## Fourier factorization with complex polarization bases in modeling optics of discontinuous bi-periodic structures

Optics Express, Vol. 17, Issue 9, pp. 7269-7274 (2009)

http://dx.doi.org/10.1364/OE.17.007269

Acrobat PDF (263 KB)

### Abstract

The coupled wave theory dealing with optics of discontinuous two-dimensional (2D) periodic structures is reformulated by using Fourier factorization with complex polarization bases, which is a generalized implementation of the fast Fourier factorization rules. The modified approach yields considerably improved convergence properties, as shown on an example of a 2D quartz grating. The method can also be applied to the calculation of 2D photonic band structures or nonperiodic cylindrical devices, and can be generalized to elements with arbitrary cross-sections.

© 2009 Optical Society of America

## 1. Introduction

3. D. Maystre, “Rigorous vector theories of diffraction gratings,” Prog. Opt. **21**, 1–67 (1984). [CrossRef]

5. K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. **65**, 3152–3155 (1990). [CrossRef] [PubMed]

6. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A **13**, 1870–1876 (1996). [CrossRef]

7. L. Li, “Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials,” J. Mod. Opt. **45**, 1313–1334 (1998). [CrossRef]

8. B. Chernov, M. Neviere, and E. Popov, “Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings,” Opt. Commun. **194**, 289–297 (2001). [CrossRef]

9. E. Popov and M. Neviere, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A **17**, 1773–1784 (2000). [CrossRef]

10. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A **14**, 2758–2767 (1997). [CrossRef]

11. K. Watanabe, R. Petit, and M. Neviere, “Differential theory of gratings made of anisotropic materials,” J. Opt. Soc. Am. A **19**, 325–334 (2002). [CrossRef]

12. K. Watanabe, “Numerical integration schemes used on the differential theory for anisotropic gratings,” J. Opt. Soc. Am. A **19**, 2245–2252 (2002). [CrossRef]

13. L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A **5**, 345–355 (2003). [CrossRef]

14. P. Boyer, E. Popov, M. Neviere, and G. Tayeb, “Diffraction theory in TM polarization: application of the fast Fourier factorization method to cylindrical devices with arbitrary cross section,” J. Opt. Soc. Am. A **21**, 2146–2153 (2004). [CrossRef]

15. N. Bonod, E. Popov, and M. Neviere, “Light transmission through a subwavelength microstructured aperture: electromagnetic theory and applications,” Opt. Commun. **245**, 355–361 (2005). [CrossRef]

16. N. Bonod, E. Popov, and M. Neviere, “Fourier factorization of nonlinear Maxwell equations in periodic media: application to the optical Kerr effect,” Opt. Commun. **244**, 389–398 (2005). [CrossRef]

17. E. Noponen and J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three dimensional profiles,” J. Opt. Soc. Am. A **11**, 2494–2502 (1994). [CrossRef]

18. S. Visnovsky and K. Yasumoto, “Multilayer anisotropic bi-periodic diffraction gratings,” Czech. J. Phys. **51**, 229–247 (2001). [CrossRef]

10. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A **14**, 2758–2767 (1997). [CrossRef]

20. A. David, H. Benisty, and C. Weisbuch, “Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape,” Phys. Rev. B **73**, 075107 (2006). [CrossRef]

21. T. Schuster, J. Ruoff, N. Kerwien, S. Rafler, and W. Osten, “Normal vector method for convergence improvement using the RCWA for crossed gratings,” J. Opt. Soc. Am. A **24**, 2880–2890 (2007). [CrossRef]

22. P. Gotz, T. Schuster, K. Frenner, S. Rafler, and W. Osten, “Normal vector method for the RCWA with automated vector field generation,” Opt. Express **16**, 17295–17301 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-22-17295. [CrossRef] [PubMed]

24. N. P. K. Cotter, T. W. Preist, and J. R. Sambles, “Scattering matrix approach to multilayer diffraction,” J. Opt. Soc. Am. A **12**, 1097–1103 (1995). [CrossRef]

25. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A **13**, 1024–1035 (1996). [CrossRef]

26. S. Kaushik, “Vector Fresnel equations and Airy formula for one-dimensional multilayer and surface-relief gratings,” J. Opt. Soc. Am. A **14**, 596–609 (1997). [CrossRef]

