Jean-Claude C Morel
email: jeanclaude.frederick.morel@gmail.com

Rabia Charef

Erwan Hamard

Antonin Fabbri

Quoc Chris Beckett

Bao Bui

Q B Bui

Earth as construction material in the circular economy context: practitioner perspectives on barriers to overcome

Keywords: soil, earth construction, earthen architecture, rammed earth, cob, circular economy

The need for a vast quantity of new buildings to address the increase in population and living standards is opposed to the need for tackling global warming and the decline in biodiversity. To overcome this twofold challenge, there is a need to move towards a more circular economy by widely using a combination of alternative low-carbon construction materials, alternative technologies and practices. Soils or earth were widely used by builders before WWII, as a primary resource to manufacture materials and structures of vernacular architecture. Centuries of empirical practices have led to a variety of techniques to implement earth known as rammed earth, cob, and adobe masonry amongst others. Earth refers to local soil with a variable composition but at least containing few percentages of clay that would simply solidify by drying without any baking. This paper discusses why and how earth naturally embeds high-tech properties for sustainable construction. Then the potential of earth to contribute to addressing the global challenge of modern architecture and the need to re-think the building practices is also explored. The current obstacles against the development of earthen architecture are examined through a survey of current earth building practitioners in Western Europe. A literature review revealed that, surprisingly, only technical barriers are being addressed by the scientific community; two-thirds of the actual barriers identified by the interviewees are not within the technical field and are almost not addressed in the scientific literature, which may explain why the earthen architecture is still a niche market despite embodying all the attributes of the best construction material to tackle the current climate and economic crisis.

Introduction

The building sector consumes approximately 48 % of global world energy use and a large volume of natural resources, extracting over 30% of the total resources [START_REF] Dixit | Life Cycle Recurrent Embodied Energy Calculation of Buildings: A Review[END_REF]. Consequently, the building sector is a major producer of greenhouse gases that contribute to climate change. As an example, sand and gravel are the most extracted group of materials [START_REF] Bendixen | Time Is Running out for Sand[END_REF] and their mining has a strong impact on biodiversity [START_REF] Park | Dramatic Decrease of Flood Frequency in the Mekong Delta Due to River-Bed Mining and Dyke Construction[END_REF]. The construction sector is also responsible for about 50 % of wastes produced in the European Union, 25% of solid waste globally, and the management of these wastes have a negative environmental impact (Benachio et al., 2020). Among these wastes, about 75 % are natural soils and stones [START_REF] Eras | Improving the Environmental Performance of an Earthwork Project Using Cleaner Production Strategies[END_REF]. Excavated soils, also called earth, are regarded as wastes in Western countries but earth has been used since, at least, the very beginning of the Neolithic revolution by human beings to build their shelters and dwellings [START_REF] Sauvage | Les Débuts de l'architecture de Terre Au Proche-Orient[END_REF]. Earthen architectures, regarded as fragile, perishable and made for the poor, are not compatible with the "ideology of progress" that prevailed during the 20 th century and fell into disuse [START_REF] Hamard | Cob, a Vernacular Earth Construction Process in the Context of Modern Sustainable Building[END_REF]. Despite biodiversity and ecosystems having been widely forgotten in the 20 th -century development schemes, however, people have started to come back to an awareness of its benefits to living quality and health. In particular, the Intergovernmental Science-Policy Platform on Biodiversity and Ecosystems Services (IPBES) has been launched jointly with governments, academia and civil society to inform policy formulation on the topic. IPBES is promoting a framework based on eighteen recognized "Nature's Contribution to People" (NCP). A major challenge today is to secure "the beneficial contribution of nature to a good quality of life for all people" [START_REF] Díaz | Assessing Nature's Contributions to People[END_REF]).

This article is part of the themed issue "The contribution of soils to the UN Sustainable Development Goals" and analyses the contribution of soils to the thirteenth Nature's Contribution to People (NCP): "Materials, companionship and labour" as defined in [START_REF] Díaz | Assessing Nature's Contributions to People[END_REF]). Soil plays a major role in NCP by providing many ecosystem services and requires careful management [START_REF] Smith | Managing the Global Land Resource[END_REF]. Most of conventional building materials are quarried from bedrocks and geological deposits, chemically modified during the manufacture process, hindering their capacity to be recycled at low energy cost. On the contrary, for earth architecture, earth is quarried from the subsoil. Raw earth materials are completely recyclable without any loss through the value chain and suit perfectly with the circular economy principles (Bui and Morel 2015). However, the development at a large scale of earth architecture, in particular areas, could have an impact on soil availability and could compete with its other NCPs. This article aims to develop why and under which conditions soils could provide a combination of alternative materials, technologies and practices compatible with the UN Sustainable Development Goals. Additionally, the current obstacles to the extensive use of earth in modern architecture will be explored, as revealed through responses from interviewed practitioners in earthen construction. Addressing those barriers would allow using soils to build millions of single-family houses or multifamily residential buildings worldwide, including in high-income countries, offering a sustainable and circular solution to the housing crisis encountered worldwide.

A high-tech multi-function material for architecture

Most of earthen building techniques have a several thousand years old history and earthen materials have been used to construct structures of qualities ranging from palaces to temporary shelters. In antiquity, means of transport were limited and builders had to adapt to locally available materials (both the raw materials for construction and those to form the temporary works, for example, formwork). Modern construction, however, boasts the benefit of ease of transport and onsite manufacture automation (Figure 1). Earth building should, therefore, be more popular now than it has ever been. However, this is not the case; rather, earth is perceived to be a low-quality material, suitable only for construction in developing countries or for bespoke architectural ventures for the wealthy. Engineers have recently examined the technical performance of these materials (section 2) but, despite considerable intake in interest from that community, little increase in the use of earthen materials in practice has emerged. To achieve that increase in interest, those latent barriers to adoption must be brought to light (section 3).

(a) Description of earth material Most earth materials are excavated from subsoil horizons and some of them are excavated from alterites or soft rock deposits [START_REF] Hamard | Rediscovering of Vernacular Adaptive Construction Strategies for Sustainable Modern Building -Application to Cob and Rammed Earth[END_REF]. Topsoils are sensitive to shrinkage and decay and are therefore unsuitable for building [START_REF] Maniatidis | A Review of Rammed Earth Construction. Report for DTI Partners in Innovation Project Developing Rammed Earth for UK Housing[END_REF]. Not all subsoil horizons are suitable for building purpose. Earth is a cohesive and frictional material in terms of geotechnique; it means that its mechanical behaviour is a combination of cohesion and friction. Cohesion is provided by colloidal particles, mainly clay minerals, and friction is provided by the contact between the particles of the granular skeleton of the earth, i.e. silts, sands, gravels and stones. Although several recommendations are available in the literature regarding the clay/silt/sand/gravel content of suitable earth for construction, most of the earth employed by past masons fell outside these recommendations, which highlights their inaccuracy (Pagliolico et al. 2010;[START_REF] Rojat | Towards an Easy Decision Tool to Assess Soil Suitability for Earth Building[END_REF].

