## Considerations on the electromagnetic flow in Airy beams based on the Gouy phase |

Optics Express, Vol. 20, Issue 21, pp. 23553-23558 (2012)

http://dx.doi.org/10.1364/OE.20.023553

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### Abstract

We reexamine the Gouy phase in ballistic Airy beams (AiBs). A physical interpretation of our analysis is derived in terms of the local phase velocity and the Poynting vector streamlines. Recent experiments employing AiBs are consistent with our results. We provide an approach which potentially applies to any finite-energy paraxial wave field that lacks a beam axis.

© 2012 OSA

## 1. Introduction

*π*/2 rads (

*π*rads for 3D waves) if they are compared with the transit of un-truncated plane waves. For aberration-free focused waves, Gouy first realized that the phase is delayed within its focal region [1]. The origin of the Gouy phase (GP) shift is ascribed to the spatial confinement of the optical beam [2

2. S. Feng and H. G. Winful, “Physical origin of the Gouy phase shift,” Opt. Lett. **26**, 485–487 (2001). [CrossRef]

3. T. D. Visser and E. Wolf, “The origin of the Gouy phase anomaly and its generalization to astigmatic wavefields,” Opt. Commun. **283**, 3371–3375 (2010). [CrossRef]

4. R. W. Boyd, “Intuitive explanation of the phase anomaly of focused light beams,” J. Opt. Soc. Am. **70**, 877–880 (1980). [CrossRef]

6. G. G. Paulus, F. Lindner, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett. **91**, 253004 (2003). [CrossRef]

7. A. Apolonski, P. Dombi, G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C. Lemell, K. Torizuka, J. Burgdörfer, T. Hänsch, and F. Krausz, “Observation of light-phase-sensitive photoemission from a metal,” Phys. Rev. Lett. **92**, 073902 (2004). [CrossRef] [PubMed]

*et al.*studied the phase behavior of Airy beams (AiBs) [8

8. X. Pang, G. Gbur, and T. D. Visser, “The Gouy phase of Airy beams,” Opt. Lett. **36**, 2492–2494 (2011). [CrossRef] [PubMed]

9. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. **32**, 979–981 (2007). [CrossRef] [PubMed]

10. M. A. Bandres, “Accelerating beams,” Opt. Lett. **34**, 3791–3793 (2009). [CrossRef] [PubMed]

12. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Natura Photon. **2**, 675–678 (2008). [CrossRef]

13. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science **324**, 229–232 (2009). [CrossRef] [PubMed]

14. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics **4**, 103–106 (2010). [CrossRef]

15. L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic Airy Beam generated by in-plane diffraction,” Phys. Rev. Lett **107**, 1–4 (2011). [CrossRef]

18. W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar, “Plasmonic Airy beam manipulation in linear optical potentials,” Opt. Lett. **36**, 1164–1166 (2011). [CrossRef] [PubMed]

## 2. Formalism

*ik*

_{0}

*z*)

*u*which evolves along the

*z*-axis with a carrier spatial frequency

*k*

_{0}=

*ω*/

*c*. Thus the wave function

*u*satisfies the paraxial wave equation 2

*i∂*+

_{ζ}u*∂*= 0 expressed in terms of the normalized spatial coordinates

_{ss}u*s*=

*x/w*

_{0}and

*ζ*=

*z/z*. Here

_{R}*w*

_{0}is the beam width and

9. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. **32**, 979–981 (2007). [CrossRef] [PubMed]

*Ai*(·) denotes the Airy function. The parameter 0 <

*a*≤ 1 is introduced in the wave function in order to provide ∫ |

*u*|

^{2}

*ds*< ∞ allowing the paraxial beam to carry a finite energy, and concurrently conserving a central lobe with parabolic shape.

