## Twin-photon confocal microscopy |

Optics Express, Vol. 18, Issue 21, pp. 22147-22157 (2010)

http://dx.doi.org/10.1364/OE.18.022147

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### Abstract

A recently introduced two-channel confocal microscope with correlated detection promises up to 50% improvement in transverse spatial resolution [Simon, Sergienko, Optics Express **18**, 9765 (2010)] via the use of photon correlations. Here we achieve similar results in a different manner, introducing a triple-confocal correlated microscope which exploits the correlations present in optical parametric amplifiers. It is based on tight focusing of pump radiation onto a thin sample positioned in front of a nonlinear crystal, followed by coincidence detection of signal and idler photons, each focused onto a pinhole. This approach offers further resolution enhancement in confocal microscopy.

© 2010 Optical Society of America

## 1. Introduction

### 1.1. Confocal microscopy

*f*and circular aperture of radius

*a*described by pupil function

*p*(

**x**), then the point spread function (PSF) for a confocal microscope is (up to normalization) where the tilde represents Fourier transform and

*J*

_{1}is a Bessel function of first order. This is to be compared to the widefield PSF for the same lens, For the confocal microscope, each passage through the lens gives a factor of

*p̃*

^{2}; since the light passes through the lens twice, this becomes squared to give the more sharply peaked function

*p̃*

^{4}. The result is a PSF for the confocal microscope which is roughly 28% narrower than that of the corresponding widefield microscope.

### 1.2. Multiple photons in confocal microscopy

*two photon confocal microscopy*[2

[2]. W. Denk, J. Strickler, and W. W. Webb, “Two-photon Laser Scanning Fluorescence Microscopy,” Science **248**, 73–76 (1990). [CrossRef] [PubMed]

[3]. W. Denk and K. Svoboda, Photon Upmanship: Why Multiphoton Imaging is More than a Gimmick, Neuron **18**, 351–357 (1997). [CrossRef] [PubMed]

*simultaneously excite*the same fluorescent molecule, thus limiting the visible region at each point of the scan to the small volume for which the intensity is large enough for two excitations to occur simultaneously with reasonable probability. Recently,

*correlation confocal microscopy*[1

[1]. D. S. Simon and A. V. Sergienko, “The Correlation Confocal Microscope,” Opt. Express **18**, 9765–9779 (2010). [CrossRef] [PubMed]

*simultaneously detected*, with a detection scheme designed such that only pairs that struck the sample within a small distance of each other have a high probability of being detected.

*three*in-focus regions, two detection and one illumination region (see fig. 1). Only points in the overlap of all three regions will be visible. If the third region is comparable to the size of the objective’s Airy disk, the overall area that can be resolved will become smaller due to the combined drop-off of the detection and illumination probabilities. This will be achieved by means of spontaneous parametric downconversion (SPDC). The two outgoing photons (signal and idler) each pass through an objective lens of focal length

*f*, and then through pinholes to photon-counting detectors. These signal and idler branches are analogous to the two halves of a standard confocal microscope. The detectors will be connected in coincidence, so that detection events will be recorded only when

*both*of the photons survive the corresponding pinhole. This alone would give us the characteristic

*p̃*

^{4}behavior of a confocal microscope. To go further, we must look at the pump beam. Typically, the pump is an approximate plane wave over an area of size much larger than the Airy disk of a microscope lens. However, it is possible to focus the pump beam to a much smaller region by means of a lens of focal length

*f*. The effect of weak focusing of the pump beam in SPDC was studied in ref. [4

_{p}[4]. T.B. Pittman, D.V. Strekalov, D.N. Klyshko, M.H. Rubin, A.V. Sergienko, and Y.H. Shih, “Two-photon geometric optics,” Phys. Rev. A **53**, 2804–2815 (1996). [CrossRef] [PubMed]

[4]. T.B. Pittman, D.V. Strekalov, D.N. Klyshko, M.H. Rubin, A.V. Sergienko, and Y.H. Shih, “Two-photon geometric optics,” Phys. Rev. A **53**, 2804–2815 (1996). [CrossRef] [PubMed]

*strongly*focusing the pump to a point-like region, and show that the effective overlap region of the signal, idler, and pump can be made small enough to noticeably enhance the resolution over that of the standard confocal microscope. In the process, we will find that the relationship of the pump beam to the signal and idler causes a second effect that introduces an additional significant spatial resolution enhancement in the lateral direction. The result is that visible-light images can be produced with resolution that normally would be possible only in the ultraviolet.

