## Nanoscale resolution for fluorescence microscopy via adiabatic passage |

Optics Express, Vol. 21, Issue 19, pp. 22139-22144 (2013)

http://dx.doi.org/10.1364/OE.21.022139

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### Abstract

We propose the use of the subwavelength localization via adiabatic passage technique for fluorescence microscopy with nanoscale resolution in the far field. This technique uses a Λ-type medium coherently coupled to two laser pulses: the pump, with a node in its spatial profile, and the Stokes. The population of the Λ system is adiabatically transferred from one ground state to the other except at the node position, yielding a narrow population peak. This coherent localization allows fluorescence imaging with nanometer lateral resolution. We derive an analytical expression to asses the resolution and perform a comparison with the coherent population trapping and the stimulated-emission-depletion techniques.

© 2013 OSA

## 1. Introduction

1. E. Abbe, “Beiträge zur theorie des mikroskops und der mikroskopischen wahrnehmung,” Arch. Mikrosk. Anat. **9**, 413 (1873). [CrossRef]

*λ*/(2NA), being

*λ*the wavelength of the addressing light, and NA the numerical aperture of the objective. Due to the reduced dimensions of the samples to investigate, frequently around few nanometers, high resolution images overcoming the diffraction limit are necessary. To this aim, one of the most extended group of techniques is based on the general concept of reversible saturable optical fluorescence transition (RESOLFT) [2

2. S. W. Hell, “Toward fluorescence nanoscopy,” Nat. Biotechnol. **21**, 1347 (2003). [CrossRef] [PubMed]

3. S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. **19**, 780 (1994). [CrossRef] [PubMed]

4. S. W. Hell and M. Kroug, “Ground-state-depletion fluorscence microscopy: a concept for breaking the diffraction resolution limit,” Appl. Phys. B **60**, 495 (1995). [CrossRef]

5. G. S. Agarwal and K. T. Kapale, “Subwavelength atom localization via coherent population trapping,” J. Phys. B: At. Mol. Opt. Phys. **39**, 3437 (2006). [CrossRef]

6. A. V. Gorshkov, L. Jiang, M. Greiner, P. Zoller, and M. D. Lukin, “Coherent quantum optical control with subwavelength resolution,” Phys. Rev. Lett. **100**, 093005 (2008). [CrossRef] [PubMed]

7. D. D. Yavuz and N. A. Proite, “Nanoscale resolution fluorescence microscopy using electromagnetically induced transparency,” Phys. Rev. A **76**, 041802(R) (2007). [CrossRef]

8. K. T. Kapale and S. Agarwal, “Subnanoscale resolution for microscopy via coherent population trapping,” Op. Lett. **35**, 2792 (2010). [CrossRef]

9. H. Li, V. A. Sautenkov, M. M. Kash, A. V. Sokolov, G. R. Welch, Y. V. Rostovtsev, M. S. Zubairy, and M. O. Scully, “Optical imaging beyond the diffraction limit via dark states,” Phys. Rev. A **78**, 013803 (2008). [CrossRef]

10. J. Mompart, V. Ahufinger, and G. Birkl, “Coherent patterning of matter waves with subwavelength localization,” Phys. Rev. A **79**, 053638 (2009). [CrossRef]

11. K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. **70**, 1003 (1998). [CrossRef]

5. G. S. Agarwal and K. T. Kapale, “Subwavelength atom localization via coherent population trapping,” J. Phys. B: At. Mol. Opt. Phys. **39**, 3437 (2006). [CrossRef]

7. D. D. Yavuz and N. A. Proite, “Nanoscale resolution fluorescence microscopy using electromagnetically induced transparency,” Phys. Rev. A **76**, 041802(R) (2007). [CrossRef]

10. J. Mompart, V. Ahufinger, and G. Birkl, “Coherent patterning of matter waves with subwavelength localization,” Phys. Rev. A **79**, 053638 (2009). [CrossRef]

12. D. Viscor, J. L. Rubio, G. Birkl, J. Mompart, and V. Ahufinger, “Single-site addressing of ultracold atoms beyond the diffraction limit via position-dependent adiabatic passage,” Phys. Rev. A **86**, 063409 (2012). [CrossRef]

