Blessing N Ezealigo

Roberto Orrù
email: roberto.orru@dimcm.unica.it

Catherine Elissalde
email: catherine.elissalde@icmcb.cnrs.fr

Hélène Debéda

U-Chan Chung

Mario Maglione

Giacomo Cao

Influence of the Spark Plasma Sintering temperature on the structure and dielectric properties of BaTi(1-x)ZrxO3 ceramics

Keywords: Dielectric permittivity, spark plasma sintering, piezoelectric, post-annealing; selfpropagating high-temperature synthesis

In this work, structural and dielectric properties of BaTi(1-x)ZrxO3 (BTZ) ceramics prepared by Spark Plasma Sintering (SPS) from powders obtained via Self-propagating High-temperature Synthesis (SHS) are shown to be strongly affected by the sintering temperature. In addition, a postannealing treatment in air of the as-prepared ceramics leads to a transition from the hexagonal to the tetragonal and cubic phases. The SPS ceramics corresponding to compositions 0.05≤x≤0.20 and obtained at a sintering temperature of 1200°C exhibit a standard ferroelectric behavior. In contrast, a diffuse phase transition is observed for the case of ceramics sintered at higher temperatures. Finally, the BTZ ceramic containing 5 at.% of Zr displays the best dielectric permittivity and piezoelectric properties as compared to the other compositions taken into account.

Introduction

To meet specific applications, the strategy commonly adopted by the scientific community to improve the electrical characteristics of ABO3 perovskite materials is to change the ceramics' chemical content. Indeed, the substitution of the A or B cation sites with other isovalent ions can modify the material's dielectric properties, and lead to changes in the related transition temperatures as well as the formation of specific phases [START_REF] Jiang | Structure and thermal stability of nanostructured iron doped zirconia prepared by high-energy ball milling[END_REF]. In this context, zirconium substituted barium titanate (BTZ) has attracted great attention in the last decade because of its wide range of applications, owing to its improved properties and potential to replace lead-based ferroelectric materials [2-3]. These properties include high dielectric permittivity, relatively low dielectric loss, large voltage tunability of the dielectric permittivity, good chemical stability, and attractive leadfree relaxor behavior [START_REF] Ahmadu | Physical properties of porous pure and Zr/Sn-doped nanocrystalline BaTiO3 ceramics[END_REF][START_REF] Garbarz-Glos | Effect of Zr Doping on the Electrical Properties of BaTiO3 Ceramics[END_REF][START_REF] Maiti | Evaluation of experimental resume of BaZrxTi1-xO3 with perspective to ferroelectric relaxor family: An overview[END_REF].

In this regard, the marked influence of Ti sites substitution with Zr on the properties of the resulting BTZ ceramics obtained by conventional sintering has been addressed by various authors Specifically, it is reported that, when the amount of Zr content is within the range 0≤x≤0.10, the ceramics show a classical ferroelectric behavior [START_REF] Kuang | Influence of Zr dopant on the dielectric properties and Curie temperatures of Ba(ZrxTi1-x)O3 (0 ≤ x ≤ 0.12) ceramics[END_REF][START_REF] Dobal | Micro-Raman scattering and dielectric investigations of phase transition behavior in the BaTiO3-BaZrO3 system[END_REF]. However, at higher Zr content, the ferroelectric phase coexists with the paraelectric one, leading to a diffuse phase transition. The latter one is promoted by the energy difference between the ferroelectric and the paraelectric phases [START_REF] Hennings | Diffuse Ferroelectric Phase Transitions in Ba(Til-y Zry)O3 Ceramics[END_REF]. Furthermore, the classical ferroelectric transitions of BaTiO3 continuously give rise to diffuse phase transition (0.10 < x ≤0.26) and relaxor state (0.26 < x ≤0.40), whose temperature of occurrence decreases to low values as the Zr content increases [START_REF] Ravez | Temperature and Frequency Dielectric Response of Ferroelectric Ceramics with Composition Ba(Ti1-xZrx)O3[END_REF]. It is worth to mention that, in addition to the latter finding, the phase transition can also be influenced by temperature variation, depolarization due to presence of porosity, slight changes in the chemical potential, and grain size effect [12, [START_REF] Frey | Grain-size effect on structure and phase transformations for barium titanate[END_REF][START_REF] Fang | Effects of Pore Morphology and Grain Size on the Dielectric Properties and Tetragonal-Cubic Phase Transition of High-Purity Barium Titanate[END_REF]. A high dielectric permittivity of about 14,000 has been reported for dense BTZ ceramics (x=0.08), which were produced by solid state reaction using high energy milling followed by conventional sintering [START_REF] Kuang | Influence of Zr dopant on the dielectric properties and Curie temperatures of Ba(ZrxTi1-x)O3 (0 ≤ x ≤ 0.12) ceramics[END_REF].

Among the synthesis methods proposed to obtain BTZ powders, Sol-gel [16-17], Sol-gel based Pechini method [START_REF] Deluca | Investigation of the composition-dependent properties of BaTi1-xZrxO3 ceramics prepared by the modified Pechini method[END_REF], solid-state reactions [START_REF] Badapanda | Electric field induced strain, switching and energy storage behaviour of lead free barium zirconium titanate ceramic[END_REF][START_REF] Mahesh | Effect of sintering temperature on the microstructure and electrical properties of zirconium doped barium titanate ceramics[END_REF], and combustion synthesis [START_REF] Julphunthong | Phase formation, microstructure and dielectric properties of Ba(Zr0.1Ti0.9)O3 ceramics prepared via the combustion technique[END_REF][START_REF] Ahmad | Synthesis of BaTiO3 and Ba(ZrxTi1-x)O3 by using the soft combustion method[END_REF] are reported. The formation of BTZ solid solution is obtained by conventional sintering in the temperature range 13001450°C after 210 h treatment. In addition to conventional sintering, only few studies have also been conducted on BTZ ceramics using Spark Plasma Sintering (SPS) [START_REF] Maiwa | Electromechanical properties of Ba(Zr0.2Ti 0.8)O 3 ceramics prepared by spark plasma sintering[END_REF][START_REF] Philippot | Continuous BaTi1-yZryO3 (0 ≤ y ≤ 1) nanocrystals synthesis in supercritical fluids for nanostructured lead-free ferroelectric ceramics[END_REF][START_REF] Liu | Enhanced dielectric strength and energy storage density in BaTi0.7Zr0.3O3 ceramics via spark plasma sintering[END_REF]. The latter one is an efficient sintering process where powders are rapidly heated by Joule effect, due to the combination of pulsed electric current and the application of uniaxial mechanical pressure [START_REF] Orrù | Consolidation/synthesis of materials by electric current activated/assisted sintering[END_REF]. In particular, the structure and electromechanical properties of Ba(Zr0.2Ti0.8)O3 ceramics obtained by SPS at 1100°C and subsequently annealed at 10001400°C for 12h has been investigated by Maiwa [START_REF] Maiwa | Electromechanical properties of Ba(Zr0.2Ti 0.8)O 3 ceramics prepared by spark plasma sintering[END_REF][START_REF] Maiwa | Dielectric and Electromechanical Properties of Ba(ZrxTi1-x)[END_REF].

In this work, SHS and SPS techniques are combined for the first time for the preparation of Zr-substituted barium titanate ceramics (BaTi(1-x)ZrxO3). Briefly, SHS is a combustion synthesis method based on the local ignition of highly exothermic reactions which can self-propagate without additional energy supply [START_REF] Varma | Combustion Synthesis of Advanced Materials: Principles and Applications[END_REF][START_REF] Cincotti | A review on combustion synthesis of novel materials: Recent experimental and modeling results[END_REF]. Temperatures in the range 20005000 K and cooling rates up to 10 3 K/min can be reached during SHS and, more importantly, high-purity products can be correspondingly obtained in a very short time (order of seconds) [START_REF] Varma | Combustion Synthesis of Advanced Materials: Principles and Applications[END_REF].