27. R. Antos, J. Pistora, J. Mistrik, T. Yamaguchi, S. Yamaguchi, M. Horie, S. Visnovsky, and Y. Otani, “Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials,” J. Appl. Phys. **100**, 054906 (2006). [CrossRef]

18. S. Visnovsky and K. Yasumoto, “Multilayer anisotropic bi-periodic diffraction gratings,” Czech. J. Phys. **51**, 229–247 (2001). [CrossRef]

## 2. Theoretical models

*H*;

_{x}*H*] are scaled)

_{y}*ε*(

*x*,

*y*) and the transverse (to the

*z*axis) components of quasi-periodic electric field

*E*

_{x(y)}

*H*,

_{x}*H*), where

_{y}*∂*=

_{x}*∂*/

*∂*etc., ∇

_{x}*= [*

_{t}*∂*,

_{x}*∂*],

_{y}*p*= sin

_{m}*ϑ*cos

^{i}*φ*+

^{i}*mλ*/Λ,

*q*= sin

_{n}*ϑ*sin

^{i}*φ*+

^{i}*nλ*/Λ, with

*λ*denoting the wavelength, Λ the grating period,

*ϑ*and

^{i}*φ*the spherical angles of the incident wave, and

^{i}*s*the eigennumber of the wave equation (1). The configuration of the optical problem is displayed in Fig. 1(a), and the principle of the Airy-like series propagation algorithm is depicted in Fig. 1(b), where the propagation matrix

**P**is determined from the eigensolutions of Eq. (1) and where the reflection and transmission matrices on interfaces

**R**

*,*

_{jk}**T**

*follow from boundary-matching conditions using Eq. (2).*

_{jk}*∂*,

_{x}*∂*become diagonal matrices -

_{y}*i*

**p**, -

*i*

**q**with particular arrangements of the

*p*,

_{m}*q*values on their diagonals [18

_{n}18. S. Visnovsky and K. Yasumoto, “Multilayer anisotropic bi-periodic diffraction gratings,” Czech. J. Phys. **51**, 229–247 (2001). [CrossRef]

*εE*(or

_{x}*εE*) can be treated by either the Laurent factorization rule [

_{y}*εE*] = [[

_{x}*ε*]][

*E*] (matrix version of the convolution rule between the Fourier coefficients of the multiplied functions) or by the inverse rule [

_{x}*εE*] = [[1/

_{x}*ε*]]

^{-1}[

*E*], where [

_{x}*f*] denotes a column vector of the Fourier coefficients of a function

*f*, while [[

*g*]] denotes a matrix composed of the Fourier coefficients of a function

*g*[18

**51**, 229–247 (2001). [CrossRef]

*εE*] = [[

_{x}*ε*]] [

*E*] and [

_{x}*εE*] = [[

_{y}*ε*]] [

*E*].

_{y}6. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A **13**, 1870–1876 (1996). [CrossRef]

*D*=

_{x}*εE*and

_{x}*D*=

_{y}*εE*, because both pairs of functions have concurrent discontinuities and both products

_{y}*D32*and

*D*(the electrical displacement) are discontinuous as well. On the other hand, by a clever change of the polarization bases at all points (using a space-dependent Jones matrix transform

_{y}**F**), we can treat independently the normal and tangential components of the fields by the correct rules, i.e., [

*D*] = [[1/

_{u}*ε*]]

^{-1}[

*E*] and [

_{u}*D*] = [[

_{v}*ε*]][

*E*], where

_{v}*E*,

_{u}*D*are field components normal to the discontinuities of the permittivity function, while

_{u}*E*,

_{v}*D*are tangential; the reason for this is because

_{v}*E*and

_{v}*D*are continuous. This idea was applied in [9

_{u}9. E. Popov and M. Neviere, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A **17**, 1773–1784 (2000). [CrossRef]

21. T. Schuster, J. Ruoff, N. Kerwien, S. Rafler, and W. Osten, “Normal vector method for convergence improvement using the RCWA for crossed gratings,” J. Opt. Soc. Am. A **24**, 2880–2890 (2007). [CrossRef]

21. T. Schuster, J. Ruoff, N. Kerwien, S. Rafler, and W. Osten, “Normal vector method for convergence improvement using the RCWA for crossed gratings,” J. Opt. Soc. Am. A **24**, 2880–2890 (2007). [CrossRef]

