Wojciech Śmigaj
email: w.smigaj@ucl.ac.uk

Boris Gralak
email: boris.gralak@fresnel.fr

Semi-analytical design of antireflection gratings for photonic crystals

Keywords: numbers: 42.70.Qs, 42.79.Wc, 42.25.Gy

This article concerns the design of antireflection structures which, placed on a photonic crystal surface, significantly diminish the fraction of energy lost to reflected waves. After a review of the classes of these structures proposed to date, a new method is presented in detail for the design of antireflection gratings operating in a wide range of angles of incidence. The proposed algorithm is illustrated by means of several examples, showing the advantages and limitations.

I. INTRODUCTION

Photonic-crystal (PC) devices may become vital ingredients of integrated optical circuits. For instance, they open new possibilities and functionalities like negative refraction [1][2][3] , supercollimation 4,[START_REF] Gralak | Metamaterials: physics and engineering explorations[END_REF] , compact isolators [START_REF] Wang | [END_REF]7 , et c. On the other hand, PCs generally present significant reflectance at their surface while, in many applications, one strives after a perfect energy transfer between the incident plane wave and the propagative PC eigenmode (or, more rarely, several such modes). An important issue is then to eliminate propagative reflected waves. In this paper we propose a semi-analytical algorithm for the design of gratings acting as wide-angle antireflection (AR) structures for two-dimensional PCs.

The article is organized as follows. In the next section we review existing AR structures for homogeneous media and PCs. Subsequently, in section III, the three steps of the new design procedure are presented in detail. This algorithm is then tested on several particular cases: a PC flat lens (section IV), a supercollimating PC (section V), and a non-reciprocal PC mirror (section VI). Finally, in section VII we discuss briefly the influence of AR gratings in the evanescent-wave regime.

II. BACKGROUND

In this section we shall review existing methods of eliminating reflections from interfaces which separate different media. We start with the simpler case of interfaces separating homogeneous media, since the basic tools used to deal with this class of systems turn out to be valuable also when more complex media, such as PCs, are involved. The existing AR solutions can be divided into three broad classes. The refractive index n of AR coatings depends only on the coordinate z perpendicular to the interface; depending on whether n(z) is piecewise-constant or not, we speak of homogeneous-layer or inhomogeneouslayer AR coatings 8 . In turn, the refractive index of AR gratings is also a function of the coordinates parallel to the interface. Figure 1 shows example AR structures belonging to these three classes.

The simplest homogeneous-layer AR coating consists of a single thin film with refractive index n and thickness d chosen so as to ensure destructive interference of waves reflected from its top and bottom surfaces, excited by a plane wave with a certain frequency ω and angle of incidence θ. The values of n and d can be obtained analytically. In order to improve the angular and frequency tolerance of the coating, one can increase the number of layers to make the transition between the refractive index of the super-and substrate more gradual 8 , thus reducing the amplitude of waves reflected on the individual discontinuities of n(z). In the limit of an infinite number of layers, one arrives at an inhomogeneous-layer coating with a continuous monotonic profile n(z). Several specific profiles have been proposed in the literature (see ref.

9 for a review). Unfortunately, these "ideal" AR coatings, even the single-layer one, cannot usually be realized because of the lack of suitable materials with the required values of n. This is the case, in particular, for air-solid interfaces, where thin films with refractive index close to unity are needed.

Short of using special ultra-low-index media, such as so-called Reststrahlen materials 8 , there are two basic ways of circumventing this problem. One consists in abandoning completely the structures based on a "continuous transition" between the super-and substrate refractive indices in favor of interference-based coatings 8 . In these systems, the total reflected wave vanishes thanks to the destructive interference of partial waves generated at interfaces between layers with contrasting values of n. For instance, so-called v-coatings consist of two layers with prescribed refractive indices n 1 and n 2 (corresponding to realistic materials) and thicknesses d 1 and d 2 adjusted so as to eliminate reflection at the desired values of ω and θ. Note that n 1 and n 2 must satisfy certain conditions in order that appropriate d 1 and d 2 can be found (ref. 10, p. 96). A disadvantage of v-type AR coatings is that their angular and frequency tolerance are usually inferior even to those of the corresponding ideal single-layer coatings (ref. 10, p. 97; ref. 11, p. 188).

The other solution consists in using subwavelength gratings to simulate AR coatings with arbitrary n(z) profiles. In many cases, the effective-medium theory of gratings can be employed to calculate the grating profile mimicking the desired n(z) dependence 9 . Several types of gratings, such as the lamellar, trapezoidal, sinusoidal, triangular and pyramidal ones, have been studied in the literature and shown to have good AR properties 9,[START_REF] Bräuer | [END_REF][13][14] . A review of the experimental methods used to fabricate such AR structures can be found in ref. 15. Let us now turn to the case of PCs. One of their distinguishing features is the dependence of their reflection coefficient on the position of their truncation plane. One could hope then that a significant reduction of a PC's reflectance could be achieved without adding any AR structure, but simply by choosing an appropriate cut. For some crystals, this has indeed proved to be possible 16,17 . In particular, Botten et al. 17 have shown that very low reflectance is a rather general feature of rod-type PCs truncated midway between successive layers of rods. For many crystals, however, no truncation plane provides a sufficiently small value of reflectance (see fig. 2). This method of reducing reflection is therefore not general enough, and one often has to resort to introducing some AR structure. Several types of them have been proposed in the literature. More often than not, they have much in common with one of the solutions developed with homogeneous materials in mind, reviewed in the previous subsection.

The simplest approach has been proposed by Li et al. 18 . It consists in placing in front of the crystal a vtype AR coating composed of two layers made of constituent materials of the crystal, usually air and a dielectric. Their thicknesses can be determined analytically or graphically as soon as the reflection coefficient of the semi-infinite uncoated PC at the selected operation frequency is known. This frequency is assumed to be low enough that only the zeroth diffraction order be propagative in the layer closest to the PC surface, so that all higher orders reflected by the crystal decay before reaching the boundary between the two layers. If it is not the case, the structure may still work, but the optimum layer thicknesses will not in general be given by the analytical formulas derived from the thin-film theory. The basic disadvantage of this otherwise very attractive approach is the relatively low angular and frequency tolerance of v-type AR coatings. Another group has studied a similar approach, in which one of the homogeneous layers is replaced by a grating of rods or holes 19,20 . The performance of the resulting AR gratings was shown to be similar to that of v-type coatings 20 . Related structures, albeit with only a single degree of freedom (the radius of the outermost rods of a rod-type PC), were also analyzed by Jin and He 21 . In turn, Zhang and Li 22 proposed a more complex wideangle AR grating for the interface between air and a PC flat lens, whose fabrication would be seriously hindered, though, by the presence of extremely thin air slits and dielectric veins.

In contrast to the above works, which concentrated on the low-frequency regime, the authors of refs. 23-25 endeavored to eliminate the reflection from an interface between a PC and a semiconductor at a frequency where multiple propagative diffraction orders existed in the latter medium. They proposed an AR grating consisting of elongated drop-shaped air holes, this nonstandard form being motivated by the desire of ensuring a gradual transition between the two media. In fact, the shape of the resulting structure resembles closely the classical trapezoidal AR gratings studied, e.g., by Raguin and Morris 14 . The improvement brought about by these gratings has subsequently been demonstrated experimentally 26 . Unfortunately, Baba et al. did not provide any analytical guidelines regarding the choice of the geometrical parameters of the gratings, resorting instead to a timeconsuming scan of the parameter space in order to find the optimum structure 23,24 .

A very important contribution was made by Lawrence et al. 27,28 , who introduced the concept of generalized matrix-valued effective immittance X of gratings and showed that their effective-medium description can be made arbitrarily accurate by allowing X to have sufficiently large dimensions. Such matrix-valued immittance cannot serve as a drop-in replacement of the corresponding scalar quantity in standard formulas derived for homogeneous media 28 . Therefore, for instance, analytical determination of the optimum parameters of an AR structure for a given PC composed of layers described by a matrix X is not possible. However, Lawrence et al. 28 derived equations similar to the classical Fresnel formulas, but involving generalized immittances, and showed that accurate values of the scattering coefficients of typical PCs could be obtained already using immittance matrices truncated to 5 × 5 elements or less. Since numerical calculations involving such small matrices are very fast, it becomes feasible to design optimum AR gratings by performing a full scan of the available parameter space. Example AR structures presented in ref. 28 include, for instance, a relatively wide-band AR coating of a superprism-type PC at a frequency where multiple propagative diffraction orders exist in the adjacent dielectric. A slight limitation of the approach of Lawrence et al. 28 is that the generalized immittance has only been defined for 2D gratings symmetrical with respect to a twofold rotation axis parallel to the direction of invariance. Therefore, it does not cover, for instance, triangular or trapezoidal gratings.

All the AR structures discussed so far are relatively compact, with thickness rarely exceeding one or two lattice constants of the underlying PC. Some authors have advocated sacrificing compactness in favor of potentially larger frequency and angular tolerance offered by thick stacks of gratings, whose geometry changes gradually so as to ensure a smooth ("adiabatic") transition of the electromagnetic field of the incident plane wave towards the Bloch mode of the semi-infinite PC. Several design principles for such gratings have been proposed 29,30 . In this work, we shall focus on compact AR structures, and therefore we omit a detailed discussion of adiabatic AR gratings.

III. DESIGN PROCEDURE

In this section we shall present an alternative method of designing AR gratings for PCs. Compared to the approaches reviewed in section II, it has the following distinguishing features. First, instead of performing a potentially time-consuming global scan of possible geometries, we use an effective-medium model of PCs to calculate analytically the geometrical parameters of a "tentative" AR grating; then, if necessary, we refine these parameters with a numerical local-minimization algorithm. The final result is a trapezoidal AR grating. Second, the AR structures obtained in the proposed way have good angular tolerance, which in some important applications of PCs is more significant than the frequency tolerance. For instance, the quality of the image produced by a PC flat lens with effective refractive index n ≈ -1 depends in the first place on the angular range of incident plane waves which are transmitted through the lens with little or no energy loss. On the other hand, frequency tolerance is not vital since n can be close to -1 only in a narrow frequency band. In our developments we draw on the results of Raguin and Morris 14 , who demonstrated that triangular and trapezoidal gratings allow a significant reduction of reflectance at interfaces between homogeneous media in a wide range of incidence angles.