### 1.3. Resolution, the Abbe limit, and blurring by material

[5]. W. Lukosz, “Optical systems with resolving powers exceeding the classical limit”, J. Opt. Soc. Am. **56**1463–1472 (1966). [CrossRef]

[6]. R. Heintzmann and M.G.L. Gustafsson, “Subdiffraction resolution in continuous samples”, Nat. Photonics **3**, 362–364 (2009). [CrossRef]

[6]. R. Heintzmann and M.G.L. Gustafsson, “Subdiffraction resolution in continuous samples”, Nat. Photonics **3**, 362–364 (2009). [CrossRef]

[7]. P. Török and T. Wilson, “Rigorous theory for axial resolution in confocal microscopes”, Opt. Commun. **137**, 127–135 (1997). [CrossRef]

[8]. T. Wilson, R. Juškaitis, and P. Higdon, “The imaging of dielectric point scatterers in conventional and confocal microscopes”, Opt. Commun. **141**, 298–313 (1997). [CrossRef]

[9]. P. Török, P. D. Higdon, and T. Wilson, “On the general properties of polarized light conventional and confocal microscopes”, Opt. Commun. **148**, 300–315 (1998). [CrossRef]

[10]. P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers”, J. Mod. Opt. **45**, 1681–1698 (1998). [CrossRef]

[11]. C. J. R. Sheppard and J. Felix Aguilar, “Electromagnetic imaging in the confocal microscope”, Opt. Commun. **180**, 1–8 (2000). [CrossRef]

## 2. The Coincidence rate and point spread function

### 2.1. Derivation of coincidence rate

*w*

_{0}and frequency

*f*to a small region at the face of a

_{p}*χ*

^{(2)}nonlinear crystal. At this face is placed a thin sample of transmittance

*t*(

**y**), where

**y**is the position in the plane transverse to the propagation direction (

*z*). Spontaneous parametric downconversion occurs inside the crystal, producing two beams, the signal (ordinary ray) and idler (extraordinary ray), of respective frequencies

*ω*and

_{o}*ω*. The angles involved in the phase-matching inside the crystal are defined in fig. 2. The

_{e}*z*-axis is the propagation axis, at angle

*ψ*from the optic axis.

[4]. T.B. Pittman, D.V. Strekalov, D.N. Klyshko, M.H. Rubin, A.V. Sergienko, and Y.H. Shih, “Two-photon geometric optics,” Phys. Rev. A **53**, 2804–2815 (1996). [CrossRef] [PubMed]

*k*and

_{p}**k**

_{⊥}are the pump momentum in the longitudinal and transverse directions, and

**r**

_{⊥}is the position within the pump beam in the transverse direction.

*Ẽ*is the Fourier transform of the pump field inside the crystal, which is given by

_{p}**53**, 2804–2815 (1996). [CrossRef] [PubMed]

**53**, 2804–2815 (1996). [CrossRef] [PubMed]

*E*″

*. Adding the sample multiplies this at each point by*

_{p}*t*(

**r**

_{⊥}).

*s*

_{0}, passes through the objective lens, then propagates another distance

*s*

_{1}before reaching a pinhole. Distances

*s*

_{0}and

*s*

_{1}satisfy the imaging condition

*k*′

*(*

_{j}*j*=

*e*,

*o*) are

*s*

_{0}+

*s*

_{1}away due to field

*j*at radial distance

**r**

_{j}_{⊥}at the crystal is given by where

*h*

_{k}_{′}

*is the amplitude for a ray launched from*

_{j}**r**

_{⊥}with momentum

*k*′

*to propagate through the lenses, survive the pinhole, and reach the detector. For the confocal system shown, this propagation factor is given up to overall normalization by The contributions to the fields at the detectors due to the downconversion fields at*