The present work is aimed to produce single-phase BaTi(1-x)ZrxO3 (x=0.050.25) ceramics by combining the two fast processes mentioned above, i.e. by following the so-called SHS-SPS approach. The influence of the sintering temperature (TD) on density, composition (structure), dielectric, and piezoelectric properties of SHS-SPS ceramics is systematically investigated. The optimal SPS, as well as post-annealing, conditions, are then identified. A novel procedure of tuning the dielectric response of the BTZ ceramics by varying the sintering temperature, regardless of the Zr content, is presented.

Experimental

BTZ powders were prepared by SHS according to the following chemical equation:

2BaO2 + (1-2x)TiO2 + 2xZrO2 +Ti → 2BaTi(1-x)ZrxO3 (1)
Hereto after, the BaZr0.05Ti0.95O3, BaZr0.10Ti0.90O3, BaZr0.15Ti0.85O3, BaZr0.20Ti0.80O3, and BaZr0.25Ti0.75O3 compositions will be indicated as BTZ005, BTZ010, BTZ015, BTZ020, and BTZ025, respectively. Details on the synthesis of the precursors and procedure can be found in

[30].
The SHS products were first converted into powders after 20 min ball milling treatment using a SPEX 8000 mixer mill (SPEX CertiPrep, Metuchen, NJ, USA). It is worth to mention that such a mild milling conditions (the ball to powder weight ratio was only equal to 2) are not expected to modify the grain size of SHS product and, in turn, affect the microstructure of the sintered ceramics.

Subsequently, 3g of the resulting powders were consolidated by SPS (515S model, Fuji Electronic Industrial Co., Ltd., Kanagawa, Japan) under vacuum conditions (down to 20 Pa) and sintering temperatures (TD) in the range 12001400 °C (heating rate of 100°C/min and holding time of 5 min) with a mechanical pressure of 40 MPa. Graphite die of 14.7 mm inner diameter, 30 mm height, and 30 mm outer diameter was used. SPS runs were carried out under temperaturecontrolled mode using an optical pyrometer (IR-AHS2, Chino, Tokyo, Japan) focused on the external surface of the die. After SPS, the bulk samples were polished on their top and bottom surfaces using progressively finer abrasive paper, to finally obtain 2 mm thick disks. The pellets were then post annealed for 3h at 1200°C (Nabertherm, mod. N60/ER, Germany) in an oxygenrich atmosphere to re-oxidize the samples and remove the carbon contamination arising from the graphite tools.

The absolute density values of the bulk specimens were determined by the Archimedes' method using distilled water as immersion medium. The crystalline phases in both the SHS powders and the SPS specimens were identified using an X-ray diffractometer (PANalytical X'pert the temperature to 450K which was measured using a K-type thermocouple. For the sake of reproducibility, each experiment was repeated at least twice.

Regarding the measurement of the piezoelectric properties, the samples were poled at a high voltage of about 2 kV (10 kV/cm) generated by a suitable supply (AET technologies, France) at 80°C under Nitrogen gas in a furnace (AET technologies, France). The applied field was kept during sample's cooling. The resonance (fr) and anti-resonance (fa) frequencies from which the electromechanical parameters were estimated, have been obtained using a network analyzer (Agilent technologies E5061B, USA).

Results

SHS powders of Zr substituted BaTiO3

The five BTZ compositions investigated in the present work all display a self-propagating character, i.e. the reaction occurred spontaneously after initiation without additional energy, regardless of the Zr content. The XRD patterns of the resulting SHS products, after being milled to powders, are shown in Fig. 1.

The hexagonal BaTiO3 phase is the only one (matched with the JCPDS card reference pattern mentioned in Fig. 1) detected by this analysis for x=0.05 and 0.10. Also, some peaks identified as c BaZrO3 and t BTZ appear in the XRD patterns of SHS products obtained for larger Zr content (x>0.10). Moreover, their intensity, and in particular that one of the main peak corresponding to the cubic BaZrO3 phase, progressively increase as the x value is augmented. It is important to note that in the powders synthesized by SHS, the desired single phase BTZ was not achieved (Fig. 1), rather an additional BaZrO3 phase was formed, which implies that at this stage

Zr is yet to be fully substituted in the B site of the BaTiO3 lattice.

In this regard, it is interesting to compare the composition and structure of BTZ powders obtained by SHS (Fig. obtained a perovskite cubic BTZ025 phase after 4h calcination at 1350℃ of powders subjected to repeated cycles of mixing and grinding.

The diverse outcomes resulting from the two synthesis methods can be motivated as follows. The formation of the metastable hexagonal phase, which is not detected in calcinated powders, could be explained by the peculiar characteristics (i.e. process rapidity, high temperatures, heating and, above all, cooling rates) of the SHS method, which evolves quite far from the equilibrium conditions. Therefore, metastable phases could be retained at room temperature. In addition, the intense milling treatments performed when considering the calcination method [START_REF] Mahesh | Effect of sintering temperature on the microstructure and electrical properties of zirconium doped barium titanate ceramics[END_REF][START_REF] Rout | Photoluminescence property of Ba(Zr0.25Ti0.75)O3 powders prepared by solid state reaction and polymeric precursor method[END_REF][START_REF] Badapanda | Structural refinement, optical and ferroelectric properties of microcrystalline Ba(Zr0.05Ti0.95)O3 perovskite[END_REF] certainly promote diffusion phenomena involved in the synthesis process, so that the formation of single-phase products is favored.

As mentioned in the Experimental section, in the present work the SHS products are mildly ball milled just to generate powders to be processed by SPS. Nonetheless, the effect of a more intense milling treatment on SHS powder and the resulting bulk BZT products characteristics will be the subject of a research work currently in progress.

Spark Plasma Sintering of SHS powders

The powders synthesized by SHS were then processed by SPS (5 min, 40 MPa, 100°C/min). The influence of the sintering temperature on the composition can be observed from room temperature XRD results depicted in (Fig. 2). Specifically, the XRD patterns of the bulk samples for the five BTZ systems are compared in Fig. 2(a)-(b) for two selected sintering temperatures 1200°C and 1400°C, respectively. It is important to note that XRD data for Zrsubstituted BaTiO3 powders sintered at TD=1300°C are very similar to those ones obtained at 1400°C, hence they are not reported here.

The XRD patterns evidence that, regardless of the Zr content, the h BaTiO3 originally present in SHS powders undergoes a gradual structural (phase) transformation during SPS.