The proposed design algorithm consists of three basic steps. In step 1 we use the value of the reflection coefficient of the PC at a particular frequency and incidence angle to calculate analytically the parameters of a homogeneous AR coating appropriate for the PC. In step 2 we convert the coating into a lamellar AR grating composed of the same materials as the PC itself. If necessary, in step 3 we adjust the shape of the grating using a numerical optimisation procedure, obtaining finally a trapezoidal AR grating. We shall now proceed to detailing these three constituent steps of the algorithm. They are summarised in the flowchart in fig. 3.

Step 1 We begin by using the classical theory of AR coatings to calculate the refractive index and thickness of a single-layer coating that should minimize the reflection from the PC surface at a fixed angle of incidence θ and frequency ω. The value of θ should lie approximately midway the desired angular operation range of the final AR structure. Consider the system shown in fig. 4, in which a propagative plane wave with frequency ω = ck 0 and wave vector k 1 = (k x , k z1 ) = (n 1 k 0 sin θ, n 1 k 0 cos θ) impinges from the dielectric 1 with refractive index n 1 on the surface of the homogeneous film 2 with thickness d 2 and refractive index n 2 coating the PC 3. Let us assume that medium 2 supports exactly one propagative diffraction order and is sufficiently thick for all the evanescent orders reflected from the PC surface to vanish at the interface between media 1 and 2. We can calculate the total amplitude of the reflected plane wave, r, by summing up Step 1

Step 2

Step 3

FIG. 3: Successive steps of the proposed AR grating design algorithm.

all the multiple reflections occurring in the system:

r = r 12 + t 12 Φr 23 Φ ∞ n=0 (r 21 Φr 23 Φ) n t 21 = r 12 + (t 12 t 21 -r 12 r 21 )r 23 Φ 2 1 -r 21 r 23 Φ 2 , (1)
where r ij and t ij denote the amplitudes of the waves reflected from the interface between media i and j and transmitted through it, respectively, and Φ ≡ e ikz2d2 with k z2 ≡ (n 2 2 k 2 0 -k 2 x ) 1/2 . The coefficients r ij and t ij are given by the Fresnel's formulas,

r ij = X j -X i X j + X i , t ij = 2X j X j + X i , (2)
with X i denoting the transverse immittance of medium i normalised to the immittance of free space. For s polarisation X i is defined as the transverse impedance of medium i:

X i ≡ Z i ≡ k 0 µ i /k z,i .
For p polarisation, it is defined as the transverse admittance of the medium:

X i ≡ Y i ≡ k 0 ǫ i /k z,i . From eqs.
(2) it immediately follows that r 21 = -r 12 and t 12 t 21 -r 12 r 21 = 1, hence

r = r 12 + r 23 Φ 2 1 + r 12 r 23 Φ 2 . ( 3
)
The parameters of the antireflection coating, n 2 and d 2 , can now be obtained by requiring the numerator of the fraction in the above equation to vanish. If the coating is lossless, so that |Φ| = 1, the numerator vanishes if and only if (i) the moduli of r 12 and r 23 are equal and (ii) the thickness d 2 is such that where m is an integer and arg z stands for the argument of the complex number z. Solving for d 2 , we get

arg r 23 + 2k z2 d 2 = arg r 12 + (2m + 1)π, (4)
d 2 = arg r 12 -arg r 23 + (2m + 1)π 2k z2 . (5)
It is usually best to choose the value of m corresponding to the smallest positive admissible value of d 2 ; otherwise, internal resonances in the coating layer can spoil its antireflective properties for some angles of incidence.

We shall now use condition (i) to determine the refractive index n 2 of the AR coating. Substituting the Fresnel's formulas (2) into the condition

|r 12 | 2 = |r 23 | 2 , we obtain (X 2 -X 1 ) 2 (X 2 + X 1 ) 2 = (X 3 -X 2 )(X * 3 -X 2 ) (X 3 + X 2 )(X * 3 + X 2 ) . (6)
Straightforward algebra leads to

X 2 2 = -Re X 3 + |X 3 | 2 /X 1 Re X 3 -X 1 X 2 1 . (7)
The s-and p-polarization cases need now to be considered separately. Assuming materials 1 and 2 to be nonmagnetic (µ 1 = µ 2 = 1), in the s-polarization case we have

X 2 = Z 2 = k 0 k z2 = k 0 n 2 2 k 2 0 -n 2 1 k 2 0 sin 2 θ , (8)
hence

n 2 2 = n 2 1 sin 2 θ + 1 Z 2 2 . ( 9
)
It is easily seen that Z 2 2 must be non-negative in order that k z2 be real, as we have assumed. For p polarization,

X 2 = Y 2 = n 2 2 k 0 k z2 = n 2 2 k 0 n 2 2 k 2 0 -n 2 1 k 2 0 sin 2 θ . ( 10
)
This leads to the quadratic equation for

n 2 2 , n 4 2 -Y 2 2 n 2 2 + Y 2 2 n 2 1 sin 2 θ = 0, (11)
which has real solutions

n 2 2 = 1 2 Y 2 2 ± Y 4 2 -4Y 2 2 n 2 1 sin 2 θ (12) provided that Y 4 2 -4Y 2 2 n 2 1 sin 2 θ ≥ 0.
It can be shown that this condition, together with the condition of realvaluedness of k z2 [for both solutions of eq. ( 12)], is fulfilled if and only if

Y 2 2 ≥ 4n 2 1 sin 2 θ. (13)
In practice, there are further constraints on the choice of the constituent material of the coating. Other experimental issues aside, n 2 is bounded from below by the refractive index of air, and from above, by the index at which a second propagative diffraction order appears at the given value of k x . In appendix A it is shown how such constraints of the general form

n 2 min ≤ n 2 2 ≤ n 2 max ( 14
)
can be transformed into equivalent constraints on the immittance of the PC, X 3 . Let us denote by Xα (α = 3, min, max) the reduced immittances Xα / X1 . It follows then that ( 14) holds if and only if one of the two following sets of conditions on X3 is satisfied:

   X3 / ∈ int P - X3 / ∈ int C min X3 / ∈ ext C max or    X3 / ∈ ext P - X3 / ∈ ext C min X3 / ∈ int C max , (15)
where P -stands for the half-plane Re X3 < 1 and C α (α = min, max) are the circles of radius 1 2 |1 -X2 α | centered at 1 2 (1 + X2 α ), 0 . The symbols int A and ext A denote the interior and exterior of a region A. To illustrate various possible geometries of the region of the complex X3 plane determined by the constraints (14) transformed into the form (15), fig. 5 shows the shape of this region for s polarization and three distinct choices of the parameters n min , n max , n 1 and θ.

Step 2 The coating obtained in step 1 is not practical, since its fabrication would call for integration of the PC with a completely different solid; moreover, a suitable material with the required value of refractive index might not be easily available. However, as noted in section II, a homogeneous thin film can often be replaced without adverse effects by a subwavelength grating: this is step 2. Such a grating could be easily etched in the same process as the underlying PC; it would then naturally be composed of the same materials as the PC, with permittivities, say, ǫ l and ǫ h (ǫ l < ǫ h ).

In order to calculate the fill factor of a binary lamellar grating mimicking a layer with refractive index n 2 obtained in the previous step, one can resort to the classical second-order effective-medium theory of gratings due to Rytov, described in ref. 14. According to this theory, the effective permittivity ǫs of a binary grating with period a and fill factor f (0 ≤ f ≤ 1), composed of materials with permittivities ǫ l and ǫ h , and operating in the s polarization is

ǫs = ǭs 1 + π 2 3 k 0 a 2π 2 f 2 (1 -f ) 2 (ǫ h -ǫ l ) 2 ǭs , (16)
where

ǭs = f ǫ h + (1 -f )ǫ l . (17)
For p polarization, the effective permittivity ǫp is

ǫp = ǭp 1 + π 2 3 k 0 a 2π 2 f 2 (1 -f ) 2 (ǫ h -ǫ l ) 2 ǭs ǭp ǫ h ǫ l 2 , (18) where ǭp
= f ǫ h + 1 -f ǫ l -1 (19)
and ǭs is given by eq. ( 17). Thus, the required fill factor can be obtained by setting ǫs or ǫp to n 2 2 in eq. ( 16) or (18) and solving it numerically for f .

It should be noted that in the domain of validity of Rytov's theory (small k 0 a/2π) the functions ǫs (f ) and ǫp (f ) are monotonically increasing from ǫ l to ǫ h . Thus, a binary grating cannot simulate a material with permittivity outside the range delimited by the permittivities of the grating's constituent materials. As a result, the bounds n 2 min and n 2 max mentioned in step 1 must fulfill n 2 min ≥ ǫ l and n 2 max ≤ ǫ h , respectively.

Step 3 The structure obtained at this stage should, in principle, ensure low reflectance for incidence angles close to θ. Nevertheless, owing to the applied approximations-neglect of higher diffraction orders excited by the PC and the AR grating-its geometrical parameters might not be precisely optimal. In addition, it is well known 13,14 that trapezoidal and triangular AR gratings have larger angular and frequency tolerance than lamellar ones. Therefore, as step 3, it is advisable to apply a numerical optimization procedure to adjust the geometry of the grating, described by some small number of parameters, so as to minimize a given objective function ρ. The geometry obtained in step 2 can be expected to provide a good starting point for a local search algorithm, such as the Nelder-Mead simplex method (ref. 31, section 10.4).

IV. APPLICATION TO A PHOTONIC CRYSTAL FLAT LENS

The first PC we shall consider is a hexagonal lattice of air holes of radius r = 0.365a, where a is the lattice constant, etched in a dielectric matrix of permittivity ǫ = 10.6. These parameters correspond to the structure whose fabrication was reported in ref. 32. For s polarization, at frequency ω = 0.311 × 2πc/a, the effective refractive index n = -1 can be attributed to the crystal, since its equifrequency contour (EFC) takes an approximately circular shape (fig. 6) with radius K ≈ ω/c and group velocity directed inwards. (Information about the techniques used to obtain the numerical results discussed in this paper can be found in appendix B.)

