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Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 7, Iss. 3 — Feb. 29, 2012
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Fast vectorial calculation of the volumetric focused field distribution by using a three-dimensional Fourier transform

J. Lin, O. G. Rodríguez-Herrera, F. Kenny, D. Lara, and J. C. Dainty  »View Author Affiliations


Optics Express, Vol. 20, Issue 2, pp. 1060-1069 (2012)
http://dx.doi.org/10.1364/OE.20.001060


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Abstract

We show that the volumetric field distribution in the focal region of a high numerical aperture focusing system can be efficiently calculated with a three-dimensional Fourier transform. In addition to focusing in a single medium, the method is able to calculate the more complex case of focusing through a planar interface between two media of mismatched refractive indices. The use of the chirp z-transform in our numerical implementation of the method allows us to perform fast calculations of the three-dimensional focused field distribution with good accuracy.

© 2012 OSA

1. Introduction

The field distribution in the focal region of a focusing lens has been studied extensively due to the presence of this element in a large number of optical systems [1

1. J. J. Stamnes, Waves in Focal Regions (Hilger, 1986).

]. The Huygens-Fresnel principle, with a few approximations to simplify the diffraction integral, is often applied in the analysis of this particular diffraction problem. For instance, the Fresnel diffraction integral can be used to calculate the focused field for a lens with low numerical aperture (NA) [2

2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

], whereas the scalar Debye integral is a good approximation for a focusing system with higher NA [3

3. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).

]. It is well known, however, that the focused field experiences a change in its polarization state under the high-NA condition [4

4. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959). [CrossRef]

]. Therefore, a vectorial calculation has to be adopted in order to evaluate all the polarization components at the focus. The vectorial Debye integral, usually known as the Debye-Wolf integral, established by Richards and Wolf [4

4. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959). [CrossRef]

], is the most widely used vectorial diffraction method. For some simple cases, this method can give a closed-form expression for the three-dimensional (3D) field distribution around the focus. Nevertheless, numerical evaluation of the integral is necessary for a more general analysis with an arbitrary field incident on the lens. Since the integrand of the Debye-Wolf integral involves highly oscillatory functions, extra caution and a long computation time are often needed to obtain accurate numerical results. Recently, Leutenegger et al. demonstrated that the 3D vectorial field distribution at the focus can be computed plane by plane under a proper transformation of the Debye-Wolf integral [5

5. M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express 14(23), 11277–11291 (2006). [CrossRef] [PubMed]

]. The field in a plane at a given axial position is obtained by a two-dimensional Fourier transform, which can be implemented efficiently on a computer with the help of algorithms such as the fast Fourier transform (FFT) and the chirp z-transform (CZT) [6

6. A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time signal processing, 2nd ed. (Prentice Hall, 1999).

].

In an alternative approach to the evaluation of the diffraction integral, McCutchen showed that the scalar Debye integral can be written as a three-dimensional Fourier transform (3D-FT) [7

7. C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. 54(2), 240–244 (1964). [CrossRef]

]. Furthermore, McCutchen argued that the method is applicable to the vectorial case if it is applied to each component of the electromagnetic (EM) field separately [8

8. C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image: erratum,” J. Opt. Soc. Am. A 19(8), 1721–1721 (2002). [CrossRef]

]. The suitability of McCutchen’s method to model focusing of complex incident polarization states has been previously analyzed [9

9. I. Iglesias and B. Vohnsen, “Polarization structuring for focal volume shaping in high-resolution microscopy,” Opt. Commun. 271(1), 40–47 (2007). [CrossRef]

]. In this work, we show the equivalence between McCutchen’s method and the Debye-Wolf integral. In other words, we show that the Debye-Wolf integral can be written as a 3D-FT, simplifying the calculation of the complete volumetric three-dimensional field distribution in the focal region of a high-NA system. The use of the CZT algorithm in the implementation of our method allows for fast and accurate calculations when compared with the results obtained from direct evaluation of the Debye-Wolf integral. We present a number of examples showing the suitability of the 3D-FT method in different scenarios.

2. Light focused into a homogenous medium

The change of polarization in the focal region of a high-NA focusing system comes from refraction at the system, which can be represented by a transformation of the electric field components at the reference surface representing a perfect converging wave. For an aplanatic system, which is the class of systems considered in this work, the reference surface is a spherical shell limited by the NA (Fig. 1
Fig. 1 Refraction of the incident electric field at the reference sphere of an aplanatic system.
).