Earth is an unsaturated porous medium, containing adsorbed water and capillary water in equilibrium with the internal and atmospheric relative humidity. In normal operating conditions, an earthen wall always contains a small amount of water. The cohesion is generated by three different attractive inter-particle surface forces that are the Van Der Waals dispersion force, capillary force, and ionic correlation forces (Van Damme and Houben 2018). Among those forces, the capillary forces, also called matric suction (when expressed as a pressure) in geotechnique, are the main contributors to cohesion. Nanoscopic adsorbed water films bridge the gap between clay minerals, and larger water menisci the gap between clay minerals and coarser aggregates (silt, sand), (Van Damme and Houben 2018). The strength of the cohesion is driven by the clay types, the clay content, the microstructure and the hydration state (water content). Clay minerals exhibit a large diversity. For example, their specific surface ranges from 20 m 2 .g -1 for a kaolinite to 850 m 2 .g -1 for a montmorillonite [START_REF] Meunier | Clays[END_REF]. For the same clay content, two different earths can have very different behaviours. Moreover, the microstructure of the earth matrix is driven by the implementation type [START_REF] Kouakou | Strength and Elasto-Plastic Properties of Non-Industrial Building Materials Manufactured with Clay as a Natural Binder[END_REF]. To propose prescriptive suitability thresholds for earthen architecture, it would require thorough, long and expensive tests. Consequently, it is not possible to predict the performance of an earthen wall from its intrinsic composition like it is possible with concrete; their performance should be measured through performance-based tests. Nevertheless, it is possible to propose decision tools for planning authorities and earthwork contractors to assess the potential of excavated soils for construction (Hamard et al. 2018;[START_REF] Rojat | Towards an Easy Decision Tool to Assess Soil Suitability for Earth Building[END_REF]. However, the technical validation of the material remains the role of a project manager specializing in earthen architecture.

(b) The different processes of manufacturing earth materials

Earth is a natural and variable material offering a myriad of building techniques that can be classified according to the water content of the earth at the time of the implementation. Three different states can be identified: a solid-state, a plastic state and a liquid state. For plastic-state techniques, earth mixture is employed in a plastic state and mechanical strength of the material is provided through drying shrinkage densification. For solid-state techniques, earth mixture is employed at an optimum water content for compaction and mechanical strength is provided through densification and drying. For the liquid-state technique, an earth slip is used to bind plant particles. However, there is no gap between these approaches due to the diversity of earthen construction techniques: rather, a continuity exists as might be described by the following list [START_REF] Kouakou | Strength and Elasto-Plastic Properties of Non-Industrial Building Materials Manufactured with Clay as a Natural Binder[END_REF].

 Rammed earth consists in compacting earth at the optimum water content, layer by layer, in formwork with a rammer (Figure 1). This technique was already prevalent in Carthage (Tunisia) in 814 B.C. It then spread around the Mediterranean and North Africa as part of the Carthaginian and Roman empires and again to Europe with the expansion of Islam in the 8 th century AD.

 Compressed Earth Block (CEB) is a quite modern technique that was developed in the middle of the nineteenth century. It consists of compacting earth at its optimum moisture content inside mould using a manual, a mechanical, or a hydraulic press. The resulting blocks are assembled with earth-based mortar to produce masonry walls.

 Cob is a building technique that emerged during the early Neolithic times. It consists in stacking clods of earth at plastic state, usually fibred, layer by layer, to build a monolithic wall. Vertical surfaces are eventually rectified by cutting off excess material after short drying time.

 Adobes are unfired bricks shaped by hand or moulded. They are made of earth mixed with water and most often with organic material such as straw or dung. They also appeared during the early Neolithic times and are one of the most widely used earth building materials.

 Extruded blocks are produced according to the same process than fired bricks except they are not baked. This type of brick is quite recent and is the result of the desire of brick makers to diversify their production.

 Earth mortars are probably the first mortars employed by mankind. They are used to lay unfired bricks (Adobes, CEB, and extruded blocks) and stones masonries.

 Wattle and Daub are made of a mixture of earth at plastic state and fibres, implemented wet, to fill a timber frame load-bearing structure. This is probably one of the oldest earth building technique as well as the technique most familiar to the public. Earth plasters are employed to coat interior and exterior wall surfaces. Plasters can be stabilized with sand, fibres, biopolymers or lime.

 Light earth is the most recent commonly used earth building technique. It arose in Germany after 1920 and was designed to improve the thermal insulation of walls by significantly increasing the fibre content of wattle and daub mixtures, to reach densities lower than 1200 kg.m -3 . It consists of binding bio-based aggregates or fibres with an earth slip (light earth). Baked clay-bricks and stabilised earth with cement or lime are excluded from the scope of the article because they require a high embodied energy and are non-reversible processes.

(c) Properties of earthen buildings

The key parameters when dealing with the mechanical performance of building materials are their compressive strength, that is the maximum axial load that can be reached when the material is submitted to uniaxial compression, and its modulus of deformability, which is the slope of the linear part of the axial stress-axial strain curve during this test. It is quite difficult to give standard values of these parameters for earthen materials since they strongly depend on the earth's nature and the implementation method. As a result, notable differences can be found on recommended design values, which are reported in the various existing codes. For example, for rammed earth, while a maximal compressive load of 0.5MPa for a thickness higher than 0.25m is prescribed in the New Zealand code (NZS 4297 1998), values of 0.4MPa to 0.6MPa are given in the Australian handbook [START_REF] Walker | The Australian Earth Building Handbook HB195[END_REF]) and eventually 0.2MPa is recommended in the French best practice guide [START_REF] Tera | Guide Bonnes Pratiques -Pisé[END_REF]. Anyway, despite their differences, these values might appear quite low when compared to conventional materials like concrete, stones, timber, etc., which generally exceed 10MPa. Nonetheless, it is sufficient to design at least two-storey building, with 50cm thick walls (TERA 2018). However, these values refer to normal operating conditions, for which the amount of water within the material remains quite low (the water content commonly ranges between 1% and 4% by dry weight, while the water content at saturation is commonly around to 20%). An increase of the water content from 2% to 12% leads to dividing the compressive strength and the stiffness by, at least, four for soils of several compositions compacted according to the Proctor procedure (energy of compaction equal to 0.6kJ/dm 3 , [START_REF] Bui | Effect Moisture Content on the Mechanical Characteristics of Rammed Earth[END_REF]. It follows that any abnormal increase of water content would induce lower strength values threatening the wall stability. Furthermore, [START_REF] Scarato | Cahier d'expert Bâti En Pisé: Connaissance, Analyse[END_REF], writing based on the analysis of more than hundreds of ancient rammed-earth buildings, identified one of the main cause of wall collapse as being an abnormal increase of the water content at the interface between the earthen wall and the basement. This can be induced by the conjunction of several factors as backfill elevation, implementation of an impermeable coating or floors, back slope near the building of road and pavements, which are composed by impermeable material like bituminous concrete…, which will lead to an increase of the capillary rises through the basement towards the earthen wall. This durability issue may be even more prejudicial when it is associated with freezing and thawing processes. For example, [START_REF] Scarato | Cahier d'expert Bâti En Pisé: Connaissance, Analyse[END_REF] reported that major collapses of earthen constructions in the Auvergne-Rhone-Alpes region in France had been recorded after thawing periods. They also underlined that earthen constructions should be avoided during winter periods because of the great risk of frost damage at an early age in which earthen material presents a high liquid saturation degree. Another durability issue caused by water is the progressive erosion caused by the wetting-drying (and possibly freezingthawing) of the wall surface due to its exposure to rainfall. The importance of these cycles has been studied experimentally by [START_REF] Bui | Durability of Rammed Earth Walls Exposed for 20 Years to Natural Weathering[END_REF]) on different types of unprotected rammed earth walls exposed during 20 years to external climatic solicitations in a wet continental climatic environment (Grenoble, France). This work eventually demonstrated an extrapolated lifetime, which is over 60 years (mean erosion of 6.4 mm, or 1.6% of the thickness of the wall after 20 years of exposure) without any mineral stabilization of the earth. In addition to these wetting-drying cycles, during their lifetime, earthen walls have to face important variations of indoor and outdoor relative humidity, that induce adsorption-desorption of the water molecules on the surface of the pores by water exchange with the surrounding air. This behaviour is due to their quite high permeability [START_REF] Fabbri | Measurement of the Relative Air Permeability of Compacted Earth in the Hygroscopic Regime of Saturation[END_REF]) caused by the existence of a network of connected macropores, combined with their high specific surface area resulting from the presence of clay minerals and, possibly, vegetal fibres. Even if these adsorption-desorption processes do not significantly change the water content of the material (generally 1% to 3% of water content increase when the relative humidity varies from 20% to 80%), they can induce noticeable strength variations and even induce shrinkage and swelling processes [START_REF] Xu | Modelling the Poroplastic Damageable Behaviour of Earthen Materials[END_REF]). These modifications in behaviour however strongly depend on the clay content and its activity. These interactions between the material and the water molecule, which drive the complex mechanical behaviour of earthen materials, also make it an excellent candidate to regulate the indoor humidity passively if it is not covered with an impermeable coating. The latent heat associated with these adsorption-desorption processes also impacts the heat transfer processes through earthen walls, which may thus modify the overall thermal behaviour of the building. An increasing number of research publications have already focused on assessing these heat, air and mass phenomena within hygroscopic walls and modelling them (for example [START_REF] Labat | Moisture Balance Assessment at Room Scale for Four Cases Based on Numerical Simulations of Heat-Air-Moisture Transfers for a Realistic Occupancy Scenario[END_REF]. Today, one of the main goals within this topic concerns the development of accurate software that can predict their impact on the building scale. This task requires the evaluation of the production and exchange of vapour within a building. And this latter may be impacted by the occupancy scenario of the inhabitants [START_REF] Bui | Impact of the Occupancy Scenario on the Hygrothermal Performance of a Room[END_REF], which is in turn potentially impacted by their feeling. The question thus is not trivial, because this feeling is not only attributable to the objective parameters of the material including its hygrothermal capabilities but also depends on elements of strongly coupled contexts (history and sensitivity of the occupant, conditions of indoor and outdoor atmospheres…). Finally, as it was pointed out by [START_REF] Soudani | Energy Evaluation of Rammed Earth Walls Using Long Term In-Situ Measurements[END_REF], the solar irradiance on the non-insulated earthen wall may have a significant impact on the thermal balance of the habitation. In particular, this study underlined that south-oriented walls can resituate with a shift around to 12h a part of the solar heat stored by the walls, even in winter, for an appropriately designed structure. With a change in that design, this shift can drop to as low as one hour [START_REF] Beckett | Measured and simulated thermal behaviour in rammed earth houses in a hot-arid climate. Part A: Structural behaviour[END_REF].