Specifically, this effect is more pronounced as the dwell temperature is increased from 1200 to 1400°C. Nonetheless, when considering the composition corresponding to x=0.05, the original hexagonal phase is still significantly present in the sintered ceramic. A direct indication of the phase transformation observed when x increases is provided, for example, by the intensity of the h BaTiO3 peak which falls in the range 26°≤2θ ≤27°. Such peak decreases and tends to disappear, particularly when TD=1400°C. XRD analysis also evidences that, when the sintering process is performed at 1400°C, the intensity of the c BaZrO3 and the h BaTiO3 peaks are also markedly reduced with respect to the starting SHS powders (Fig. 1). It is thus possible to conclude that, depending on the x value, the content of t BTZ or the c BTZ phases is enhanced (Fig. 2(b)) during SPS when the sintering temperature is high enough. This is confirmed by the shift in 2θ diffraction peak at about 31° towards a lower angle (Fig. 2(c)). However, a residual presence of c BaZrO3 is still detected by XRD. In the literature, the tetragonal phase is generally reported for x ≤ 0.08

Post-annealing of SPS samples

From the results described in section 3.2, it is found that the bulk specimens obtained by S1) revealed that, after the post-annealing process, the sintered product obtained by SPS at TD= 1200°C is mainly composed of the P4mm tetragonal phase (about 87 wt. %) with a small amount of the cubic Pm-3m phase (approximately 13 wt.%). As the sintering temperature was augmented to 1400°C, the content of the cubic phase in the postannealed ceramic increased at the expenses of the tetragonal one. Nonetheless, based on the Rietveld analysis, the BTZ005 sample obtained for TD= 1400°C still consists of a mixture of tetragonal P4mm (about 70 wt.%) and Pm-3m (about 30 wt.%) phases (see Supplementary Table S1). The phases' coexistence could arise from the segregation of Zr when high sintering temperature (1400°C) is applied on heterogeneous starting powders. As a result, ceramics containing Zr rich (cubic) and Zr poor (tetragonal) phases could be obtained [START_REF] Benaballah | Linking large piezoelectric coefficients to highly flexible polarization of lead free BaTiO3-CaTiO3-BaZrO3 ceramics[END_REF].

XRD data of post-annealed SPS samples corresponding to x=0.25 are shown in Fig. 3(c).

Residual BaZrO3 is reduced during post heat treatment in air, to finally provide single-phase c BTZ for ceramics processed by SPS at TD=1400°C, as confirmed by the Rietveld analysis (see Supplementary Table S1). It should be noted that the compositional changes observed when considering x=0.10, 0.15, and 0.20, not reported here for the sake of brevity, fall within those ones described above in the case of x=0.05 and 0.25.

Another effect observed after the post-annealing treatment is the color change of SPS samples. Indeed, as shown in Fig. 4a, the resulting disks are dark, after the sintering process.

Regarding the latter finding, it should be noted that the dark color in barium titanate-as well as other oxide-based ceramics is general observed when the related synthesis and/or sintering processes are conducted under reducing environments. In the present work, both the SHS and SPS routes were performed under such condition. As far as the SHS powders are concerned, it was recently shown that their dark color is due to the fact that the synthesized products are oxygen deficient [START_REF] Ezealigo | Annealing effects on the structural and optical properties of undoped and Zr-doped Ba titanate prepared by selfpropagating high-temperature synthesis[END_REF]. In addition, a heat-treatment of these powders in air at increasing temperatures (up to 1200 °C) was found to determine a gradual color change from the initial dark to light yellow [START_REF] Ezealigo | Annealing effects on the structural and optical properties of undoped and Zr-doped Ba titanate prepared by selfpropagating high-temperature synthesis[END_REF]. The same motivation is also valid to justify the dark color of bulk BTZ samples obtained in this work by SPS, since the adopted reducing conditions (vacuum) likely lead to the formation of oxygen vacancy. Accordingly, in order to restore the expected BTZ stoichiometry and also make the obtained ceramics insulating, a post-annealing treatment in air was carried out. The latter process performed at various temperature/time conditions evidences that, at relatively lower temperatures (8001000°C), the samples exhibited several cracks on their surface, and the transformation from dark to white color was not obtained. In contrast, such drawback was avoided, or strongly limited, when the annealing process was conducted for 3 h at 1200°C. The optical images of the various BTZ samples correspondingly obtained are shown in Fig. 4b. The annealing process induces a significant color change from dark to white-grey in all BTZ ceramics. Thus, a post-annealing at 1200°C for 3h appears as the optimal condition to be adopted. BTZ005 sintered at 1200°C exhibits an inhomogeneous microstructure with several dense zones made of grains with a melted aspect that make an accurate evaluation of the corresponding grain size difficult (Fig. 5a). These areas are still present in ceramics sintered at 1400°C, while the microstructure is globally more homogeneous with well-defined grains and grain boundaries (Fig. 5b). The observed transgranular fracture mode reflects weak grain boundaries. As far as the more homogeneous region of the BTZ005 sample sintered at 1400°C is concerned, grain size estimated using Image J software [START_REF] Schneider | NIH Image to Image J: 25 years of image analysis[END_REF] is in the range 24 µm, so that no exaggerated grain growth is observed despite the high sintering temperature. The microstructure of BTZ025 ceramics sintered at 1200°C is characterized by a significant level of porosity and the presence of cracks (Fig. 5c). This agrees with the low value of density obtained (Fig. S1). Similar to BTZ005, when the sintering temperature is increased up to 1400°C, the microstructure becomes denser, in good agreement with the values reported in Fig. S1. The fracture remains transgranular and equiaxed grains in the size range 13 µm are observed. As previously mentioned for the BTZ005 product sintered at 1400°C, the corresponding grain size values were also estimated by considering the more homogeneous BTZ025 sample region where grain boundaries can be clearly identified.

Microstructural characterizations

SEM

Dielectric study

Effect of sintering temperature on dielectric behavior

The variation of the dielectric permittivity as a function of temperature for the BTZ ceramics sintered at different TD and post-annealed for 3h at 1200ºC are shown in Fig. 6.

For a fixed composition, both the permittivity values and the transition temperatures are affected by the sintering temperature. Higher TD leads to a decrease of the temperature corresponding to the maximum of permittivity value (Tm) as well as an increase in permittivity levels, especially in the vicinity of the transition (Fig. 7; Table S2 (Suppl. Material)). In particular, it is worth noting that, for the composition corresponding to x= 0.05, a sintering temperature increase from 1200°C to 1400°C also induces a change from a classical ferroelectric to a more diffuse phase transition.

A decrease of the Curie Temperature and an increase of the Rhombohedral (R)Orthorhombic (O)

[RO] and Orthorhombic (O)Tetragonal (T) [OT] phase transitions were correspondingly observed. When the Zr content is increased, the transition temperature is decreased, as expected, and for ceramics sintered at temperatures higher than 1200°C, the ferroelectric behavior changes from classical (x=0.05) to a diffuse mode (x>0.10). The dielectric study as a function of temperature (Fig. 6) confirms the diffuse (not relaxor) behavior of the BTZ ceramics obtained by SHS-SPS, due to the absence shift in Tm as a function of frequency for 0.05≤x<0.20 (Fig. S2 (Suppl. Material)). Whatever the Zr content and the sintering temperature, the dielectric loss at 1 kHz remains stable and mostly lower than 8%. The comparison of dielectric behavior with regards to TD for each Zr composition is shown in Fig. S3 (Suppl. Material).

To investigate the behavior in the paraelectric region, the Curie constant, C, was estimated from the Curie-Weiss law:

) ( o T T C    ; (T>Tc) ( 2
)
where Tc is the Curie point (transition temperature).

The inverse of the dielectric permittivity (1/) plotted as a function of temperature was used to estimate the Curie-Weiss temperature (To) and the Curie constant Fig. 8 and Table S3 (Suppl.

Material).

Regarding the investigated range of Zr content, the Curie constant decreases monotonically with an increase of the x value for a given TD. Also, the Curie-Weiss temperature was found to decrease with TD and Zr content.

The modified Curie-Weiss (C-W) plot can be used to describe the deviation from linearity due to the diffuse character of the transition. The modified Curie-Weiss equation is given by:

C T T m m    ) ( 1 1    ; (T>Tm) ( 3
)
where  is the real permittivity, m is the maximum dielectric constant, Tm is the temperature at the maximum dielectric permittivity,  is the diffuseness coefficient. For normal ferroelectrics =1, while for relaxor =2.