10. J. Mompart, V. Ahufinger, and G. Birkl, “Coherent patterning of matter waves with subwavelength localization,” Phys. Rev. A **79**, 053638 (2009). [CrossRef]

12. D. Viscor, J. L. Rubio, G. Birkl, J. Mompart, and V. Ahufinger, “Single-site addressing of ultracold atoms beyond the diffraction limit via position-dependent adiabatic passage,” Phys. Rev. A **86**, 063409 (2012). [CrossRef]

## 2. Model

*(*

_{S}*x*,

*t*) ≡

*μ*_{32}·

*E*_{S}(

*x*,

*t*)/

*ħ*and Ω

*(*

_{P}*x*,

*t*) ≡

*μ*_{12}·

*E*_{P}(

*x*,

*t*)/

*ħ*, respectively, where

*E*_{S}(

*E*_{P}) is the electric field amplitude of the Stokes (pump),

*μ*_{32}(

*μ*_{12}) is the electric dipole moment of the |3〉↔|2〉 (|1〉↔|2〉) transition, and

*ħ*is the reduced Planck constant. The excited level |2〉 has a decay rate

*γ*

_{21}(

*γ*

_{23}) to the ground state |1〉 (|3〉). If the two-photon resonance condition is fulfilled, one of the eigenstates of the Hamiltonian, the so-called dark-state, takes the form: where Ω(

*x*,

*t*) = (|Ω

_{P}(

*x*,

*t*)|

^{2}+ |Ω

_{S}(

*x*,

*t*)|

^{2})

^{1/2}. Note that |

*D*(

*x*,

*t*)〉 does not involve the excited state |2〉. In our scheme, and in order to adiabatically follow the dark state, both pulses are sent in a counterintuitive temporal sequence, applying first the Stokes and, with a temporal delay

*T*, the pump [see bottom of Fig. 1(a)]. In addition, to ensure no coupling between the different energy eigenstates, a global adiabatic condition must be fulfilled, which imposes Ω(

*x*)

*T*≥

*A*, where

*A*is a dimensionless constant that for optimal Gaussian profiles and delay times takes values around 10 [11

11. K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. **70**, 1003 (1998). [CrossRef]

_{S0}(Ω

_{P0}) is the peak Rabi frequency of the Stokes (pump) pulse,

*J*

_{1}(

*υ*) is the first order Bessel function and

*υ*= (2

*πx*NA)/

*λ*is the optical unit corresponding to the Cartesian coordinate

*x*in the focal plane,

*σ*is the temporal width of the pulses,

*T*=

*t*

_{P}−

*t*

_{S}is the temporal delay between the pulses, which is proporcional to

*σ*, and

*δ*= 1.22

*π*is the offset with respect to

*υ*= 0 corresponding to the first cutoff of

*F*(

*υ*, 0). It is possible to obtain an analytical expression for the FWHM of the final population peak in |1〉,

*p*

_{1}(

*x*), by considering that the global adiabaticity condition is reached for |

*υ*| ≈ FWHM, assuming a Gaussian population peak profile and considering |

*υ*| ≪

*δ*. Thus, from the spatial profiles in Eqs. (2)–(3), the FWHM of the population distribution is where

*R*≡ (Ω

_{P0}/Ω

_{S0})

^{2}is the intensity ratio between the pump and the Stokes pulses, and

*k*≡ Ω

_{S0}

*T/A*must fulfill 0 <

*k*< 1. In such a SLAP-based fluorescence microscope, the effective PSF is given by the product

*h*

_{exc}(

*υ*)

*p*

_{1}(

*υ*) normalized to 1, where

*h*

_{exc}(

*υ*) is the PSF of the E pulse. If we assume that all the localized population leads to fluorescence, the lateral resolution is determined by Eq. (4), where the first factor accounts for diffraction, the second one is 1.22, and the third one can be less than 1 depending on the adiabaticity of the process, and tends to zero in the limit

*R*→ ∞, i.e., when Ω

_{P0}≫ Ω

_{S0}. Other dark-state techniques proposed to obtain atomic localization [5

5. G. S. Agarwal and K. T. Kapale, “Subwavelength atom localization via coherent population trapping,” J. Phys. B: At. Mol. Opt. Phys. **39**, 3437 (2006). [CrossRef]