(d) The benefit to people Intrinsically, earth materials are embodying the holistic approach by being unalterable and respecting the natural cycle of life. Earthen buildings abandoned or unusable any more will simply disintegrate and merge with their environment, becoming ruins that are completely harmless. All earth materials can be recovered and returned to earth without any action, being top materials in the circular cycle, or cradle to cradle. Earthen materials are perfectly in line with the Circular Economy (CE) principles through their infinite reuse and recyclability with the only downside to add, if required, recyclable components. The use of earth as construction materials have several immediate benefits. First, a significant amount of excavation waste ends up in landfills (e.g. representing 50% of the total inert waste in France, De Larrard and Colina 2018) whereas they can be revalued as construction materials, having as seen above the potential to regulate the indoor air quality (IAQ) and the hydrothermal comfort. Earth material contains zero volatile organic compounds (VOCs) compared to many other materials (paint, adhesives, sealants among others) affecting the IAQ and exposing occupants to health risks through the building life span, [START_REF] Akom | The indoor environmental quality performance of green low-income single-family housing[END_REF]. It was recently scientifically established (although already well known by end-users) that earthen architecture, when properly implemented in the construction phase, generates an excellent IAQ especially due to its passive capacity of moisture buffering [START_REF] Mcgregor | A review on the buffering capacity of earth building materials[END_REF]. Moreover, the clay particles remove the pollutants [START_REF] Darling | Impacts of a clay plaster on indoor air quality assessed using chemical and sensory measurements[END_REF]) due to the properties of clay nanoparticles. In the context of the 2020 COVID-19 outbreak, the IAQ is crucial because people will spend much more time at home during the lockdown rules (Abouleish 2020) and it is established that a poor IAQ worsens underlying health conditions [START_REF] Derbez | Indoor air quality and comfort in seven newly built, energy-efficient houses in France[END_REF], making people less resistant to COVID-19 [START_REF] Chow | Preliminary Estimates of the Prevalence of Selected Underlying Health Conditions Among Patients with Coronavirus Disease 2019-United States[END_REF]). Moreover, even after the outbreak, it is suspected that in the long term, people will prefer to work from home more than ever.

The obstacles against modern earthen architecture

With the increasing demand of society about more sustainable materials, numerous scientific investigations have focused on earth materials during the last two decades. Examining publications related to earth from 1998 to 2019, in Scopus's database, the result showed that more than 98% of the publications related to earth are classified in the field of engineering [START_REF] Morel | What are the barriers affecting the use of earth as a modern construction material in the context of circular economy?[END_REF]. Therefore, the search of publications on steel, concrete and earth was performed in the Engineering subject area, search limited to title, keywords and abstract. Since there are different techniques of earth materials, the keywords for the earth-based structures are limited to some specific techniques ("earth blocks" OR "rammed earth" OR "cob house").

Figure 2 illustrates the number of publications divided by the publications published in 1998, respectively for each material investigated. The result shows that while there are linear increases of publications on steel and concrete, there is an exponential growth of the publications related to earth material, which shows the particular scientific attraction of earthbased materials in the last years. The question therefore arises as to why academic interest in these materials has increased so rapidly, and seems set to do so for some time, but why earthen construction has made so little impact in the construction sector. To examine this, interviews were conducted with professionals from Western Europe to identify what barriers they are facing in their day-to-day practice for the implementation of their circular approaches and the use of reclaimed materials.

Figure 2: Rate of growth for the last two decades of publications on dominant structures (concrete and steel) compared to the publications related to earthen architecture found in Scopus ©.

Interview framework

The data presented in this section is extracted from a large-scale research project on the current CE approaches in the construction sector, reported by authors in two journal papers, where the methodology adopted is deeply explained (Charef et al., 2021;Charef and Emmitt, 2021). To strengthen the results and increase the transferability, interviewees having international experiences have been selected. Moreover, the generalizability has been addressed by seeking experts, playing keys roles in the asset lifecycle and having worked internationally. The interviewees' responses indicated that the earthen architecture faces many obstacles, e.g staying marginal within the construction sector, reputed to be fragmented, having a lack of efficiency and damaging the environment. The necessity of having a holistic view of the building is claimed by construction experts, particularly, in terms of cost, management and become compulsory to embrace a circular economy approach. Buildings must be designed mindfully to make sure that their components are reusable or recyclable and no more waste is generated through the lifespan of the buildings. Six categories of impediments were identified: (i) economical, (ii) organizational, (iii) sociological, (iv) technical, (v) political and (vi) environmental, which are each explored in detail below. Technical barriers to the use of earth as a building material have aroused the interest of researchers since the 90s and have multiplied since (Figure 2). Whereas the other categories of impediments, whether social, economic, organizational or political are almost not specifically addressed in the literature except recently by Charef et al. (2021) who reported approaches embracing the CE principles, including earthen architecture among others. Nonetheless, as shown in the Sankey diagram summarising the interview results (Figure 3), those barriers were highlighted by the interviewees, in almost similar importance (the number of barriers identified (n=36) for technical aspect and in the comparable range for other aspects).