_{j}**r**

_{j}_{⊥}are where

*A*

_{1}contains all constants and

**y**is the displacement of the microscope or sample during scanning.

*δ*(

*ω*+

_{e}*ω*–

_{o}*ω*), so inserting eq. (12) into eq. (9):

_{p}**53**, 2804–2815 (1996). [CrossRef] [PubMed]

*ω*= Ω

_{o}_{0}+

*ν*′,

*ω*= Ω

_{e}*+*

_{e}*ν*, with Ω

*+ Ω*

_{o}*=*

_{e}*ω*. We have also defined Furthermore, we may write

_{p}*δ*(

*ω*+

_{e}*ω*–

_{o}*ω*) =

_{p}*δ*(

*ν*+

*ν*′) and decompose the momentum integration measures according to

*A*

_{4}. We have also defined the time difference

*T*

_{12}=

*T*

_{1}–

*T*

_{2}and the inverse group velocity difference

*ν*and

*z*integrations, we have where Π(

*T*

_{12}) is the unit step function which is only nonzero for 0 <

*T*

_{12}<

*DL*; the presence of Π(

*T*

_{12}) simply expressed the fact that both photons must be created simultaneously and within the crystal. Using

*ω*= Ω

_{p}*+ Ω*

_{o}*, the amplitude now reduces to*

_{e}*d*≈

*f*, so that

_{p}*r*

_{⊥}. Performing a shift of integration variable

### 2.2. Lateral PSF and numerical results

*t*(

**r**

_{⊥}+

**y**) =

*δ*

^{(2)}(

**r**

_{⊥}+

**y**). We will henceforth also assume

*T*

_{12}<

*DL*. Taking the absolute square of

*A*, the lateral or transverse PSF is then: The PSF is narrowed relative to that of the standard confocal microscope as a result of two items: (i) the exponential factor

*p̃*. The exponential factor is due to the focusing by the pump lens. The factors of 2, however, appear even if the pump is not focused; they arise in the following way. In a standard confocal microscope, the Fourier transform of the pupil function arises because

*p*(

**r**

_{⊥}) is multiplied by a phase factor

*e*

^{−ik⊥·r⊥}as the photon propagates in the transverse direction from the point

**r**

_{⊥}in the focused spot to the axis at the pinhole. Integrating over

**r**

_{⊥}then gives the Fourier transform. In our case, both signal and idler exhibit such phase shifts; however, the pump itself has a radially-dependent phase. The pump photon thus contributes an additional phase factor equal in size to the sum of the phases gained by the signal and idler. So the phase is doubled, and the argument of the Fourier transformed pupil function is also doubled. Note that in the degenerate case Ω

*= Ω*

_{o}*=*

_{e}*ω*/2, we can interpret this in the following manner: although we are viewing the signal or idler at frequency

_{p}*ω*/2, the resolution is being determined by the properties of the pump, which has twice the frequency and thus higher resolution. Thus, if the pump is ultraviolet, with signal and idler in the visible range, we will end up with visible-light images that have UV-level resolution.

_{p}*r*

_{0}is comparable to or smaller than the size of the Airy disk, at which point the exponential factors begin to introduce additional narrowing. To get an idea of the sizes of

*r*

_{0}and

*R*we can insert some typical values. Assume a pump of wavelength

_{airy}*λ*= 351

_{p}*nm*, with signal and idler wavelengths

*λ*=

_{o}*λ*= 2

_{e}*λ*, and suppose that all lenses have radius

_{p}*a*= 2

*cm*and focal length

*f*= 2

*cm*. Then for pump beam of radius

*w*

_{0}= 1

*mm*, we find: Note that it is easy to reduce

*r*

_{0}if necessary: simply place a beam expander into the path of the pump beam. By increasing the radius of the beam, we fill a larger portion of the focusing lens, thereby effectively increasing its numerical aperture and allowing the beam to be focused to a smaller spot. We see from eq. (22) that

*r*

_{0}shrinks by the same factor by which the pump radius is expanded. As we increase the pump radius

*w*

_{0}, the resolution should remain approximately constant until

*r*

_{0}and

*R*are roughly equal; continued increase in

_{airy}*w*

_{0}beyond this point should then show improving resolution. For the example above, setting

*r*

_{0}=

*R*and using eq. (26) shows that this occurs around

_{airy}*w*

_{0}= 3.7

*mm*.