**F**, whose columns are the basis polarization vectors

*= [*

**u***ξ*,;

*ζ*],

*= [-*

**v***ζ*

^{*};

*ξ*

^{*}] (mutually orthogonal [23]), can be defined as the rotation by a polar angle

*ϕ*, i.e.,

*ξ*= cos

*ϕ*,

*ζ*= sin

*ϕ*, within a single cell where we define polar coordinates

*re*=

^{iϕ}*x*+

*i*, and then by periodic repeating over the entire 2D space. The distribution of the basis vector

_{y}*in the periodic cell is depicted in Fig. 2(a), from where it is obvious that the matrix function*

**u****F**(

*x*,

*y*) has no concurrent discontinuities with the electric field, so that we can use the Laurent rule for the transformation of polarization

**F**(

*x*,

*y*) has a singularity at the center of the periodic cell and other discontinuities on the cell boundaries, which slows down the convergence of the numerical implementation, as will be evidenced below.

**F**(

*x*,

*y*) becomes completely continuous. We still use Eqs. (5) and (6) but with

*, as depicted in Fig. 2(b), with*

**u***R*denoting the radius of the cylindrical element and

*D*(

*ϕ*)= Λ/2max(∣cos

*ϕ*∣, ∣sin

*ϕ*∣) being the distance from the cell’s center to its edge. In Eq. (7) the Jones vector on the right represents a polarization ellipse (with ellipticity

*η*) oriented along the

*x*coordinate, the matrix in the middle rotates this polarization by the azimuth

*θ*, and the factor

*e*preserves the continuity of the phase at the center and on the boundaries of the cell. Similarly to Model B, the azimuth of the polarization ellipse is constant on the lines coming from the cell’s center, but now the ellipticity is zero (corresponding to a linear polarization) only on the dot’s edges, has the maximum value (

^{iθ}*π*/4 for the circular polarization) at the cell’s center and on its boundaries, and is continuously varying (with a smooth sine dependence) in the intermediate ranges [Fig. 2(c)]. The continuity at the center and boundaries of the periodic cell can be easily checked by evaluating lim

_{r→0}

*= lim*

**u**_{r→D(ϕ)}

*= (1/2)*

**u**^{1/2}[1;

*i*], which is independent of

*ϕ*. Hence we obtain a smooth and completely continuous matrix function

**F**(

*x*,

*y*) corresponding to Model C.

## 3. Numerical example and discussion

*R*= 200 nm, depth

*d*= 100 nm, square periodicity Λ = 300 nm, an incident plane wave with the wavelength

*λ*= 500 nm, the angle of incidence

*ϑ*= 60°, and the angle of the plane of incidence

^{i}*φ*= 0 [the configuration of Fig. 1(a)], with the values of permittivity

^{i}*ε*

_{a}= 1 for vacuum inside holes and

*ε*

_{b}= 2.138 for quartz [28].

*s*and

*p*polarizations, denoted

*R*

_{ss}^{(m,n)}and

*R*

_{pp}^{(m,n)}for the [

*m*,

*n*] order. Note that in the configuration of Fig. 1(a) the mixed-polarization (

*sp*and

*ps*) efficiencies of the two diffracted orders of interest are zero. We display the dependences of the quantities according to the maximum indices

*M*= max ∣

*m*∣ and

*N*= max ∣

*n*∣ of the Fourier harmonics retained inside the periodic medium. For simplicity we keep

*M*=

*N*, so that the order of the [[

*ε*]]-like matrices is (2

*N*+ 1)

^{2}. Due to the limited memory of the computer used, calculations were possible to perform for

*N*≤ 16.

^{-4}for

*N*≥ 7, Model C yields precision of about 10

^{-5}for the same

*N*(except the sensitive case of

*R*

_{pp}^{(0,0)}with particularly low reflection). The values yielded by Model A are even one order worse than those of Model B, so that they are not visible in Figs. 3(a)–3(c); therefore, insets are included with longer ranges for comparison.

## 4. Conclusion

*N*= 4 (in the presented configuration), which considerably saves the computer memory and the time of calculations. It is worth pointing out that the essential difference between the presented Model C and the previous Model B is (from the mathematical viewpoint) the fact that the matrix transformation of polarization in the former method contains complex-valued elements, so that this generalization is surprisingly simple.

