## Spectral analysis of three-dimensional photonic jets

Optics Express, Vol. 16, Issue 18, pp. 14200-14212 (2008)

http://dx.doi.org/10.1364/OE.16.014200

Acrobat PDF (796 KB)

### Abstract

The spatial and spectral properties of three-dimensional photonic jets are studied in a framework employing rigorous Lorentz-Mie theory. The contributions to the field from each spectral component are studied quantitatively and highlight the distinctive features of photonic jets. In particular, the role of evanescent field in photonic jets generated by small spheres is investigated. Secondary lobes in the propagative frequency distribution are also singled out as a distinctive property of photonic jets. It is shown that these differences lead to angular openings of photonic jets at least twice as small as those in comparable ‘Gaussian’ beams.

© 2008 Optical Society of America

## 1. Introduction

1. Z. Chen, A. Taflove, and V. Backman, “Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique,” Opt. Express **12**, 1214–1220 (2004). [CrossRef] [PubMed]

3. Z. Chen, A. Taflove, X. Li, and V. Backman, “Superenhanced backscattering of light by nanoparticles,” Opt. Lett. **31**, 196–198 (2006). [CrossRef] [PubMed]

1. Z. Chen, A. Taflove, and V. Backman, “Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique,” Opt. Express **12**, 1214–1220 (2004). [CrossRef] [PubMed]

2. X. Li, Z. Chen, A. Taflove, and V. Backman, “Optical analysis of nanoparticles via enhanced backscattering facilitated by 3-D photonic nanojets,” Opt. Express **13**, 526–533 (2005). [CrossRef] [PubMed]

2. X. Li, Z. Chen, A. Taflove, and V. Backman, “Optical analysis of nanoparticles via enhanced backscattering facilitated by 3-D photonic nanojets,” Opt. Express **13**, 526–533 (2005). [CrossRef] [PubMed]

4. A. Heifetz, K. Huang, A. V. Sahkian, X. li, A. Taflove, and V. Backman, “Experimental confirmation of backscattering enhancement induced by a photonic jet,” Appl. Phys. Lett. **89**, 221118 (2006). [CrossRef]

5. P. Ferrand, J. Wenger, M. Pianta, H. Rigneault, A. Devilez, B. Stout, N. Bonod, and E. Popov, “Direct imaging of photonic nanojet,” Opt. Express **16**, 6930–6940 (2008). [CrossRef] [PubMed]

6. D. Grojo, P. Delaporte, and A. Cros, “Removal of particles by impulsional laser,” Journal de Physique IV **127**, 145–149 (2005). [CrossRef]

8. W. Guo, Z. B. Wuang, L. Li, D. J. Whitehead, B. S. Luk’yanchuk, and Z. Liu, “Near-field laser parallel nanofabrication of arbitrary-shaped patterns,” Appl. Phys. Lett. **90**, 243101 (2007). [CrossRef]

1. Z. Chen, A. Taflove, and V. Backman, “Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique,” Opt. Express **12**, 1214–1220 (2004). [CrossRef] [PubMed]

3. Z. Chen, A. Taflove, X. Li, and V. Backman, “Superenhanced backscattering of light by nanoparticles,” Opt. Lett. **31**, 196–198 (2006). [CrossRef] [PubMed]

9. S. Lecler, S. Haacke, N. Lecong, O. Crégut, J. L. Rehspringer, and C. Hirlimann, “Photonic jet driven non-linear optics: example of two-photon fluorescence enhancement by dielectric microspheres,” Opt. Express **15**, 4935–4942 (2007). [CrossRef] [PubMed]

8. W. Guo, Z. B. Wuang, L. Li, D. J. Whitehead, B. S. Luk’yanchuk, and Z. Liu, “Near-field laser parallel nanofabrication of arbitrary-shaped patterns,” Appl. Phys. Lett. **90**, 243101 (2007). [CrossRef]

10. J. Kofler and N. Arnold, “Axially symmetric focusing as a cuspoid diffraction catastrophe: Scalar and vector cases and comparison with the theory of Mie,” Phys. Rev. B **73**, 235401 (2006). [CrossRef]

