## Near-field effects on coherent anti-Stokes Raman scattering microscopy imaging

Optics Express, Vol. 15, Issue 7, pp. 4118-4131 (2007)

http://dx.doi.org/10.1364/OE.15.004118

Acrobat PDF (317 KB)

### Abstract

We introduce a numerical approach, the finite-difference time-domain (FDTD) method, to study the near-field effects on coherent anti-Stokes Raman scattering (CARS) microscopy on nanoparticles. Changes of the induced nonlinear polarization, scattering patterns, and polarization properties against different diameters of spherical nanoparticles are calculated and discussed in detail. The results show that due to near-field effects, the induced nonlinear polarization is significantly enhanced at the water-particle interface, with 1.5-fold increase in intensity compared to that inside the particles, and the near-field enhancement increases with decreasing diameters of nanoparticles. The enhanced scattering dominates over the scattering contribution from the particles when the nanoparticle size decreases down to the scale of less than a half wavelength of excitation light. Further studies show that near-field effects make the induced perpendicular polarization of CARS signals being strictly confined within the nanoparticles and the particle-water interface, and this perpendicular polarization component could contribute approximately 20% to the backward scattering. The ratio values of the perpendicular polarization component to the total CARS signals from nanoparticles sizing from 75 nm to 300 nm in backward scattering are approximately 3 to 5 times higher than those in forward scattering. Therefore, near-field effects can play an important role in CARS imaging. Employing the perpendicular polarization component of CARS signals can significantly improve the contrast of CARS images, and be particularly useful for revealing the fine structures of bio-materials with nano-scale resolutions.

© 2007 Optical Society of America

## 1. Introduction

_{p}, a Stokes beam of frequency ω

_{s}, and a CARS signal at the anti-Stokes frequency of ω

_{as}=2ω

_{p}-ω

_{s}generated in the phase matching direction [1]. The vibrational contrast in CARS microscopy is created when the frequency difference ω

_{p}-ω

_{s}between the pump and the Stokes beams is tuned to be resonant with a Raman-active molecular vibration of samples. CARS signal is free of autofluorescence background due to the anti-Stokes shifting, and is more efficient than conventional Raman scattering due to stimulated generation of signal. Recently, CARS technique has received great interest in imaging live cells [2–10

2. P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced
polarization third order in the electric field strength,”
Phys. Rev. **137**,A801 (1965). [CrossRef]

11. J. X. Cheng and X. S. Xie, “Green’s function formulation for third-harmonic
generation microscopy,” J. Opt. Soc. Am. B **19**,1604 (2002). [CrossRef]

12. C. Liu and S. H. Park, “Anisotropy of near-field speckle
patterns,” Opt. Lett. **30**,1602 (2005). [CrossRef] [PubMed]

13. T. Ichimura, N. Hayazawa, M. Hashimoto, Y. Inouye, and S. Kawata, “Tip-enhanced coherent anti-Stokes Raman scattering
for vibrational nano-imaging,” Phys. Rev. Lett. **92**,220801 (2004). [CrossRef] [PubMed]

14. K.S. Yee “Numerical solution of initial boundary value problem
involving Maxwell equations in isotropic media,” IEEE
Trans. Antennas Propagat. **14**,302 (1966). [CrossRef]

## 2. Finite-difference time-domain (FDTD) method for simulations

14. K.S. Yee “Numerical solution of initial boundary value problem
involving Maxwell equations in isotropic media,” IEEE
Trans. Antennas Propagat. **14**,302 (1966). [CrossRef]

_{n-2},x) and the spatial finite-difference of the magnetic fields of surrounding grids at time t

^{n-1}. The magnetic field H(t

_{n-2},x) can be obtained using the H(t

_{n-1},x) and the finite-difference of the electric field of surrounding grids at time t

_{n}. After a number of iterations on electric and magnetic fields, the calculated electromagnetic field will converge to an explicit value anticipated by Maxwell equations.

