## Photonic crystal diffraction gratings

Optics Express, Vol. 8, Issue 3, pp. 209-216 (2001)

http://dx.doi.org/10.1364/OE.8.000209

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### Abstract

It is shown from numerical results deduced from a rigorous theory of diffraction that diffraction gratings made with two-dimensional dielectric photonic crystals may present blazing effects. Since these structures are lossless, efficiencies of 100% in the -1st order can be obtained in polarized light. Efficiency curves in Littrow mount are shown.

© Optical Society of America

## 1. Introduction

1. R. Petit, *Electromagnetic theory of gratings* (Springer-Verlag, 1980). [CrossRef]

5. D. Maystre, “Rigorous vector theories of diffraction gratings” in *Progress in Optics Volume XXI*, E. Wolf ed.(North-Holland,Amsterdam, 1984) [CrossRef]

1. R. Petit, *Electromagnetic theory of gratings* (Springer-Verlag, 1980). [CrossRef]

1. R. Petit, *Electromagnetic theory of gratings* (Springer-Verlag, 1980). [CrossRef]

7. E. Yablonovitch, “Photonic crystals,” J. of Modern Optics **41**, 173–194 (1994) [CrossRef]

10. D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. **3**, 975–993 (1994) [CrossRef]

## 2. Presentation of a photonic crystal grating.

_{g}grids separated by a distance d√3/2 (N

_{g}=6 in the figure)

_{g}=Md (M positive integer) placed at a distance h from the upper grid of the photonic crystal, the rods having a radius R′ and the same index ν as the rods of the photonic crystal.

_{b}of the lowest grid. Throughout the paper, the period d of the photonic crystal is equal to 1 and the index of the dielectric material to 3.

10. D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. **3**, 975–993 (1994) [CrossRef]

^{st}order Littrow mount, at least in the region where blazing effects are expected. Let us recall that in the - 1

^{st}order Littrow mount, the -1st diffracted order and the incident wave are propagating in opposite directions. From the classical grating formula, it can be shown easily that such a mount requires the following relation between the grating parameters to be satisfied:

_{g}is the grating period, θ the incidence angle and λ the wavelength.

_{g}/3, 2 d

_{g}), since in that range, two orders only are diffracted, the -1

^{st}and 0

^{th}orders. It is in this mount and in this range of wavelength that blazing effects are obtained even though, due to their particular geometry, echelette gratings allow one to get blazing effects outside that range as well. As a consequence, when the period d

_{g}of the grating coincides with the period d of the photonic crystal, the blazing region is placed below the forbidden band gaps of the photonic crystal. In other words, when λ/d

_{g}is greater than 2, the only reflected order is the zeroth order which has no interest in spectroscopy. This conclusion does not hold for the gratings of fig.1, where the period d

_{g}of the grating is a multiple of the period d of the photonic crystal. In that case, the Littrow relation becomes:

## 3. The theory in outline.

1. R. Petit, *Electromagnetic theory of gratings* (Springer-Verlag, 1980). [CrossRef]

5. D. Maystre, “Rigorous vector theories of diffraction gratings” in *Progress in Optics Volume XXI*, E. Wolf ed.(North-Holland,Amsterdam, 1984) [CrossRef]

10. D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. **3**, 975–993 (1994) [CrossRef]

1. R. Petit, *Electromagnetic theory of gratings* (Springer-Verlag, 1980). [CrossRef]

5. D. Maystre, “Rigorous vector theories of diffraction gratings” in *Progress in Optics Volume XXI*, E. Wolf ed.(North-Holland,Amsterdam, 1984) [CrossRef]

^{-4}, thanks to the circular shape of the rods.

_{g}of the top grid. Consequently, the discretization points used in the solution of the integral equations must be placed on M rods, and not on 1 rod only as usual. Of course, this feature increases the size of the matrices to handle in the calculations by a multiplicative factor of M. Nevertheless, the computation time remains moderate since it does not exceed 30 seconds on a PC (processor Intel 450MHz) for a single wavelength calculation on the diffraction grating represented in figure 1 with M=3 and n

_{g}=5.

## 4. Numerical results.

### 4.1 Efficiency for s-polarized light

**3**, 975–993 (1994) [CrossRef]

_{g}of grids in the photonic crystal is equal to 5. The distance h between the photonic crystal and the top grid has been taken equal to √3/2 (i.e. the same value as the distance between two grids of the photonic crystal). The red curves in figure 2 give the energy reflected by the photonic crystal (without the top grid) for the same incidence as in Littrow mount. One can see that the main gap extends from λ=2.0 to λ=3.6. Inside this gap, the reflectivity always exceeds 99% and culminates at 99.98%, but higher values could be reached by increasing N

_{g}.