The modified Curie-Weiss plots and the corresponding data are provided in Fig. S4 and Table S3 (Suppl. Material), respectively.

To illustrate the change of behavior from classical to diffuse one as a function of the Zr content, the variation of Tc (Tm) and diffuseness coefficient are presented in Fig. 9 for ceramics sintered at 1400°C.

Piezoelectric study

In Fig. 10 the results of the piezoelectric study of BTZ are presented. The piezoelectric charge coefficient (d31) and the planar electromechanical coupling factor (kp) were significantly influenced by the SPS temperature (TD). All samples show good mechanical quality factor (Qm), consistent with the low dielectric loss (see Supplementary

          2 2 2 2 1 r a a m r m f f f C Z f Q  (4) a r a p f f f k 2 2   , ( 5
)           r a r a r a f f f f f f k 2 tan 2 1 2 31    , ( 6
) 2 2 11 4 1 w f s r E   , ( 7
) E s k d 11 31 31   , ( 8
)
where, in addition to the parameters (d31, kp, and Qm) previously defined, fr, fa, Zm, C, k31, , E s 11 , , , and w are resonance frequency, anti-resonance frequency, minimum impedance, Capacitance (in Farads), electromechanical coupling factor in the length extensional mode, elastic coefficient at constant electric field, dielectric permittivity at room temperature, density in Kg/m 3 , and width of the ceramics, respectively.

Discussion

Structural analysis of BTZ ceramics: from SHS powders to SPS and post-annealed samples. Evolution of their composition

The presence of the high-temperature stable h BaTiO3 phase in the SHS product with a small contribution of peaks attributed to t BTZ is likely ascribed to the severe processing conditions (high temperature, heating, and cooling rates) inherent to the synthesis process as well as to the reducing environment (Argon gas) correspondingly adopted. The additional c BaZrO3 phase is also detected at Zr content ≥ 15 wt.%, i.e. (BTZ015, BTZ020, and BTZ025), proving that a solid solution has not formed between the BaTiO3 and BaZrO3 phases, while a co-existence of the individual phases is obtained. When x≤ 0.10, no BaZrO3 phase was detected due to the low concentration of Zr. It should be noted that the detailed quantitative analysis of the XRD pattern (Rietveld method) for the BTZ015 powder also evidenced the presence of a small amount of c

BaTiO3 along with traces of Ba2TiO4 [START_REF] Ezealigo | Annealing effects on the structural and optical properties of undoped and Zr-doped Ba titanate prepared by selfpropagating high-temperature synthesis[END_REF]. Nonetheless, due to the relatively low intensity of the corresponding peaks, these two phases cannot be unequivocally identified in the XRD patterns shown in Fig. 1, so that they are not considered in the present work.

After SPS, a decrease in the h BaTiO3 peak intensity is observed with respect to the starting SHS powders. Moreover, according to the Zr content, the h BaTiO3 phase is gradually transformed into t BTZ or c BTZ, as the TD value is progressively increased. The formation of c BTZ phase is favored by higher Zr amounts and when the sintering temperature is set to 1400ºC (Fig. 2b-2c).

Twofold effects are noticeable for both the highest Zr content (i.e x=0.25) and sintering temperature (i.e. TD=1400°C), respectively: the formation of the c BTZ phase concomitant with the disappearance of t BTZ. In parallel, the secondary c BaZrO3 phase (c BZ) decreases progressively as the sintering temperature is increased from 1200 to 1400°C. The overall XRD study shows that the targeted BaTi(1-x)ZrxO3 solid solution can only be obtained by SHS-SPS under specific composition and temperature conditions. A high-temperature treatment of the SPS product is required to obtain the targeted single-phase BTZ.

The decisive role of the post-annealing treatment on the evolution of the phases present in SPS ceramics is emphasized here. Beyond its re-oxidizing role, post-annealing, when performed at sufficiently high temperature, allows for the elimination of the h BaTiO3 phase still present in some of the SPS ceramics. However, it should be noted that, during the post-annealing process, the hexagonal and tetragonal phases are partially or fully converted into the cubic one (Supplementary Table S1). This finding agrees with the work of Jha and Jha Besides, from the secondary c BaZrO3 phase evolution, which depends on the thermal cycle, it is clear that the procedure to eliminate such unwanted secondary phase is to combine the SPS process (TD=1400°C) with a post-thermal treatment carried out at 1200°C or higher temperatures (Fig. 3).

In addition, whatever the Zr content, the post-annealing must be performed at least at 1200°C to efficiently re-oxidize the samples. It is worth noting that an annealing duration of 3 h is not only efficient enough, but also much shorter than the one reported in the literature [23, 27].

Dielectric behavior: Effect of sintering temperature and composition

For a fixed Zr content (x=0.05), a normal ferroelectric behavior is observed. However, for TD =1400ºC, the OrthorhombicTetragonal [OT] phase transition becomes less marked compared with ceramics obtained at TD ≤1300ºC, and a broad transition is observed at Tc. This effect can be correlated with the XRD analysis that shows a transition, albeit uncomplete, from t BTZ to c BTZ, when increasing the sintering temperature. Generally, all BTZ ceramics compositions sintered at TD =1200ºC, showed a normal ferroelectric behavior except for the x=0.25 composition. In the latter case, the very diffuse phase transition associated with low permittivity values is correlated with the low value of density obtained at 1200°C. The microstructural investigation of BTZ025 sintered at 1200°C confirms a poorly densified ceramic and the presence of cracks. It is worth noting that no significant shift in TO-T and Tc was observed from x= 0.05 to x= 0.20, thus demonstrating that a sintering temperature of 1200°C is not sufficient to induce the Zr-Ti substitution.

When sintering is performed at 1400°C, a continuous decrease of Tc is observed when the composition was varied from x=0.10 to x=0.25 and the transition becomes more and more diffuse.

A relaxor behavior starts to be detected for x=0.25 (Tm = 8 from 100 Hz to 500 kHz), for which c BTZ is the only phase observed on the room temperature XRD pattern (Fig. 3(c)).

As a result, a change from classical to diffuse ferroelectric behavior in SHS-SPS ceramics can be induced by increasing not only the Zr content but also the sintering temperature at a fixed composition. Considering the microstructural analysis, such dielectric behavior is not correlated to the microstructure, while, as expected, depends rather on compositional effects at the atomic scale.

High diffusivity parameter (γ) was observed at low Zr content. The reason for this finding could be related to the coexistence of tetragonal and cubic phases seen in Fig. 3

Piezoelectric study

Both the composition and the sintering temperature have an impact on the piezoelectric performances. The piezoelectric charge coefficient decreases with increasing x values (Fig. 10(a)), which confirms that high Zr content does not favor the obtainment of good piezoelectric properties.

This result is effectively in agreement with the improved piezoelectric properties reported for BTZ ceramics with x ≤0.08 [44, 46 47, 49]. In addition, our study also evidenced that, when operating under optimal SPS temperature (TD) conditions to obtain higher values of the dielectric permittivity, the piezoelectric properties are correspondingly enhanced, which is in line with

Conclusion

BTZ ceramics have gained a high interest due to the possibility to tailor their ferroelectric properties through a proper composition control. Different BTZ ceramics are produced in this work using a novel approach based on the combination of the fast SHS and SPS processes, followed by an air-thermal treatment of relatively short duration (3 h), compared to the annealing time (10 h) typically considered in the literature. Here, the possibility of tuning the dielectric response (from classical to diffuse phase transition or relaxor) by varying, for a given composition, the sintering temperature, was clearly evidenced. In this regard, it should be noted that, to the best of our knowledge, no study was reported so far in the literature on the influence of the SPS temperature on the dielectric behavior of BTZ ceramics, regardless of the method used for powder synthesis.