C min C max (a) 1 0 1 Im Q Z 3 0 1 2 Re Q Z 3 C min C max (b) 0 1 2 Re Q Z 3 C min C max (c) 0 1 2 3 Re Q Z 3
Veselago [START_REF] Veselago | [END_REF] predicted that a slab of material with n = -1 should act as a flat lens: an image of an object placed near one of the surfaces of the slab should be produced on the other side of the slab. Figure 7 shows the map of the modulus of the electric field generated by a wire source with current 1 A located above a slab of the PC in question. The parts (a) and (b) refer to slabs truncated in the ways shown in figs. 8(a) and (b), respectively; from now on, these two structures will be referred to as S1 and S2. In accordance with the theoretical predictions, images are formed below the slabs. However, their amplitude is low (67 and 79 V/m for structures S1 and S2, respectively) and intense beams reflected from the top of the lenses are visible in the upper part of the plots. This suggests that only a small fraction of energy is transmitted through the lenses. Indeed, as shown in fig. 8(a), the reflectance of structure S1, |r 0 (θ)| 2 , where r 0 is the specular reflection coefficient, exceeds 29% for all angles of incidence. Structure S2 performs better for low incidence angles, but degrades quickly with increasing θ. We shall now apply the algorithm presented in section III to design an AR grating for this PC, which ideally should function regardless of the angle of incidence.

As discussed in refs. 38 and 39, the effective-medium model of PCs tends to be more accurate for crystals truncated along a plane with constant permittivity profile, as is the case, for instance, for structure S1. Therefore in the first step of the design procedure we shall calculate the effective transverse impedance Z 3 of this structure. We consider two ways of obtaining this quantity. First, we calculate it in the framework of the model presented in ref. 38 using eq. ( 18) derived there. At frequency ω = 0.311 × 2πc/a and angle of incidence θ = 45 • (corresponding to k x = 0.220 × 2π/a) we get Z 1 = 1.414 and Z 3 = 0.319. We should now check whether Z3 ≡ Z 3 /Z 1 = 0.225 lies within the region determined by the conditions (15) equivalent to the constraints (14) with n min = 1 and n max = 2.51 (the maximum index of a medium in which only a single propagative diffraction order exists). Figure 9, in which the value of Z3 cited above is marked with point A, shows that this is indeed the case. Therefore eqs. ( 5) and ( 9) can be used to calculate the parameters of the AR coating of the crystal: refractive index n 2 = 1.649 and thickness d 2 = 0.540a. The geometry of this structure, called S3 from now on, is shown in fig. 10(a) and its reflectance is plotted in fig. 10(e) with a solid black line. It can be seen that the application of the coating reduces significantly the reflectance of the crystal, especially for small angles of incidence. However, the parameters of S3 are certainly not optimal, since its reflectance at the "design angle" θ = 45 • is as large as 9%. This is due to the relatively large error introduced by the single-mode approximation for negative-refraction PC bands, as pointed out in ref. 38. We shall evaluate, therefore, an alternative method of obtaining the impedance Z 3 ≡ X 3 , which consists in calculating it directly from the rigorous specular reflection coefficient r 0 of the uncoated crystal at the chosen ω and k x . In other words, we assume that r 0 can be expressed in the form r 0 = (X 3 -X 1 )/(X 3 +X 1 ) [cf. eq. ( 2)] and invert this formula to obtain

X 3 = X 1 1 + r 0 1 -r 0 . ( 20
)
Of course, the effective immittance defined in this way depends on the material properties of medium 1. Nevertheless, at least for the PC in question, this dependence is weak for sufficiently small n 1 : we obtain Z 3 = 0.258+0.175i for n 1 = 1 and the effective impedance does not change by more than 10% up to n 1 = 2.25. As shown in fig. 9, the reduced impedance Z3 = 0.182 + 0.124i corresponding to the above value of Z 3 (marked with point B) also lies within the allowed region of the Z3 plane. Taking this value of Z 3 , from eqs. ( 5) and ( 9) we get n 2 = 1.884 and d 2 = 0.565a. The angular dependence of the reflectance of the PC covered with this coating, shown in fig. 10(b) and called S4 in the following, is plotted in fig. 10(e) with a solid gray line. It is evident that this structure has much better angular tolerance than S3; moreover, its reflectance at θ = 45 • is only 0.05%. Therefore we choose S4 as a basis for the further steps of the algorithm. Numerical inversion of eq. ( 16) gives the fill factor f = 0.192 of the binary grating mimicking a medium with n = 1.884. Since we would like the angular dependence of r 0 to be symmetric with respect to θ = 0, the grating should be positioned so as to preserve the vertical mirror symmetry axes of the underlying PC. This can be done in two ways, shown in figs. the other hand, grating S6 behaves better in the high-θ region.

The lamellar gratings obtained in step 2 provide already a remarkable improvement over the uncoated PC and, in contrast to the AR coatings from step 1, should be manufacturable. Nevertheless, their geometry can be further ameliorated. To this end, as mentioned in the last paragraph of section III, we use the Nelder-Mead simplex algorithm to find the optimum values of the dimensions w i , w o , h i , and h o parametrising the trapezoidal grating shown in fig. 11. The objective function ρ is defined as the average of the numerically calculated reflectance of the given structure over the desired angular tolerance interval [θ min , θ max ],

ρ = 1 θ max -θ min θmax θmin |r 0 (θ)| 2 dθ. (21)
The integral in eq. ( 21) is calculated with the 20-point Gauss-Legendre quadrature algorithm (ref. 31, section 4.5), whose typical relative accuracy, ∼10 -5 , is better than that of the reflectance calculations, ∼10 -3 . The initial shape of the grating is taken to correspond to one of the lamellar gratings obtained in step 2, i.e., w i = w o = f a, h i = 0, and

h o = d 2 .
The search routine is terminated when the size of the simplex, defined as the average distance of its vertices from its geometric centre, falls below 10 -5 . The final values of the geometrical parameters of the grating are determined by selecting the best among the 16 structures obtained by rounding each of the parameters delivered by the simplex algorithm upwards or downwards to a multiple of 0.01a. Application of this procedure with θ min = 0 • , θ max = 90 • to grating S5 yields grating S7 shown in fig. 12(a). The plot in fig. 12(e) (solid black line) demonstrates the excellent antireflective properties of this structure (note the scale of the vertical axis). Its average reflectance is as low as 2.8%; in fact, |r 0 (θ)| 2 does not exceed 5.5% until θ = 87 • . The structure does not seem to present special fabrication difficulties-e.g., acute angles-except possibly for the relatively thin dielectric veins separating the circular holes from the surface. Should this pose a real experimental difficulty, one can increase the value of h i at the expense of a slight performance deterioration. For example, grating S8 with h i = 0.08a [fig. 12(b)] has average reflectance of 4.8%. Figure 13 shows the map of the modulus of the electric field produced by a point source placed above a PC slab coated with AR gratings of type S7 from above and below. The comparison with fig. 7 reveals the significant improvement brought about by the AR grating: not only are the reflected beams prominent in the upper part of the latter figure suppressed, but the amplitude of the image formed by the lens grows to 159 V/m, two times more than in fig. 7(b).

Optimization of structure S6 leads to gratings with average reflectance comparable to that of S7 and S8 but composed of "narrower" trapezoids (w i +w o ≈ 0.2a), thus less suitable for fabrication. Therefore we omit the detailed discussion of these structures.

As a complement, we mention that in refs. of the specular reflection coefficient r 0 ) calculated with a less accurate quadrature algorithm. The average reflectance of structure S9 in the full 0 • -90 • range, 2.8%, matches that of S7; in a more restricted range, say, 0 • -80 • , the performance of grating S9 is even slightly better. Nonetheless, its disadvantage lies in the presence of very thin dielectric veins at the surface. Grating S10, with h i = 0.08a, is devoid of this problem. However, it is superseded by structure S8 with identical h i , which has somewhat lower average reflectance.

Finally, a word about tolerance to fabrication imperfections. A fabrication process invariably perturbs the geometrical parameters of the manufactured structure. To assess the sensitivity of the proposed gratings to fabrication errors, we have determined the maximum perturbation of each of the four geometrical parameters of grating S7 for which the grating's average reflectance in the angular range 0 ≤ θ ≤ θ max did not exceed 5%. Three values of θ max were considered: 90 • , 80 • , and 60 • . The results of this test are summarized in table I. It can be seen that the grating is more sensitive to variations of the height of the trapezoids (via the h o and h i parameters) than of their width (w o and w i ). The constraints for θ max = 90 • and θ max = 80 • are rather stringent and unlikely to be met in practice. In contrast, fabrication of a structure satisfying the constraints for θ max = 60 • seems well within reach of current technology. We have also tested the frequency tolerance of grating S7, finding that the its average reflectance stays below 5% for 0.3094 ≤ ωa/2πc ≤ 0.3113 (θ max = 90 • ), 0.3048 ≤ ωa/2πc ≤ 0.3122 (θ max = 80 • ), and 0.2511 ≤ ωa/2πc ≤ 0.3173 (θ max = 60 • ). This tolerance seems quite sufficient for applications related to lensing.

V. APPLICATION TO A SUPERCOLLIMATING PHOTONIC CRYSTAL

The second example to be considered is a PC composed of a square lattice of air holes of radius r = 0.3a, where a is the lattice constant, etched in a dielectric matrix of permittivity ǫ = 12.25. Near the frequency ω = 0.265 × 2πc/a its EFCs for p polarization take a square-like shape (cf. fig. 14). In consequence, supercollimated beams 42,43 can propagate in the crystal.

The solid and dashed black curves in fig. 15(e) show the angular dependence of the reflectance of this PC at ω = 0.265×2πc/a and for two different truncation planes, corresponding to the structures shown in figs. 15(a)-(b) and called S11 and S12 in the following. It is seen that the crystal cut through hole centers has fairly low reflectance: about 10% at normal incidence and decreasing for larger angles up to θ ≈ 65 • . This level of power losses might in fact be already sufficient for practical applications. Nevertheless, for the sake of illustration, we shall present the design procedure of AR gratings that help to decrease even further the reflectance of the PC in question.