_{g}is increased. The relative width of the best efficiency curve (fig. 2f) is comparable to the widths of the best efficiency curves obtained for echelette, sinusoidal or lamellar gratings in s-polarized light. However, it must be noticed that the shape of the efficiency curve in fig. 2f is very peculiar, with two blaze wavelengths (λ=2.2 and λ=2.5) separated by a small interval where the drop of efficiency does not exceed 15%. In the same curve, a third blazing effect is obtained at a wavelength of 1.5, but that one is separated from the other ones by a strong anomaly where the efficiency drops to a value less than 5%.

### 4.2 Efficiency for p-polarized light

**3**, 975–993 (1994) [CrossRef]

_{t}between the upper interface and the top grid is equal to 0.433 in all the calculations, in order to permit the radius R of the top grid to reach large values. With these parameters, the main gap is located between 3d and 4.2d, thus we have been led to adopt M=3, in order to get the gap of the photonic crystal and the blazing region of the grating close to each other.

_{g}.

## 5. Conclusion.

11. R.C. McPhedran and D. Maystre, “A detailed theoretical study of the anomalies of a sinusoidal diffraction grating,” Optica Acta **21**, 413–421 (1994). [CrossRef]

12. F. Gadot, A. Chelnokov, A. De Lustrac, P. Crozat, and J.M. Lourtioz, “Experimental demonstration of complete photonic band gap in graphite structure,” Appl.Phys. Lett. **71**, 1780–1782 (1997) [CrossRef]

## References and links

1. | R. Petit, |

2. | M. C. Hutley, |

3. | D. Maystre, |

4. | E.G. Loewen and E. Popov, |

5. | D. Maystre, “Rigorous vector theories of diffraction gratings” in |

6. | D. Maystre, “General study of grating anomalies from electromagnetic surface modes,” in |

7. | E. Yablonovitch, “Photonic crystals,” J. of Modern Optics |

8. | C.M. Soukoulis, |

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

10. | D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. |

11. | R.C. McPhedran and D. Maystre, “A detailed theoretical study of the anomalies of a sinusoidal diffraction grating,” Optica Acta |

12. | F. Gadot, A. Chelnokov, A. De Lustrac, P. Crozat, and J.M. Lourtioz, “Experimental demonstration of complete photonic band gap in graphite structure,” Appl.Phys. Lett. |

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(050.1960) Diffraction and gratings : Diffraction theory

(300.6550) Spectroscopy : Spectroscopy, visible

**ToC Category:**

Focus Issue: Photonic bandgap calculations

**History**

Original Manuscript: November 10, 2000

Published: January 29, 2001

**Citation**

Daniel Maystre, "Photonic crystal diffraction gratings," Opt. Express **8**, 209-216 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-3-209

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### References

- R. Petit, Electromagnetic theory of gratings (Springer-Verlag, 1980). [CrossRef]
- M. C. Hutley, Diffraction gratings (Academic press, 1982)
- D. Maystre, Diffraction gratings (SPIE Milestones series, 1992).
- E.G. Loewen and E. Popov, Diffraction gratings and applications (Marcel Dekker, 1997).
- D. Maystre, "Rigorous vector theories of diffraction gratings" in Progress in Optics Volume XXI,E. Wolf ed.(North-Holland, Amsterdam, 1984). [CrossRef]
- D. Maystre, "General study of grating anomalies from electromagnetic surface modes," in Electromagnetic surface modes, A.D. Boardman ed.(John Wiley &sons, 1982).
- E. Yablonovitch, "Photonic crystals," J. of Modern Optics 41, 173-194 (1994). [CrossRef]
- C.M. Soukoulis, Photonic band gap materials (Kluwer Academic publishers, 1995).
- J. D. Joannopoulos, R. D. Meade and J. N. Winn, Photonic crystals (Princeton university press, 1995).
- D. Maystre, "Electromagnetic study of photonic band gaps," Pure Appl. Opt. 3, 975-993 (1994). [CrossRef]
- R.C.McPhedran and D. Maystre, "A detailed theoretical study of the anomalies of a sinusoidal diffraction grating," Optica Acta 21, 413-421 (1994). [CrossRef]
- F. Gadot, A. Chelnokov, A. De Lustrac, P. Crozat and J.M. Lourtioz, "Experimental demonstration of complete photonic band gap in graphite structure," Appl. Phys. Lett. 71, 1780-1782 (1997). [CrossRef]

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