*u*= (2

*π*)

^{−1}∫

*ũ*(

*k*)exp(

*iψ*)

*dk*, being

*ψ*=

*ks*−

*k*

^{2}

*ζ*/2 the phase distribution of a plane-wave spectral component with transverse spatial frequency

*k*. The phase

*ϕ̃*+

*ψ*of the integrand is highly oscillating, where

*ϕ̃*= arg(

*ũ*). However it reaches a stationary point, that is

*∂*[

_{k}*ϕ̃*+

*ψ*] = 0, for two frequencies

*k*

_{±}satisfying The principle of stationary phase [19] (PSP) establishes that the FK diffraction integral has a predominant contribution of frequencies in the vicinities of

*k*

_{±}. Inside the geometrical shadow

*s*>

*a*

^{2}+

*ζ*

^{2}/4 no stationary points may be found. In the light area, constructive interference is attained if the phase

*ϕ̃*+

*ψ*at frequencies

*k*

_{+}and

*k*

_{−}differs by an amount 2

*πm*, where

*m*is an integer. This condition provides the locus of points with peaks in intensity, which leads to These curves describe perfect parabolas.

*k*represents a particular light “ray” with linear trajectory (3) in the

*sζ*-plane. The envelope (caustic) of this family of rays results from the solution given in Eq. (4) for

*m*= 0, providing the ballistic signature of an AiB.

*u*| and the phase

*ϕ*= arg(

*u*) of the wave function given in Eq. (1) corresponding to an AiB of

*a*= 0.1. The accelerating behavior of the AiB is limited, and out of the near field the interference-driven parabolic peaks fade away. To establish the boundaries of the near field, we point out that |

*ũ*| falls off less than a half of its maximum in the interval

*k*is no more than the far field beam angle in the normalized coordinates, and

_{max}*k*= 2.6 in Fig. 1. As a consequence, the length of the caustic is finite and it is observed in |

_{max}*ζ*| < 2

*k*. Along the beam waist

_{max}*ζ*= 0 the energy is mostly localized in the region

*ζ*| < 5.2 and |

*s*| < 6.9, where 4 interference peaks are clearly formed.

*s*. This effect is not caused by the finite energy of the beam since it is observed for

*a*= 0, but it occurs by a non-even symmetry of its spatial spectrum (2).

*k*is of relevance. The resultant Fraunhofer pattern is

*u*→ (2

*πiζ*)

^{−1/2}

*ũ*(

*k*)

*u*(

_{PW}*s*,

*ζ*), valid in the limit |

*ζ*| ≫

*k*for points of the contour

_{max}*C*≡

*s*=

*kζ*+

*a*

^{2}−

*k*

^{2}taken from Eq. (3). Note that the paraxial wave field

*u*= exp (

_{PW}*iks*−

*ik*

^{2}

*ζ*/2) corresponds to a non-truncated plane wave.

## 3. The Gouy phase

*φ*(

_{G}*ζ*) is commonly estimated as the cumulative phase difference between a given paraxial field and a plane wave also traveling in the +

*z*direction [4

4. R. W. Boyd, “Intuitive explanation of the phase anomaly of focused light beams,” J. Opt. Soc. Am. **70**, 877–880 (1980). [CrossRef]

*u*around

*s*= 0, the dephase

*φ*(

_{G}*ζ*) may be derived analytically as the difference

*ϕ*(

*ζ*) − lim

_{ζ→−∞}

*ϕ*evaluated along the beam axis. Due to the particular acceleration of the AiB, however, one cannot encounter a beam axis in this case.

*u*given in (1) and a plane wave

*u*= exp (

_{PW}*iψ*) with a given tilt

*k*≠ 0. Following the discussion given above, the phase fronts of

*u*and

*u*become parallel in the far field around the light ray (3). In order to obtain

_{PW}*φ*(

_{G}*ζ*) for the normalized angle

*k*we employ

*ϕ*−

*ks*+

*k*

^{2}

*ζ*/2 instead of

*ϕ*, which is now evaluated at points of the contour

*C*. In Fig. 2(a) we plot

*φ*(

_{G}*ζ*) for different values of the zenith angle

*k*. As expected, the value of the GP varies rapidly in the near field. Well beyond the near field, in the limit

*ζ*→ +∞, the GP shift approaches −

*π*/2.