Here, we are mostly interested in coupling the incoming θmax wi wo hi ho 90 • 0.378-0.424a ( 22 nm) 0.151-0.170a ( 9 nm) 0.045-0.055a ( 5 nm) 0.543-0.557a ( 7 nm) 80 • 0.330-0.463a ( 63 nm) 0.132-0.186a (26 nm) 0.033-0.062a (14 nm) 0.530-0.569a (19 nm) 60 • 0.192-0.533a (162 nm) 0.084-0.208a (59 nm) 0 -0.082a (39 nm) 0.497-0.586a (42 nm) TABLE I: Ranges of geometrical parameters of grating S7 for which its average reflectance at frequency 0.311 × 2πc/a in the angular range 0 ≤ θ ≤ θmax does not exceed 5%. The numbers in parentheses are the lengths of the tolerance intervals for a = 476 nm, which corresponds to operation wavelength λ = a/0.311 = 1530 nm. Note that the tolerance intervals correspond to perturbations of one parameter at a time (not all parameters simultaneously). light to modes lying on the flat horizontal part of the PC's EFC. As shown in fig. 14, at frequency ω = 0.265×2πc/a this corresponds roughly to the range |θ| ≤ 45 • , i.e., |k x | ≤ 0.187 × 2π/a. Therefore we choose θ = 22.5 • as the design angle of the AR coating. As in the previous section, we test two different ways of calculating the effective immittance (in this case, admittance) of the crystal cut along a constant-permittivity plane, i.e., structure S11. The effective-medium model presented in ref. 38 yields Y 3 = 6.138. In turn, the effective admittance calculated from the rigorous reflection coefficient of structure S11 embedded in air is Y 3 = 6.075 -1.191i. Figure 16 shows that the reduced admittances corresponding to both these values lie within the region of the complex Ỹ3 plane determined by the conditions (15) equivalent to the constraints (14) with n 1 = n min = 1, n max = 3.391 (the refractive index for which the second propagative diffraction order appears) and θ = 22.5 • . The parameters of the AR coatings determined from these two values of Y 3 are (n 2 = 2.548, d 2 = 0.374) and (n 2 = 2.595, d 2 = 0.391), respectively. Figures 15(c)-(d) show the geometry of these coatings, henceforth referred to as S13 and S14, whereas the angular dependence of their reflectance is plotted with solid and dashed gray lines in fig. 15(e). As in the PC lens case, the AR coating S14 designed using the value of Y 3 obtained from the rigorous reflection coefficient of the crystal performs slightly better than the other one. Therefore structure S14 shall be used in the subsequent design step.

From numerical inversion of eq. ( 18) it follows that the fill factor of the binary grating mimicking a medium with n = 2.595 for p polarization is f = 0.812. Figures 17(a)-(b) show the geometry of the two gratings, called S15 and S16, with this fill factor and a mirror symmetry plane perpendicular to the direction of periodicity. From the juxtaposition of their reflectance curves [fig. 17(c)] it follows that structure S15 has somewhat better performance than S16. In fact, there is some similarity between the geometry of grating S15 and the truncated crystal S12, which also exhibited fairly low reflectance: the surface of both these structures contains "teeth" shifted by 1 2 a in the horizontal direction with respect to the positions of the circular holes. Therefore, one could view the crystal S12 as an imperfect realization of the AR grating S15.

The lamellar grating S15 can be further ameliorated by adjusting its thickness d 2 and fill factor f to minimize the objective function ρ defined in eq. ( 21). We take θ min = 0, θ max = 45 • and, as before, perform the optimization with the Nelder-Mead simplex algorithm. This leads to structure S17 with d 2 = 0.37a and f = 0.73, shown in fig. 17(c). Its reflectance curve is plotted in fig. 17(d) (solid line). In the angular range 0 ≤ θ ≤ 45 • , the reflectance never exceeds 0.6%, on average amounting to only 0.12%. The structure does not seem to present any special fabrication problems. It is possible to continue the grating's optimization by allowing it to take a trapezoidal rather than a lamellar shape; however, in view of its already very good AR properties, this appears unnecessary.

VI. APPLICATION TO A NON-RECIPROCAL MIRROR

In this section, the procedure is applied to a unidirectional mirror similar to that proposed in ref. 44. This device has the form of a slab of the PC shown in the inset of fig. 18(a). The PC consists of a hexagonal lattice of non-centrosymmetric motifs etched in a magneto-optical matrix characterized by a gyrotropic permittivity tensor

ǫ =   (2.25) 2 0 0.1i 0 (2.25) 2 0 -0.1i 0 (2.25) 2   . ( 22
)
The presence of this magneto-optical material lifts the time-reversal symmetry of Maxwell's equations. Since the spatial inversion symmetry is also broken, the crystal becomes nonreciprocal: the usual property of the dispersion relation, ω(k) = ω(-k) (ref. 45, pp. 22-23), no longer holds. The p-polarization EFC of this crystal at frequency ω = 0.4547 × 2πc/a is shown in fig. 18(a).

It can be seen that plane waves impinging at the angle of about 44 • (corresponding to k x = 0.315 × 2π/a) on the bottom surface of a slab made from this PC will be coupled to its propagative mode, and thus will be partially transmitted. However, waves travelling in the opposite direction, incident from the top, will be totally reflected, since the crystal has no propagative modes with -0.323 × 2π/a ≤ k x ≤ -0.308 × 2π/a. This behaviour justifies the name unidirectional mirror. The problem with the presented device is its large forward loss: even waves propagating in the "allowed" direction undergo a significant reflection on the surfaces of the slab. This can be seen in fig. 18(b), where the reflectance curve of the crystal truncated as in fig. 18(a) is juxtaposed with the relevant fragment of the EFC. Clearly, to be useful in practice, the unidirectional mirror needs to be coated with some AR structure. Unfortunately, the design of an appropriate AR grating using the procedure described in section III turns out to be impossible. For instance, at k x = 0.315 × 2πc/a, near the centre of the region where the unidirectional mirroring effect occurs, the reflection coefficient of the uncoated PC is r 0 = 0.890 -0.055i, which corresponds to Ỹ3 = 13.5 -7.3i. As evidenced by fig. 18(d), this lies far outside the region of the complex Ỹ3 plane determined by the conditions (15) equivalent to the constraints (14) with n 1 = n min = 1 and n max = 1.50 (the refractive index of the most optically dense coating in which the second propagative diffraction order would not yet appear). In fact, if we blindly apply eqs. ( 7) and ( 12) to calculate the refractive index of the optimum AR coating, we obtain n 2 = 5.795 or 0.698. It is obvious that none of these indices can be simulated by any binary grating composed of the constituent materials of the PC. Therefore, an AR structure for the unidirectional mirror will probably need to be designed with some purely numerical method. In particular, Lawrence et al. 28 have shown their approach to give good results for a superprism that, uncoated, has an extremely high reflectance (|r 0 | 2 = 0.996). We wish to stress, however, that the breakdown of the design procedure in this case is not due to a failure of the applied approximations. Covering the crystal by an AR coating with parameters obtained from eqs. ( 5) and ( 12)-refractive index 0.698 and thickness 7.16a-does make the reflectance drop from 79% to barely 1.5%. This shows that the approximations are justified, and practical realisation of the design is inhibited by material constraints rather than theoretical flaws.

VII. EVANESCENT WAVES REGIME

The behaviour of evanescent waves is approached through the example of the PC flat lens described in section IV. In that case, the amplification of these waves is of vital importance to obtain subwavelength resolution 2 . In a PC flat lens, two mechanisms may lead to an amplification of evanescent waves 46 : the single interface resonances and the overall resonances (see ref. 46 for a detailed discussion). While the original flat lens 2 exploits the former mechanism, in the PC context solely the latter has been investigated. This situation can be changed thanks to the new models introduced recently 28,38,47 and based on the reflectance of semi-infinite crystals.

The present approach is based on the following conjecture: assuming that a nearly constant effective index is obtained over all the range of wavevectors corresponding to propagative waves, it can be expected that this remains true when this range is continued in the evanescent regime. For instance, this behaviour has already been observed in multilayers 48 . For the photonic crystal flat lens, the refractive index is close to -1 at ω = 0.311 × 2πc/a (from the dispersion law of Fig. 6) and, with the AR grating, it can be considered that the effective index is nearly -1 for the range of propagating waves. Now, if the effective index remains nearly -1 for evanescent waves, then it can be expected that single-interface resonances are present around the working frequency ω = 0.311×2πc/a. Fig. 19 shows the single-interface resonances of structure S7 below the light cone. It is observed that the dispersion curve of these resonances starts from the in- tersection between the bands of the bulk PC and the the light cone at ω = ck x = 0.311 × 2πc/a. This shows that the effective index has no jump from propagative to evanescent waves, and thus confirms the conjecture proposed above, as well as the result found in ref. 48 in the one-dimensional case. To our knowledge, no model can predict this continuity of the effective index at this change of regime. This might be a subject for further investigations.

VIII. DISCUSSION AND CONCLUSIONS

In this article we have presented a new method of designing gratings that, superimposed on surfaces of PCs crystals, will minimize their reflectance. The design algorithm consists of three steps. First, the parameters of a homogeneous-layer AR coating are calculated from an effective-medium approximation of the PC in question. Second, an analytical effective-medium theory of gratings is used to find the parameters of a binary lamellar grating composed solely of the constituent materials of the crystal and approximating the coating obtained in the previous step. Third, the shape of the grating is refined with a numerical local-search routine so as to minimize the reflectance of the structure in the desired angular or frequency range. Thanks to this last step, the approximations made in the analytical derivations used in the first two steps of the procedure can be compensated for.

The approximation errors mentioned above have two sources. The first is the application of an effectivemedium model to describe the reflection on the surface of the crystal. In the form presented in ref. 38, this model relies on the single-mode approximation: the incident plane wave is assumed to excite only one PC mode. This approximation is the better, the more the profile of the electromagnetic fields of the dominant PC mode on the crystal surface resembles that of a single plane wave. Thus the accuracy of the single-mode approximation can be estimated by calculating how much of the energy of the electric and magnetic field of the dominant mode on the crystal surface is contained in their zeroth Fourier harmonics.