Consider a non-paraxial incident monochromatic wave having three orthogonal local polarization components along the directions depicted by unit vectors eρ, eϕ, and ez shown in Fig. 1. At the reference surface, vectors eρ and ez transform into eθ and er, respectively, upon refraction, while eϕ remains the same. Thus, for incident linearly polarized light, the transformation results in a space-variant polarization state that generates polarization components other than the original ones. This transformation can be described as follows
Eref=[ExrefEyrefEzref]=A0cosθM1[ExinEyinEzin]
(1)
where Emin and Emref, for m=x,y,z, are the components of the incident and refracted fields, and the transformation matrix, M1, is given by

M1=12[[(cosθ+1)+(cosθ1)cos2ϕ](cosθ1)sin2ϕ2sinθcosϕ(cosθ1)sin2ϕ[(cosθ+1)(cosθ1)cos2ϕ]2sinθsinϕ2sinθcosϕ2sinθsinϕ2cosθ].
(2)

The factor A0cosθ, with A0 a constant, is the apodization function to account for energy conservation [4

4. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959). [CrossRef]

]. The transformation given by M1 in Eq. (2) is a matrix representation of the transformation required to calculate the strength factor in the Debye-Wolf integral. The propagation after refraction of the three orthogonally polarized components (in x-, y-, and z-direction) is governed by the individual scalar Debye integral
Emf(P)=iAλΩEmref(Q)exp(ikqR)dΩ,m=x,y,z
(3)
where λ is the wavelength of the monochromatic light, in the corresponding medium, and k is the wave number. Figure 2
Fig. 2 Geometry for the calculation of the focused field distribution for an individual polarization state. q is the unit vector in the direction OQ and R is the position vector for point P in the focal region.
is a diagram showing the geometry of the problem. The resultant vectorial field distribution around the focus is the superposition of the focused fields from the three polarization components.

It has been shown that the scalar Debye integral can be rewritten in the form of a scaled 3D-FT [7

7. C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. 54(2), 240–244 (1964). [CrossRef]

,10

10. J. Lin, X.-C. Yuan, S. S. Kou, C. J. R. Sheppard, O. G. Rodríguez-Herrera, and J. C. Dainty, “Direct calculation of a three-dimensional diffracted field,” Opt. Lett. 36(8), 1341–1343 (2011). [CrossRef] [PubMed]

]. That is, Eq. (3) can be rewritten as

Ef(P)=iAλVEref(Q)S(Q)exp(ikqR)dV.
(4)

In Eq. (4) the pupil function represents an infinitesimally thin spherical shell defined as S(Q)=δ(r1), with δ() the Dirac delta distribution and r=x2+y2+z2.

Now, as stated by Richards and Wolf [4

4. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959). [CrossRef]

], the Debye-Wolf integral for the electric field can be written as
E(P)=ik2πΩa(sx,sy)szexp(ik[Φ(sx,sy)+srp])dsxdsy
(5)
where a(sx,sy) is the so-called strength factor, which can be calculated using the matrix in Eq. (2), Φ(sx,sy) is the aberration function, s is a unit vector pointing from a point in the spherical pupil to the focus, rp is the vector from the focus to the point P, where the field is to be calculated, and Ω is the solid angle subtended by the aperture of the lens as seen from the focal point. Vectors s and rp are given by

s=(sinθcosϕ,sinθsinϕ,cosθ)rp=(rpsinθpcosϕp,rpsinθpsinϕp,rpcosθp).
(6)

Considering a perfect lens, i.e. with no aberrations, and substituting s and rp from Eq. (6) in Eq. (5), we find that

E(P)=ik2π02π0πa(θ,ϕ)exp(ikrp[sinθpsinθcos(ϕpϕ)+cosθpcosθ])sinθdθdϕ.
(7)

Noting that the strength factor, a(θ,ϕ), is only defined on the surface of a sphere of radius k, since s is a unitary vector, it is apparent that Eq. (7) can be written as the following triple integral
E(P)=i2πk×02π0π0A(θ,ϕ)exp(ikrp[sinθpsinθcos(ϕpϕ)+cosθpcosθ])ρ2sinθdθdϕ
(8)
with A(θ,ϕ)=a(θ,ϕ)δ(ρk) for ρ the distance, from the focus, in the direction of the unit s vector. Equation (8) was derived explicitly from the Debye-Wolf integral in Eq. (5), and is the 3D-FT of A(θ,ϕ) in spherical coordinates, except for a constant factor. Therefore, from Eqs. (4) and (8), the equivalence between the Debye-Wolf integral and McCutchen’s method is clear. This equivalence, of course, is also valid for the corresponding integral for the magnetic field in the focal region.