*w*

_{0}, with all other parameter values as given in the previous paragraph. We see that the PSF decreases in width compared to the standard confocal microscope by 50%, 61%, and 68% respectively, for pump radii of 1 mm, 8 mm, and 12 mm. In the limiting case where the beam completely fills the focusing lens (

*w*

_{0}=

*a*= 2

*cm*, not shown on the graph), the maximum resolution improvement over the standard confocal microscope is about 77.3%. At the opposite extreme of small beam radius, the PSF remains constant (given by the dashed green curve in fig. 3) as

*w*

_{0}decreases below about 4 mm, consistent with the estimate given above. As always, of course, increasing the numerical aperture of the lenses or increasing the frequency of the light will further improve the resolution.

*α*and numerical aperture

*NA*=

*n*sin

*α*, the pump beam will have an angular spread of

*α*both entering and leaving the focused region. The downconversion will increase the outgoing angular spread by a few additional degrees, depending on the downconversion parameters. So if the objective lens has acceptance angle roughly the same as that of the focusing lens in the pump, then the objective lens will be always be filled. Thus, there will generally be no problem filling a high numerical aperture lens to provide good confocal imaging.

*same*lens and separated by a beam splitter after the pinhole. This not only reduces the number of lenses needed, but should make alignment significantly easier. In addition, this version has the advantage of increased counting rate, since the full azimuthal angle around the propagation axis is now covered by the lens.

## 3. Discussion

*A*in which a small object is effectively visible to the detectors at each moment, and a plot of its intensity traces out the form of the point spread function for the detection system. This PSF is largest near the particle location and then drops off rapidly with distance. The more rapid the drop-off, the more precisely the position of the object can be localized. The confocal microscope resolves small particles and structures better than a conventional widefield microscope because the drop-off with distance is much sharper.

_{vis}*A*, is still the same size as before, but there will be nothing to see if the particle is slightly off-axis and therefore not illuminated by the focused pump. Now change the scanning method: hold the detection apparatus fixed and move the particle instead. As the particle is scanned across the visible region, there is darkness (zero coincidence rate) until the particle crosses the origin, when the coincidence counter lights up briefly, followed by darkness again as the particle moves out of the pump. Therefore, although the optical system through which the outgoing light passes has finite resolution and can localize positions only to within the region

_{vis}*A*, the extra information given by the localization of the pump beam to a point allows us to localize the particle position

_{vis}*to infinite precision*.

*A*. But the principle is still the same. The localization of the pump gives extra information about the position of the particle beyond what the outgoing imaging system provides, since now we will see nothing unless the object is within the intersection of

_{pump}*A*and

_{pump}*A*. Thus, convolving the illumination amplitude in

_{vis}*A*in with the detection amplitude in

_{pump}*A*, we obtain a combined spot which is

_{vis}*effectively*smaller. By this we mean that, although the radius of the region visible to the detection system is the same size, the drop off in coincidence rate as the particle moves away from the origin is more rapid due to the combined drop-off of detection and illumination. The location of the half-maximum moves inward toward the center, giving a smaller full-width at half-maximum and thus improved resolution. The standard confocal microscope has improved localization ability over the widefield microscope due to the convolution of one illumination branch and one detection branch; the twin-photon microscope microscope thus goes further, achieving additional localization via the convolution of one illumination branch with the product of

*two*detection branches.