When the contribution of higher-order modes to the surface field becomes too large, the "empirical" definition of the crystal's immittance X 3 (20) can be used to rescue the analytical Fresnel-like formula expressing r 0 as a function of the immittance X 1 of the adjacent homogeneous medium. Higher-order crystal modes are then no longer assumed not to be excited at all. Instead, we effectively freeze their excitation amplitudes at the values obtained for the particular value of X 1 used in the calculation of X 3 . The idea is that as long as the contribution of these modes to the surface field remains moderate, the effect induced by the change of their amplitudes will be even smaller, and so justifiably negligible. In many practical situations this approximation gives excellent results. For instance, for the crystals studied in sections IV and V, it allows us to recover to within a few percent the change in the PC's reflection coefficient from the reference value used to calculate X 3 as the refractive index of the homogeneous medium is varied almost in the whole range corresponding to a single propagative diffraction order. In the superprism-type crystal analysed in section VI the first evanescent mode's contribution to the surface field is as important as that of the propagative mode, and even the effective-immittance-based formula for r 0 does not give better accuracy than just assuming the reflection coefficient to be independent from the refractive index of the medium adjacent to the crystal. However, this suffices in fact to find the parameters of an efficient AR coating, even though the latter cannot be converted into a practically realisable grating.

The second source of error is the Rytov's approximation.

The frequency region interesting for PC applications-ωa/2πc typically from 0.1a to 0.5a-lies close to the limit of applicability of Rytov's effectivemedium model of gratings. However, the numerical results presented in this paper indicate that the reflectivity curves of typical binary gratings designed with this model do not differ much from those of the original homogeneous coatings. The simple numerical post-optimisation procedure employed in step 3 seems fully capable of compensating the error due to Rytov's approximation.

The algorithm of AR grating design presented in this paper can be viewed as complementary to the method proposed by Lawrence et al. 27,28 . While their approach is based on an exhaustive scan of the whole parameter space (made very efficient by the application of the matrixvalued effective immittance of gratings), ours rests on approximate analytical considerations used to find a starting point for a local search procedure.

The proposed method has been applied to three example crystals with EFCs of different curvature: a supercollimating crystal with a very flat EFC, a crystal exhibiting negative refraction, with almost circular EFC, and a PC superprism, whose EFC has a kink. In the two first cases, the design process succeeded in producing AR gratings ensuring very low reflectance in a wide angular range. The obtained structures are quite compact and apparently rather straightforward to fabricate. In the last case, the procedure broke down owing to the violation of the constraints (15) on the effective immittance of the crystal that must be satisfied in order that the AR coating produced in the first step can be approximated with a binary grating made of realistic materials. The existence of these constraints is the basic limitation of the presented procedure. 14) together with the constraint (13) for at least one of the solutions (12) of eq. ( 11). The symbol n 2

x denotes n 2 1 sin 2 θ.

FIG. 1 :

 1 FIG. 1: Example AR structures belonging to the three principal classes described in the text. Darker areas denote regions with higher refractive index. (a) A homogeneous-layer AR coating. (b) An inhomogeneous-layer AR coating. (c) A (binary lamellar) AR grating.

FIG. 2 :

 2 FIG. 2:The dependence of the reflectance |r0| 2 of the PC shown in the inset, placed in air, on the location of the truncation plane z = z0. The PC consists of a hexagonal lattice of air holes with radius 0.365a, where a is the lattice constant, etched in a dielectric matrix with permittivity ǫ = 10.6. The impinging wave is normally incident, s-polarized and has frequency ω = 0.311 × 2πc/a. It can be seen that |r0| 2 does not fall under 0.13 for any truncation plane.

2 FIG. 4 :

 24 FIG.4: System considered in step 1 of the AR grating design procedure.

FIG. 5 :FIG. 6 :

 56 FIG. 5: Shaded: regions of the complex Z3 plane determined by the condition (15) equivalent to the constraint (14) for s polarization and (a) n1 = 1, nmin = 1.5, nmax = 3, θ = 0, (b) n1 = 1, nmin = 1, nmax = 3, θ = 30 • , (c) n1 = 1.5, nmin = 1, nmax = 3, θ = 0. The circles Cmin and Cmax are defined in the text after eq. (15); note that in the case (b) Cmax degenerates into the point (1, 0).

FIG. 7 :FIG. 8 :

 78 FIG. 7: Modulus of the electric field generated by an spolarized wire source with current 1 A located above a slab of the PC studied in section IV truncated along a plane (a) lying midway between two neighboring rows of holes, (b) crossing the centers of holes.

3 FIG. 9 :

 39 FIG. 9: Shaded circle: region of the complex Z3 plane determined by the condition (15) equivalent to the constraint (14) for s polarization, n1 = nmin = 1 and nmax = 2.51. Points A and B: reduced impedances Z3 of structure S1 calculated in two different ways described in the text.

  10(c) and (d). The reflectance of these two structures, called S5 and S6, is plotted in fig. 10(e). Clearly, grating S5 reproduces fairly faithfully the original reflectance curve of the AR coating S4. On

2 FIG. 10 :

 210 FIG. 10: (a)-(b) Geometry of AR coatings S3 and S4, characterised by refractive index n2 and thickness d2 specified next to the drawings. (c)-(d) Geometry of binary lamellar AR gratings S5 and S6, characterised by fill factor f and thickness d2 specified next to the drawings. (e) Angular dependence of the reflectance of the structures shown in parts (a)-(d).

FIG. 11 :

 11 FIG.11: Definition of the geometrical parameters wi, wo, hi, and ho of a trapezoidal grating superposed on the surface of structure S1.

  40 and 41 two other trapezoidal AR gratings, here denoted S9 and S10, were presented. Their geometrical parameters, shown in figs. 12(c) and (d), were obtained by minimizing the objective function 2 π π/2 0 |r 0 (θ)| dθ (average modulus

FIG. 12 :FIG. 13 :

 1213 FIG. 12: (a)-(d) Geometry of AR gratings S7-S10 characterized by parameters wi, wo, hi, and ho specified next to the drawings. (e) Angular dependence of the reflectance of the structures shown in parts (a)-(d). To help visualize the details of the |r0(θ)| 2 dependence, the y axis has been truncated at |r0| 2 = 0.2.

FIG. 14 :

 14 FIG.14: EFC of the PC studied in section V at frequency ω = 0.265 × 2πc/a. The shaded region corresponds to the range θ ≤ 45 • (|kx| ≤ 0.187 × 2π/a), where the EFC is approximately flat and for which the minimization of the PC's reflectance is made.

FIG. 15 :

 15 FIG. 15: (a)-(b) Geometry of the PC studied in section V truncated along a plane (a) lying midway between two neighboring rows of holes, (b) crossing the centers of holes. (c)-(d) Geometry of AR coatings S13 and S14, characterized by refractive index n2 and thickness d2 specified next to the drawings. (e) Angular dependence of the reflectance of the structures shown in parts (a)-(d).

3 FIG. 16 :

 316 FIG.16: Shaded circle: region of the complex Ỹ3 plane determined by the condition (15) equivalent to the constraint(14) for p polarization, n1 = nmin = 1 and nmax = 3.391. Points A and B: reduced admittances Ỹ3 of structure S11 calculated in two different ways described in the text.

FIG. 17 :

 17 FIG. 17: (a)-(c) Geometry of binary lamellar AR S15, S16 S17 characterized by fill factor and thickness d2 specified next to the drawings. (d) Angular dependence of the reflectance of the structures shown in parts (a)-(c).

3 FIG. 18 :

 318 FIG. 18: (a) p-polarization EFC of the magneto-optical PC shown in the inset at frequency ω = 0.4547 × 2πc/a. (b) Solid line: kx-dependence of the reflectance |r0| 2 of this crystal, placed in air and truncated in the way indicated in the inset of part (a). Dashed line: a fragment of the EFC from part (a). The shaded region indicates the range of kx for which the crystal has a non-reciprocal gap, i.e. there are no propagative modes with the x component of the Bloch vector equal to -kx. (c) Shaded circle: region of the complex Ỹ3 plane determined by the condition (15) equivalent to the constraint (14) for p polarization, n1 = nmin = 1 and nmax = 1.50. The reduced admittance of the crystal, Ỹ3 = 13.5 -7.3i, lies far beyond the range of the graph.

FIG. 19 :

 19 FIG. 19: Single-interface resonances of the structure S7. The shaded areas are the bulk bands of the underlying PC.

TABLE II :

 II Minimum and maximum bounds on Y 2 2 sufficient and necessary for fulfillment of the condition (

	Range of n 2 min , n 2 max and n 2 x		Y 2 min	Y 2 max
	0 < n 2 x ≤	n 2 min 2	,	n 2 min ≤ n 2 max		n 4 min n 2 min -n 2 x	n 4 max n 2 max -n 2 x
	n 2 min 2	< n 2 x < n 2 min ≤ n 2 max ≤ 2n 2 x		n 4 max n 2 max -n 2 x	n 4 min n 2 min -n 2 x
	n 2 min 2	< n 2 x < n 2 min ,	2n 2 x < n 2 max ≤	n 2 min n 2 x n 2 min -n 2 x	4n 2 x	n 4 min n 2 min -n 2 x
	n 2 min 2	< n 2 x < n 2 min ,	n 2 min n 2 x n 2 min -n 2 x	< n 2 max	4n 2 x	n 4 max n 2 max -n 2 x
	n 2 min ≤ n 2 x < n 2 max ≤ 2n 2 x		n 4 max n 2 max -n 2 x	∞
	n 2 min ≤ n 2 x ,	2n 2 x < n 2 max		4n 2 x	∞

Acknowledgments

W. Śmigaj acknowledges a fellowship from the Triangle de la Physique.

APPENDIX A: CONSTRAINTS ON X3

We begin by noting that the conditions (14) can always be rewritten in the form

with appropriate X min and X max . Specifically, for s polarization, the formulas for Z 2 min and Z 2 max follow readily from eq. ( 9):

For p polarization, due to the more complex form of eq. ( 12) and the presence of the supplementary condition (13), several cases must be considered. The final formulas for Y 2 min and Y 2 max are given in table II.