## 4. Physical interpretation

*k⃗*= ∇

*ϕ*− ∇

*ψ*and

*C*represents a contour of integration (3) with startpoint at

*ζ*→ −∞. The form is exactly the same as that encountered when we calculate the work done by a resultant

*force*Δ

*k⃗*that varies along the path

*C*. Here Δ

*k⃗*is understood as the difference of the local wave vector

*k⃗*=

_{i}*k*

_{0}

*z*̂ + ∇

*ϕ*of the AiB and that corresponding to the reference plane wave,

*k⃗*=

_{PW}*k*

_{0}

*z*̂ + ∇

*ψ*. Note that

*k⃗*approaches

_{PW}*k*

_{0}

*z*̂ + (

*k/w*

_{0})

*x*̂ to order

*k*, and that

*k⃗*‖

_{PW}*dr⃗*over the contour

*C*. This is of relevance since many physical processes, like the generation of curved plasma channels [13

13. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science **324**, 229–232 (2009). [CrossRef] [PubMed]

12. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Natura Photon. **2**, 675–678 (2008). [CrossRef]

*k⃗*, which is in direct proportion to the electromagnetic momentum and the time-averaged flux of energy [20

_{i}20. H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express **16**, 9411–9416 (2008). [CrossRef] [PubMed]

*r⃗*to

*r⃗*+

*dr⃗*over

*C*leads to a nonnegative contribution of the line integral (5) if (a) the wave vectors

*k⃗*and

_{PW}*k⃗*are nonparallel, and if (b) the wavenumber

_{i}*k*

_{0}of the reference plane wave and that

*k*=

_{i}*|k⃗*of the field

_{i}|*u*are different. This is illustrated in Fig. 2(b). For a Gaussian beam

*φ*= −

_{G}*π*/4 − arctan(

*ζ*)/2 at

*k*= 0, where

*k⃗*‖

_{i}*k⃗*but

_{PW}*k*<

_{i}*k*

_{0}. In AiBs, however, both angular and modular detuning of

*k⃗*are produced.

_{i}*k⃗*graphically by means of the Poynting vector streamlines (PSLs). Commonly employed with vector fields, the PSLs are tangent to the vector

_{i}*k⃗*and consequently they satisfy the differential equation

_{i}*k⃗*×

_{i}*dr⃗*= 0, that is

*dx/dz*= (

*k⃗*̂)/(

_{i}· x*k⃗*·

_{i}*z*̂). The PSLs indicate the direction of wave propagation since they are perpendicular to the phase fronts. Under the paraxial approximation the equation for the PSLs reduces to

*ds*/

*dζ*=

*∂*in normalized coordinates. For an AiB we finally have where

_{s}ϕ*Ai*′(

*α*) =

*∂*(

_{α}Ai*α*). The exact solution

*s*=

*s*

_{0}+

*ζ*

^{2}/4 is obtained for

*a*= 0.

*C*of light rays (dashed lines), the PSLs hold

*ds*/

*dζ*= 0 at

*ζ*= 0. Therefore, PSLs approach a parabola

*s*=

*s*

_{0}+

*s*

_{0}″

*ζ*

^{2}/2 in the neighbourhood of the beam waist, where

*s*″

_{0}=

*as*

_{0}−

*aAi*′(

*s*

_{0})

^{2}/

*Ai*(

*s*

_{0})

^{2}+ 1/2. If

*a*≪ 1 and

*s*

_{0}≥ 0 then

*s*

_{0}″ ≈ 1/2 featuring a regular parabolic trajectory. However for sufficiently large values of −

*s*

_{0}≫ 1 then

*s*

_{0}″ < 0 revealing a concavity inversion along the semi-axis. Moreover, in the far field

*ds*/

*dζ*→

*k*leading to exact solutions in the form of Eq. (3) as |

*ζ*| → ∞. These straight lines represent the asymptotes of the PSLs.