**P**

^{(3)}(

**r**,t)is the third-order nonlinear polarization,

*n*is the refractive index of the medium for the signal field, and

*c*is the velocity of light in vacuum. The third-order polarization at the anti- Stokes frequency of

*ω*= 2

_{as}*ω*-

_{p}*ω*can be written as

_{s}^{(3)}

*is the third-order nonlinear coefficient of an isotropic medium;*

_{ijkl}*E*(

^{P}_{j}**r**,

*ω*,

_{p}*t*),

*E*(

^{P}_{k}*r*,

*ω*,

_{p}*t*),and

*E*(

^{S*}_{l}**r**,

*ω*,

_{s}*t*) are the time-dependant amplitudes of the pump and the Stokes beams in

*j*,

*k*, and

*l*directions, respectively. If the distributions of the pump and the Stokes light fields, and the third-order nonlinear coefficient are known, the third-order nonlinear polarization at different directions can be determined using Eq. (2), and then the CARS distribution could be derived by solving Eq. (1). However, when the near-field effects are taken into account, such as the presence of non-propagating evanescent waves, it is difficult to directly determine the excitation light fields to obtain the explicit solutions of Eq. (1) due to the complex boundary conditions introduced by the near-field effects. With the advantage of without the need to consider the boundary conditions, the FDTD method is employed in this study to simulate the CARS generation and propagation, by solving the Maxwell equations numerically with leapfrog approach.

14. K.S. Yee “Numerical solution of initial boundary value problem
involving Maxwell equations in isotropic media,” IEEE
Trans. Antennas Propagat. **14**,302 (1966). [CrossRef]

## 3. Simulation results

### 3.1. Geometry and parameters for FDTD simulations

*λ*

_{p}40 (or

*λ*

_{s}/40) at each step, whereby

*λ*

_{p}(750 nm) is the wavelength of the pump beam, and

*λ*

_{s}(852 nm) is the wavelength of the Stokes beam. The excitation volume of the incident light fields was calculated using the vector formulas [18

18. S. Y. Hasegawa, N. Aoyama, A. Futamata, and T. Uchiyama, “Optical tunneling effect calculation of a solid
immersion lens for use in optical disk memory,” Appl.
Opt. **38**,2297 (1999). [CrossRef]

*x*-direction and an axial length of 1400 nm in

*z*-direction, which are almost the same as the values calculated using the conventional vector formulas [18

18. S. Y. Hasegawa, N. Aoyama, A. Futamata, and T. Uchiyama, “Optical tunneling effect calculation of a solid
immersion lens for use in optical disk memory,” Appl.
Opt. **38**,2297 (1999). [CrossRef]

*π*- phase change occurs between different orders of the lobes. The above results confirmed the validity of using the FDTD method to study the nonlinear optical process taking place within the excitation volume of the high N. A. objective.

### 3.2 Distribution of the excitation light fields surrounding nanoparticles

19. K. Takeda, Y. Ito, and C. Munakata, “Simultaneous measurement of size and refractive
index of a fine particle in flowing liquid,” Meas. Sci.
Technol. **3**,27 (1992). [CrossRef]

^{-1}[4

4. J. X. Cheng and X. S. Xie, “Coherent anti-Stokes scattering microscopy:
instrumentation, theory, and application,” J. Phys. Chem.
B **108**,827 (2004). [CrossRef]

*E*along the x- and y- directions [dotted cross lines in Fig. 2(b)]. One notes that the Stokes light field yields a similar distribution in the nanoparticles as compared to the pump beam (data not shown). In comparison with the light field distribution in the focal volume in water without spherical nanoparticles [blue curve in Fig. 2(c)], the illumination field distribution inside the nanoparticle becomes stronger [red and green curves in Fig. 2(c)], and the light field surrounding the nanoparticle (i.e., at the particle-water interface) is also significantly enhanced [red curve in Fig. 2(c)], especially in the x-direction. This phenomenon can be explained as a result of near-field effect, i.e., the generation of the non-propagating evanescent waves, due to the index mismatch at the nanoparticle-water interface. The refractive index of the nanoparticle

*n*is larger than the index of surrounding water

_{particle}*n*, the conservation of the electric displacement

_{water}*D*at the interface leads to an enhanced electric field in water by a factor of

*n*/

_{particle}*n*(1.59/1.33=1.19) [20

_{water}20. M. Born and E Wolf, *Principles of Optics: Electromagnetic Theory of Propagation,
Interference and Diffraction of Light (7th Edition)*
(Cambridge University Press, New York,
2002). [PubMed]

12. C. Liu and S. H. Park, “Anisotropy of near-field speckle
patterns,” Opt. Lett. **30**,1602 (2005). [CrossRef] [PubMed]

*virtual*’ shell surrounding the particles as shown in Fig. 2(c), and therefore, Raman radiation generated from this ‘

*virtual*’ shell is expected to significantly contribute to the CARS signals.

