferometry,” Phys. Rev. Lett. **101**233603 (2008). [CrossRef] [PubMed]

## 5. Reduction to One Sample.

*e*

^{+iϕ(y)}. But then the beam is split at

*BS*

_{1}, after passing through the detection pinhole; half of the beam continues onward to the second beam splitter

*BS*

_{2}unaltered, while the other half is deflected downward to a phase conjugating mirror. The mirror reverses the sign of the phase and deflects the beam back to

*BS*

_{1}. The half of this beam that is transmitted through

*BS*

_{1}at this second encounter then recombines with the unaltered beam at

*BS*

_{2}; from this point on, all is as it was for the two-branch version of the previous section, assuming

*t*(

**y**) is real. In the lower branch between

*BS*

_{1}and

*BS*

_{2}, either a high-density optical filter or another 50/50 beam splitter must be inserted in order to equalize the intensity in the upper and lower branches.

*t*(

**y**) is complex, one copy of it will also be complex conjugated at the mirror. The factor

*t*(

**y**

^{′}+

**y**)

*t*(

**y**

^{″}+

**y**) inside the integrals of Eq. (18) now becomes

*t*(

**y**

^{′}+

**y**)

*t*

^{*}(

**y**

^{″}+

**y**); since sample-induced aberrations may be viewed as extra position-dependent phases appearing in

*t*, this change, when combined with the suppression of pairs (

**y**

^{′},

**y**

^{″}) that are far apart, will partially cancel the sample-induced aberrations, but aberrations introduced by the lenses are unaffected.

## 6. Thick Samples and Axial Resolution.

*ϕ*(

**y**) occurs is the same as the object plane; in other words, we have taken the sample to be of negligible thickness and the phase modulation plane to be right up against it. But for the more realistic case of a sample of finite thickness, with the plane to be imaged in the sample’s interior, the phase modulation obviously can not be done in the object plane. Instead, the phase modulation plane must be moved a small distance

*ζ*

_{2}away from the object plane, to the exterior of the sample. The distance from the lens to this plane is

*ζ*

_{1}=

*z*

_{2}-

*ζ*

_{2}(see Fig. 9). We apply a phase shift

*ϕ*

^{′}(

**x**

^{″}) =

*c*-

**b**

^{′}·

**x**

^{″}in this plane, which we still assume to vary only in the radial direction. Then the propagation factor from the source pinhole to the object plane [Eq. (4)] can be shown to now be

**b**

^{′}has been altered and the position of each point in the diffraction spot has been shifted in the argument of

*p*̃. Since

**b**

^{′}is a radial vector, each point of the Airy disk moves radially a distance

**b**

^{′}= ∣

**b**

^{′}∣. The result is that the Airy disk expands in radius by ∣Δ

*y*∣. This is clearly undesirable, so it is necessary to choose

*ζ*

_{2}such that ∣Δ

*y*∣ <<

*R*, making the radial shift negligible. Inserting realistic values, this means that we

_{airy}*ζ*

_{2}must be at most of order 10

^{-4}

*m*. Thus, our ability to view the interiors of thick samples will be limited to depths of less than 100

*μm*.

*y*compared to

*y*, we have

**b**

^{″}=

**b**(or, equivalently,

**b**and

*c*are the original parameter values used in the previous sections.

**y**= 0) at a distance

*z*=

*z*

_{2}+

*δz*from the lens.

*z*

_{2}is the distance to the confocal plane

*δz*is the defocusing distance. We assume that ∣

*δz*∣ <<

*z*

_{2}and study how the axial point spread function varies as a function of defocus. It is well known [32

32. R.H. Webb, “Confocal Optical Microscopy,” Rep. Prog. Phys. **59**, 427–471 (1996). [CrossRef]

*x*) = sin(

*x*)/

*x*.

*δz*and

_{a}*δz*, one for each photon. Using Eq. (22) in place of Eq. (4) and allowing for defocus in each arm, a calculation similar to that of Section 4 leads to the amplitude

_{b}*J*

_{0}(

*x*) is the Bessel function of zeroth order.

*L*and

*M*are known as Lommel functions and can be calculated by means of series expansions [44].

*A*

_{3}gives us the coincidence rate. If the object

*t*is taken to be constant in the

*z*direction and a delta function in the transverse (x and y) directions, then this coincidence rate at

**y**= 0 will give us the axial PSF. Up to an overall normalization constant, we then have

## 7. Conclusions.

*BS*

_{2}by the same output port (port 3), we could look for coincidences

*between*the two possible output ports (one photon each in ports 3 and 4). This would simplify the apparatus; however it can be shown that the resolution in this variation is not as good as the one we have described in the earlier sections, although it is still better than the standard confocal microscope. To be specific, the improvement factor of [1-cos(2

*ϕ*(

*y*))] = sin

^{2}(

*ϕ*(

*y*)) in Eq. (20) would be replaced by sin(2

*ϕ*(

*y*)), which decreases more slowly away from the axis for the same choice of phase function.

