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Optics Express

Optics Express

  • Editor: J. H. Eberly
  • Vol. 8, Iss. 3 — Jan. 29, 2001
  • pp: 209–216
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Photonic crystal diffraction gratings

Daniel Maystre  »View Author Affiliations


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

Diffraction gratings [1

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

5

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

] have played a vital role in many fields of science and technology, including spectroscopy, astronomy, beam-sampling for high power lasers, filtering, etc. The most commonly used grating is the metallic reflection grating. One of the main requirements for this kind of grating is the so-called “blazing effect”, viz. the concentration of the reflected light in one scattered beam only, at a given wavelength [1

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

4

4. E.G. Loewen and E. Popov, Diffraction gratings and applications (Marcel Dekker, 1997)

]. It is well known that ruled and holographic gratings present blaze effects, however, in the visible or near infrared regions, the power in the blazed scattered beam is always significantly lower than the incident power, due to losses in the metal coating. In order to improve the scattered power, many techniques have been proposed, namely the deposit of a stack of reflective dielectric layers on top of the metal. Unfortunately, dielectric coated gratings may present strong Wood anomalies [1

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

4

4. E.G. Loewen and E. Popov, Diffraction gratings and applications (Marcel Dekker, 1997)

,6

6. D. Maystre, “General study of grating anomalies from electromagnetic surface modes,” in Electromagnetic surface modes, A.D. Boardman ed.(John Wiley &sons, 1982)

], i.e. strong absorption phenomena caused by the excitation of guided waves in the stack of dielectric layers.

The aim of the paper is to show that the use of dielectric photonic crystals [7

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

9

9. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic crystals (Princeton university press, 1995)

] could initiate the development of a new kind of diffraction grating, the photonic crystal diffraction grating. Such a grating is made by carving adequately a dielectric two-dimensional photonic crystal (for instance a crystal made of circular dielectric cylinders with hexagonal symmetry). It is well known that photonic crystals allow a total control of light propagation: inside some ranges of wavelengths (transmission gaps), such a structure does not permit light propagation. Thus, it is a natural idea to use reflection properties of photonic crystals inside the gaps to realize lossless diffraction gratings.

Numerical data is obtained from a generalization of a rigorous integral formalism devoted to the calculation of the field reflected and transmitted by a stack of dielectric periodic grids [10

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

] to the case where the grids have not the same periods (however, the periods must remain a multiple of each other). This theory reduces the scattering problem to the solution of a set of integral equations and leads to the inversion of linear systems of equations.

After a brief presentation of this theory, it will be shown from rigorous numerical results that a photonic crystal grating can be blazed inside the transmission gap of the photonic crystal, as metallic gratings. However, its remarkable peculiarity lies in the fact that it can concentrate in the blazed scattered order the totality of the incident power, at least for polarized light. This amazing property could lead to a break-through in grating technology, provided the construction of such gratings can be achieved. Efficiency curves in Littrow mount will be compared with those of classical metallic gratings for both polarizations.

2. Presentation of a photonic crystal grating.

Figure 1 shows the structure of two photonic crystal gratings. At the left-hand side, the grating is made of dielectric rods of radius R and index ν placed in air while the one at the right-hand side is made by a dielectric substrate pierced by air holes (inverted contrast). It is worth noting that these gratings are composed by two different parts:

• The lowest part is a two-dimensional photonic crystal with hexagonal symmetry and period d. This photonic crystal has an infinite extension in x, while the extension in y is limited to Ng grids separated by a distance d√3/2 (Ng=6 in the figure)

• The upper part is a grid of period dg=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.

In the case of the inverted contrast (right hand side), the top of the dielectric substrate is placed at a distance ht of the top grid whilst the bottom of the substrate is located at the distance hb 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.