To arrive at the form of the constraints on X 3 , we substitute eq. ( 7) into inequality (A1) and introduce reduced immittances Xα ≡ X α /X 1 (α = 3, min, max), obtaining

This expression can be rewritten as

It follows that the constraints ( 14) are equivalent to the set of conditions written out in eq. ( 15).

APPENDIX B: NUMERICAL METHODS

A word about the different methods used for the numerical simulations presented in this article. Almost all the calculations whose results are presented in sections IV and VII have been made with the differential method [START_REF] Vincent | Electromagnetic theory of gratings[END_REF][START_REF] Popov | [END_REF][START_REF] Nevière | Light propagation in periodic media. Differential theory and design[END_REF][START_REF] Śmigaj | Master's thesis[END_REF] . The exceptions are the field maps shown in figs. 7 and 13, produced with the finite-element method using the RF module of the COMSOL package. The numerical results reported on in section V have been obtained with the finite-difference frequencydomain method. Finally, the simulations of the magnetooptical PC discussed in section VI have been made with a code based on the Fourier modal method 49 and implementing the factorisation rules for anisotropic materials derived in ref. 50.


Christoph S Garbe

Andre Butz

Isabelle Dadou

Boris Dewitte

Veronique Garçon

Serena Illig

Aurélien Paulmier

Joel Sudre

Hussein Yahia

V Garc ¸on

Climatically-Active Gases In The Eastern Boundary Upwelling And Oxygen Minimum Zone (Omz) Systems

Keywords: Air-Sea Interactions, Fluxes, Green House Gases, Satellite Retrieval, Partial differential equations

published or not. The documents may come

INTRODUCTION

The EBUS (Eastern Boundary Upwelling Systems) and OMZs (Oxygen Minimum Zone) contribute very significantly to the gas exchange between the ocean and the atmosphere [START_REF] Waldron | Carbon export and sequestration in the southern benguela upwelling system: lower and upper estimates[END_REF][START_REF] Hales | Atmospheric co2 uptake by a coastal upwelling system[END_REF], notably with respect to the greenhouse gases (hereafter GHG). Invasion or outgasing fluxes of radiatively-active gases at the air-sea interface result in coupled or decoupled sink and source configurations [START_REF] Paulmier | The oxygen minimum zone (omz) off chile as intense source of co 2 and n 2 o[END_REF]. From the in-situ ocean measurements, for example, the uncertainty of the net global ocean-atmosphere CO 2 fluxes is between 20 and 30%, and could be much higher in the EBUS-OMZ. Off Peru, very few in-situ data are available presently, which justifies alternative approaches for assessing the fluxes, and notably satellite, earth-observation data, and the outputs of modeling systems as well. Here we will focus on the Humboldt system off Peru (see Figure 1).

GHG vertical column densities (VCD) can be extracted from satellite spectrometers. The accuracy of these VCDs need to be highly accurate in order to make extraction of sources feasible. To this accuracy is extremely challenging, This project is funded by ESA through the STSE OceanFlux initiative (http://www.oceanflux-upwelling.eu). Fig. 1. Surface oceanic circulation together with Sea Surface Temperature, annual mean. Following [START_REF] Montes | On the pathways of the equatorial subsurface currents in the eastern equatorial Pacific and their contributions to the Peru-Chile Undercurrent[END_REF].

particularly above water bodies, as water strongly absorbs infra red (IR) radiation. To increase the amount of reflected light, specular reflections (sun glint) can be used on some instruments such as GOSAT [START_REF] Butz | Toward accurate CO 2 and CH 4 observations from GOSAT[END_REF]. Also, denoising techniques from image processing may be used for improving the signalto-noise ratio (SNR).

GHG air-sea fluxes determination can be inferred from inverse modeling applied to VCDs, using state of the art modeling, at low spatial resolution [START_REF] Garbe | Optimal Control in Image Processing[END_REF][START_REF] Garbe | Modeling of atmospheric transport of chemical species in the polar regions[END_REF]. Due to the long life-time of GHGs in the atmosphere, significant contributions of the VCDs may have been transported from far afield. in correctly choosing the boundary conditions of the inverse modeling approach, these effects can be separated. Only through this sep-6150 978-1-4673-1159-5/12/$31.00 ©2012 IEEE IGARSS 2012

aration can the sources be determined accurately. These approaches will be tested on simulated data, which will make an assessment of the accuracy, limitations and requirements of the satellite instrument feasible.

For accurately linking sources of GHGs to EBUS and OMZs, the resolution of the source regions needs to be increased. This task develops on new non-linear and multiscale processing methods for complex signals [START_REF] Turiel | Microcanonical multifractal formalism -a geometrical approach to multifractal systems: Part i. singularity analysis[END_REF] to infer a higher spatial resolution mapping of the fluxes and the associated sinks and sources between the atmosphere and the ocean. Such an inference takes into account the cascading properties of physical variables across the scales in complex signals. The use of coupled satellite data (e.g. SST and/or Ocean color) that carry turbulence information associated to ocean dynamics is taken into account at unprecedented detail level to incorporate turbulence effects in the evaluation of the air-sea fluxes.

Based on a parameterization, the transfer velocities K can be computed from satellite products. This leads to a measurement of p ocean GHG , which in turn will be validated by the output of a coupled physical biogeochemical model and in-situ measurements.

CONSTRAINING SURFACE FLUXES BY EO DATA

In the atmosphere, the temporal evolution of a tracer c is given by the equation :

∂c ∂t = -u∇c + 1 ρ ∇ (ρT d ∇c) + 1 ρ g + F, (1)
with the concentration c, the wind field u, the density of air ρ, the turbulent diffusivity tensor T d , the chemical transformation rate g and the net flux at the air-sea interface F . Using optimal control and inverse problem modeling, in this contribution we will derive the mapping of F using Earth Observation and satellite data [START_REF] Garbe | Optimal Control in Image Processing[END_REF][START_REF] Garbe | Modeling of atmospheric transport of chemical species in the polar regions[END_REF]. It will estimate sources and sinks of gases, based on vertical column densities (VCD), at the spatial resolution of VCD data. For the retrieval of VCD of GHGs, we employ the approach detailed in [START_REF] Butz | Retrievals of atmospheric CO 2 from simulated spaceborne measurements of backscattered near-infrared sunlight: accounting for aerosol effects[END_REF]. This methods yields very accurate results due to its ability to simultaneously retrieve gas concentrations and particle scattering properties of the atmosphere using an efficient radiative transfer (RT) model [START_REF] Otto | Efficient calculation of intensity and polarization spectra in vertically inhomogeneous scattering and absorbing atmospheres[END_REF].

Following [START_REF] Garbe | Modeling of atmospheric transport of chemical species in the polar regions[END_REF], the underlying problem can be formulated as

∂c ∂t = ∇ • (D ∇c) -u • ∇c + F, c(t 0 , x) = c 0 (x), c(t, x) = c in (t, x) on Γ I , D ∂c ∂ν = 0 on Γ O , D ∂c ν -β c = γ on Γ G .
(

) 2
Since the problem is ill-posed, we regularize it by introducing the Tikhonov functional given by

J(q, c) = c -C(c) Z + α 2 F Q . ( 3
)
The first term corresponds to the data term and includes given observations. Here C(c) corresponds to the VCDs. The second term is the regularization term with a small parameter α ≥ 0 . Thus, we aim at the solution of the following optimal control problem: Minimize J(q, c) (

subject to the state equation [START_REF] Hales | Atmospheric co2 uptake by a coastal upwelling system[END_REF]. The inverse problem is discretized using a dG(0)cG(1) Galerkin discretization. Thus, for the temporal integration we employ the backward Euler method which corresponds to the discontinuous Galerkin method. A detailed description of the discretization scheme can be found in [START_REF] Eriksson | Computational differential equations[END_REF]. For the discretization in space we use piecewise trilinear functions. This corresponds to Q 1 elements. To increase computational efficiency, the problem can be solved with a goal oriented adaptive grid refenement [START_REF] Garbe | Modeling of atmospheric transport of chemical species in the polar regions[END_REF].

The relation between the net flux F and the partial pressure p GHG of a GHG is of the form (for typical GHG such as CO 2 , N 2 O, CH 4 ):

F = αK p air GHG -p ocean GHG , (5)
α being the gas solubility (which depends mostly of SST and salinity) and K is a function of wind, salinity, temperature, sea state, which can all be obtained from satellite data. Since p air GHG can be assumed to be constant over the upwelling region under study, the scalar p ocean GHG can then be obtained from inverse modeling and satellite data at VCD resolution.

The quantity p ocean GHG is a complex signal depending, at any spatial resolution, on sea surface temperature, salinity, chlorophyll concentration, dissolved inorganic carbon, alkalinity and nutrients concentrations. Both the biological pump, with chlorophyll a as a proxy, and the physical pump, driven by the temperature and salinity (e.g. solubility, water mass), govern the evolution of pGHG, when dealing with CO 2 for instance, in the surface ocean. For other GHG such as N 2 O, the partial pressure of N 2 O will also depend on the evolution of the various chemical forms of nitrogen and on the activity of the bacteria involved directly or indirectly in the major N transformation processes such as nitrification, denitrification, anammox, etc. It is therefore a competitive research challenge to use high resolution satellite data (at various spatial resolutions) coupled with low spatial resolution model outputs for p ocean GHG to derive a higher spatial resolution mapping of p ocean GHG . Recent advances in non-linear and multiscale processing of complex and turbulent signals [START_REF] Yahia | Motion analysis in oceanographic satellite images using multiscale methods and the energy cascade[END_REF] allow to make use of cascading properties relevant to turbulence to assess high spatial resolution of scalar or vectorial quantities from low resolution satellite data and inference across the scales. The methodology makes use of the framework of reconstructible systems and optimal wavelets, which shows a particular ability to perform fusion of satellite data acquired at different spatial resolutions [START_REF] Pottier | Inferring missing data in satellite cholorophyll maps using turbulent cascading[END_REF], thus making it particularly appropriate for merging different kinds of satellite products. High resolution maps (resolving submesoscale features) will be provided at different timescales of interest, from intraseasonal to seasonal over the upwelling under study.