Specifically, it was observed that the TD value and the Zr content significantly impact the structure, microstructure, dielectric, and piezoelectric properties of the SHS-SPS prepared ceramics. The BaZrO3 impurity phase formed during SHS, and still present after SPS, was eliminated thanks to the post-annealing treatment of the sintered ceramics. Increasing the sintering temperature above 1200°C leads to improved density, more homogeneous microstructure, as well as enhanced dielectric permittivity. The ceramic containing 5 at.% of Zr sintered at TD=1300ºC

showed the best piezoelectric properties. Better performances are expected if a single-phase ceramic is obtained after SPS and the microstructure of the final ceramics is suitably refined. Work is underway along this direction.

[ 7 - 9 ]

 79 . The BZT potential for application in tunable microwave devices has been demonstrated [10].

[ 8 , 34 ]

 834 while the cubic and/or rhombohedral phases are found for x≥ 0.10 [18, 20, 35-36].

  SPS still contain undesired secondary phases, particularly h BaTiO3 and/or c BaZrO3, whatever the Zr content. The effect of a post-heat treatment in air, conducted for 3 h at 1200°C, on the composition of the BTZ ceramics produced by SPS at TD= 1200, 1300, and 1400°C, was then investigated. The room temperature XRD patterns of the corresponding post-annealed samples are shown in Fig.3(a)-3(b), for the compositions corresponding to x=0.05 and 0.25, i.e. the lowest and highest Zr content examined in this work.Let us consider first the ceramics obtained with the lowest Zr content (Fig.3(a)-3(b)). The post-annealing treatment appears to remove the h BaTiO3 phase initially present in SPS sintered products (Fig.2(a)-2(b)). Also, t BTZ is the major phase detected by XRD at room temperature in post-thermal treated SPS ceramics obtained at TD=1200 and 1300°C. In particular, the Rietveld analysis (see Supplementary Table

  images of the fracture surfaces of BTZ005 and BTZ025 ceramics obtained by SPS at TD=1200 and 1400° and annealed at 1200℃ for 3h are shown in Fig.5(a)-5(d). The densities of post-annealed ceramics are reported in Fig.S1(Suppl. material).

  [42] focused on BaZr0.05Ti0.95O3 ceramics obtained by conventional technique, where the sintering process conducted at higher temperatures caused the tetragonal to cubic phase transformation. The formation of the c BTZ phase for Zr content corresponding to x=0.100.25 is also consistent with similar results reported in the literature for BTZ ceramics obtained by conventional solid-state reactions [36].

  results reported in the literature for BTZ ceramics produced by alternative techniques [49-50]. When considering BTZ005, sintering at 1200°C results in moderate density values and poor quality of grain boundaries, that are not convenient for improving piezoelectric properties. On the other hand, BTZ005 sintered at 1300°C exhibits the optimal piezoelectric properties, which is in agreement with the literature [39-40, 51]. However, the obtained piezoelectric coefficient remains relatively low when compared to the state of the art [23, 47, 51]. Improvements based on microstructure (density and grain size) and on the poling conditions would allow reaching better performances.

FiguresFigure 1 :

 1 Figures

Figure 2 :

 2 Figure 2: XRD patterns of BTZ ceramics obtained by SPS at (a) TD= 1200ºC, (b) TD= 1400°C, (c)

Figure 3 :

 3 Figure 3: XRD patterns of (a)-(b) BTZ005, and (c) BTZ025 ceramics obtained by SPS (TD=1200,

Figure 4 :

 4 Figure 4: Images of BaTi(1-x)ZrxO3 (x=0.050.25) ceramics sintered by SPS at 1400°C before (a)

Figure 5 :

 5 Figure 5: Fractured surfaces of BTZ ceramics obtained by SPS and subsequently annealed at

Figure 6 :Figure 7 :

 67 Figure 6: Dielectric permittivity and loss (1kHz) of BaTi(1-x)ZrxO3 (x=0.050.25) ceramics: (a)

Figure 8 :

 8 Figure 8: Curie-Weiss plots of BaTi(1-x)ZrxO3 (x=0.05 and 0.25) ceramics sintered at 1400°C and

Figure 9 :

 9 Figure 9: Plot of Tc (Tm) at 1 kHz and diffuseness coefficient (γ) as a function of Zr content for

Figure 10 :

 10 Figure 10: Piezoelectric properties of BTZ ceramics annealed at 1200ºC for 3h. (a) Piezoelectric

Figure S1 :

 S1 Figure S1: Density of post-annealed BTZ ceramics (TD=12001400℃) as a function of Zr

Figure S2 :

 S2 Figure S2: Curie temperature as a function of frequency for different BTZ compositions (TD

Fig. S3 :

 S3 Fig. S3: Dielectric constant and loss of BaZrxTi(1-x)O3 (x=0.05-0.25) ceramics at 1kHz. (O=

Fig. S5 :

 S5 Fig. S5: Impedance as a function of frequency, showing the resonance (fr) and antiresonance (fa)

31].

  The relative content of the different phases was determined taking advantage of a Rietveld analytical procedure [

	Fracture surfaces of BTZ ceramics were examined by high resolution scanning electron
	microscopy (HRSEM) (mod. S4000, Hitachi, Tokyo, Japan) equipped with a UltraDry EDS
	Detector (Thermo Fisher Scientific, Waltham, MA, USA). Microstructural analysis (grain size,
	porosity) were conducted on SEM images using the open source software Image J (version 1.54a
	for Windows, 64 bit, National Institutes of Health, Bethesda, MD, USA).
	The dielectric measurements were carried out after depositing by sputtering gold electrodes
	on both sides of the dense ceramics (Technics Hummer JR, USA). The LCR meter (Wayne Kerr
	4300, London) recorded the capacitance and imaginary permittivity (i) values from which the data
	acquisition program estimated the dielectric permittivity and dielectric loss. Liquid nitrogen was
	used to decrease the set-up temperature to about 85 K and a furnace (Eurotherm 3508, UK) raised

Pro diffractometer, Netherlands; operating at 40 kV and 40 mA, using the X'Celerator detector) equipped with a Ni filtered CuKα radiation (λ = 1.5405 Å).

Table S4

 S4

	). A graph showing impedance
	as a function of frequency, from which resonance and antiresonance frequencies can be obtained,
	is reported in Fig S5 (Supp. Material) for the case of BTZ005 ceramics sintered at 1300℃.

BTZ025 samples could not be poled due to high current leakage arising from their low density. Hence, no piezoelectric data were obtained for this system. The following equations were used to obtain the electromechanical parameters [41]:

  (b). However, at high Zr content, the obtained γ parameter is in the range of the values reported in the literature,

	which generally depend on the adopted synthesis process [35, 43-44]. In addition, similar to the
	works by Mahesh et al. [20] and Cai et al. [45], higher sintering temperature induces diffuse
	transition and increases the γ parameter (see Supplementary Table S3). The Tc (Tm) values obtained
	for the various compositions are in good agreement with the literature for BTZ ceramics obtained
	by other methods [46-48].