11. A. V. Itagi and W. A. Challener, “Optics of photonic nanojets,” J. Opt. Soc. Am A **22**, 2847–2858 (2005). [CrossRef]

**12**, 1214–1220 (2004). [CrossRef] [PubMed]

## 2. Simulations parameters

*λ*

_{v}=525

*nm*, is adopted. At this wavelength, latex spheres have a refractive index of

*N*=1.6. The surrounding medium is considered to be either air,

_{s}*N*

_{ο}=1, or water,

*N*

_{ο}=1.33. The index contrast will then be respectively equal to

*ρ*=1.6, or

*ρ*=1.2. For a radius

*R*=1

*µm*, the corresponding size parameters are respectively

*k*

_{ο}

*R*=2π

*RN*

_{ο}/

*λ*

_{v}=12 and

*k*

_{ο}

*R*=16.

*µm*in water illuminated by a plane wave is simulated in Fig. 1(a). The photonic jet can roughly be described with 4 parameters displayed in Fig. 1(b): the ‘focal’ distance

*f*from the surface of the sphere to the point of maximum intensity, the intensity enhancement

*I*

_{max}, at the ‘focus’, the beam width

*w*

_{0}at the focus and finally the ‘diffraction length’

*z*

_{r}. These last two parameters will be precisely defined in the next paragraph analogously to Gaussian beam parameters.

*z*and a transverse axis

*x*at

*z*=

*f*. An example is presented in Fig. 2, for

*R*=1

*µm*and

*ρ*=1.2. It has been found that the intensity enhancement along the propagation axis z, displayed in Fig. 2(a), can be fitted in its decreasing part by a Lorentzian distribution. At the same time, the transversal intensity enhancement, displayed in Fig. 2(b) can be predominantly fitted by a Gaussian distribution. Therefore, we define the diffraction length,

*z*

_{r}, as the half width of the photonic jet at the half maximum of the Lorentzian fit of the axial intensity distribution, and the beam width,

*w*(z) as the transverse width at

*I*(

*z*)/

*e*

^{2}of the Gaussian fitted distribution. The width at the diffraction focus,

*w*

_{0}, of the photonic jet is then defined as the width at

*z*=

*f*.

10. J. Kofler and N. Arnold, “Axially symmetric focusing as a cuspoid diffraction catastrophe: Scalar and vector cases and comparison with the theory of Mie,” Phys. Rev. B **73**, 235401 (2006). [CrossRef]

10. J. Kofler and N. Arnold, “Axially symmetric focusing as a cuspoid diffraction catastrophe: Scalar and vector cases and comparison with the theory of Mie,” Phys. Rev. B **73**, 235401 (2006). [CrossRef]

## 3. Spectral study of photonic jets

**M**

*and*

_{n,m}**N**

*[12*

_{n,m}12. B. Stout, M. Nevière, and E. Popov, “Light diffraction by three-dimensional object: differential theory,” J. Opt. Soc. Am. A **22**, 2385–2404 (2005). [CrossRef]

13. O. Moine and B. Stout, “Optical force calculations in arbitrary beams by use of the vector addition theorem,” J. Opt. Soc. Am. B **22**, 1620–1631 (2005). [CrossRef]

*f*

*are the Mie coefficients for scattered field and*

_{n,m}*k*

_{ο}=2

*πN*

_{ο}/

*λ*

_{v}is the propagation wave vector in the host medium. The details of our notations are given in Appendix A. Expressing the transverse wavevector

**K**⃗ as:

*r*

_{‖},

*ϕ*,

*z*) are cylindrical coordinates and

*K*=0).

*m*=1 (arbitrary polarization can of course be obtained by superpositions with circularly left polarized waves,

*m*=-1). Using the Cartesian vector spherical harmonics, the scattered field can be written as:

*A*,

*B*, and

*C*. We focus our attention on the total spectral amplitude

*S*=(

*AA**+

*BB**+

*CC**)

^{1/2}.

*K*/

*k*

_{ο}for a sphere of radius

*R*=1

*µm*at the position

*ϕ*=0,

*z*=1.05

*µm*.