(a) Technical barriers

The interviewees noted that to gain more momentum, earthen architecture has to fight against technical obstacles, whether related to buildings, data, materials or technologies. Earthen buildings have the particularity to use less finishing works (e.g. plaster or paint) and to allow users to benefit from the aesthetic purity of the structural walls. Similar to conventional construction, the earthen architecture follows the shearing layers concept (site, structure, skin, services, space plan and stuff), having different timescales [START_REF] Brand | How buildings learn: What happens after they're built[END_REF], except that the layer of the finishing work may not exist. Therefore, this requires attention during the design and arrangement phases. Specific requirements compulsive by the material itself must be known by the construction team but also by the client and the end-users for the maintenance of the building. The complexity and size of buildings are also obstacles for the use of earth materials for their construction. Moreover, the size and access of the site are central, whether earth is processed onsite or off-site, due to the considerable size of the blocks (masonry units or rammed earth wall units) that can be handled using machines. When earth excavations come from the site where the construction is planned, sufficient space is required to store them. Assessing the technical ability for construction of excavated earth can appear, at a first glance, a barrier easily overcome by doing simple civil engineering tests, either in laboratory or onsite. However, the picture is not so simple. One main reason is the difficulty to establish the link between the parameters which are measured in laboratory and the real on-site performance of the material. For example, earthen walls exhibit, in service, quite important heterogeneities and variations in water content. Since, this parameter significantly impacts the mechanical performance of earthen materials, the use of the compressive strength value obtained on dry and homogeneous samples to design earthen constructions may appear quite questionable. To solve this problem more researches are notably needed to have a better assessment of water dynamic within the material. One other sound example concerns the lack of consensus to evaluate some major durability issues such as fire and freezing-thawing resistances. At last, the impact of hygrothermal processes on inhabitant comfort are not yet totally understood by the scientific community, which leads to some difficulties in order to properly assess thermal performances of earthen buildings.

Regarding the sanitary quality, it is achieved by checking if the source site has hosted polluting activities. Conversely, it is more complicated when the materials, from different sites, are sent and piled up in landfills. Indeed, the earth waste must be tested in a laboratory to make sure that the material was not polluted. The easiest and best way is to use excavations' earth from the construction site itself or coming from nearby sites to limit transportation and their related environmental impacts and access to the site history. This makes it possible to know and plan in advance the quantities, availability and location of materials (b) Organizational barriers Earthen architecture has to overcome a set of organizational barriers, requiring to apply changes in programming, design, construction and "in use" phases. The fragmented nature of the construction sector appears as a brake and has to be addressed differently to increase the efficiency and improve the communication between the stakeholders, considered as crucial by the interviewees. As an example, the involvement of masons during the design phase is central to adapt the design according to the material limitations, such as the thickness of the walls, and the avoidance of finishing work leading to think upfront about the finished surface not being obscured by artefacts or services (water, heating, electricity, furniture, etc.). Moreover, earthen construction systems are different compared to common architecture and must be studied with the stakeholders concerned. End-users must also be warned on how they can use their earth walls and maintain the facade of the building.

The earthen architecture sector is also facing a lack of skills, education and training, particularly among designers and builders. As discussed above masons play a central role in earthen projects, specifically during the design phase. As a result, the overall budget of the project is distributed differently. Similarly, the responsibility of each stakeholder must be adapted and contractually agreed. The contract must have been, upfront updated according to the specific needs for earthen architecture and projects developed within the CE framework.

(c) Political barriers

In the CE context, in addition to the individual level responsibilities, the lack of responsibilities at a territorial level was also raised by the interviewees as an important concern that must be addressed. Incentivising measures should be set up to encourage local governments to develop their territory according to the CE principles and switch to the use of more circular materials. Therefore policies and regulations must be strengthened to support the move towards a circular-thinking society.

Several political shortfalls and regulations' complexity related to earthen buildings have been pointed out by the interviewees, as being critical barriers for the growth of the use of "non-common" materials, such as earth. Lack of regulations for non-common techniques, lack of appropriate standardization and incentives are examples of policies' weaknesses that must be addressed urgently to decline drastically the environmental impact of the construction sector and improve human well-being; often earthen architecture has the potential to achieve these goals and should be encouraged (see section 2). The difficulty to get the insurance for the use of reclaimed materials, including excavation earth is also a struggle faced by practitioners willing to adopt a circular approach and most of the time they must get a technical certification. Moreover, getting technical certification is a "veritable obstacle course "for applicants, a costly, time consuming and sometimes biased process due to the unawareness of certification bodies. (d) Socio-Economic barriers Earth construction must emerge in a complex economic context and an unbalanced market where the recovered materials are not plebiscited. There is a lack of interest and demand for earthen construction, although experts note some emulsion around this material. The absence of a structured market obliges one to prepare the market upstream of projects, particularly when the project requires specific materials with specific expectations in terms of reuse and recycling. Also, a lack of awareness of the benefits of this material is keeping earthen construction behind. Real estate financing enforces clients to focus on their budget and to have a very short vision of the building leading them to make decisions based on that. Except for very few clients, profit-seeking and consume linearly is mainly the motto of our current society. In parallel, the lack of automation and regulations keep earthen architecture more expensive and time-consuming, compared to other types of architecture. Wherefore, earthen architecture struggles to bloom and with the lack of a holistic view, clients are not ready upfront for using more circular materials and embracing the CE approach.

To stimulate earthen architecture demand, some false beliefs must be corrected, and exemplary projects (like the one in Figure 1) require more advertisements to build up a better and modern social image of earthen architecture and show its beneficial and intrinsic response for humans and environment needs. Notably, the interviews' results indicated that environmental concerns permeated all of the assessed categories but that environmental barriers specifically were not considered to be a significant factor preventing the uptake of earthen construction. This is reasonable, as an environmental benefit should, by its nature, be expressed in context of the relevant activity rather than existing in isolation. The increase of awareness, understanding and potentials of earth as a construction material will create, as a result, an emulation around it and address the scepticism, lack of trust, and lack of concerns for the end-of-life of buildings.

Figure 3: The barriers for the use of earth as a construction material in the circular economy context, (n) stands for the number of occurrences reported by the interviewees, in the categories, subcategories or sub-subcategories

Conclusion

This paper presents an overview of the role of soils in the provision of materials for construction, in the context of an urgent need for the construction sector to switch from the linear to a circular economy. First, descriptions about the characteristics of earth-based materials were presented, then obstacles against modern earthen architecture were also developed. It has been shown that the five main types of barriers for the application of earth-based materials in the circular economy context were: economical; organizational; sociological; political; environmental; and technical related-topics.

Although the technical area is the most studied, there are still numerous topics needing further investigations, such as how to assess properly the material performance of earth-based materials in which the tests on representative elementary volumes are complicated (eg. cob and rammed earth walls with the presence of big elements like gravels, stones, fibres).

Similarly, the real impact of earth-based materials on building thermal behaviour and indoor air quality must be further explored. Research is also needed in the areas of control and prediction of the drying kinetics of the wall and its impact on mechanical behaviour (creep, strength) and on durability (freezing-thawing and fire resistance). Moreover, the identification of the conditions leading to durability issues requires particular attention and tools to control them should be developed.

The assessment of the inhabitant comfort and the impact of earthen walls on it are also one of the many technical uncovered areas that should be explored in the future. The scientific results obtained on the technical aspect will surely bring useful information for the studies raising the barriers from other non-technical aspects.

As the Sankey diagram illustrates, there are other four categories of obstacles faced by earthen architecture: organizational, sociological political and economic barriers. In the current context, tackling these non-technical obstacles is crucial and urgent to address the climate and economic crisis. Surprisingly, they have almost not been studied although they are equally important. Therefore, in parallel, those grey areas should be prioritized and investigated. The lack of exploration of those areas could explain why the earthen architecture is still a niche market although it embodies numerous attributes of a sustainable construction material to tackle the current climate, economic and societal crisis. Notably, the sixth category "environment", although not being addressed directly, was found to permeate all other categories.