7. D. D. Yavuz and N. A. Proite, “Nanoscale resolution fluorescence microscopy using electromagnetically induced transparency,” Phys. Rev. A **76**, 041802(R) (2007). [CrossRef]

**39**, 3437 (2006). [CrossRef]

*D*(

*υ*)〉|

^{2}= 1/2 for

*υ*= FWHM/2. Using the profiles given in (2) and (3), the FWHM for CPT is

*R*for different values of

*k*. From the figure, we can see that in the whole range of parameters considered, the peak obtained with SLAP is significantly narrower than the one with CPT. Note that the adiabatic nature of the SLAP technique allows to increase the final resolution by increasing the time delay

*T*, while fixing the intensities.

## 3. Numerical results

*t*. Typical values for Δ

*t*are some hundreds of ps, larger than the vibrational relaxation time

*τ*(∼ps) but much shorter than the fluorescence time

*τ*

_{fl}(∼ns) of the transition

*n*

_{1}→

*n*

_{0}. First, the E field excites all the population to state

*n*

_{1}. Next, the D field, which has a doughnut-like spatial profile, produces a spatial depletion of the population in

*n*

_{1}by stimulated emission. Out of the node, the excited population is removed resulting in fluorescence inhibition, and reducing the width of the effective PSF. Note here that the main distinctive feature of SLAP with respect to general RESOLFT techniques, e.g., STED, is the adiabatic nature of the state transfer process, which as discussed below, confers robustness and flexibility on our method.

*A*= 20 are represented. For the simulations, we have used the density-matrix formalism for a Λ-system with degenerated ground states with the following parameter setting:

*γ*

_{21}=

*γ*

_{23}= 2

*π*× 6.36 GHz, Ω

_{S0}/

*γ*

_{21}= 1.5,

*σ*= 100 ps,

*T*= 1.5

*σ*, NA = 1.4, and

*λ*= 490 nm. Figure 3(c) shows the final population

*p*

_{1}(

*x*) in |1〉 using the SLAP technique for

*R*= 10 (dashed line),

*R*= 50 (dotted line) and

*R*= 300 (dotted-dashed line), marked with vertical arrows in Fig. 3(b). In addition, the population

*n*

_{1}(

*x*) using the STED technique (black solid line) is also shown. For the STED simulation, we have used rate equations for the Rhodamine B dye with intensity profiles of the exciting and depletion pulses corresponding to Eq. (2) and Eq. (3), respectively, and typical values [3

3. S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. **19**, 780 (1994). [CrossRef] [PubMed]

*σ*

_{cs}= 10

^{−17}cm

^{2}, peak intensity of the depletion laser

*τ*= 1 ps,

*τ*

_{fl}= 2 ns, Δt = 90 ps,

*σ*= 100 ps, NA = 1.4,

*λ*

_{E}= 490 nm, and

*λ*

_{D}= 600 nm, obtaining FWHM = 65.2 nm. Note that the final peak in STED does not reach unity due to the loss of population by fluorescence while depletion acts. This does not occur in SLAP, since the final peak corresponds to the population in a ground state.

## 4. Conclusions and perspectives

_{S0}is kept constant. This behavior is similar in STED microscopy, whose resolution improves by increasing the intensity of the depletion laser. Then, we have performed a numerical comparison between the STED technique using typical parameter values and the SLAP technique with Rabi frequencies of the order of GHz, obtaining a similar resolution in both cases.

14. X. Michalet, F. F. Pinaud, L. A. Bentolila, J. M. Tsay, S. Doose, J. J. Li, G. Sundaresan, A. M. Wu, S. S. Gambhir, and S. Weiss, “Quantum dots for live cells, in vivo imaging, and diagnostics,” Science **307**, 538 (2005). [CrossRef] [PubMed]

15. U. Hohenester, F. Troiani, E. Molinari, G. Panzarini, and C. Macchiavello, “Coherent population transfer in coupled semiconductor quantum dots,” Appl. Phys. Lett. **77**, 1864 (2000). [CrossRef]

16. J. Fabian and U. Hohenester, “Entanglement distillation by adiabatic passage in coupled quantum dots,” Phys. Rev. B **72**, 201304(R) (2005). [CrossRef]