In order to make efficient diffraction gratings with photonic crystals, two wavelength constraints must be simultaneously satisfied: one that characterizes the blaze configuration and another that ensures the existence of a photonic band gap, enabling the back plane to act as a mirror supporting only the single, specular, propagating order. The presence of the top grid is justified by scattering requirements. In general, photonic crystals present gaps in a range of wavelengths greater than the period d of the crystals, the center of the gap being placed typically at a value equal to 2d or 3d [9

9. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic crystals (Princeton university press, 1995)

,10

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

]. As a consequence, a grating having the same period d cannot be used in -1st order Littrow mount, at least in the region where blazing effects are expected. Let us recall that in the - 1st 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:

λdg=2sin(θ)
(1)

where dg is the grating period, θ the incidence angle and λ the wavelength.

The most interesting range of wavelength is the “blazing region”, viz. the interval (2 dg/3, 2 dg), since in that range, two orders only are diffracted, the -1st and 0th 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 dg 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/dg 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 dg of the grating is a multiple of the period d of the photonic crystal. In that case, the Littrow relation becomes:

λd=2Msin(θ)
(2)

in such a way that λ/d can be greater than 2.

Fig.1. Structure of a photonic crystal grating. Left: grating made with dielectric rods in air, right: grating made by air galleries in a dielectric substrate (inverted contrast).

3. The theory in outline.

We have used an integral theory of gratings. This theory has been developed in our laboratory since the 70’s [1

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

,3

3. D. Maystre, Diffraction gratings (SPIE Milestones series, 1992)

,5

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

] and its last version permitted us to deal with rod gratings as well as relief gratings, of arbitrary shapes and in arbitrary ranges of wavelengths [10

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

]. The peculiarity of this code lies in the fact that it can deal with classical gratings used in spectroscopy as well as two-dimensional photonic crystals. Thus, it constitutes a valuable tool for the analysis of diffraction gratings made of photonic crystals.

The theory is based on the use of the notion of scattering matrix. This matrix expresses the linear relations between the plane waves propagating upwards and downwards on both sides of an horizontal grid. Computing successively the scattering matrix of each grid, from the bottom to the top by solving an integral equation, then taking into account the radiation conditions for the scattered field leads to the solution of a linear system of equations of infinite size. The validity of the numerical truncation of the matrix is checked by the convergence of the results as the size of the truncated matrix is increased.

This integral theory has been widely checked by extensive comparisons with experimental data [1

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

5

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

]. From classical tests (energy balance, reciprocity,…), it appears that the results have in general a relative precision better than 1% but for photonic crystal diffraction gratings studied in this paper, the relative precision is of the order of 10-4, thanks to the circular shape of the rods.

It is worth noting that the particular shape of photonic crystal diffraction gratings required a generalization of the theory described in ref. 10. Indeed, in that case, the field in the region of the photonic crystal has not the period d of the grids but the period dg 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 ng=5.

4. Numerical results.

4.1 Efficiency for s-polarized light

The main conclusion that can be drawn from figure 2 is the existence of a blazing effect as soon as R′/d reaches 0.15. It is worth noting that the width of the efficiency curves is limited by the width of the gap of the photonic crystal. The maximum efficiency culminates at values greater than 99.96% in fig. 2c, 2d, 2e and 2f. Since these values are nothing but the reflectivity of the photonic crystal, it can be deduced that the maximum efficiency will tend to unity as Ng 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%.

Since the efficiency curves of figure 2 have been obtained without any optimization process, it can be expected that further improvements of the efficiency curves could be obtained, by varying h for instance, even though some calculations have shown that this parameter is not a crucial one for this polarization.

4.2 Efficiency for p-polarized light

For this polarization, it has been shown that photonic crystals with inverted contrast are the best [9

9. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic crystals (Princeton university press, 1995)

,10

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

], thus we will consider photonic crystals represented at the right hand side of figure 1. With an index of 3, interesting gaps are obtained for radii R greater than 0.325, and we have adopted this value in order to keep a moderate size of the air holes. The distance ht 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.

The efficiency curves obtained in figure 3 show that, in contrast with the results obtained for s-polarization, the efficiency obtained without any optimization of h is quite small. No blaze effect is obtained and, even though the efficiency obtained in fig.3c culminates at a value close to 90%, the width of the peak is small. In order to improve the efficiency, three changes have been introduced in the parameters:

• Keeping M=3, we have introduced two rods per period (instead of 1) in the top grid. It emerges that such a change does not improve the efficiency, thus it has been abandoned.