*S*is normalized to 1 at

*K*=0, and plotted for two different index contrasts

*ρ*=1.2 in Fig. 3(a) and

*ρ*=1.6 in Fig. 3(b). The radial spectrum has the following features: a generally decreasing and oscillating features in the region 0 <

*K*/

*k*

_{ο}< 1 corresponding to propagative fields, a singularity originating from the homogeneous medium Green function at

*K*/

*k*

_{ο}=1 and a monotonically decreasing behaviour when

*K*>

*k*

_{0}for evanescent field contributions.

**73**, 235401 (2006). [CrossRef]

*S*as a function of the position along the propagation axis

*z*and the radial frequencies

*K*/

*k*

_{ο}for refraction index contrasts

*ρ*=1.2 in (a) and

*ρ*=1.6 in (b). For both index contrasts, the evanescent field is considerably attenuated after distances of more than 500

*nm*. Consequently, for low index contrasts, like

*ρ*=1.2 where

*f*=1.6

*µm*, the evanescent field will not bring any significant contribution to the width properties of the photonic jet. On the other hand, for

*ρ*=1.6, the photonic jet is on the surface of the sphere so that contribution of the evanescent field may be important. This point and the influence of the high propagative spatial frequencies will be discussed in the next section.

## 4. Photonic jet analysis

### 4.1 Direct space analysis of evanescent and propagative spectral contributions

*K*>

*k*

_{ο}) contributions. The photonic jet intensity along the propagation axis is plotted in red in Fig. 5(a)) while the intensity with the evanescent field removed is plotted in green. The same procedure is studied in Fig. 5(b)) for the transverse intensity at

*z*=

*f*. One remarks that when the evanescent field is removed, the maximum intensity position of the photonic jet moves away from the surface, the maximum field enhancement drops by half its value and the size of the width at diffraction focus is increased by 10 percent (from 230

*nm*to 256

*nm*). This example demonstrates that the evanescent field plays a significant role in the field distribution of photonic jets close to the surface of the sphere, which is the case when index contrasts are large. Nevertheless, the evanescent field cannot be held responsible for the principal photonic jet features, particularly when refractive index contrast is low.

*ρ*=1.6). It shows the angles corresponding to the first few maxima (red) and the minima (black) of

*S*(cf. Fig. 3(b)). The maxima correspond to high intensity angles while the minima correspond to angles of low intensity regions. The maxima in the spectral distribution can therefore be associated with the presence of secondary lobes in the direct field structure. The first secondary lobes tend to confine the central lobe into a low divergent beam, while the secondary lobes with high transverse components tend to reduce the length and the width of photonic jets.

### 4.2 Origin of the secondary lobes

*f*

^{(h)}

_{n,1}| and |

*f*

^{(e)}

_{n,1}| in this expansion are displayed in Fig. 7, where we recall that scattered field in the VSWF basis is expressed:

*n*is approximatively equal to the size parameter

*kR*. This corresponds to rays passing close to the edge of the sphere in the context of Van de Hulst’s localization principle [17]. The scattered intensity of the term of VSWFs of order

*n*=11, which slightly dominates is displayed in Fig. 8. |

*f*

^{(h)}

_{11}

**M**

_{11,1}|

^{2}is displayed in Fig. 8(a) and |

*f*

^{(e)}

_{11}

**N**

_{11,1}|

^{2}in Fig. 8(b). The VSWFs waves have intensity distributions comprised of number of ‘beams’ (there are in fact 2

*n*‘beams’ in a VSWF of order

*n*). The secondary beams seen in Figs. 1 and 6 are therefore intrinsically present in the VSWFs.

*n*=11 (cf. Fig. 7) is displayed in Fig. 9.

**M**and

**N**) tends to enhance the principal (forward direction) beam and reduce the intensity of the secondary beams, especially in the backward direction. However, the forward secondary lobes still contain non negligible intensity.

*λ*~

*R*) of the scatterers which requires an expansion on VSWFs. These secondary beams thus appear as the principal distinguishing feature of photonic jets with respect to the Gaussian beam behaviour generated in a paraxial optics approximation.