*k*

_{0}. In fact, modular detuning of

*k⃗*implies a local deviation of the phase velocity

_{i}*v*=

_{i}*ω*/

*k*with respect to the speed of light

_{i}*c*=

*ω*/

*k*

_{0}in vacuum [19]. Taking into account the paraxial regime, the local wavenumber is given by

*k*=

_{i}*k*

_{0}+

*κ*/

*z*, where Moreover we may obtain a simple expression for

_{R}*v*by using Δ

_{i}*v/c*≈ −Δ

*k/k*

_{0}= −

*κ*/

*k*

_{0}

*z*, where Δ

_{R}*v*=

*v*−

_{i}*c*and Δ

*k*=

*k*−

_{i}*k*

_{0}. In Fig. 3 we plot the parameter

*κ*operating as a trend indicator of the spatial variation of the phase velocity of the AiB depicted in Fig. 1. In the far field

*κ*vanishes leading to a wave field with wavenumber

*k*=

_{i}*k*

_{0}and phase velocity

*v*=

_{i}*c*. However

*κ*presents a more complex behavior in the near field. Out of the geometrical shadow,

*κ*< 0 and it drops near the peaks of intensity. This effect is associated with superluminality, which is well known in Gaussian beams and other kind of focused beams [21

21. M. A. Porras, C. J. Zapata-Rodríguez, and I. Gonzalo, “Gouy wave modes: undistorted pulse focalization in a dispersive medium,” Opt. Lett. **32**, 3287–3289 (2007). [CrossRef] [PubMed]

22. C. J. Zapata-Rodríguez and M. A. Porras, “Controlling the carrier-envelope phase of few-cycle focused laser beams with a dispersive beam expander,” Opt. Express **16**, 22090–22098 (2008). [CrossRef] [PubMed]

*κ*≈ 0 and it strictly vanishes if

*a*= 0. On the contrary,

*κ*grows sharply around the valleys of intensity, and for

*a*= 0 it diverges due to the presence of phase singularities.

## 5. Conclusions

*force*Δ

*k⃗*that varies along the given path. Here Δ

*k⃗*is understood as the difference of the local wave vector of the AiB, which is in direct proportion to the electromagnetic momentum and the time-averaged flux of energy, and that corresponding to the reference plane wave. The equations describing the Poynting vector streamlines and the spatial variation of the phase velocity for these beams provide a general platform for exploring the flow of electromagnetic energy. These ideas can be used for a variety of applications. For example, our procedure facilitates a means to optically control the movement of particles in air and fluid [12

12. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Natura Photon. **2**, 675–678 (2008). [CrossRef]

13. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science **324**, 229–232 (2009). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” Compt. Rendue Acad. Sci. (Paris) |

2. | S. Feng and H. G. Winful, “Physical origin of the Gouy phase shift,” Opt. Lett. |

3. | T. D. Visser and E. Wolf, “The origin of the Gouy phase anomaly and its generalization to astigmatic wavefields,” Opt. Commun. |

4. | R. W. Boyd, “Intuitive explanation of the phase anomaly of focused light beams,” J. Opt. Soc. Am. |

5. | A. E. Siegman, |

6. | G. G. Paulus, F. Lindner, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett. |

7. | A. Apolonski, P. Dombi, G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C. Lemell, K. Torizuka, J. Burgdörfer, T. Hänsch, and F. Krausz, “Observation of light-phase-sensitive photoemission from a metal,” Phys. Rev. Lett. |

8. | X. Pang, G. Gbur, and T. D. Visser, “The Gouy phase of Airy beams,” Opt. Lett. |

9. | G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. |

10. | M. A. Bandres, “Accelerating beams,” Opt. Lett. |

11. | Y. Hu, G. Siviloglou, P. Zhang, N. Efremidis, D. Christodoulides, and Z. Chen, “Self-accelerating Airy beams: generation, control, and applications,” in |

12. | J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Natura Photon. |

13. | P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science |

14. | A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics |

15. | L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic Airy Beam generated by in-plane diffraction,” Phys. Rev. Lett |

16. | A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, “Generation and near-field imaging of Airy surface plasmons,” Phys. Rev. Lett |

17. | P. Zhang, S. Wang, Y. Liu, X. Yin, C. Lu, Z. Chen, and X. Zhang, “Plasmonic Airy beams with dynamically controlled trajectories,” Opt. Lett. |

18. | W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar, “Plasmonic Airy beam manipulation in linear optical potentials,” Opt. Lett. |

19. | M. Born and E. Wolf, |

20. | H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express |

21. | M. A. Porras, C. J. Zapata-Rodríguez, and I. Gonzalo, “Gouy wave modes: undistorted pulse focalization in a dispersive medium,” Opt. Lett. |