MULTISCALE PROPERTIES OF THE SIGNAL USED

As described in the previous section, p ocean GHG is a complex signal depending on temperature, salinity, chorophyll, dissolved inorganic carbon, alkalinity and nutrients concentrations. Such a signal is therefore expected to feature cascading, multiscale and other characteristic properties found in turbulent signals. We will present preliminary results which prove that a simulated p ocean GHG signal possesses the key signatures characteristic of complex multiscale signals, notably through a study of log-histograms and singularity spectrums. These characteristics prove that p ocean GHG signals are prone to be studied with the help of multiscale methods [START_REF] Turiel | Microcanonical multifractal formalism -a geometrical approach to multifractal systems: Part i. singularity analysis[END_REF] to infer information along the scales.

The multiscale properties of a signal are well understood though the study of singularity spectrum and singularity exponents which give access to the determination of geometric subsets responsible of cascading properties [START_REF] Arneodo | Multifractales et Turbulence[END_REF][START_REF] Turiel | Microcanonical multifractal formalism -a geometrical approach to multifractal systems: Part i. singularity analysis[END_REF]. Figure 2 shows a computation of the singularity exponents on a simulated p ocean CO2 signal obtained from the ROMS coupled physical/biogeochemical simulation model [START_REF] Penven | A numerical study of the southern Benguela circulation with an application to fish recruitment[END_REF][START_REF] Penven | Generation of cyclonic eddies by the agulhas current in the lee of the agulhas bank[END_REF][START_REF] Veitch | The benguela: A laboratory for comparative modeling studies[END_REF][START_REF] Vu | Physical and biogeochemical processes maintaining the oxygen minimum zone in the benguela upwelling system using an eddy resolving model[END_REF]. The computation is performed using the FluidExponents software developed at INRIA. The spatial distribution of the singularity exponents underlines the complex distribution of fronts and vortices displayed by the p ocean CO2 signal, which possesses turbulent characteristics. For comparison, we show in the same image the singularity exponents computed on the SST signal.

CONCLUSION

In this contribution we present a concerted effort to assess the influence Eastern Boundary Upwelling Systems on gas exchange between ocean and atmosphere. We link a coupled physical-biochemical model to parameters extracted from satellite remote sensing. A multiscale approach based on the singularity exponents is used to increase the resolution of our approach. We will present a framework as described above for determining sources and sinks of GHG from satellite remote sensing with the Peru OMZ as a test bed. First results of this analysis are very promising.

Fig. 2 .

 2 Fig. 2. Left : singularity exponents computed on simulated SST signal from the ROMS simulation model. Right: singularity exponents computed on simulated p ocean CO2 signal from the ROMS simulation model. Note the similarity of coherent structures and the turbulent character of the p ocean CO2 signal. These images are computed using the ROMS model and FluidExponents software. The p ocean CO2 and SST signals used in the computation of the two images are averaged on a 5 days time interval.


Yoann Gabillon
email: yoann.gabillon@imag.fr

Mathieu Petit
email: mathieu.petit@imag.fr

Gaëlle Calvary
email: gaelle.calvary@imag.fr

Humbert Fiorino
email: humbert.fiorino@imag.fr

Automated planning for User Interface Composition

Keywords: H.5.2 [User Interfaces]: Ergonomics, Graphical user interfaces (GUI), Prototyping, User-centered design. D2.2 [Software Engineering]: Design Tools and Techniques, User-Interfaces Design, Human factors, Algorithms User Interfaces composition, Semantic models, Automated task planning, Context of use

In ubiquitous computing, both the context of use and the users' needs may change dynamically with users' mobility and with the availability of interaction resources. In such changing environment, an interactive system must be dynamically composable according to the user need and to the current context of use. This article elicits the degrees of freedom User Interfaces (UI) composition faces to, and investigates automated planning to compose UIs without relying on a predefined task model. The composition process considers a set of ergonomic criterions, the current context of use, and the user need as inputs of a planning problem. The user need is specified by the end-user (e.g., get medical assistance). The system composes a UI in turn by assembling fragments of models along a planning process.

INTRODUCTION

Pushed forward by new information technologies, Weiser's vision of ubiquitous computing comes to reality [START_REF] Weiser | The computer for the 21st century[END_REF]. His definition of ambient computing implies 1) a global knowledge of an information system context, and 2) adaptation processes to comply with a given context of use. The context of use is usually defined as a <user, platform, environment> triplet. Unpredictable contexts of use might affect users' interactive behaviors and task organization. Therefore, each User Interface (UI) design option from the task model to the final UI is highly contextual and might be decided at runtime. Therefore, most of the ubiquitous design frameworks consider variations of the context of use as inputs to select UI options (i.e., plastic design [START_REF] Thevenin | Plasticity of user interfaces: Framework and research agenda[END_REF], automatic generation [START_REF] Paternò | ConcurTaskTrees: A Diagrammatic Notation for Specifying Task Models[END_REF], mashups [START_REF] Brodt | Context-aware mashups for mobile devices[END_REF]). However, to the best of our knowledge, the user task variation is usually left out. This article outlines an approach, based on automated planning, to support task as well as UI variations in an integrated framework for UI composition. In the following, section 2 exemplifies multilevel UI composition on a medical support case study. Section 3 elicits the degrees of freedom UI composition faces to. Section 4 introduces automated planning and highlights the UI composition process. Section 5 presents an integrative framework for UI composition by planning. The focus is set on the composition of models (Model-based composer) and code (Code composer). Section 6 summarizes our contributions and draws some perspectives.

RUNNING CASE STUDY

Victor is a New-York citizen on vacation in Philadelphia. After spending his day tasting the rich local food, Victor feels bloated at night and needs to find the doctor on duty. Using his PDA, he specifies his need in general terms: "I would like to get medical support".

According to Victor's need and to the available interaction resources and existing information, the system abstracts the goal, plans a task model, and composes one possible UI. The composition process is not fully autonomous: it requires additional information from Victor. The negotiation UIs (Figure 1) are composed by the system as well. Given Victor's current location, the system asks Victor whether he prefers to return home or to find assistance in Philadelphia (Figure 1a). Victor chooses to consult a local doctor. The system therefore finds and provides him with possible local contact information: the nearest hospital or doctor on duty, a medical hotline, or the firemen (Figure 1b). Victor selects the doctor on duty. The systems provides him with contact and location information. The UI layout matches the current user platform:

Smartphone. If Victor prefers to keep information at hand, a UI is generated for his Smartphone. With respect to the limited screen resolution, pieces of information are tabbed and no additional data is provided (Figure 2). Desktop Wall. If a desktop wall is available, the system generates a single pane UI allowing to contact and/or to get route information to the doctor's office. Additional information about close services, like the nearest all-night chemist, is also provided (Figure 3).

MODELS ARE KEY

This section goes back to model based design in Human Computer Interaction (HCI), and claims for keeping these models at runtime so that to support dynamic adaptation.

Model based design

UIs are modeled along several levels of abstraction. For example, the CAMELEON reference framework identifies four main levels of design decisions [START_REF] Calvary | A unifying reference framework for multi-target user interfaces[END_REF]. The task model (TM) describes how a given user task can be carried out; the abstract UI (AUI) delineates task-grouping structures (i.e., workspaces); the concrete UI (CUI) selects and layouts the interaction elements (i.e., interactors) into the workspaces; at last, the final UI (FUI) is about the code. Mappings relate these models to each other. For example, a task should be mapped to one workspace of the AUI at least.

In a dynamic context of use, any of these UI design decisions and their subsequent models and mappings might be updated at runtime to match the current context of use. As long as these adaptations satisfy the usability and utility properties, the UI is said to be plastic [START_REF] Thevenin | Plasticity of user interfaces: Framework and research agenda[END_REF]. In Victor's case study, every design decision might be adapted in a plastic way. For example, the task "Find nearest chemist" may be removed from the task model. The AUI model associated to the Smartphone favors the "Call the office" subtask whilst the desktop wall version gives a simultaneous access to the two subtasks ("Call the office" and "Find route information"). Variations at the CUI level are not exemplified in the case study. We could imagine a switch from a route display to a list of directions so that to fit with the Smartphone display. Such adaptations might be seen as a transformation between two graphs of models.

Graph of models to support adaptation

Earlier work defined principles for UI plasticity [START_REF] Sottet | Model-driven adaptation for plastic user interfaces[END_REF]. The authors structured the CAMELEON reference framework as a network of models and mappings (Figure 4), and claimed for keeping this graph alive at runtime so that to support adaptation. Their model allows UI designers to check out replaceable (i.e. functionally equivalent) units at run-time. For example, a given layout of interactors at the CUI level might be switched to another one depending on the desired ergonomic properties [START_REF] Scapin | Ergonomic criteria for evaluating the ergonomic quality of interactive systems[END_REF]. We propose to replace these hand-made choices by predicates dependent of the context of use, and manipulated by the system.

Figure 5 illustrates the design process along the models and mappings proposed in [START_REF] Sottet | Model-driven adaptation for plastic user interfaces[END_REF] and the replaceable options described in [START_REF] Demeure | The COMETS Inspector: towards run time plasticity control based on a semantic network[END_REF]. For example, at the task level (TM), two options exist for T2 depending on the context of use (Figure 5 b&c).

In Figure 5, within a level of abstraction, units relate to each other according to a consumer-provider relationship (Figure 5:

p c  link).
For example, at the TM level, one of the options for the task T2 relies on the occurrence of a provider leaf option 1 for the task T3 (Figure 5a). Therefore, as T2 "consumes" T3, this option will be triggered if and only if T3 is satisfied. Depending on the current context of use, consumer-provider links behave like "opened" or "closed" transistors. In a given p c  relationship, the status of a transistor depends on the contextual requirements of the provider (p). For example, at the TM level in Figure 5, one of the task T2 options is possible only for experienced users (Figure 5d).

Fig. 5. Example of a TM options graph.