*ε*

^{-1}∇ × (∇ ×

*) = (*

**E***ω*/

*c*)

^{2}

*], nonperiodic cylindrical devices (by using a large periodic cell), etc., and can be straightforwardly generalized to elements with arbitrary cross-sections [by replacing the constant value of the radius*

**E***R*in Eq. (9) with a function

*R*(

*ε*) and by generalizing the azimuth

*θ*(

*r*,

*ε*) so that the linear polarization on the edge of the element becomes again normal to it].

## Acknowledgments

## References and links

1. | R. Petit (ed.), |

2. | M. Neviere and E. Popov, |

3. | D. Maystre, “Rigorous vector theories of diffraction gratings,” Prog. Opt. |

4. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, |

5. | K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. |

6. | L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A |

7. | L. Li, “Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials,” J. Mod. Opt. |

8. | B. Chernov, M. Neviere, and E. Popov, “Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings,” Opt. Commun. |

9. | E. Popov and M. Neviere, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A |

10. | L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A |

11. | K. Watanabe, R. Petit, and M. Neviere, “Differential theory of gratings made of anisotropic materials,” J. Opt. Soc. Am. A |

12. | K. Watanabe, “Numerical integration schemes used on the differential theory for anisotropic gratings,” J. Opt. Soc. Am. A |

13. | L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A |

14. | P. Boyer, E. Popov, M. Neviere, and G. Tayeb, “Diffraction theory in TM polarization: application of the fast Fourier factorization method to cylindrical devices with arbitrary cross section,” J. Opt. Soc. Am. A |

15. | N. Bonod, E. Popov, and M. Neviere, “Light transmission through a subwavelength microstructured aperture: electromagnetic theory and applications,” Opt. Commun. |

16. | N. Bonod, E. Popov, and M. Neviere, “Fourier factorization of nonlinear Maxwell equations in periodic media: application to the optical Kerr effect,” Opt. Commun. |

17. | E. Noponen and J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three dimensional profiles,” J. Opt. Soc. Am. A |

18. | S. Visnovsky and K. Yasumoto, “Multilayer anisotropic bi-periodic diffraction gratings,” Czech. J. Phys. |

19. | R. Antos, S. Visnovsky, J. Mistrik, and T. Yamaguchi, “Magneto-optical polar-Kerr-effect spectroscopy on 2D-periodic subwavelength arrays of magnetic dots,” International Journal of Microwave and Optical Technology |

20. | A. David, H. Benisty, and C. Weisbuch, “Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape,” Phys. Rev. B |

21. | T. Schuster, J. Ruoff, N. Kerwien, S. Rafler, and W. Osten, “Normal vector method for convergence improvement using the RCWA for crossed gratings,” J. Opt. Soc. Am. A |

22. | P. Gotz, T. Schuster, K. Frenner, S. Rafler, and W. Osten, “Normal vector method for the RCWA with automated vector field generation,” Opt. Express |

23. | R. M. A. Azzam and N. M. Bashara, |

24. | N. P. K. Cotter, T. W. Preist, and J. R. Sambles, “Scattering matrix approach to multilayer diffraction,” J. Opt. Soc. Am. A |

25. | L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A |

26. | S. Kaushik, “Vector Fresnel equations and Airy formula for one-dimensional multilayer and surface-relief gratings,” J. Opt. Soc. Am. A |

27. | R. Antos, J. Pistora, J. Mistrik, T. Yamaguchi, S. Yamaguchi, M. Horie, S. Visnovsky, and Y. Otani, “Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials,” J. Appl. Phys. |

28. | E. D. Palik (ed.), |

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(050.1755) Diffraction and gratings : Computational electromagnetic methods

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: March 6, 2009

Revised Manuscript: April 10, 2009

Manuscript Accepted: April 16, 2009

Published: April 17, 2009

**Citation**

Roman Antos, "Fourier factorization with complex polarization bases in modeling optics of discontinuous bi-periodic structures," Opt. Express **17**, 7269-7274 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-9-7269