### 3.3. Properties of the induced nonlinear polarization

*χ*, while the scatterer (i.e, nanoparticle) has both resonant and nonresonant components,

^{NR}_{water}*χ*

^{(3)}

*=*

_{sca}*χ*

^{(3)R}

*+*

_{sca}*χ*

^{(3)NR}

*; with the nonresonant susceptibilities of both water and scatterers being frequency independent, then the resonant susceptibilities of*

_{sca}*χ*

^{(3)R}

*can be written as [21]:*

_{sca}_{R}, ω

_{R}, and A

_{R}are the half-width at the half-maximum of the resonant peak, , the resonant vibrational frequency of the Raman mode, and the spontaneous Raman scattering cross-section, respectively.

*χ*

^{(3)NR}

_{1122}=

*χ*

^{(3)NR}

_{1212}=

*χ*

^{(3)NR}

_{1221}=

*χ*

^{(3)NR}

_{1111}/3 . The values of Λ

_{R}=4.1 cm

^{-1}, and

*A*/

_{R}*χ*

^{(3)NR}

*=2.0 cm*

_{sca}^{-1}are chosen in terms of the CARS spectrum of polystyrene bead at the resonant Raman shift (

*ω*= 1600 cm

_{R}^{-1}) and the ratio of nonresonant susceptibilities of water to polystyrene beads (

*χ*

^{(3)NR}

*/*

_{water}*χ*

^{(3)NR}

*=0.6) [5*

_{sca}5. J. X. Cheng, A. Volkmer, and X. S. Xie, “Theoretical and experimental characterization of
coherent anti-Stokes Raman scattering microscopy,” J. Opt.
Soc. Am. B **19**,1363 (2002). [CrossRef]

*χ*

^{(3)R}

_{1122}/

*χ*

^{(3)R}

_{1111}=3/4 [1]. The induced nonlinear cubic polarization is calculated using [1]

*χ*

^{(3)NR}

_{1122}and

*χ*

^{(3)NR}

_{1221}are 0.2 each for water, and 0.333 for polystyrene beads [1].

*x-z*plane, reflecting that the phase changes of the induced polarizations occur at the interface between the particle and the surrounding medium (i.e., water).

### 3.4. Scattering of CARS signals from nanoparticles

4. J. X. Cheng and X. S. Xie, “Coherent anti-Stokes scattering microscopy:
instrumentation, theory, and application,” J. Phys. Chem.
B **108**,827 (2004). [CrossRef]

_{p}/2. However, with further increasing the particle size (larger than λ

_{p}/2), the scattering from particles increases dramatically and dominates over the total scattering. The above results accord with the calculations shown in Fig. 3, whereby the local light field at the ‘

*virtual*’ shell generated due to the near-field effect is much stronger than that inside the particle when the particle sizes are small, and thus the near-field enhanced scattering is stronger than that from the scatterer itself. Meanwhile, with further increasing the particle size (intensity profiles in Fig. 3), the electromagnetic field inside the scatterer and the excitation volume on the scatters increase rapidly, while the enhanced intensity at the ‘

*virtual*’ shell structure (i.e., water-particle interface) drops off quickly with the increasing size, and hence the scattering from the surrounding water outside the particle becomes relatively weak.