Table S1 :

 S1 Phases and quantitative phase analysis results (Rietveld procedure) of post annealed

	BTZ005 (TD=1200°C and TD=1400°C) and BTZ025 (TD=1400°C) samples.
		BTZ005 (TD=1200°C)
	Phase	a	c	Wt.
		(Å)	(Å)	(%)
	Tetragonal P4mm (BaTiO3)	3.998	4.022	87.5
	Cubic Pm-3m (BaTiO3)	4.056	/	12.5
		BTZ005 (TD=1400°C)
	Phase	a	c	Wt.
		(Å)	(Å)	(%)
	Tetragonal P4mm (BaTiO3)	3.995	4.024	70.3
	Cubic Pm-3m (BaTiO3)	4.017	/	29.7
		BTZ025 (TD=1400°C)
	Phase	a	c	Wt.
		(Å)	(Å)	(%)
	Cubic Pm-3m (BaTiO3)	4.040	/	100

Table S2 :

 S2 Dielectric characteristics at 1kHz of BaZrxTi(1-x)O3 (x=0.050.25) ceramics. TD values are indicated below samples ID.

	Sample	Dielectric	Dielectric	tan δ at	tan δ at
	ID	constant at room	constant at Curie	room	Curie
		temperature	temperature	temperature	temperature
	BTZ005	2142±73	3521±56	0.041	0.031
	1200℃
	BTZ005	2524±57	4518±228	0.039	0.016
	1300℃
	BTZ005	2749±63	5195s±172	0.044	0.020
	1400℃
	BTZ010	1954±15	3269±218	0.044	0.021
	1200℃
	BTZ010	2199±42	3775±82	0.048	0.024
	1300℃
	BTZ010	1903±256	3145±465	0.053	0.015
	1400℃
	BTZ015	1606±67	2525±129	0.043	0.023
	1200℃
	BTZ015	2234±45	3541±61	0.046	0.027
	1300℃
	BTZ015	2503±26	4721±181	0.059	0.030
	1400℃
	BTZ020	1703±75	2510±182	0.031	0.020
	1200℃
	BTZ020	2215±86	3057±150	0.039	0.028
	1300℃
	BTZ020	2514±83	4216±314	0.058	0.032
	1400℃
	BTZ025	635±17	799±21	0.067	0.078
	1200℃
	BTZ025	1866±127	2097±57	0.055	0.048
	1300℃
	BTZ025	2579±480	3128±525	0.063	0.052
	1400℃

Table S3 :

 S3 Curie temperature, Curie-Weiss temperature, temperature of deviation from Curie-Weiss law, Curie constant, and diffuseness coefficient values of BaZrxTi(1-x)O3 ceramics. TD values are indicated below samples ID.

	Sample	Tc (Tm) (K)	To (K)	Tcw (K)	Curie constant (K -1 )	
	ID				x10 5
	BTZ005	386	361	399	1.22	1.42
	1200℃
	BTZ005	386	370	401	1.39	1.69
	1300℃
	BTZ005	378	368	401	1.23	1.82
	1400℃
	BTZ010	390	373	408	1.02	1.71
	1200℃
	BTZ010	369	362	403	1.31	1.83
	1300℃
	BTZ010	366	360	403	1.11	1.85
	1400℃
	BTZ015	393	367	415	0.98	1.81
	1200℃
	BTZ015	370	374	410	1.12	1.86
	1300℃
	BTZ015	350	361	402	1.30	1.83
	1400℃
	BTZ020	396	371	417	0.97	1.66
	1200℃
	BTZ020	347	360	406	0.94	1.86
	1300℃
	BTZ020	341	356	397	1.14	1.9
	1400℃
	BTZ025	374	322	421	0.69	1.92
	1200℃
	BTZ025	338	337	401	0.94	1.93
	1300℃
	BTZ025	331	314	360	1.04	1.95
	1400℃

Table S4 :

 S4 Piezoelectric characteristics and density values of BaZrxTi(1-x)O3 ceramics prepared bySHS and sintered by SPS (TD values are indicated below samples ID). Literature data are also reported for the sake of comparison

	Sample	Density (g/cm 3 )	Qm	kp (%)	-d31 (pC/N)	Ref
	ID
	BTZ005	5.43	181	11.5±0.5 48.5±4.5	This work
	1200℃
	BTZ005	5.89	81	20±2	87.5±2	This work
	1300℃
	BTZ005	5.57	79	17.5±0.5 84±1	This work
	1400℃
	BTZ010	5.51	137	14	58±5	This work
	1200℃
	BTZ010	5.64	181	13±1	56.5±3.5	This work
	1300℃
	BTZ010	5.47	57	19±1	79.5±3.5	This work
	1400℃
	BTZ015	5.65	130	12±2	38.5±2.5	This work
	1200℃
	BTZ015	5.77	196	11.5±0.5 41.5±1.5	This work
	1300℃
	BTZ015	5.78	138	13.5±05	56±1	This work
	1400℃
	BTZ020	5.38	193	10.5±0.5 30±1	This work
	1200℃
	BTZ020	5.50	128	12.5±1.5 43±6	This work
	1300℃
	BTZ020	5.84	178	11.5±0.5 37.5±1.5	This work
	1400℃
	BTZ020	5.83	312	25	126	[23]
	1100℃
	BTZ005	5.80	-	-	64	[39]
	1500℃
	BTZ004	5.86	180	42.1	90	[40]
	1450℃

Acknowledgments

One of the authors (B.N.E) performed her activity in the framework of the International PhD in Innovation Sciences and Technologies at the University of Cagliari and is also grateful for the ERASMUS Placedoc program for allowing mobility to France. The valuable contribution of Dr. S. Garroni and Dr. A. Iacomini (University of Sassari, Italy) for Rietveld's analysis and useful discussion is gratefully acknowledged.

Supplementary Material

Influence of the Spark Plasma Sintering temperature on the structure and dielectric properties of BaTi(1-x)ZrxO3 ceramics Blessing N. Ezealigo 1 , Roberto Orrù


Luis Alemán Castañeda

Miguel A Alonso
email: miguel.alonso@rochester.edu

Luis A Alemán-Castañeda

Reflector design and study via conformal mappings

Keywords: Reflector Design, Illumination optics, Reflector design, Reflectors, Conformal mappings

come

Reflector design and study via conformal mappings

INTRODUCTION

The field of illumination optics has seen great progress [START_REF] Winston | Selected Papers on Nonimaging Optics[END_REF][START_REF] Winston | Nonimaging optics: a tutorial[END_REF][START_REF] Wu | Design of freeform illumination optics[END_REF] in both numerical and analytical implementations of the irradiance and/or intensity calculations given any source and a target. Early work designed reflectors through source-target maps, i.e. the correspondence between the ray angles from the source to the reflector and the target direction, and through parameters describing the system, such as the acceptance angle. [START_REF] Winston | Selected Papers on Nonimaging Optics[END_REF] Modern techniques have shifted to assessing/obtaining a given irradiance or radiant intensity distributions. [START_REF] Miñano | New method of design of nonimaging concentrators[END_REF][START_REF] Ries | Tailored freeform optical surfaces[END_REF][START_REF] Oliker | Mathematical aspects of design of beam shaping surfaces in geometrical optics[END_REF][START_REF] Canavesi | Target flux estimation by calculating intersections between neighboring conic reflector patches[END_REF][START_REF] Wester | Designing optical free-form surfaces for extended sources[END_REF][START_REF] Hirst | Fast, deterministic computation of irradiance values using a single extended source in 3d[END_REF] The general 3D problem is governed by a nonlinear second-order Monge-Ampere equation, that in many cases has no analytical solution so numerical methods must be used. However, when there is either transnational or rotational symmetry, the system reduces to a 2D problem and analytic methods can then be used, e.g, the so-called "edge-ray", [START_REF] Gordon | Tailored edge-ray concentrators as ideal second stages for fresnel reflectors[END_REF][START_REF] Davies | Edge-ray principle of nonimaging optics[END_REF][START_REF] Ries | Tailored edge-ray reflectors for illumination[END_REF][START_REF] Rabl | Reflector design for illumination with extended sources: the basic solutions[END_REF][START_REF] Ong | Tailored edge-ray designs for illumination with tubular sources[END_REF] and"flow-line" [START_REF] Winston | Nonimaging optics: a tutorial[END_REF][START_REF] Jiang | Thermodynamic origin of nonimaging optics[END_REF][START_REF] Jiang | Flow line asymmetric nonimaging concentrating optics[END_REF] methods. The approach described here falls in this category.