## 5. Photonic jet propagation

*w*

_{ο}, and diffraction length

*z*

_{r}are linked by the relation 2

*z*

_{r}=

*k*

_{ο}

*w*

_{ο}

^{2}. According to this formula, low angular openings

*θ*

*≈*

_{d}*w*

_{ο}/

*z*

_{r}are obtained for large waists. But, it has been observed that the principal beam (i.e. the photonic jet) is extremely narrow (cf Fig. 2). This fact is explained by the presence of oscillatory high spatial frequencies which enrich the angular spectrum and confine the field transversally. Using similarities with Gaussian beam of section 1, longitudinal confinement of photonic jets and Gaussian beams have been compared. For low index contrasts, it has been calculated that for identical waists, photonic jets present a Rayleigh length twice greater than classical Gaussian beams. Thus, its angular opening can be estimated to be twice lower than the Gaussian one. The oscillatory high spatial frequencies permit to generate a gaussian-like principal beams with low angular openings and extreme transversal confinement. A microsphere can easily create narrow beams with angular openings at least twice smaller than for a Gaussian beam created by current high numerical aperture optical lenses. This confirms what was observed in the reciprocal space in previous sections. The beam created by a microsphere illuminated by an optical plane wave has a rich spectrum, leading to unique propagation features in direct space.

## 6. Conclusion

## Appendix A: Numerical simulations

**M**

_{n,m}and

**N**

_{n,m}[12

12. B. Stout, M. Nevière, and E. Popov, “Light diffraction by three-dimensional object: differential theory,” J. Opt. Soc. Am. A **22**, 2385–2404 (2005). [CrossRef]

*k*

_{ο}=2

*πN*

_{ο}/

*λ*

_{v}where

*λ*

_{v}is the incident vacuum wavelength and

*N*

_{ο}is the refractive index of the surrounding medium. The

**Y**,

**Z**, and

**X**in eq.(1) are vector spherical harmonics (VSHs) defined in spherical coordinates by [13

13. O. Moine and B. Stout, “Optical force calculations in arbitrary beams by use of the vector addition theorem,” J. Opt. Soc. Am. B **22**, 1620–1631 (2005). [CrossRef]

*Rg*, wherein the outgoing spherical Hankel functions

*h*are replaced by spherical Bessel functions,

_{n}*j*:

_{n}*k*

_{s}=2

*πN*/

_{s}*λ*

_{v}=

*ρk*

_{ο}and

*N*

_{s}the refractive index of the sphere with

*ρ*=

*N*

_{s}/

*N*

_{ο}=

*k*

_{s}/

*k*

_{ο}the refractive index contrast. The

*a*

*and*

_{nm}*s*are respectively the incident and internal coefficients in the field expansion.

_{nm}*E*

_{ο}is a real parameter determining the incident field amplitude. For plane waves, these coefficients are:

_{i}is the incident field polarization. The EM wave is circularly polarized to facilitate the analysis:

*f*

_{n,m}are the scattered coefficients in the field expansion. The boundary conditions at the surface of the sphere,

*r*=

*R*, enable one to link the scattered field coefficients with the incident field coefficients via the ‘Mie’ coefficients:

*ψ*(

_{n}*x*)≡

*xj*(

_{n}*x*) and

*ξ*(

_{n}*x*)≡

*xh*

_{n}^{(+)}(

*x*) are the Ricatti Bessel functions. The ‘Mie’ coefficients only depend on the size parameter

*k*

_{0}

*R*and the refractive index contrast between the two media,

*ρ*. It is usually sufficient to adopt the Wiscombe criterion for the truncation of convergence of numerical simulations as a limit for the truncation of the basis [18

18. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt.19, 1505–1509 (1980). [CrossRef] [PubMed]

## Appendix B: Spectral expansion of the electromagnetic field

**X**

_{n,m,}

**Y**

_{n,m}and

**Z**

_{n,m}are the vector spherical harmonics and

*h*

_{n}is the spherical outgoing Hankel function.