^{6}W/cm

^{2}for |

**| ≃ 10**

*μ*^{−28}C m [17

17. P. G. Eliseev, H. Li, A. Stintz, G. T. Liu, T. C. Newell, K. J. Malloy, and L. F. Lester, “Transition dipole moment of InAs/InGaAs quantum dots from experiments on ultralow-threshold laser diodes,” Appl. Phys. Lett. **77**, 262 (2000). [CrossRef]

18. S. E. Irvine, T. Staudt, E. Rittweger, J. Engelhardt, and S. W. Hell, “Direct light-driven modulation of luminescence from Mn-Doped ZnSe quantum dots,” Angew. Chem. Int. Ed. **47**(14), 2685–8, (2008). [CrossRef]

## Acknowledgments

## References and links

1. | E. Abbe, “Beiträge zur theorie des mikroskops und der mikroskopischen wahrnehmung,” Arch. Mikrosk. Anat. |

2. | S. W. Hell, “Toward fluorescence nanoscopy,” Nat. Biotechnol. |

3. | S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. |

4. | S. W. Hell and M. Kroug, “Ground-state-depletion fluorscence microscopy: a concept for breaking the diffraction resolution limit,” Appl. Phys. B |

5. | G. S. Agarwal and K. T. Kapale, “Subwavelength atom localization via coherent population trapping,” J. Phys. B: At. Mol. Opt. Phys. |

6. | A. V. Gorshkov, L. Jiang, M. Greiner, P. Zoller, and M. D. Lukin, “Coherent quantum optical control with subwavelength resolution,” Phys. Rev. Lett. |

7. | D. D. Yavuz and N. A. Proite, “Nanoscale resolution fluorescence microscopy using electromagnetically induced transparency,” Phys. Rev. A |

8. | K. T. Kapale and S. Agarwal, “Subnanoscale resolution for microscopy via coherent population trapping,” Op. Lett. |

9. | H. Li, V. A. Sautenkov, M. M. Kash, A. V. Sokolov, G. R. Welch, Y. V. Rostovtsev, M. S. Zubairy, and M. O. Scully, “Optical imaging beyond the diffraction limit via dark states,” Phys. Rev. A |

10. | J. Mompart, V. Ahufinger, and G. Birkl, “Coherent patterning of matter waves with subwavelength localization,” Phys. Rev. A |

11. | K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. |

12. | D. Viscor, J. L. Rubio, G. Birkl, J. Mompart, and V. Ahufinger, “Single-site addressing of ultracold atoms beyond the diffraction limit via position-dependent adiabatic passage,” Phys. Rev. A |

13. | A. P. Alivisatos, “Semiconductor clusters, nanocrystals, and quantum dots,” Science, New Series , |

14. | X. Michalet, F. F. Pinaud, L. A. Bentolila, J. M. Tsay, S. Doose, J. J. Li, G. Sundaresan, A. M. Wu, S. S. Gambhir, and S. Weiss, “Quantum dots for live cells, in vivo imaging, and diagnostics,” Science |

15. | U. Hohenester, F. Troiani, E. Molinari, G. Panzarini, and C. Macchiavello, “Coherent population transfer in coupled semiconductor quantum dots,” Appl. Phys. Lett. |

16. | J. Fabian and U. Hohenester, “Entanglement distillation by adiabatic passage in coupled quantum dots,” Phys. Rev. B |

17. | P. G. Eliseev, H. Li, A. Stintz, G. T. Liu, T. C. Newell, K. J. Malloy, and L. F. Lester, “Transition dipole moment of InAs/InGaAs quantum dots from experiments on ultralow-threshold laser diodes,” Appl. Phys. Lett. |

18. | S. E. Irvine, T. Staudt, E. Rittweger, J. Engelhardt, and S. W. Hell, “Direct light-driven modulation of luminescence from Mn-Doped ZnSe quantum dots,” Angew. Chem. Int. Ed. |