_{p}/2. This phenomenon is due to the fact that the phase change occurs at the water-particle interface as shown in Fig. 3 (bottom panel). Owing to the refractive index mismatch (

*Δn*~0.32) at the nanoparticle-water interface, the maximum phase difference between the scatterings from the particle and the water can be

_{2πΔnd /λCARS + φ}and

_{4πΔnd/λCARS + φ}in the forward and backward scatterings, respectively. Where

*d*is the diameter of the particle, and

*φ*is the initial phase difference as shown in Fig. 3. As for the forward scattering (F-CARS), the phase difference is always smaller than π/2, thereby resulting in a constructive interference of scatterings from the particles and the surrounding water for generating strong CARS. In the backward scattering (E-CARS), the phase difference can be smaller than π/2 for smaller particles (less than λ

_{p}/2), thus the constructive interference between the small particles and the water can be established for generating a relatively high CARS signal. However, the phase difference in E-CARS can also be larger than π/2 for larger particles (larger than λ

_{p}/2), resulting in a destructive interference for generating relatively small CARS signals [Fig. 5(b)].

### 3.5 Effect of the induced perpendicular polarization on CARS signals

*x-y*plane, whereby Figs. 6(a), 6(b), and 6(c) represent the distributions for

*x*-component, z-component, and

*y*-component, respectively. Obviously, the amplitude of the perpendicular component [

*y*-component, Fig. 6(c)] is approximately 20 times weaker than that of the

*x*-component [Fig. 6(a)], and this may be the main reason why the perpendicular polarization component was always neglected in most of CARS studies.

*x-y*plane, whereby Figs. 6(d), 6(e), and 6(f) correspond to the distributions of

*x*-component,

*z*-component,

*y*-component, respectively. Compared to the polarization distribution from pure water [Figs. 6(a) and 6(b)], there is only a relatively small change in the amplitudes of

*x*- and

*z*- components from nanoparticles [Figs. 6(d) and 6(e)]. However, the perpendicular component (i.e.,

*y*-component) of the induced polarization from nanoparticles [Fig. 6(f)] is found to be almost 6 times stronger than that from pure water [Fig. 6(c)]. Further calculations show that the enhancement of the perpendicular component only takes places in the regions at the particle-water interface and inside the particle. Therefore, with the advantages of the perpendicular polarization which is strictly confined to the nanoparticle-water interface, the perpendicular polarization CARS signal can be potentially useful for revealing the fine structures of living cells (e.g., membrane).

## 4. Discussion

**14**,302 (1966). [CrossRef]

22. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II:
Structure of the image field in an aplanatic system,”
Proc. Roy. Soc. A **253**,358 (1959). [CrossRef]

4. J. X. Cheng and X. S. Xie, “Coherent anti-Stokes scattering microscopy:
instrumentation, theory, and application,” J. Phys. Chem.
B **108**,827 (2004). [CrossRef]

*virtual*’ shell structure surrounding the surface of the scatterer (Fig. 3). Simulation results also show that the CARS radiation from the virtual ‘shell’ structures can be much stronger than that from the nanoparticles, and it will become dominant in the backward scattering when the particle size is small than the half wavelength of excitation light [Fig. 5(b)]. These simulation results are in agreement with the observations of E-CARS experiments on cells or tissues [4

**108**,827 (2004). [CrossRef]

_{p}/2, the F-scattering is approximately 5 times stronger than the backward scattering in the perpendicular polarization direction, however, its F-scattering of parallel polarization is almost 20 times stronger than the corresponding backscattering (Fig. 5). This phenomenon is probably due to the different scattering volumes responsible for both parallel and perpendicular polarizations. In CARS experiments, both the scatterers and the surrounding solvent can radiate CARS signals in parallel polarization, thus the effective excitation volume is relatively large, leading to a large CARS radiation in parallel polarization. As most of total scattering energy is scattered in forward direction because of a relative large scattering volume (Fig. 4) or constructive interference with phase difference <π/2 (Fig. 3), this enables the forward scattering to be much stronger than that of backscattering. This coincides with most CARS experiments [5

5. J. X. Cheng, A. Volkmer, and X. S. Xie, “Theoretical and experimental characterization of
coherent anti-Stokes Raman scattering microscopy,” J. Opt.
Soc. Am. B **19**,1363 (2002). [CrossRef]

24. T. Wilson and J. B. Tan, “Finite sized coherent and incoherent detectors in
confocal microscopy,” J. Microsc. **182**,61 (1995). [CrossRef]