*within*the focused spot.

## Acknowledgments

## References and links

1. | M. Minsky, U.S. Patent # 3013467, |

2. | M. Minsky, “Memoir on Inventing the Confocal Scanning Microscope,” Scanning , |

3. | R. Pecora, |

4. | K.S. Schmitz, |

5. | B.J. Berne and R.J. Pecora, |

6. | D. Magde, E. Elson, and W.W. Webb, “Thermodynamic Fluctuations in a Reacting System-Measurement by Fluorescence Correlation Spectroscopy,” Phys. Rev. Lett. |

7. | W.W. Webb, “Fluorescence Correlation Spectroscopy: Inception, Biophysical Experimentations, and Prospectus,” Appl. Opt. |

8. | W. Denk, J. Strickler, and W.W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science |

9. | W. Denk and K. Svoboda, “Photon Upmanship: Why Multiphoton Imaging is More than a Gimmick,” Neuron |

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23. | R.S. Bennink, S.J. Bentley, and R.W. Boyd, “Two-Photon Coincidence Imaging with a Classical Source,” Phys. Rev. Lett. |

24. | R.S. Bennink, S.J. Bentley, R.W. Boyd, and J.C. Howell, “Quantum and Classical Coincidence Imaging,” Phys. Rev. Lett. , |

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26. | Y.J. Cai and S.Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E |

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28. | G. Scarcelli, V. Berardi, and Y.H. Shih, “Can Two-Photon Correlation of Chaotic Light Be Considered as Correlation of Intensity Fluctuations?,” Phys. Rev. Lett. |

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32. | R.H. Webb, “Confocal Optical Microscopy,” Rep. Prog. Phys. |

33. | J.C. Mertz, |

34. | P.F. Carcia, R.H. French, M.H. Reilly, M.F. Lemon, and D.J. Jones, “Optical superlattices-a strategy for designing phase-shift masks for photolithographys at 248 and 193 nm: Application to AIN/CrN”, Appl. Phys. Lett |

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38. | L.-H. Ou and L.-M. Kuang, Ghost imaging with third-order correlated thermal light, J. Phys. B: At. Mol. Opt. Phys. |

39. | D.-Z. Cao, J. Xiong, S.-H. Zhang, L.-F. Lin, L. Gao, and K. Wang, Enhancing visibility and resolution in Nthorder intensity correlation of thermal light, Appl. Phys. Lett. |

40. | I. N. Agafonov, M. V. Chekhova, T. Sh. Iskhakov, and A. N. Penin, High-visibility multiphoton interference of Hanbury Brown-Twiss type for classical light, Phys. Rev. A |

41. | Q. Liu, X.-H. Chen, K.-H. Luo, W. Wu, and L.-A. Wu, Role of multiphoton bunching in high-order ghost imaging with thermal light sources, Phys. Rev. A |

42. | K. W. C. Chan, M. N. OSullivan, and R. W. Boyd, High-Order Thermal Ghost Imaging, Opt. Lett. |

43. | T.M. Cover and J.A. Thomas, |

44. | K.D. Mielenz, “Algorithms for Fresnel Diffraction at Rectangular and Circular Apertures,” J. Res. Nat. Inst. Stand. Technol. , |

**OCIS Codes**

(180.1790) Microscopy : Confocal microscopy

(180.5810) Microscopy : Scanning microscopy

**ToC Category:**

Microscopy

**History**

Original Manuscript: February 11, 2010

Revised Manuscript: March 30, 2010

Manuscript Accepted: April 5, 2010

Published: April 26, 2010

**Virtual Issues**

Vol. 5, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

D. S. Simon and A. V. Sergienko, "The correlation confocal microscope," Opt. Express **18**, 9765-9779 (2010)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-18-10-9765

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### References

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- G. Scarcelli, V. Berardi, and Y. H. Shih, “Can Two-Photon Correlation of Chaotic Light Be Considered as Correlation of Intensity Fluctuations?” Phys. Rev. Lett. 96, 063602 (2006). [CrossRef] [PubMed]
- F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-Resolution Ghost Image and Ghost Diffraction Experiments with Thermal Light,” Phys. Rev. Lett. 94, 183602 (2005). [CrossRef] [PubMed]
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