Implementing the 3D-FT in either Eq. (4) or Eq. (8), after transformation to Cartesian coordinates, using the FFT algorithm would result in an output space with the same transverse dimension and sampling size as the aperture. However, the region of interest (ROI) in the high-NA case is only a small portion at the center (~2μm × 2μm) as compared with the aperture size of the lens (typically a few mm in radius). To sufficiently resolve details of the ROI, zero padding is often used to increase the output sampling size. Since the ROI only accounts for a small part of the whole output, most of the additional output sampling points brought in by the padding will fall out of the ROI. Furthermore, the zero padding inevitably increases the size of the 3D input matrix by tens of times, increasing the demand for computational resources. Therefore, we opted for the CZT algorithm [6

6. A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time signal processing, 2nd ed. (Prentice Hall, 1999).

] to evaluate the 3D-FT. At first, the CZT algorithm appears slower than the FFT in dealing with an input with size of a power of two. Nevertheless, this algorithm allows us to avoid the use of zero padding and select only the ROI as the output and, hence, turns out to be much faster than the FFT algorithm in our calculations. For a size of the 3D output space of 128×128×128 pixels, the calculation time for our method is typically a few seconds on a personal computer, whereas it takes tens of minutes for the direct integration of the Debye-Wolf integral. The gain in computation time does not reduce the accuracy of the results. Figure 3
Fig. 3 Comparison between the x-component of the electric field obtained by direct evaluation of Debye-Wolf integral (red solid line) and the 3D-FT method (blue circles) along different lines: (a) & (b) y = z = 0; (c) & (d) x = z = 0; (e) & (f) x = y = 0.
is a comparison between different cross sections of the Ex component in the focal region of an aplanatic system, with NA=0.95, for incident light linearly polarized in the x-direction, λ0=532nm, as obtained by direct evaluation of the Debye-Wolf integral and the 3D-FT method. The relative error in the focal volume between the two methods was calculated as [11

11. P. Török, P. R. T. Munro, and E. E. Kriezis, “Rigorous near- to far-field transformation for vectorial diffraction calculations and its numerical implementation,” J. Opt. Soc. Am. A 23(3), 713–722 (2006). [CrossRef] [PubMed]

],
ε=i=1N|EDW(ri)E3DFT(ri)|2i=1N|EDW(ri)|2
(9)
where EDW and E3D-FT are the complex amplitudes of the electric field vector obtained with the Debye-Wolf integral and the 3D-FT method, respectively, and N is the number of pixels compared. The complex amplitudes in Eq. (9) are normalized with respect to the maximum intensity obtained in each method. The relative error calculated in this case is ε=1.512×103, which shows the overall good accuracy obtained with the 3D-FT method. This value, however, does not offer information about where, within the focal volume, the comparison between both methods is better or worse. Figure 4a
Fig. 4 (a) Axial distribution of ε for planes parallel to the focal plane. Cross-section of the magnitude of the difference between the two methods, Δ, along the (b) x- and (c) y-direction in the focal plane.
shows values of ε over planes parallel to the focal plane for different axial positions, z. This figure shows that the accuracy of the 3D-FT method is best at the focal plane, reduces as we move away in the z-direction, and slightly improves at the edge of the volume analyzed. As can be expected, the accuracy is worse at planes close to the minima in the focused field axial distribution. This behavior reflects the role of computational precision rather than a limitation of the 3D-FT method.