• Keeping M=3 and adopting R′=0.4 as in figure 3c, h has been varied. It appears that this parameter is a crucial one and that the efficiency is improved as h is decreased. Taking into account the constraints (two rods cannot interpenetrate…), h=0.41 (instead of h=0.866 in figure 3) has been chosen. That way, a blazing effect is obtained, as it will be seen, but a further improvement cannot be reached since the radius R′ cannot be increased notably.

• Adopting M=3 and h=0.41, the size of the rods has been increased by replacing the circular shape by an elliptic one. The length a of the semi-axis a parallel to the y axis remains equal to 0.4 but the length b of the semi-axis parallel the x axis takes greater values, from b=0.4 (circular shape) to b=1.2.

Fig.2. Efficiency of a photonic crystal grating (see the left hand side of fig.1) in a -1st order Littrow mount and s-polarization. From (a) to (f), the values of R′/d are respectively equal to 0.1, 0.125, 0.15, 0.175, 0.2, 0.25.
Fig.3. Efficiency of photonic crystal gratings with inverted contrast for h=d√3/2. From a to c, R′ is equal to 0.25, 0.325 and 0.4.

Fig. 4 shows the result of this last optimization. Obviously, blaze effects are obtained, despite a strong anomaly which moves away from the right to the left of the gap as the horizontal semi-axis is increased. As a consequence, the best efficiency curves are obtained for values of b less than 0.8. For this polarization, the relative width of the efficiency curve around the blaze wavelength is significantly lower than that obtained with metallic gratings. The efficiency in fig.4 a, b, c, d, e culminates at values exceeding 99.5%, i.e. the reflectivity of the photonic crystal. Larger peak efficiencies could be obtained by increasing Ng.

Fig. 4.. Efficiency of photonic crystal gratings with inverted contrast for h=0.41 The top grid is made of elliptic rods with a semi-axis along the vertical (parallel to the y axis) direction equal to 0.4. From a to f, the horizontal (parallel to the x axis) semi-axis is equal to 0.4, 0.5, 0.6, 0.8, 1., 1.2.

5. Conclusion.

References and links

1.

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

2.

M. C. Hutley, Diffraction gratings (Academic press, 1982)

3.

D. Maystre, Diffraction gratings (SPIE Milestones series, 1992)

4.

E.G. Loewen and E. Popov, Diffraction gratings and applications (Marcel Dekker, 1997)

5.

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

6.

D. Maystre, “General study of grating anomalies from electromagnetic surface modes,” in Electromagnetic surface modes, A.D. Boardman ed.(John Wiley &sons, 1982)

7.

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

8.

C.M. Soukoulis, Photonic band gap materials (Kluwer Academic publishers, 1995)

9.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic crystals (Princeton university press, 1995)

10.

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

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]

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

  1. R. Petit, Electromagnetic theory of gratings (Springer-Verlag, 1980). [CrossRef]
  2. M. C. Hutley, Diffraction gratings (Academic press, 1982)
  3. D. Maystre, Diffraction gratings (SPIE Milestones series, 1992).
  4. E.G. Loewen and E. Popov, Diffraction gratings and applications (Marcel Dekker, 1997).
  5. D. Maystre, "Rigorous vector theories of diffraction gratings" in Progress in Optics Volume XXI,E. Wolf ed.(North-Holland, Amsterdam, 1984). [CrossRef]
  6. D. Maystre, "General study of grating anomalies from electromagnetic surface modes," in Electromagnetic surface modes, A.D. Boardman ed.(John Wiley &sons, 1982).
  7. E. Yablonovitch, "Photonic crystals," J. of Modern Optics 41, 173-194 (1994). [CrossRef]
  8. C.M. Soukoulis, Photonic band gap materials (Kluwer Academic publishers, 1995).
  9. J. D. Joannopoulos, R. D. Meade and J. N. Winn, Photonic crystals (Princeton university press, 1995).
  10. D. Maystre, "Electromagnetic study of photonic band gaps," Pure Appl. Opt. 3, 975-993 (1994). [CrossRef]
  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]

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