Recently, the construction sector engaged in its fourth industrial revolution, embracing its digitalisation by changing how buildings are designed, constructed, operated and maintained. Technologies such as the use of Building Information Modelling, Internet of Things, Artificial Intelligence, Blockchain, automation and robotic are emerging technologies having a huge potential to support the entire construction industry. However, how the earthen architecture could benefit from the digitalisation and automation of the construction sector are areas needing investigation.

Figure 1 :

 1 Figure 1: the "Orangerie" rammed earth building during its construction in the centre of Lyon (top, © Fabrice Fouillet). The arches are load-bearing structure holding the timber floor and the flat roof. The timber floors are ensuring safety against earthquakes by tying the facades together and connecting them to the vertical timber structure around the lift (in the middle, of the photo). The partial on-site automation of the construction enabled to deliver the project with reasonable cost and time. Down left: A single-family house in rammed earth, Burgundy, France (© Nicolas Meunier).

Acknowledgement:

This work was supported by an Institutional Links grant, ID 429388357, under the Newton Fund partnership. The grant is funded by the UK Department for Business, Energy and Industrial Strategy and the Ton Duc Thang University in Vietnam and delivered by the British Council. This paper has benefited from researches coordinated by the Rilem Technical Committee 274-TCE: Testing and characterisation of earth-based building materials and elements.


Noama Ouchani
email: noama03@yahoo.fr

Abdelaziz El Moussaouy

Hassan Aynaou

Youssef El Hassouani

El Houssaine

El Boudouti

Bahram Djafari-Rouhani

Abdelaziz El Moussaouy

Youssef El Hassouani

Optical transmission properties of an anisotropic defect cavity in one-dimensional photonic crystal

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Photonic Crystals (PCs) have attracted a great deal of attention during the last two decades [1][2][3] due to their potential applications in optoelectronic and optical communications [4][5][6][7][8][9][10][11][12][13][14][15][16]. They can be used in the fabrication of lasers [4,5], optical diodes [6,7], waveguides [8,9], filters [10], dielectric reflectors [11][12][13][14], sensors [15,16], etc. Research in PCs has known a tremendous expansion and covers a wide range of electromagnetic spectrum from microwaves to the visible. Several technological difficulties restrict the fabrication of three-dimensional (3D) PCs in the visible (VIS) and infrared (IR) regions, due to their small lattice constant, which should be comparable to the wavelength [START_REF] Inoue | Photonic Crystals: Physics, Fabrication and Applications[END_REF]. However, the com-plication associated with 3D PCs lead to the investigation of onedimensional (1D) periodic structures, which can be easily produced for this range of wavelengths, by using thin-film deposition techniques.

For most applications, a defective PCs are more desired than the perfect ones. The physical properties of the localized or defect modes within the photonic band gaps (PBGs) has been increasingly studied. According to such combination, several materials are used as the defect layer in the design of the photonic devices. Several works have investigated the optical and microwave properties of defect modes in 1D isotropic dielectric PCs. Liu et al. [10] have studied a 1D PC structure consisting of alternate layers of Ta 2 O 5 and MgF 2 films with a defect layer in the visible region. They have shown that the number and the frequencies of the defect modes can be controlled in the visible range by adjusting the thickness of the defect layer. This structure can be useful in the design of blue-

We investigate theoretically the possibility to control the optical transmission in the visible and infrared regions by a defective one dimensional photonic crystal formed by a combination of a finite isotropic superlattice and an anisotropic defect layer. The Green's function approach has been used to derive the reflection and the transmission coefficients, as well as the densities of states of the optical modes. We evaluate the delay times of the localized modes and we compare their behavior with the total densities of states. We show that the birefringence of an anisotropic defect layer has a significant impact on the behavior of the optical modes in the electromagnetic forbidden bands of the structure. The amplitudes of the defect modes in the transmission and the delay time spectrum, depend strongly on the position of the cavity layer within the photonic crystal. The anisotropic defect layer induces transmission zeros in one of the two components of the transmission as a consequence of a destructive interference of the two polarized waves within this layer, giving rise to negative delay times for some wavelengths in the visible and infrared light ranges. This property is a typical characteristic of the anisotropic photonic layer and is without analogue in their counterpart isotropic defect layers. This structure offers several possibilities for controlling the frequencies, transmitted intensities and the delay times of the optical modes in the visible and infrared regions. It can be a good candidate for realizing high-precision optical filters. ,d). The solid and dashed lines correspond to the perfect and defective structures, respectively. The black dots on the λ-axis represent the edges of the band gaps of the infinite isotropic SL. The incidence angle of the incoming light is θ i = 0 • . green color filters. The green and red color filters were studied by Xiang et al. [START_REF] Xiang | [END_REF].

Superconductor material and liquid crystal are among potential materials to serve as defect layers in tunable photonic devices, as their optical properties can be controlled by external excitation. Indeed, Wu and Gao [16] have studied the effect of the temperature on the defect modes in a 1D dielectric PC heterostructure with a superconducting defect. They have shown that this structure has very high temperature sensitivity. Dadoenkova et al. [19] have investigated PBG spectra of a 1D dielectric PC with a complex defect layer, consisting of ultrathin superconducting and dielectric sublayers. Recently, some research works [20][21][22][23] have investigated the effect of an electric field on the defect modes induced by a liquid crystal defect layer (anisotropic medium) or by an Ag/LiNbO 3 /Ag sandwich structure into 1D PC [22,23]. Tang el al. [24] have explored the influence of the anisotropy of the singlenegative material on photonic band gaps and tunneling modes of 1D PCs containing dispersive anisotropic single-negative metamaterials. Penninck et al. [25] have studied the emission of light in uniaxially anisotropic thin film devices. This emission is performed through a dipole antenna inserted inside a thin isotropic film sandwiched between two anisotropic layers. The whole structure is placed between glass and air.

1D birefringent photonic crystal based on anisotropic porous silicon layers has been designed to control simultaneous enhancement of second and third harmonic generation [26]. The structure consists of 12 quarter wavelength pairs of layers separated by a half wavelength cavity spacer. Recently the chiral PCs with defects have been of particular interest due to their rich optical properties. Various types of defects have been considered, such as: a thin isotropic [27][28][29] or anisotropic [30][31][32] defect layers inserted between two chiral PCs. The properties of the defect modes induced by anisotropic dielectric layer in 1D birefringent photonic crystal, have been investigated theoretically by some of us [33].

In all the above works, only the reflectance and/or the transmittance of defect modes have been explored. In this work, we are interested to some optical transmission properties of the defect modes such as: the transmission delay times and the behavior of total densities of states (DOS) in the visible and infrared regions of a defective one dimensional photonic crystal constituted by a combination of a finite isotropic superlattice (SL) and a dielectric anisotropic defect layer. In particular, we analyze the delay times of the defect modes and compare their behavior with the total DOS. More especially, we show the possibility of existence of transmission zeros induced by the anisotropic defect layer, giving rise to negative delay times for some wavelengths in the visible and infrared regions. Let us recall that the existence of transmission zeros has been shown in 2D photonic crystals [34,35], however the possibility of existence of such forbidden transmission regions in 1D lamellar systems has not been investigated before. This property is a characteristic of the anisotropic defect layer and is without analogue in their counterpart isotropic layer. It is worth mentioning that transmission zeros and the corresponding negative delay times have been demonstrated in quasi-one dimensional circuits made of coaxial cables in the radio-frequency domain [36]. Also, we highlight the impact of the orientation of the principal axes of the defect layer with respect to the laboratory axes, on the behavior of the optical modes in the forbidden band-gap of the system. The effect of the position of the cavity layer within the photonic crystal on the transmittance and the delay time of the defect modes will be also investigated. In general, the main theoretical method adopted for the analysis of the 1D multilayered PCs, is the transfer matrix method [10,12,14]. In this paper, we use the Green's function approach which enables to derive the transmission and reflection coefficients, the dispersion curves, the delay times as well as the total and local DOS of electromagnetic modes propagating through the structure.