## 5. Conclusions

## Acknowledgments

## References and links

1. | R. J. H. Clark and R. E. Hester, |

2. | P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced
polarization third order in the electric field strength,”
Phys. Rev. |

3. | Y. R. Shen, |

4. | J. X. Cheng and X. S. Xie, “Coherent anti-Stokes scattering microscopy:
instrumentation, theory, and application,” J. Phys. Chem.
B |

5. | J. X. Cheng, A. Volkmer, and X. S. Xie, “Theoretical and experimental characterization of
coherent anti-Stokes Raman scattering microscopy,” J. Opt.
Soc. Am. B |

6. | H. Wang, Y. Fu, P. Zickmund, R. Shi, and X. J. Cheng, “Coherent anti-Stokes Raman scattering imaging of
axonal myelin in live spinal tissues,” Biophys.
J. |

7. | E. O. Potma, C. L. Evans, and X. S. Xie, “Heterodyne coherent anti-Stokes Raman scattering
(CARS) imaging,” Opt. Lett. |

8. | J. P. Ogilvie, E. Beaurepaire, A. Alexandrou, and M. Joffre, “Fourier-transform coherent anti-Stokes Raman
scattering microscopy,” Opt. Lett. |

9. | D. Oron, N. Dudovich, and Y. Silberberg, “Single-pulse phase-contrast nonlinear Raman
spectroscopy,” Phys. Rev. Lett. |

10. | E. R. Andresen, H. N. Paulsen, V. Birkedal, J. Thϕgersen, and S. R. Keiding, “Broadband multiplex coherent anti-Stokes Raman
scattering microscopy employing photonic-crystal fibers,”
J. Opt. Am. B |

11. | J. X. Cheng and X. S. Xie, “Green’s function formulation for third-harmonic
generation microscopy,” J. Opt. Soc. Am. B |

12. | C. Liu and S. H. Park, “Anisotropy of near-field speckle
patterns,” Opt. Lett. |

13. | T. Ichimura, N. Hayazawa, M. Hashimoto, Y. Inouye, and S. Kawata, “Tip-enhanced coherent anti-Stokes Raman scattering
for vibrational nano-imaging,” Phys. Rev. Lett. |

14. | K.S. Yee “Numerical solution of initial boundary value problem
involving Maxwell equations in isotropic media,” IEEE
Trans. Antennas Propagat. |

15. | A. Taflove, |

16. | R. W. Boyd, |

17. | S. Mukamel, |

18. | S. Y. Hasegawa, N. Aoyama, A. Futamata, and T. Uchiyama, “Optical tunneling effect calculation of a solid
immersion lens for use in optical disk memory,” Appl.
Opt. |

19. | K. Takeda, Y. Ito, and C. Munakata, “Simultaneous measurement of size and refractive
index of a fine particle in flowing liquid,” Meas. Sci.
Technol. |

20. | M. Born and E Wolf, |

21. | H. Lotem, R. T. Lynch, and N. Blombergen, “Interference between Raman resonances in four-wave
difference mixing,” Phys. Rev. |

22. | B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II:
Structure of the image field in an aplanatic system,”
Proc. Roy. Soc. A |

23. | L. Novotny and B. Hecht, |

24. | T. Wilson and J. B. Tan, “Finite sized coherent and incoherent detectors in
confocal microscopy,” J. Microsc. |

**OCIS Codes**

(180.6900) Microscopy : Three-dimensional microscopy

(190.3970) Nonlinear optics : Microparticle nonlinear optics

(300.6230) Spectroscopy : Spectroscopy, coherent anti-Stokes Raman scattering

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: February 5, 2007

Revised Manuscript: March 13, 2007

Manuscript Accepted: March 13, 2007

Published: April 2, 2007

**Virtual Issues**

Vol. 2, Iss. 5 *Virtual Journal for Biomedical Optics*

**Citation**

Cheng Liu, Zhiwei Huang, Fake Lu, Wei Zheng, Dietmar W. Hutmacher, and Colin Sheppard, "Near-field effects on coherent anti-Stokes Raman scattering microscopy imaging," Opt. Express **15**, 4118-4131 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-7-4118