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### References

- R. Petit (ed.), Electromagnetic Theory of Gratings (Springer, 1980).
- M. Neviere and E. Popov, Light Propagation in Periodic Media: Diffraction Theory and Design (Marcel Dekker, New York, 2003).
- D. Maystre, "Rigorous vector theories of diffraction gratings," Prog. Opt. 21, 1-67 (1984). [CrossRef]
- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ., 1995)
- K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990). [CrossRef] [PubMed]
- L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A 13, 1870- 1876 (1996). [CrossRef]
- L. Li, "Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials," J. Mod. Opt. 45, 1313-1334 (1998). [CrossRef]
- B. Chernov, M. Neviere, and E. Popov, "Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings," Opt. Commun. 194, 289-297 (2001). [CrossRef]
- E. Popov and M. Neviere, "Grating theory: new equations in Fourier space leading to fast converging results for TM polarization," J. Opt. Soc. Am. A 17, 1773-1784 (2000). [CrossRef]
- L. Li, "New formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. Soc. Am. A 14, 2758-2767 (1997). [CrossRef]
- K. Watanabe, R. Petit, and M. Neviere, "Differential theory of gratings made of anisotropic materials," J. Opt. Soc. Am. A 19, 325-334 (2002). [CrossRef]
- K. Watanabe, "Numerical integration schemes used on the differential theory for anisotropic gratings," J. Opt. Soc. Am. A 19, 2245-2252 (2002). [CrossRef]
- L. Li, "Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors," J. Opt. A 5, 345-355 (2003). [CrossRef]
- P. Boyer, E. Popov, M. Neviere, and G. Tayeb, "Diffraction theory in TM polarization: application of the fast Fourier factorization method to cylindrical devices with arbitrary cross section," J. Opt. Soc. Am. A 21, 2146- 2153 (2004). [CrossRef]
- N. Bonod, E. Popov, M. Neviere, "Light transmission through a subwavelength microstructured aperture: electromagnetic theory and applications," Opt. Commun. 245, 355-361 (2005). [CrossRef]
- N. Bonod, E. Popov, M. Neviere, "Fourier factorization of nonlinear Maxwell equations in periodic media: application to the optical Kerr effect," Opt. Commun. 244, 389-398 (2005). [CrossRef]
- E. Noponen and J. Turunen, "Eigenmode method for electromagnetic synthesis of diffractive elements with three dimensional profiles," J. Opt. Soc. Am. A 11, 2494-2502 (1994). [CrossRef]
- S. Visnovsky and K. Yasumoto, "Multilayer anisotropic bi-periodic diffraction gratings," Czech. J. Phys. 51, 229-247 (2001). [CrossRef]
- R. Antos, S. Visnovsky, J. Mistrik, and T. Yamaguchi, "Magneto-optical polar-Kerr-effect spectroscopy on 2Dperiodic subwavelength arrays of magnetic dots," International Journal of Microwave and Optical Technology 1, 905-909 (2006).
- A. David, H. Benisty, and C. Weisbuch, "Fast factorization rule and plane-wave expansion method for twodimensional photonic crystals with arbitrary hole-shape," Phys. Rev. B 73, 075107 (2006). [CrossRef]
- T. Schuster, J. Ruoff, N. Kerwien, S. Rafler, andW. Osten, "Normal vector method for convergence improvement using the RCWA for crossed gratings," J. Opt. Soc. Am. A 24, 2880-2890 (2007). [CrossRef]
- P. Gotz, T. Schuster, K. Frenner, S. Rafler, and W. Osten, "Normal vector method for the RCWA with automated vector field generation," Opt. Express 16, 17295-17301 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-22-17295. [CrossRef] [PubMed]
- R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, 1997).
- N. P. K. Cotter, T. W. Preist, and J. R. Sambles, "Scattering matrix approach to multilayer diffraction," J. Opt. Soc. Am. A 12, 1097-1103 (1995). [CrossRef]
- L. Li, "Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings," J. Opt. Soc. Am. A 13, 1024-1035 (1996). [CrossRef]
- S. Kaushik, "Vector Fresnel equations and Airy formula for one-dimensional multilayer and surface-relief gratings," J. Opt. Soc. Am. A 14, 596-609 (1997). [CrossRef]
- R. Antos, J. Pistora, J. Mistrik, T. Yamaguchi, S. Yamaguchi, M. Horie, S. Visnovsky, and Y. Otani, "Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials," J. Appl. Phys. 100, 054906 (2006). [CrossRef]
- E. D. Palik (ed.), Handbook of Optical Constants of Solids (Academic, 1998).

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