This paper is organized as follows: in section 2 we present the theoretical model and the method to derive the expressions of the reflection and transmission coefficients, the delay time and the densities of states, when S or P polarized waves is incident upon the interface separating the substrate and the SL. Section 3 gives the numerical results of the transmittance and the delay time of the optical modes with respect to the wavelength and their dependence on the azimuthal angle and the position of the defect layer within the photonic crystal. The spatial behavior of electromagnetic modes along the z-axis of the SL, has been investigated through the local density of states. Finally, a summary of this work is presented in section 4.

Theoretical model

The geometrical structure is schematically illustrated in Fig. 1.

It is denoted by s 1 /(AB) N C (BA) N /s 2 , where s i (i = 1o r 2) means the semi-infinite substrates and N is the number of the AB bilayers (periods) at each side of the anisotropic defect layer C . The unit cell of the structure is composed of two dielectric isotropic layers A and B. All the interfaces of the layers are taken to be parallel to the (XY) plane of a cartesian (laboratory) coordinate system and the Z axis is along the normal to the interfaces (Fig. 1). The thicknesses of the layers A, B and C are denoted d A , d B and d C , respectively. d = d A + d B is the period of the SL. The materials constituting the layers are assumed to be homogeneous and non-magnetic. The isotropic layers A and B are characterized by the dielectric constants ǫ A and ǫ B respectively, whereas the anisotropic layer is characterized by the diagonal dielectric tensor ǫ = (ǫ x , ǫ y , ǫ z ), where ǫ x , ǫ y and ǫ z are the dielectric constants in the corresponding principal axes (xyz) of the crystal. For a given orientation of the crystal, its dielectric tensor with respect to the fixed (XYZ) coordinate system is described by using the Euler angles [START_REF] Goldstein | Classical Mechanics[END_REF][START_REF] Landry | [END_REF] θ, φ and ψ . For the sake of simplicity and without loss of generality, we investigate in this study the effect of only the azimuthal angle φ and omit the effect of the two other angles (θ = ψ = 0). The axes of the birefringent layer C are shown in Fig. 1.

Among different techniques used to study the propagation of electromagnetic waves in periodic structures, one can cite the transfer matrix [START_REF] Yeh | Optical Waves in Layered Media[END_REF][START_REF] Yeh | Optical Waves in Crystals: Propagation and Control of Laser Radiation[END_REF] and the Green's function methods [START_REF] Dobrzynski | [END_REF]. Both techniques enable to calculate dispersion relations and transmission and reflection coefficients. However, the Green's function presents the advantage to deduce easily the densities of states as well as different scattering properties of the system [START_REF] Dobrzynski | [END_REF]. An interesting connection between Green's functions and transfer matrices has been established by Velasco et al. [START_REF] Velasco | [END_REF] and El Boudouti et al. [43]. This combination yields to study of the physical properties of any composite system. In this work, we use a simple formulation of the Green's function called interface response theory of continuous media [44]. This technique is suitable for treating composite systems containing a large number of interfaces [START_REF] Dobrzynski | [END_REF]. In this theory, the Green's function of a composite system can be written as [44] g

(DD) = G(DD) + G(DM){[G(MM)] -1 g(MM)[G(MM)] -1 -[G(MM)] -1 }G(MD), (1)
where D and M are respectively, whole space and the space of the interfaces in the lamellar system. G is a block-diagonal matrix in which each block G i corresponds to the bulk Green's function of the subsystem i. All the matrix elements g(DD) of the composite material can be obtained from the knowledge of the matrix elements g(MM) in the interface space M. g(MM) is calculated from its inverse g -1 (MM). The latter is formed out by a linear superposition of the surface matrix elements g -1 i (MM) of any independent film bounded by perfectly free interfaces with appropriate boundary conditions. The matrix elements g -1 i (MM) for an anisotropic medium are given in previous works [33,45].

Within this theory, the reflected and transmitted waves u(D), resulting from a uniform plane wave U (D) incident upon a plane boundary between two different media, are given by [44]

u(D) = U (D) + G(DM) [G(MM)] -1 g(MM)[G(MM)] -1 -[G(MM)] -1 U (M).
(

) 2
Let us mention that the incident wave, generated in the substrate s, can have two different polarizations, namely, S or P . In general, each wave entering the anisotropic medium, generates two transmitted waves and two reflected waves with different polarizations. Let us call E iS and E iP the amplitudes of the S and P components of the incident field respectively. Then, the amplitudes of the reflected and transmitted fields can be written as [33,45]:

E RS (z) = r SS E iS (z) + r SP E iP (z) , (3)
E TS (z) = t SS E iS (z) + t SP E iP (z) , (4)
E RP (z) = r PS E iS (z) + r PP E iP (z) , (5)
E TP (z) = t PS E iS (z) + t PP E iP (z) . ( 6
)
The expressions of r ij and t ij in these equations (i, j = S or P ) for an anisotropic structure are given in the previous works [33,45]. Therefore, the total reflection coefficients for the P and S modes are given respectively by

R P = R PP + R SP =| r PP | 2 +|r SP | 2 , (7) R S = R SS + R PS =| r SS | 2 +|r PS | 2 , ( 8
)
and the total transmission coefficients for the P and S modes are given respectively by

T P = T PP + T SP =| t PP | 2 +|t SP | 2 , ( 9
)
T S = T SS + T PS =| t SS | 2 +|t PS | 2 . ( 10
)
From the expressions of the r ij and t ij , one can deduce also the phases of the transmission and reflection coefficients. The derivatives of these phases with respect to the pulsation ω, yield to an important physical quantity. This latter called the delay time, indicates the times needed by the wave to complete the transmission or reflection process through the structure. Four different delay times can be defined for the transmission coefficients, namely

τ ss = dφ ss dω ; τ ps = dφ ps dω ; τ pp = dφ pp dω ; τ sp = dφ sp dω ( 11
)
where φ ss , φ ps , φ pp , φ sp are the phases of the transmission coefficients t ss , t ps , t pp , t sp . As mentioned above, the Green's function enables to calculate the density of states, especially, one can determine the variation of the density of states n between the structure s 1 /(AB) N C (BA) N /s 2 and a reference system formed out of the same volumes of the bulk semi-infinite substrates s 1 and s 2 and the finite SL. This quantity is given by [46]:

n(ω) = 1 π d dω arg det g(M 0 M 0 ) [g s 1 (0, 0)g s 2 (L, L)] 1/2 , ( 12
)
where g(M 0 M 0 ) is the Green's function of the whole system at its both extremities M 0 ={0, L}, whereas g s 1 (0, 0) and g s 2 (L, L) are the elements of the Green's functions at the surfaces 0 and L of the two substrates (air).

In a previous work, some of us [47] have shown analytically a relation between the transmission delay time τ T and the variation of the density of states n(ω) for isotropic layered media, namely τ T = π n(ω). In the case of anisotropic media, the expressions of the different delay times (Eq. ( 11)) and n(ω) (Eq. ( 12)) are more complicated. Therefore, we limit ourselves to a numerical comparison.