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### References

- R. J. H. Clark and R. E. Hester, Advances in Nonlinear Spectroscopy (Wiley, New York, 1988).
- P. D. Maker and R. W. Terhune, "Study of optical effects due to an induced polarization third order in the electric field strength," Phys. Rev. 137, A801 (1965). [CrossRef]
- Y. R. Shen, Principles of Nonlinear Optics (Wiley, New York, 1984).
- J. X. Cheng and X. S. Xie, "Coherent anti-Stokes scattering microscopy: instrumentation, theory, and application," J. Phys. Chem. B 108, 827 (2004). [CrossRef]
- J. X. Cheng, A. Volkmer and X. S. Xie, "Theoretical and experimental characterization of coherent anti-Stokes Raman scattering microscopy," J. Opt. Soc. Am. B 19, 1363 (2002). [CrossRef]
- H. Wang, Y. Fu, P. Zickmund, R. Shi and J. X. Cheng, "Coherent anti-Stokes Raman scattering imaging of axonal myelin in live spinal tissues," Biophys. J. 89, 581 (2005). [CrossRef] [PubMed]
- E. O. Potma, C. L. Evans and X. S. Xie, "Heterodyne coherent anti-Stokes Raman scattering (CARS) imaging," Opt. Lett. 31, 241 (2006). [CrossRef] [PubMed]
- J. P. Ogilvie, E. Beaurepaire, A. Alexandrou and M. Joffre, "Fourier-transform coherent anti-Stokes Raman scattering microscopy," Opt. Lett. 31, 480 (2006). [CrossRef] [PubMed]
- D. Oron, N. Dudovich and Y. Silberberg, "Single-pulse phase-contrast nonlinear Raman spectroscopy," Phys. Rev. Lett. 89, 273001 (2002). [CrossRef]
- E. R. Andresen and H. N. Paulsen, V. Birkedal, J. Thøgersen and S. R. Keiding, "Broadband multiplex coherent anti-Stokes Raman scattering microscopy employing photonic-crystal fibers," J. Opt. Soc. Am. B 22, 1934 (2005). [CrossRef]
- J. X. Cheng and X. S. Xie, "Green’s function formulation for third-harmonic generation microscopy," J. Opt. Soc. Am. B 19, 1604 (2002). [CrossRef]
- C. Liu and S. H. Park, "Anisotropy of near-field speckle patterns," Opt. Lett. 30, 1602 (2005). [CrossRef] [PubMed]
- T. Ichimura, N. Hayazawa, M. Hashimoto, Y. Inouye and S. Kawata, "Tip-enhanced coherent anti-Stokes Raman scattering for vibrational nano-imaging," Phys. Rev. Lett. 92, 220801 (2004). [CrossRef] [PubMed]
- YeeK.S. "Numerical solution of initial boundary value problem involving Maxwell equations in isotropic media," IEEE Trans. Antennas Propagat. 14, 302 (1966). [CrossRef]
- A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, 1995).
- R. W. Boyd, Nonlinear Optics (Academic, Boston, Mass., 1992).
- S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University Press, New York, 1995).
- S. Y. Hasegawa, N. Aoyama, A. Futamata and T. Uchiyama, "Optical tunneling effect calculation of a solid immersion lens for use in optical disk memory," Appl. Opt. 38, 2297 (1999). [CrossRef]
- K. Takeda, Y. Ito and C. Munakata, "Simultaneous measurement of size and refractive index of a fine particle in flowing liquid," Meas. Sci. Technol. 3, 27 (1992). [CrossRef]
- M. Born and E Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (7th Edition) (Cambridge University Press, New York, 2002). [PubMed]
- H. Lotem, R. T. Lynch and N. Blombergen, "Interference between Raman resonances in four-wave difference mixing," Phys. Rev. 126, 1977 (1962).
- B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems II: Structure of the image field in an Aplanatic system," Proc. Roy. Soc. A 253, 358 (1959). [CrossRef]
- L. Novotny and B. Hecht, Principles of Nano-optics (Cambridge University Press, New York, 2006).
- T. Wilson and J. B. Tan, "Finite sized coherent and incoherent detectors in confocal microscopy," J. Microsc. 182, 61 (1995). [CrossRef]

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