*E*

_{ο}is a real parameter determining the incident field amplitude. It can be shown in the scalar case that:

*Y*

_{n,m}are the scalar spherical harmonics set as :

*P*

^{m}

_{n}are the associated Legendre polynomials and

*c*

_{n,m}is a normalization factor:

*x̂*,

*ŷ*,

*ẑ*are the cartesian unit vectors, and using the Cartesian vector spherical harmonics defined by:

*J*

_{n}are the cylindrical Bessel functions Proceeding further, we obtain:

**M**

_{n,1}(

*k*

_{ο}

*r*∥,

*ϕ*,

*z*) and

**N**

_{n,1}(

*k*

_{ο}

*r*∥

*ϕ*,

*z*) contribute to the field expansion and

## Acknowledgments

## References and Links

1. | Z. Chen, A. Taflove, and V. Backman, “Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique,” Opt. Express |

2. | X. Li, Z. Chen, A. Taflove, and V. Backman, “Optical analysis of nanoparticles via enhanced backscattering facilitated by 3-D photonic nanojets,” Opt. Express |

3. | Z. Chen, A. Taflove, X. Li, and V. Backman, “Superenhanced backscattering of light by nanoparticles,” Opt. Lett. |

4. | A. Heifetz, K. Huang, A. V. Sahkian, X. li, A. Taflove, and V. Backman, “Experimental confirmation of backscattering enhancement induced by a photonic jet,” Appl. Phys. Lett. |

5. | P. Ferrand, J. Wenger, M. Pianta, H. Rigneault, A. Devilez, B. Stout, N. Bonod, and E. Popov, “Direct imaging of photonic nanojet,” Opt. Express |

6. | D. Grojo, P. Delaporte, and A. Cros, “Removal of particles by impulsional laser,” Journal de Physique IV |

7. | S. M. Huang, M. H. Hong, B. Luk’yanchuk, and T. C. Chong, “Nanostructures fabricated on metal surfaces assisted by laser with optical near-field effects,” Appl. Phys. A: Mater. Sci. Process. |

8. | W. Guo, Z. B. Wuang, L. Li, D. J. Whitehead, B. S. Luk’yanchuk, and Z. Liu, “Near-field laser parallel nanofabrication of arbitrary-shaped patterns,” Appl. Phys. Lett. |

9. | S. Lecler, S. Haacke, N. Lecong, O. Crégut, J. L. Rehspringer, and C. Hirlimann, “Photonic jet driven non-linear optics: example of two-photon fluorescence enhancement by dielectric microspheres,” Opt. Express |

10. | J. Kofler and N. Arnold, “Axially symmetric focusing as a cuspoid diffraction catastrophe: Scalar and vector cases and comparison with the theory of Mie,” Phys. Rev. B |

11. | A. V. Itagi and W. A. Challener, “Optics of photonic nanojets,” J. Opt. Soc. Am A |

12. | B. Stout, M. Nevière, and E. Popov, “Light diffraction by three-dimensional object: differential theory,” J. Opt. Soc. Am. A |

13. | O. Moine and B. Stout, “Optical force calculations in arbitrary beams by use of the vector addition theorem,” J. Opt. Soc. Am. B |

14. | M. Born, E. Wolf, M. Born, and E. Wolf Pergamon press, 1986), pp. 556–592. |

15. | M. Born, E. Wolf, M. Born, and E. Wolf (Pergamon press, 1986), pp. 611–664. |

16. | L. Mandel, E. Wolf, L. Mandel, and E. Wolf (Cambridge University press, 1995), pp. 92–146. |

17. | H. C. Van de Hulst and H. C. Van de Hulst (Dover publication, 1981), pp. 200–227. |

18. | W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt.19, 1505–1509 (1980). [CrossRef] [PubMed] |

19. | M. Born, E. Wolf, M. Born, and E. Wolf (Pergamon press, 1986), pp. 370–458. |

**OCIS Codes**

(230.3990) Optical devices : Micro-optical devices

(260.2110) Physical optics : Electromagnetic optics

(290.4020) Scattering : Mie theory

(290.5850) Scattering : Scattering, particles

**ToC Category:**

Optical Devices

**History**

Original Manuscript: May 1, 2008

Revised Manuscript: June 25, 2008

Manuscript Accepted: July 10, 2008

Published: August 27, 2008

**Virtual Issues**

Vol. 3, Iss. 11 *Virtual Journal for Biomedical Optics*

**Citation**

Alexis Devilez, Brian Stout, Nicolas Bonod, and Evgueni Popov, "Spectral analysis of three-dimensional photonic jets," Opt. Express **16**, 14200-14212 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-18-14200