## 4. Conclusions

## Acknowledgments

## References and links

[1]. | D. S. Simon and A. V. Sergienko, “The Correlation Confocal Microscope,” Opt. Express |

[2]. | W. Denk, J. Strickler, and W. W. Webb, “Two-photon Laser Scanning Fluorescence Microscopy,” Science |

[3]. | W. Denk and K. Svoboda, Photon Upmanship: Why Multiphoton Imaging is More than a Gimmick, Neuron |

[4]. | T.B. Pittman, D.V. Strekalov, D.N. Klyshko, M.H. Rubin, A.V. Sergienko, and Y.H. Shih, “Two-photon geometric optics,” Phys. Rev. A |

[5]. | W. Lukosz, “Optical systems with resolving powers exceeding the classical limit”, J. Opt. Soc. Am. |

[6]. | R. Heintzmann and M.G.L. Gustafsson, “Subdiffraction resolution in continuous samples”, Nat. Photonics |

[7]. | P. Török and T. Wilson, “Rigorous theory for axial resolution in confocal microscopes”, Opt. Commun. |

[8]. | T. Wilson, R. Juškaitis, and P. Higdon, “The imaging of dielectric point scatterers in conventional and confocal microscopes”, Opt. Commun. |

[9]. | P. Török, P. D. Higdon, and T. Wilson, “On the general properties of polarized light conventional and confocal microscopes”, Opt. Commun. |

[10]. | P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers”, J. Mod. Opt. |

[11]. | C. J. R. Sheppard and J. Felix Aguilar, “Electromagnetic imaging in the confocal microscope”, Opt. Commun. |

**OCIS Codes**

(180.1790) Microscopy : Confocal microscopy

(180.5810) Microscopy : Scanning microscopy

(190.4970) Nonlinear optics : Parametric oscillators and amplifiers

**ToC Category:**

Microscopy

**History**

Original Manuscript: August 30, 2010

Manuscript Accepted: September 19, 2010

Published: October 5, 2010

**Virtual Issues**

Vol. 5, Iss. 14 *Virtual Journal for Biomedical Optics*

**Citation**

D. S. Simon and A. V. Sergienko, "Twin-photon confocal microscopy," Opt. Express **18**, 22147-22157 (2010)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-18-21-22147

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### References

- [] D. S. Simon and A. V. Sergienko, ”The Correlation Confocal Microscope,” Opt. Express 18, 9765-9779 (2010). [CrossRef] [PubMed]
- [] W. Denk, J. Strickler, and W. W. Webb, ”Two-photon Laser Scanning Fluorescence Microscopy,” Science 248, 73-76 (1990). [CrossRef] [PubMed]
- [] W. Denk and K. Svoboda, Photon Upmanship: Why Multiphoton Imaging is More than a Gimmick, Neuron 18, 351-357 (1997). [CrossRef] [PubMed]
- [] T.B. Pittman, D.V. Strekalov, D.N. Klyshko, M.H. Rubin, A.V. Sergienko, Y.H. Shih, ”Two-photon geometric optics,” Phys. Rev. A 53, 2804-2815 (1996). [CrossRef] [PubMed]
- [] W. Lukosz, ”Optical systems with resolving powers exceeding the classical limit”, J. Opt. Soc. Am. 561463-1472 (1966). [CrossRef]
- Q1[] R. Heintzmann, M.G.L. Gustafsson, ”Subdiffraction resolution in continuous samples”, Nat. Photonics 3, 362-364 (2009). [CrossRef]
- [] P. Török, T. Wilson, ”Rigorous theory for axial resolution in confocal microscopes”, Opt. Commun. 137, 127-135 (1997). [CrossRef]
- [] T. Wilson, R. Ju?skaitis, and P. Higdon, ”The imaging of dielectric point scatterers in conventional and confocal microscopes”, Opt. Commun. 141, 298-313 (1997). [CrossRef]
- [] P. Török, P. D. Higdon, T. Wilson, ”On the general properties of polarized light conventional and confocal microscopes”, Opt. Commun. 148, 300-315 (1998). [CrossRef]
- [] P. Török, P. D. Higdon, T. Wilson, ”Theory for confocal and conventional microscopes imaging small dielectric scatterers”, J. Mod. Opt. 45, 1681-1698 (1998). [CrossRef]
- [] C. J. R. Sheppard and J. Felix Aguilar, ”Electromagnetic imaging in the confocal microscope”, Opt. Commun. 180, 1-8 (2000). [CrossRef]

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