In UI design, mappings link together options of different levels of abstraction. For example, interactors from the CUI level are usually mapped onto workspaces of the AUI level. These mappings, presented in Figure 4, or the definitional links in [START_REF] Demeure | The COMETS Inspector: towards run time plasticity control based on a semantic network[END_REF] constitute abstracting-reifying relationships between the options of distinct CAMELEON levels of abstraction (Figure 6: For example, the TM level presented in Figure 5 might be reified into several options of an AUI level (Figure 6). In Figure 6

   links).

UI COMPOSITION BY PLANNING

This section presents the core principles of planning and shows how this approach is valuable for UI composition.

Principles of automated planning

An automated planning algorithm derives a temporal sequence of actions into a plan to accomplish a given goal [START_REF] Nau | Automated Planning: Theory & Practice[END_REF]. For example, in the previous case study, the sequence {"Call the doctor"→"Find route information"} is a plan made of two actions. A Planning algorithm pipes syntactic processes to perform symbolic computations. Such logical reasoning is formally described by a finite-state machine where actions are transitions between possible states of the world. Actions are defined by sets of pre/post-conditions. Pre-conditions specify the run-time dependencies of an action while post-conditions are met after executing the action. For example, Victor's Smartphone should be connected (pre-condition) to display a location map (action). When this action is executed, the map is eventually displayed (post-condition) on the Smartphone. An updated state of the world integrates these new post-conditions, therefore enabling further actions.

Automated planning for UI composition

A planning solver algorithm computes a transition graph between an initial state of the world and a final state corresponding to the system/user goal. Currently, such algorithms are mainly applied to service composition [START_REF] Traverso | Automated Composition of Semantic Web Services into Executable Processes[END_REF]. However, as illustrated in our case study, context-dependent UI composition and automated planning strongly relate. Thus, we propose to address UI composition by planning where:  "Actions" are "User interfaces options". Existing components (e.g., the UI associated to the task "Call the office") are actions for the planner;  The "State of the world" is made of the current "Context of use" and the "Ergonomic properties" to be satisfied. For example, the fact "Victor owns a Smartphone" is a predicate of the state of the world;  The "selected plan" is the "composed UI". For example, the UI displayed on the Smartphone is a concretization of the plan {"Choose the city"→"Choose the doctor" →"Contact the doctor"→{"Call the office"→"Find the route information"→ "Find the nearest pharmacy"}} computed by the planner. Even if several challenges still need to be worked out to bridge the gap between automated planning and UI composition, next section presents "Compose", a first framework for rapidly prototyping UIs by planning. Its use by end-users belongs to the future.

THE COMPOSE FRAMEWORK

Compose is a proof of concept of UI composition by planning. It has been built on top of several functional Java-coded components (Figure 7).

Fig. 7. Functional decomposition of Compose.

The Context of use and quality in use managers translate the required ergonomic criteria and the current context of use into predicates. These assertions define the current state of the world. For example, the predicate Has("User","Desktop Wall") is true when Victor stands nearby a managed desktop wall.

The User requirements manager expresses a user need as a goal to be met. For example, Victor's need would be to "Get medical support".

The Model-based composer and the code composer are the core components of Compose. The model-based composer handles the planning process, whilst the code composer translates a resulting plan into a FUI. In the current prototype, planning is applied to the task level only. Once the TM level is composed, mappings are made with a generic purpose graphic toolkit called COMET [START_REF] Demeure | K. COMET(s), A Software Architecture Style and an Interactors Toolkit for Plastic User Interfaces[END_REF]. COMETs are reusable context-aware widgets defined at the task level and reified along the CAMELEON reference framework. The next sections focuses on the core components of Compose.

Model-based composer

The model based composer takes actions as inputs and structures them into a plan. This planning process is twofold: at first, the user task modeling is composed by collating predefined subtasks (Figure 8(p1)); next, each task (i.e.: the planner actions) is mapped onto a UI (Figure 8(p2)). These selections bring out a composed UI (i.e., the selected plan) whose properties match the current state of the world. The resulting plan is a semantic description of the UI to be composed.

Fig. 8. Compose planner instantiation.

In Victors' case study, Compose waits for a user need specification (i.e. "Get medical support"). The composer tries to find a corresponding TM level entry point. The option "Get Medical Support" is selected. The planning algorithm then explores the semantic network of p c  relationships between the task options of the TM level (Figure 9). For each uncovered task option, Compose checks whether it is possible or not to map the task onto a COMET and render the UI. These mappings are derived according to the current state of the world. For example, leaf task options like "Choose the city" or "Choose the doctor" might be mapped onto a UI as soon as Victor's platform is available whatever the characteristics of the platform are (in Figure 9: t1 & t2). Other task options like "Call the office" rely on carrier capabilities at the platform level (in Figure 9: t3). "Contacting the doctor" option distinguishes between several screen sizes and resolutions (Figure 9: t4 & t5). When a large screen is available, such a sub-task option involves tree leaf options (Figure 9: u1), while on a Smartphone display, solely two of them are displayed (Figure 9: u2).

Once all contextual pre-requisites of a provider option are met, the relationships to his consumers turn green and each of them might in turn be checked-out. After a provider/consumer relationship status has been specified, the state of the world is updated with the new facts the providing option concurs to establish. For example, when "Choose the city" pre-requisites are met, the composer knows for sure that Victor will be able to specify his searching location and the fact "The location has been set" is added to the state of the world. Such contextualized semantic UI model highlights the appropriate task factorization in a given context of use. When a green path of provider-consumer relationship is established from the provided objective to the leafs task options, a task tree has been found to achieve the user goal. In such case, the code composer is provided with the planned task tree. Subsequent mappings are made between tasks and COMETs to derive the final UI.

Code composer

The Code composer derives the UI code from the graph of models at the task level. At design time, the options of the task level have been statically associated to COMETS. Therefore, in Compose, each action of the plan is reified by a contextualized COMET. For example, the option "Get medical support" is mapped to a COMET laying out a sequence of frames on the desktop wall. The Code Composer brings these pieces of UI together in a unified layout. For instance, the desktop wall task tree provided by the model-based composed is mapped to the COMET presented in Figure 10. For example, the action "Get medical assistance" is mapped to a "COMET C7" laying out a sequence of frames on the desktop wall. These frames contain several sub-COMETs ("COMET {C3, C1, C4}") to map the task options "Choose the city", "Choose the doctor" and "Contact the doctor". In turn, the mapping "COMET C4", that reifies the task "Contact the doctor", contains several vertically aligned sub-COMETs. These sub-COMETs ("COMET {C2, C5, C6}") are mapped in the same way.

CONCLUSION AND FUTURE WORK

This article outlines a work in progress to support opportunistic user needs. A UI is composed by selecting a path in a graph of models according to the current context of use and the ergonomic properties to be satisfied. UI composition is seen as a planning problem. So far, the focus has been set on the model-based composer whatever the time is: design time for the designer thus providing a rapid prototyping tool, or runtime for the end-user as an intelligent assistant.

Future works include improvements of planners to fully support UI composition. This means (1) generating trees (i.e., tasks structures) instead of sequences, (2) defining appropriate functional and implementational software architectures for general-purpose ubiquitous computing, (3) taking non functional properties into account (i.e., returning the best plan instead of the first one). Thus, beyond perspectives in HCI, this work has challenged planning for ubiquitous computing.

  (a) Possible locations. (b) Possible options.

Fig. 1 .

 1 Fig. 1. Automatically composed UI.

Fig. 2 .

 2 Fig. 2. The generated UIs for a Smartphone.

Fig. 3 .

 3 Fig. 3. The UI generated for a desktop wall display.

Fig. 4 .

 4 Fig. 4. Semantic graph of models of an interactive system [8].The graph expresses and maintains multiple perspectives on a system. For example, a UI may include a task model, a concept model, an AUI model and a CUI model linked by mappings. In turn, the UI components are mapped onto items of the Functional Core, whereas the CUI interactors are mapped onto the input and output (I/O) devices of the platform. Although such a model provides a helpful organizational view on the elements and relationships involved when designing a plastic interactive software, the proposed mappings between the context of use and the other components hardly describe contextual choices inside each model (TM, CUI, AUI, etc.). Demeure et.al. provide a complementary semantic graph of models to control UI plasticity within each design option level[START_REF] Demeure | The COMETS Inspector: towards run time plasticity control based on a semantic network[END_REF]. Their model allows UI designers to check out replaceable (i.e. functionally equivalent) units at run-time. For example, a given layout of interactors at the CUI level might be switched to another one depending on the desired ergonomic properties[START_REF] Scapin | Ergonomic criteria for evaluating the ergonomic quality of interactive systems[END_REF]. We propose to replace these hand-made choices by predicates dependent of the context of use, and manipulated by the system.

Fig. 6 .

 6 Fig.6. Abstracting-reifying relationships between two design options at the TM and AUI levels of abstraction.

  , a task option T1 is reified into a workspace layout "W3" of the AUI level. Like the p c  relationship,    relationship between levels of abstraction makes sense in a given context of use only. For example, Figure 6 depicts a runtime configuration where the workspace layout W3 cannot reify the task T2 given the current context of use (Figure 6 a). The relationships we propose (    and p c  ) for modeling software can easily be explored automatically. The next section investigates automated planning.

Figure 9

 9 Figure 9 outlines the status of the p c  relationship between the task options after Compose has explored and checked-out a state of the world wherein Victor interacts on a desktop wall display.

Fig. 9 .

 9 Fig. 9. Possible TM level planning when a desktop wall is available.

Fig. 10 .

 10 Fig. 10. The "Desktop Wall" planned task tree. Each task is reified by a pre-defined COMET.

A leaf option has no relationship for neither providing nor reifying options.Copyright is held by the author/owner(s) SEMAIS'11, Feb 13 2011, Palo Alto, CA, USA

Copyright is held by the author/owner(s) SEMAIS'11, Feb 13 2011, Palo Alto, CA, USA

ACKNOWLEDGMENTS

This work has been mainly founded by the "Informatique, Signal, Logiciel Embarqué" research cluster of the Rhône-Alpes region. It has also been supported by the french "ANR MyCitizSpace" and the european ITEA2 UsiXML projects.