22. | C. J. Zapata-Rodríguez and M. A. Porras, “Controlling the carrier-envelope phase of few-cycle focused laser beams with a dispersive beam expander,” Opt. Express |

**OCIS Codes**

(260.1960) Physical optics : Diffraction theory

(350.5030) Other areas of optics : Phase

(070.7345) Fourier optics and signal processing : Wave propagation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: July 10, 2012

Revised Manuscript: August 6, 2012

Manuscript Accepted: August 6, 2012

Published: September 28, 2012

**Citation**

Carlos J. Zapata-Rodríguez, David Pastor, and Juan J. Miret, "Considerations on the electromagnetic flow in Airy beams based on the Gouy phase," Opt. Express **20**, 23553-23558 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-21-23553

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### References

- L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” Compt. Rendue Acad. Sci. (Paris)110, 1251–1253 (1890).
- S. Feng and H. G. Winful, “Physical origin of the Gouy phase shift,” Opt. Lett.26, 485–487 (2001). [CrossRef]
- T. D. Visser and E. Wolf, “The origin of the Gouy phase anomaly and its generalization to astigmatic wavefields,” Opt. Commun.283, 3371–3375 (2010). [CrossRef]
- R. W. Boyd, “Intuitive explanation of the phase anomaly of focused light beams,” J. Opt. Soc. Am.70, 877–880 (1980). [CrossRef]
- A. E. Siegman, Lasers (University Science Books, Mill Valley, 1986).
- G. G. Paulus, F. Lindner, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett.91, 253004 (2003). [CrossRef]
- A. Apolonski, P. Dombi, G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C. Lemell, K. Torizuka, J. Burgdörfer, T. Hänsch, and F. Krausz, “Observation of light-phase-sensitive photoemission from a metal,” Phys. Rev. Lett.92, 073902 (2004). [CrossRef] [PubMed]
- X. Pang, G. Gbur, and T. D. Visser, “The Gouy phase of Airy beams,” Opt. Lett.36, 2492–2494 (2011). [CrossRef] [PubMed]
- G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett.32, 979–981 (2007). [CrossRef] [PubMed]
- M. A. Bandres, “Accelerating beams,” Opt. Lett.34, 3791–3793 (2009). [CrossRef] [PubMed]
- Y. Hu, G. Siviloglou, P. Zhang, N. Efremidis, D. Christodoulides, and Z. Chen, “Self-accelerating Airy beams: generation, control, and applications,” in Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and R. Morandotti, eds. (Springer, 2012), vol. 170, pp. 1–46.
- J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Natura Photon.2, 675–678 (2008). [CrossRef]
- P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science324, 229–232 (2009). [CrossRef] [PubMed]
- A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics4, 103–106 (2010). [CrossRef]
- L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic Airy Beam generated by in-plane diffraction,” Phys. Rev. Lett107, 1–4 (2011). [CrossRef]
- A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, “Generation and near-field imaging of Airy surface plasmons,” Phys. Rev. Lett107, 116802 (2011). [CrossRef] [PubMed]
- P. Zhang, S. Wang, Y. Liu, X. Yin, C. Lu, Z. Chen, and X. Zhang, “Plasmonic Airy beams with dynamically controlled trajectories,” Opt. Lett.36, 3191–3193 (2011). [CrossRef] [PubMed]
- W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar, “Plasmonic Airy beam manipulation in linear optical potentials,” Opt. Lett.36, 1164–1166 (2011). [CrossRef] [PubMed]
- M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University Press, 1999).
- H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express16, 9411–9416 (2008). [CrossRef] [PubMed]
- M. A. Porras, C. J. Zapata-Rodríguez, and I. Gonzalo, “Gouy wave modes: undistorted pulse focalization in a dispersive medium,” Opt. Lett.32, 3287–3289 (2007). [CrossRef] [PubMed]
- C. J. Zapata-Rodríguez and M. A. Porras, “Controlling the carrier-envelope phase of few-cycle focused laser beams with a dispersive beam expander,” Opt. Express16, 22090–22098 (2008). [CrossRef] [PubMed]

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