**OCIS Codes**

(020.0020) Atomic and molecular physics : Atomic and molecular physics

(110.0180) Imaging systems : Microscopy

(270.0270) Quantum optics : Quantum optics

**ToC Category:**

Microscopy

**History**

Original Manuscript: June 17, 2013

Revised Manuscript: August 16, 2013

Manuscript Accepted: August 16, 2013

Published: September 12, 2013

**Virtual Issues**

Vol. 8, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

Juan Luis Rubio, Daniel Viscor, Veronica Ahufinger, and Jordi Mompart, "Nanoscale resolution for fluorescence microscopy via adiabatic passage," Opt. Express **21**, 22139-22144 (2013)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-21-19-22139

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### References

- E. Abbe, “Beiträge zur theorie des mikroskops und der mikroskopischen wahrnehmung,” Arch. Mikrosk. Anat.9, 413 (1873). [CrossRef]
- S. W. Hell, “Toward fluorescence nanoscopy,” Nat. Biotechnol.21, 1347 (2003). [CrossRef] [PubMed]
- S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett.19, 780 (1994). [CrossRef] [PubMed]
- S. W. Hell and M. Kroug, “Ground-state-depletion fluorscence microscopy: a concept for breaking the diffraction resolution limit,” Appl. Phys. B60, 495 (1995). [CrossRef]
- G. S. Agarwal and K. T. Kapale, “Subwavelength atom localization via coherent population trapping,” J. Phys. B: At. Mol. Opt. Phys.39, 3437 (2006). [CrossRef]
- A. V. Gorshkov, L. Jiang, M. Greiner, P. Zoller, and M. D. Lukin, “Coherent quantum optical control with subwavelength resolution,” Phys. Rev. Lett.100, 093005 (2008). [CrossRef] [PubMed]
- D. D. Yavuz and N. A. Proite, “Nanoscale resolution fluorescence microscopy using electromagnetically induced transparency,” Phys. Rev. A76, 041802(R) (2007). [CrossRef]
- K. T. Kapale and S. Agarwal, “Subnanoscale resolution for microscopy via coherent population trapping,” Op. Lett.35, 2792 (2010). [CrossRef]
- H. Li, V. A. Sautenkov, M. M. Kash, A. V. Sokolov, G. R. Welch, Y. V. Rostovtsev, M. S. Zubairy, and M. O. Scully, “Optical imaging beyond the diffraction limit via dark states,” Phys. Rev. A78, 013803 (2008). [CrossRef]
- J. Mompart, V. Ahufinger, and G. Birkl, “Coherent patterning of matter waves with subwavelength localization,” Phys. Rev. A79, 053638 (2009). [CrossRef]
- K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys.70, 1003 (1998). [CrossRef]
- D. Viscor, J. L. Rubio, G. Birkl, J. Mompart, and V. Ahufinger, “Single-site addressing of ultracold atoms beyond the diffraction limit via position-dependent adiabatic passage,” Phys. Rev. A86, 063409 (2012). [CrossRef]
- A. P. Alivisatos, “Semiconductor clusters, nanocrystals, and quantum dots,” Science, New Series, 271, 933–937 (1996).
- X. Michalet, F. F. Pinaud, L. A. Bentolila, J. M. Tsay, S. Doose, J. J. Li, G. Sundaresan, A. M. Wu, S. S. Gambhir, and S. Weiss, “Quantum dots for live cells, in vivo imaging, and diagnostics,” Science307, 538 (2005). [CrossRef] [PubMed]
- U. Hohenester, F. Troiani, E. Molinari, G. Panzarini, and C. Macchiavello, “Coherent population transfer in coupled semiconductor quantum dots,” Appl. Phys. Lett.77, 1864 (2000). [CrossRef]
- J. Fabian and U. Hohenester, “Entanglement distillation by adiabatic passage in coupled quantum dots,” Phys. Rev. B72, 201304(R) (2005). [CrossRef]
- P. G. Eliseev, H. Li, A. Stintz, G. T. Liu, T. C. Newell, K. J. Malloy, and L. F. Lester, “Transition dipole moment of InAs/InGaAs quantum dots from experiments on ultralow-threshold laser diodes,” Appl. Phys. Lett.77, 262 (2000). [CrossRef]
- S. E. Irvine, T. Staudt, E. Rittweger, J. Engelhardt, and S. W. Hell, “Direct light-driven modulation of luminescence from Mn-Doped ZnSe quantum dots,” Angew. Chem. Int. Ed.47(14), 2685–8, (2008). [CrossRef]

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