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### References

- Z. Chen, A. Taflove, and V. Backman, "Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique," Opt. Express 12, 1214-1220 (2004). [CrossRef] [PubMed]
- X. Li, Z. Chen, A. Taflove, and V. Backman, "Optical analysis of nanoparticles via enhanced backscattering facilitated by 3-D photonic nanojets," Opt. Express 13, 526-533 (2005). [CrossRef] [PubMed]
- Z. Chen, A. Taflove, X. Li, and V. Backman, "Superenhanced backscattering of light by nanoparticles," Opt. Lett. 31, 196-198 (2006). [CrossRef] [PubMed]
- A. Heifetz, K. Huang, A. V. Sahkian, X. li, A. Taflove, and V. Backman, "Experimental confirmation of backscattering enhancement induced by a photonic jet," Appl. Phys. Lett. 89, 221118 (2006). [CrossRef]
- P. Ferrand, J. Wenger, M. Pianta, H. Rigneault, A. Devilez, B. Stout, N. Bonod, and E. Popov, "Direct imaging of photonic nanojet," Opt. Express 16, 6930 - 6940 (2008). [CrossRef] [PubMed]
- D. Grojo, P. Delaporte, and A. Cros, "Removal of particles by impulsional laser," Journal de Physique IV 127, 145-149 (2005). [CrossRef]
- S. M. Huang, M. H. Hong, B. Luk�??yanchuk, and T. C. Chong, "Nanostructures fabricated on metal surfaces assisted by laser with optical near-field effects," Appl. Phys. A: Mater. Sci. Process. 77, 293-296 (2003).
- W. Guo, Z. B. Wuang, L. Li, D. J. Whitehead, B. S. Luk�??yanchuk, and Z. Liu, "Near-field laser parallel nanofabrication of arbitrary-shaped patterns," Appl. Phys. Lett. 90, 243101 (2007). [CrossRef]
- S. Lecler, S. Haacke, N. Lecong, O. Crégut, J. L. Rehspringer, and C. Hirlimann, "Photonic jet driven non-linear optics: example of two-photon fluorescence enhancement by dielectric microspheres," Opt. Express 15, 4935-4942 (2007). [CrossRef] [PubMed]
- J. Kofler and N. Arnold, "Axially symmetric focusing as a cuspoid diffraction catastrophe: Scalar and vector cases and comparison with the theory of Mie," Phys. Rev. B 73, 235401 (2006). [CrossRef]
- A. V. Itagi and W. A. Challener, "Optics of photonic nanojets," J. Opt. Soc. Am A 22, 2847-2858 (2005). [CrossRef]
- B. Stout, M. Nevière, and E. Popov, "Light diffraction by three-dimensional object: differential theory," J. Opt. Soc. Am. A 22, 2385-2404 (2005). [CrossRef]
- O. Moine and B. Stout, "Optical force calculations in arbitrary beams by use of the vector addition theorem," J. Opt. Soc. Am. B 22, 1620-1631 (2005). [CrossRef]
- M. Born and E. Wolf, "Rigorous diffraction theory," in Principles of optics, M. Born and E. Wolf (Pergamon press, 1986), pp. 556-592.
- M. Born and E. Wolf, "Optics of metals," in Principles of optics, M. Born and E. Wolf (Pergamon press, 1986), pp. 611-664.
- L. Mandel and E. Wolf, "Some useful mathematical techniques," in Optical coherence and quantum optics, L. Mandel and E. Wolf (Cambridge University press, 1995), pp. 92-146.
- H. C. Van de Hulst, "Very large spheres," in Light scattering by small particles, H. C. Van de Hulst (Dover publication, 1981), pp. 200-227.
- W. J. Wiscombe, "Improved Mie scattering algorithms," Appl. Opt. 19, 1505-1509 (1980). [CrossRef] [PubMed]
- M. Born and E. Wolf, "Elements of the theory of diffraction," in Principles of optics, M. Born and E. Wolf (Pergamon press, 1986), pp. 370-458.

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