Numerical results and discussion

The perfect photonic crystal s 1 /(AB) N /s where λ 0 = 1550 nm, which leads to d A = 164.89 nm and d B = 265.41 nm, respectively. λ 0 = 1550 nm is chosen for its use in optical communications. The number N of the AB bilayers (periods) of the PC is chosen equal to N = 6, for which the PBGs are well defined. Both substrates surrounding the PC are simply assumed to be air. The periodicity of the perfect SL is broken by inserting the defect layer C in the center of the structure as depicted in Fig. 1. The defect layer is made of an anisotropic dielectric material: SbSI, with its principal optical indices [START_REF] Yeh | Optical Waves in Layered Media[END_REF]:

n (C) x = 2.7, n (C) y = 3.2, n (C) z = 3.8.
The orientation of the principal axes of the layer C is characterized by the azimuthal angle φ C , as illustrated in Fig. 1, and the thickness of this layer is chosen equal to d C = 263.50 nm, for which the defect modes are well defined in PBGs. In most of our numerical results, we study the behavior of the optical modes in a defective 1D PC when the incoming light has S or P polarization. After multiple reflection and transmission in the anisotropic layer C , the incoming wave produces two reflected and two transmitted waves with S and P polarizations. For special cases when the principal axes of the defect layer are parallel or perpendicular to the fixed axes, the S and P polarized waves remain uncoupled [45]. Therefore, a single incident S or P polarized plane wave generates a single reflected R S or R P polarized plane wave and a single transmitted T S or T P polarized plane wave, respectively.

We display in Fig. 2 the transmittances T P (Fig. 2(a), (b)) and T S (Fig. 2(c),(d)) of the optical modes as a function of the wavelength λ for a perfect PC (solid lines) and a defective PC (dashed lines). In this illustration, we assume that the incident electromagnetic wave is launched normally (θ i = 0 • ) to the SL, and the principal axes of the defect layer C are parallel φ C = 0 • (Fig. 2(a),(c)), or perpendicular φ C = 90 • (Fig. 2(b),(d)) to the fixed axes of the coordinates system (XY Z). It is shown that for these two values of the azimuthal angle φ C , the defect modes appear inside the PBGs of the transmission spectrum. The localized modes in the electromagnetic forbidden bands of the structure arise due to the constructive interference of incident and reflected waves in the defect layer. From the analysis of Fig. 2, it can be observed that when the principal axes of anisotropic defect layer were parallel to the fixed axes of the coordinates system, the defect modes (dashed curves) are localized at λ VIS P 0 = 535.81 nm, λ IR P 0 = 1611.47 nm in the visible (VIS) and infrared (IR) regions, respectively, for transmission spectrum of P -polarized wave (Fig. 2(a)). Moreover, when the azimuthal angle of the layer C is equal to 90 • , both modes shift to shorter wavelengths and appear respectively at λ VIS P 90 = 500.22 nm and λ IR P 90 = 1497.83 nm (Fig. 2(b)). The comparison of the results

for S (Fig. 2(c,d)) and P (Fig. 2(a,b)) polarized incident waves, The black dots on the λ-axis represent the edges of band gaps of the infinite isotropic SL and for normal incidence.

shows that the S-polarized defect modes for φ C = 0 • have the same behavior as the P -polarized defect modes when φ C = 90 • , i.e., λ VIS S 0 = λ VIS P 90 and λ IR S 0 = λ IR P 90 . Moreover, the same behavior of the localized modes in PBGs has been observed when the principal axes of defect layer are perpendicular to the fixed axes and the incoming light has S polarization, i.e., λ VIS S 90 = λ VIS P 0 and λ IR S 90 = λ IR P 0 . These results are in good accordance with those presented in Ref. [START_REF] Fesenko | Advances in Photonic Crystals[END_REF]. In addition, the transmission amplitudes of these peaks reach approximately unity for φ C = 0 • and φ C = 90 • . The black dots on the λ-axis indicate the limits of the band gaps of the infinite isotropic SL. These findings clearly show that the orientation of the principal axes of the anisotropic defect layer changes the properties of the defect modes within the VIS and IR regions band gaps. In order to understand the spatial distribution of the S-polarized defect modes labeled 1, 2i n Fig. 2(c) along the Z -axis of the SL, we have plotted in Fig. 3 the local density of states versus the reduced position Z /d for the azimuthal angle φ C = 0 • of the defect layer. This quantity reflects the spatial behavior of the square modulus of the electric field inside the structure. Figs. 3(a) and (b) show a strong localization of the defect modes 1 and 2i n the vicinity of the defect layer C and a decaying behavior on both sides of the interfaces of this layer. Similar behavior of the localized mode can be observed for other orientations of the principal axes of the defect layer and also for P -polarized waves.

In the following, we consider (Fig. 4) the effect of the incidence angle on the optical properties of the defect modes within the forbidden band gaps in the VIS and IR regions for φ C = 0 • and φ C = 90 • . Fig. 4 gives the dependence of the peak wave- lengths on the angle of incidence, when the input light has P (Fig. 4(a,b)) or S (Fig. 4(c,d)) polarization. One can notice that when the incidence angle increases, the position of P and S polarized defect modes shift towards the lower wavelengths inside the PBGs (hatched areas) as illustrated in Fig. 4. In addition, when the incidence angle is greater than 35 • , the defect branch within the first gap for P polarization and φ C = 0 • (Fig. 4(a)) and for S polarization and φ C = 90 • (Fig. 4(d)), disappear at the higher edge of the forbidden gap and falls in the allowed band. Furthermore, the transmission amplitudes of these modes remain almost unity whatever the incidence angle for both cases (φ C = 0 • and φ C = 90 • ).

Till now, we have depicted only the properties of the defect modes in the transmission spectrum of the defective 1D PC when the orientation of the principal axes of defect layer are parallel or perpendicular with respect to the laboratory axes. Let us now consider the effect of other orientations of the axes of the layer C on the behavior of the defect mode when the incident electromagnetic wave is launched normally to the SL. We present in Fig. 5(a,b), the evolution of the transmission spectra T SS and T PS of S-polarized wave as a function of the wavelength λ for the azimuthal angle φ C = 60 • . It is clear that the birefringence of the defect layer has a strong impact on the number and the magnitudes of the defect modes inside the PBGs in the VIS and IR regions. For this azimuthal angle (φ C = 60 • ) of the layer C , two defect modes appear inside each PBGs as every incident plane wave with S or P polarization generates two reflected and two transmitted plane waves containing both S and P polarized plane waves. These two propagating waves are coupled, thus each mode is composed of two components of the transmission (T SS and T PS ) spectrum for the same wavelength. These results correlated with the results in our previous work [33], when the PC is composed only by anisotropic media.

Moreover, the intensities of the defect modes depend strongly on the azimuthal angle of the anisotropic defect cavity as illustrated in Fig. 5(c,d), where we have plotted the dependence of peak heights T SS (Fig. 5(c)) and T PS (Fig. 5(d The Green's function method enables us to derive other physical properties of the composite system under study, through the analysis of the transmission delay time τ and the total density of states n. The delay time describes the time needed for photons to complete the transmission process and provides information about the interaction of an incident wave with the modes confined in the anisotropic defect layer. In the following, we study these two quantities and give a comparison between the total DOS and the delay time of the optical waves. We display in Fig. 6, the variation of DOS (π n) (Fig. 6(a,b)) and the delay time τ (Fig. 6(c,d)) as a function of the wavelength for S-polarized incident wave and when the principal axes of defect layer are parallel or perpendicular to the fixed ones. It is clear that the delay time of the transmitted peaks in the VIS and IR regions are very sensitive to the azimuthal angle of the anisotropic defect layer. The frequencies of the defect modes obtained from the total DOS (Fig. 6(a,b)) coincide with those given by the delay time (Fig. 6(c,d)) and the transmission coefficient (see Fig. 3(c,d)). In addition, the delay time τ s exhibits the same behavior as the DOS (π n) for φ C = 0 • (Figs. 6(c) and (a)) and for φ C = 90 • (Figs. 6(d) and (b)). For these particular orientations of the principal axes of the defect layer, the density of states is directly proportional to the transmission delay time, i.e., τ s = π n(ω), in accordance with the results demonstrated for isotropic PC [47]. These results remain also valid for P -polarized waves. However, for φ C = 0 • and φ C = 90 • , different transmission delay times (τ ss , τ ps ) and (τ pp , τ sp ) for S and P transmitted polarized waves, respectively, show different behaviors in comparison with the DOS as it is illustrated in Fig. 7. One can notice that the delay times spectrum for both S and P transmitted polarized waves show negatives peaks for some wavelengths in the VIS and IR regions. We have checked that the corresponding transmission amplitudes reach almost zero as a consequence of a destructive interference between S and P waves inside the anisotropic defect layer. The negative peaks in τ pp (Fig. 7(b)) and τ ss (Fig. 7(c)) spectra are localized between the two defect modes in both VIS and IR regions, whereas those in τ sp and τ ps (Fig. 7(d)) appear at the vicinities of the PBG edges. It is worth mentioning that the negative delay times in 1D systems are generally observed by taking into account the absorption in these systems [36,[START_REF] Hassouani | [END_REF]. However, in the present system, the negative delta peaks are intrinsically broadened. Moreover, because of the symmetry of the structure, the delay time τ sp presents the same behavior as τ ps . Indeed, rendering the structure asymmetric, by moving for example the cavity layer C inside the PC, gives rise to τ ps = τ sp . Let us notice that the existence of transmission zeros in one of the two components of the transmission, may give rise also to Fano and electromagnetic induced transparency resonances [35] even though these properties have not been explored here and postponed to a future work.