3. Light focused through an interface

4. Conclusion

We have shown the equivalence between McCutchen’s method and the Debye-Wolf integral by rewriting the latter as a three-dimensional Fourier transform. Thus, the three-dimensional vectorial field distribution in the focal region of a high-NA system predicted by Debye-Wolf integral can be accurately calculated with the 3D-FT method. This method is able to deal with an arbitrary input field, including vortex and cylindrical beams. We have also shown that the method is applicable to focusing through an interface between two media of mismatch refractive indices by introducing the spherical aberration function due to the interface. The method can be extended to a stratified medium [18

18. P. Török and P. Varga, “Electromagnetic diffraction of light focused through a stratified medium,” Appl. Opt. 36(11), 2305–2312 (1997). [CrossRef] [PubMed]

] and other scenarios where the Debye-Wolf integral is applicable. In this case, the transformation matrix has to be modified accordingly. Note that, however, this modification is also required in the calculation of the strength factor for the direct integration of the Debye-Wolf integral.

The proposed method facilitates the fast vectorial calculation of a focal field. One important advantage is that the calculation time of the 3D-FT method does not depend on the complexity of the incident field, whereas the calculation time to evaluate the Debye-Wolf integral is largely determined by the input. The integration over the azimuthal angle in the Debye-Wolf integral can be done analytically for uniform input polarization (e.g. linearly polarized light). This leaves only one integral to be performed numerically, for each component of the field (x, y, and z), which can speed up the calculation. However, even in these favorable conditions for direct integration of the Debye-Wolf integral, the 3D-FT method is significantly faster. We compared the speed using a volume of 4λ×4λ×4λ around the focal point, for cubic pixels with side λ/30 (121×121×121 points) using MATLAB on a PC with an Intel Core2 Duo 2.53 GHz processor and 4 GB of RAM. The typical calculation time for direct evaluation of the Debye-Wolf integral is around 23 minutes. The same result obtained with the 3D-FT method takes approximately 2 seconds, and the accuracy achieved, measured as the relative error, is in the order of 10−3. The maximum benefit of the method presented here is obtained when the field in the whole 3D focal volume is desired. If the focused field is required only in a small number of points (e.g. a single plane at the focus) direct integration of Debye-Wolf integral may be sufficiently fast.

In conclusion, the 3D-FT method, and its implementation using the CZT algorithm, allows for fast, efficient, and accurate calculation of the volumetric focused field distribution when compared with direct integration of the Debye-Wolf integral.

Acknowledgments

J. Lin acknowledges the fellowship support from the Agency for Science, Technology and Research (A*STAR), Singapore. O. G. Rodríguez-Herrera, F. Kenny, and J. C. Dainty are grateful to Science Foundation Ireland for its support through grant 07/IN.1/I906.

References

1.

J. J. Stamnes, Waves in Focal Regions (Hilger, 1986).

2.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

3.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).

4.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959). [CrossRef]

5.

M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express 14(23), 11277–11291 (2006). [CrossRef] [PubMed]

6.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time signal processing, 2nd ed. (Prentice Hall, 1999).

7.

C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. 54(2), 240–244 (1964). [CrossRef]

8.

C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image: erratum,” J. Opt. Soc. Am. A 19(8), 1721–1721 (2002). [CrossRef]

9.

I. Iglesias and B. Vohnsen, “Polarization structuring for focal volume shaping in high-resolution microscopy,” Opt. Commun. 271(1), 40–47 (2007). [CrossRef]

10.

J. Lin, X.-C. Yuan, S. S. Kou, C. J. R. Sheppard, O. G. Rodríguez-Herrera, and J. C. Dainty, “Direct calculation of a three-dimensional diffracted field,” Opt. Lett. 36(8), 1341–1343 (2011). [CrossRef] [PubMed]

11.

P. Török, P. R. T. Munro, and E. E. Kriezis, “Rigorous near- to far-field transformation for vectorial diffraction calculations and its numerical implementation,” J. Opt. Soc. Am. A 23(3), 713–722 (2006). [CrossRef] [PubMed]

12.

D. Ganic, X. Gan, and M. Gu, “Focusing of doughnut laser beams by a high numerical-aperture objective in free space,” Opt. Express 11(21), 2747–2752 (2003). [CrossRef] [PubMed]

13.

J. Lin, X.-C. Yuan, S. H. Tao, and R. E. Burge, “Variable-radius focused optical vortex with suppressed sidelobes,” Opt. Lett. 31(11), 1600–1602 (2006). [CrossRef] [PubMed]

14.

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1(1), 1–57 (2009). [CrossRef]

15.