In many general schemes, point sources are used for designing reflectors with extended sources, such as the aforementioned edge-ray method. In addition, the supporting paraboloids [START_REF] Kochengin | Determination of reflector surfaces from near-field scattering data[END_REF][START_REF] Canavesi | Direct calculation algorithm for two-dimensional reflector design[END_REF] and linear programming [START_REF] Canavesi | Target flux estimation by calculating intersections between neighboring conic reflector patches[END_REF][START_REF] Canavesi | Observations on the linear programming formulation of the single reflector design problem[END_REF] methods have been successful in generating the desired intensity distributions with good numerical performance, even when optimizing for extended sources. In these two methods, the reflector is subdivided into patches of paraboloids so that, based on the properties of conics, a far-field intensity distribution is tailored. Although these properties have been used extensively in non-imaging optics, the connection between their shape and the direction of light after reflection has been restricted mainly to piecewise parabolic reflectors. In a recent work, [START_REF] Alemán-Castañeda | Study of reflectors for illumination via conformal maps[END_REF] we provided using conformal mappings a conceptual connection valid also in the continuous, smooth case, hence yielding a simple, visual tool for the assessment of any reflector with translational or rotational symmetry.

In this manuscript we will first motivate and introduce the conformal map studying a simple piecewise parabolic reflector. Then, in order to illustrate the simplicity and usefulness of the transformation proposed, we present a series of examples of piecewise parabolic and continuous smooth reflectors. Furthermore, we display how by employing this map, a piecewise parabolic reflector can be easily designed. Lastly, we show how extended sources can be also studied via translations.

THE QUADRATIC CONFORMAL MAP

Let us consider a parabolic reflector of focal length f whose focus is at the origin and whose axis is at an angle β with respect the x-axis. If a point source were placed at its focus, light coming from the source would be redirected parallel to the parabola's axis. Therefore, we would like to find a map that transforms such parabolas into straight lines whose directions are related to that of the redirected light. To find this map, it is useful to write the general expressions of both the parabola and a straight line with impact parameter (distance to the origin) d and angle β:

parabola: r(θ) = f csc 2 θ -β 2 , line: r( θ) = d csc θ -β , (1)
where an overbar is used to label quantities that refer to the straight line, to distinguish them from those of the parabola. It follows that the transformations

θ = 2 θ, r = r2 , (2)
provide the desired map that converts a line with parameters ( √ f , β/2) into a parabola with parameters (f, β). In other words, if we express the reflector's 2D coordinates (x, y) as a complex number z = x + iy = r exp (iθ), the conformal map √ z = √ r exp (iθ/2) = z = x + iȳ exactly transforms parabolas with focus at the origin into straight lines, and vice-versa for z2 . This map has been used in other physical contexts, e.g. to describe the caustics of structured Gaussian beams. [START_REF] Alonso | Ray-optical poincare sphere for structured gaussian beams[END_REF] Furthermore, this map has also been used to connect Keplerian motion whose orbits are ellipses or hyperbolas centered at one of the foci, with the classical motion in a symmetric 2D quadratic (attractive or repulsive) potential whose orbits are ellipses or hyperbolas centered at the potential's center. [START_REF] Rockmore | On the isotropic oscillator and the hydrogenic atom in classical and quantum mechanics[END_REF][START_REF] Mittag | Conformal transformations and the application of complex variables in mechanics and quantum mechanics[END_REF][START_REF] Grant | Classical orbits in power-law potentials[END_REF] The case considered here of a parabola being mapped onto a straight line corresponds to the limiting case where the quadratic potential vanishes (free motion) and the Keplerian path is an escape/capture orbit.

We can then define an abstract light space with coordinates (x, ȳ) in which a light curve exists, in contrast with the real space with coordinates (x, y) where the reflector exists. These spaces are linked by the map just described. As shown in Fig. 1, a reflector composed of confocal, parabolic, contiguous segments -as in the supporting paraboloids method-is transformed into a light curve which is simply a chain of straight line segments. The angle that each segment makes with the x-axis is just half the angle of the reflected light with respect to the x-axis, and the angle subtended from the origin by each segment (not its length) is proportional to the corresponding redirected power. There is a subtlety to be discussed: while the conformal map z2 is well defined, the inverse map √ z is not. The problem arises due to the modular nature of the polar angle which makes √ z to be multi-valued. To avoid this, a branch cut must be chosen, that means, selecting a range for θ. For example, when choosing θ ∈ [0, 2π), the positive real line becomes the branch cut (where the discontinuity happens). It is convenient for continuity of the light curve to choose the branch cut as the average target direction. However, the choice of branch changes neither the direction nor the subtended angles and hence is not critical.

PIECEWISE REFLECTORS

Reflector Design

When designing a reflector, one must choose the relationship between the ray(s) angle coming from the source toward the reflector, α, and the target direction, β, i.e., select the so-called source-target mapping. Both angles are measured with respect the optic axis which we will generally select as our x axis in our 2D representation.

It is important to mention that a source-target mapping doesn't determine a unique reflector, but a family of solutions. To have a unique solution an extra parameter is needed: the distance with respect the source (origin) at a given point.

The inverse map z = z2 can be used to design reflectors from their respective light curves without explicitly tracing any rays. For a piecewise parabolic reflector the procedure is the following:

1. Choose the source range.

2. In light space this defines a range subtending just half the angle, which must be subdivided into sections depending on the fraction of energy desired for each, taking into account the source intensity distribution (e.g. if it is uniform or not).

3. In each section, draw straight-line segments oriented at half the desired target angle -this becomes the light curve.

4. Obtain the reflector by applying the map z = z2 .

Note that connecting at their ends the light curve segments leads to a continuous reflector, while disconnected segments lead to Fresnel-type reflectors. In Fig. 2 we construct a continuous reflector following the former procedure.

Reflector assessment

There are two main basic geometries of reflectors: convergent (where rays cross after reflection), and divergent (where they do not cross). Depending on the system's specifications, each geometry has advantages and disadvantages. The proposed conformal map reveals these two geometries directly, as illustrated for the two piecewiseparabolic continuous reflectors shown in Fig. 3 (with equal target and source ranges). A concave/convex light curve towards the origin implies a convergent/divergent reflector, refer to Appendix B for the proof and further discussion. Note that light curves consisting of straight segments in the same directions, each subtending the same angles, yield after the z = z2 map different reflectors with the same source and target ranges, not necessarily purely convergent or divergent, as in Fig. 2.

-1 Let us now consider discontinuous reflectors such as piecewise-parabolic Fresnel-type reflectors, commonly used for collimation when thin reflectors are preferred due to packaging or construction limitations (although they tend to be longer). The light curve is discontinuous, like the reflector. Additionally, since typically all reflector segments have a common holder, the light curve segments should also remain attached to the mapped holder. For example, if we consider that all segments start at a line oriented at θ c and with impact factor d, then the light curves all depart from a rectangular hyperbola (with perpendicular asymptotes) oriented at θ c /2 -π/4 and semi-axis √ d, into which the line is transformed, see Appendix C.