To widen the scope of our findings, we have also investigated the effect of the position of the cavity layer within the PC on the optical properties of defect modes in the PBGs of the structure. We have plotted in Fig. 8, the transmittance Fig. 8(a,b) and the delay time Fig. 8(c,d) of the localized modes in the IR region for different positions of the cavity layer within the PC and for φ C = 90 • .

The PC is composed of 13 cells and the cavity layer can be inserted at different positions labeled n inside the PC (0 ≤ n ≤ 12).

The transmission spectra are displayed for S-polarized waves. One can notice that the transmittance and the delay time of the defect modes depend strongly on the position of the cavity layer within the finite SL. The magnitudes of the filtered modes in the PBGs, reach their maximum value when the defect cavity is inserted in the middle of the structure (i.e., n = 6) and decreases gradually when the defect layer is moved far from the middle of the PC, as shown in Figs. 8(b) and (d). The transmittance and the delay time almost vanish when the cavity layer is placed at both extremities of the structure. These results remain also valid for other orientations of the principal axes of the defect layer and for P -polarized waves.

Conclusion

In summary, we have investigated theoretically the effect of an anisotropic defect cavity on the optical transmission properties in one-dimensional isotropic photonic crystal. This study is performed by means of the Green's function approach, through an investigation of the transmission coefficients, the delay times and the DOS for electromagnetic modes in the finite structure. The transmittance and the delay times are discussed as a function of the azimuthal angle and the position of the anisotropic defect cavity within the PC. We have presented, for the first time to our knowledge, a comparison between the total DOS and different transmission delay times in such systems. We have shown that the delay time reproduces exactly the DOS when the principal axes of the defect layer are parallel or perpendicular to the laboratory axes. However, far from these directions, the birefringence of the cavity layer induces, in addition to mode conversion, transmission zeros in one of the two components of the transmission, for some wavelengths in the VIS and IR regions, giving rise to negative delta peaks in the delay times. These properties are without analogue in their counterpart isotropic defect layers. This structure can be of potential interest in the fields of photonics and optoelectronics as a high-precision optical filter.

Fig. 1 .

 1 Fig. 1. Schematic representation of a finite 1D superlattice composed of 2N cells with an anisotropic defect cavity: SbSI (layer C ) in the middle of the structure. The unit cell of the perfect structure is composed by TiO 2 (layer A) and SiO 2 (layer B). The layers A, B and C are characterized by their thicknesses d A , d B and d C , respectively. The principal axes of the defect layer are oriented by the azimuthal angle φ C with respect to the laboratory axes (XY Z). d = d A + d B is the period of the SL. The input and output isotropic media are air and the incident electromagnetic wave is launched with an angle θ i with respect to the normal to the superlattice.

Fig. 2 .

 2 Fig. 2. Variation of the transmittances T P (a, b) and T S (c, d) as a function of the wavelength λ for φ C = 0 • (a, c) and φ C = 90 • (b, d). The solid and dashed lines correspond to the perfect and defective structures, respectively. The black dots on the λ-axis represent the edges of the band gaps of the infinite isotropic SL. The incidence angle of the incoming light is θ i = 0 • .

Fig. 3 .

 3 Fig. 3. Spatial representation of the local density of states (in units of d/c) of the modes labeled 1a n d 2i n Fig. 2(c) at λ = 500.22 nm (a) and λ = 1497.83 nm (b), respectively. The input S-polarized wave is launched normally to the SL and the azimuthal angle of the layer C is φ C = 0 • .

Fig. 4 .

 4 Fig. 4. Dependence of the peak wavelengths of the transmitted P -a n d S-polarized waves for φ C = 0 • (a, c) and φ C = 90 • (b, d) versus the incidence angle θ i . The black triangles and dots denote the defect modes in PBGs. The dotted curves represent the edges of the band gaps (hatched areas) of the infinite isotropic SL.

Fig. 5 .

 5 Fig. 5. Variation of the transmissions T SS (a) and T PS (b) versus the wavelength λ for φ C = 60 • . (c) and (d) represent the dependence of peak heights T SS and T PS of the defect modes labeled 1a n d 2i n (a) and (b) on the azimuthal angle φ C , respectively. The incoming S-polarized light is launched normally to the SL.

Fig. 6 .

 6 Fig. 6. Variation of the density of states n (in units of d/c) (a, b) and the delay time τ S (c, d) of S incident polarized waves versus the wavelength λ for φ C = 0 • (a, c) and φ C = 90 • (b, d). The black dots on the λ-axis represent the edges of the band gaps of the infinite isotropic SL at normal incidence.

Fig. 7 .

 7 Fig. 7. Variation of the density of states n (in units of d/c) (a) and different delay times: τ PP (b), τ SS (c) and τ PS = τ SP (d), as a function of the wavelength λ for φ C = 60 • .

Fig. 8 .

 8 Fig. 8. Variation of the transmittance T S and the delay time (τ S ) of S polarized modes versus the wavelength for different positions of the cavity layer n (0 ≤ n ≤ 12) inside a finite PC composed of 13 cells. The curves are sketched in IR region and for φ C = 90 • . (b) and (d) represent the dependence of the transmission amplitude and the delay time of the defect modes as a function of the defect cavity position n.

  )) of the modes labeled 1 and 2i n Figs. 5(a) and (b) versus the azimuthal angle φ C . The results of Fig. 5(c) show that by adjusting φ C , the intensities of the defect modes exhibit different behaviors inside the PBGs of the structure. Indeed, by increasing φ C from 0 • to 90 • , the transmission amplitudes of the mode 2 (indicated by the black dots) increase from 0 to 1, while the amplitudes of the mode 1 (indicated by black triangles) decrease from 1 to 0. However, the transmitted P -polarized modes (Fig. 5(d)), exhibit similar behavior. Their intensities gradually increase by rotating the azimuthal angle of the defect layer, reach a maximum value around φ C = 45 • and then decrease to vanish for φ C = 90 • . Also, both transmitted P -and S-polarized defect modes have identical magnitudes (0.25) around φ C = 45 • .

  The refractive indices of these materials are, respectively, n A = 2.35 and n B = 1.46. The thicknesses d A and d B of these layers are chosen in accordance with the quarter wave stack condition, i.e., n A d A = n B d B = λ 0 4

2 consists of two alternating isotropic media: TiO 2 (layer A) and SiO 2 (layer B).

Acknowledgements

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.