C. J. R. Sheppard and M. Gu, “Axial imaging through an aberration layer of water in confocal microscopy,” Opt. Commun. 88(2-3), 180–190 (1992). [CrossRef]

16.

P. Török, P. Varga, Z. Laczik, and G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12(2), 325–332 (1995). [CrossRef]

17.

S. H. Wiersma, P. Török, T. D. Visser, and P. Varga, “Comparison of different theories for focusing through a plane interface,” J. Opt. Soc. Am. A 14(7), 1482–1490 (1997). [CrossRef]

18.

P. Török and P. Varga, “Electromagnetic diffraction of light focused through a stratified medium,” Appl. Opt. 36(11), 2305–2312 (1997). [CrossRef] [PubMed]

OCIS Codes
(050.1960) Diffraction and gratings : Diffraction theory
(110.0180) Imaging systems : Microscopy
(180.6900) Microscopy : Three-dimensional microscopy
(260.1960) Physical optics : Diffraction theory
(260.5430) Physical optics : Polarization
(070.7345) Fourier optics and signal processing : Wave propagation

ToC Category:
Physical Optics

History
Original Manuscript: August 23, 2011
Revised Manuscript: October 9, 2011
Manuscript Accepted: October 11, 2011
Published: January 4, 2012

Virtual Issues
Vol. 7, Iss. 3 Virtual Journal for Biomedical Optics

Citation
J. Lin, O. G. Rodríguez-Herrera, F. Kenny, D. Lara, and J. C. Dainty, "Fast vectorial calculation of the volumetric focused field distribution by using a three-dimensional Fourier transform," Opt. Express 20, 1060-1069 (2012)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-20-2-1060


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References

  1. J. J. Stamnes, Waves in Focal Regions (Hilger, 1986).
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  3. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).
  4. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci.253(1274), 358–379 (1959). [CrossRef]
  5. M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express14(23), 11277–11291 (2006). [CrossRef] [PubMed]
  6. A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time signal processing, 2nd ed. (Prentice Hall, 1999).
  7. C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am.54(2), 240–244 (1964). [CrossRef]
  8. C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image: erratum,” J. Opt. Soc. Am. A19(8), 1721–1721 (2002). [CrossRef]
  9. I. Iglesias and B. Vohnsen, “Polarization structuring for focal volume shaping in high-resolution microscopy,” Opt. Commun.271(1), 40–47 (2007). [CrossRef]
  10. J. Lin, X.-C. Yuan, S. S. Kou, C. J. R. Sheppard, O. G. Rodríguez-Herrera, and J. C. Dainty, “Direct calculation of a three-dimensional diffracted field,” Opt. Lett.36(8), 1341–1343 (2011). [CrossRef] [PubMed]
  11. P. Török, P. R. T. Munro, and E. E. Kriezis, “Rigorous near- to far-field transformation for vectorial diffraction calculations and its numerical implementation,” J. Opt. Soc. Am. A23(3), 713–722 (2006). [CrossRef] [PubMed]
  12. D. Ganic, X. Gan, and M. Gu, “Focusing of doughnut laser beams by a high numerical-aperture objective in free space,” Opt. Express11(21), 2747–2752 (2003). [CrossRef] [PubMed]
  13. J. Lin, X.-C. Yuan, S. H. Tao, and R. E. Burge, “Variable-radius focused optical vortex with suppressed sidelobes,” Opt. Lett.31(11), 1600–1602 (2006). [CrossRef] [PubMed]
  14. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon.1(1), 1–57 (2009). [CrossRef]
  15. C. J. R. Sheppard and M. Gu, “Axial imaging through an aberration layer of water in confocal microscopy,” Opt. Commun.88(2-3), 180–190 (1992). [CrossRef]
  16. P. Török, P. Varga, Z. Laczik, and G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A12(2), 325–332 (1995). [CrossRef]
  17. S. H. Wiersma, P. Török, T. D. Visser, and P. Varga, “Comparison of different theories for focusing through a plane interface,” J. Opt. Soc. Am. A14(7), 1482–1490 (1997). [CrossRef]
  18. P. Török and P. Varga, “Electromagnetic diffraction of light focused through a stratified medium,” Appl. Opt.36(11), 2305–2312 (1997). [CrossRef] [PubMed]

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