1 2 3 4 x -2 -1 1
On the other hand, for a collimator, which usually can be visualized as situated between two lines, its corresponding segments in light space lie between the two rectangular hyperbolas obtained from mapping them.

Figure 4 shows two reflectors and their respective light curves. All light curve segments are parallel but disconnected, i.e. their impact parameters differ. The later was expected since Fresnel-type collimators are constituted by confocal equally-oriented parabolas, but of different focal lengths. Lastly, note that while the light line segments follow a hyperbola (a convex curve), the geometry of the reflector is recovered from the angular change of the light curve: convergent if co-rotating with respect to the origin, divergent if counter-rotating, and collimating if constant. As Fig. 4 (b) shows, the light curve segments co-rotate, indicating the convergence of rays after reflection as seen in Fig. 4 (a).

CONTINUOUS SMOOTH REFLECTORS

This approach is valid not only for segmented piecewise-parabolic reflectors, but also for continuous, smooth reflectors, which can be regarded as the limit of an infinite number of small parabolic segments (refer to Appendix A for the rigorous proof that both the reflector and the light curve are connected by the proposed transformations). In fact, for these reflectors (or segments of) that are purely converging or diverging, a polar diagram of the resulting radiant intensity can be generated from the light curve (x, ȳ). Let these coordinates be parametrized by a variable τ . The length of a segment of this curve that spans a given infinitesimal range of directions is proportional to the local radius of curvature. This length, multiplied by an obliquity factor and divided by the distance to the origin, gives the infinitesimal angle subtended by this segment. The expression in polar coordinates is then given by

(I, θ) = ẋ2 + ẏ2 x2 + ȳ2 |x ẏ -ẋȳ| | ẋÿ -ẍ ẏ| , 2 arctan ẏ ẋ = | ż| 2 Im( ż z * ) |z| 2 Im( z ż * ) , 2 arg( ż) , (3)
where the dot denotes a derivative in τ . Note that I diverges at the light curve's inflection points, which correspond to far-field caustics. These points typically separate segments that are convergent (which form caustics between the reflector and the far field) and divergent (which form no real caustics).

In the following subsections, we will study two examples of continuous, smooth reflectors. The first consists in a comparison of circular reflectors used as collimators, in which we will use the intensity expression previously derived. The second illustrates how this method can be used also for extended sources via translations.

Study of Collimators

Consider a circular reflector of radius R, when used as a collimator at different distances from the source. This reflector is paraxially equivalent to an ideal parabolic collimator when the light source is placed half a radius away from its center (f = R/2). For comparison, the focal length of the parabola is set to f = R/2. Figure 5 shows (red curves) that the light curve in this case is considerably flat within the central region, mimicking the straight line for the ideal parabolic collimator (dashed line) within a range of source angles. Within ±45 • of the x-axis, (green dashed lines), this profile at d = R/2 has good performance paraxially as a collimator. Collimation degrades at higher angles, e.g. at ±90 • (red dashed lines). For a small translation, the collimation range can be extended to ≈ 67.5 • . For closer positions there is clear variation from divergent to convergent behavior, the crosses indicating the inflection points (far-field caustics). (c) Logarithmic radiant intensity diagrams. These become multivalued for non-purely convergent/divergent reflectors due to the appearance of far-field caustics (cusps on the plot).

This light curve is concave away from the central region, indicating convergent behavior. The convergence at the edges can be compensated by slightly reducing the distance between the source and the reflector (this can be understood as using defocus to balance spherical aberration) at the cost of introducing divergent behavior near the axis. This can be appreciated from the light curves in Fig. 5(b) for two locations closer than the collimating paraxial position. On the other hand, the behavior is purely convergent for circular reflectors further away from the paraxial collimating position, for which Eq. (3) provides a continuous intensity diagram, as shown in Fig. 5(c).

Extended Sources

Let us finish with an example corresponding to an extended source/target: the Compound Parabolic Concentrator (CPC), [START_REF] Winston | Light collection within the framework of geometrical optics *[END_REF][START_REF] Winston | Principles of solar concentrators of a novel design[END_REF] which achieves the ideal concentration limit. [START_REF] Winston | Nonimaging optics: a tutorial[END_REF][START_REF] Miñano | Effect of restricting the exit angle on the limit of concentration for cylindrical concentrators[END_REF] A CPC consists of a planar source/concentrator and two parabolic reflectors that redirect the light from/to the opposite end of the source/concentrator at π ∓ θ c , where θ c is the acceptance angle, i.e., the minimum angle from one of the corners of the source/concentrator to the upper end of the parabolic reflector (see Fig. 6). Due to the symmetry of the problem, only one of the reflectors needs to be considered.

The procedure is as follows: given the reflector shape, the origin is translated from the original focus along the extended source/concentrator, and the map z = √ z of the reflector is then applied to reveal the direction in which light is directed after reflection. Figure 6 shows the results for six points of the extended planar concentrator and an acceptance angle of θ c = π/3. The first point, corresponding to the opposite end of the concentrator, leads to a perfectly straight light line at θ c /2. As the point approaches the reflector, the direction angles change mostly for small source angles (small ȳ) to a divergent geometry, whereas for larger source angles (near the edge of the reflector) the light is directed at angles close to the acceptance angle. The direction for all the positions is bounded by θ c /2, which is achieved only at the initial position. This was to be expected, since the construction of the CPC can be derived using the edge-ray technique.

CONCLUSIONS

The conformal maps z2 and √ z provide a simplified understanding of the relation between the reflector's shape and the light direction distribution after reflection, for both segmented and continuous, smooth reflectors. Additionally, light space can be used to design reflectors starting from their light curves, which are particularly simple for piecewise parabolic reflectors. Furthermore, from this new graphic representation, general reflector characteristics can be perceived easily even without explicitly tracing rays. For example, convergent/divergent/collimating geometries are mapped onto the convexity/concavity/straightness of the light curve. Nevertheless, light curves do not fully replace rays. For instance, cases when rays hit a reflector more than once cannot be visualized directly.

The approach is based on point-like sources which is a useful approximation in many cases and is widely used by several already-known algorithms. However, it can be applied also to extended sources by treating each point as a point source and translating the reflector before applying the map. This method is currently limited to systems whose symmetries reduce the problem to 2D, meaning that skew rays cannot yet be considered. Future work could focus on a generalization to 3D. A first (not yet successful) attempt in this direction was the use of quaternions to generalize the complex map, leading to a 4D space from which a 3D subspace can be chosen. Another possible extension is to reflectors that generate desired irradiances at finite distances, for which the map should not only reveal the direction of light but also the distance at which it focuses. . Schematic construction of a curve given a correspondence between its tangent angle and its polar coordinate: (left) for a reflector with source-target map β(α), the tangent angle is equal to (α + β)/2; (right) for the light curve with polar angle ᾱ, the tangent angle is β. Notice that for the reflector, β is negative, i.e., light is going downwards.

APPENDIX B. REFLECTORS GEOMETRY REVEALED BY THE LIGHT CURVE

Let us consider a reflector whose source-target map is given by β(α), where α is the source angle, which in our case is the polar angle. The corresponding light curve's local direction is then given by β = β/2. The light curve, or some of its segments, are said to be concave/convex/straight if the angular derivative of its direction, ∂ β/∂ ᾱ, is positive/negative/zero. It follows from relation between the reflector and its light curve that

∂ β ∂ ᾱ = 1 2 ∂β ∂ ᾱ . (10)
Then, from the relation between the light space and the real space, remember α = 2ᾱ, we finally obtain

∂ β ∂ ᾱ = 1 2 ∂β ∂α ∂α ∂ ᾱ = ∂β ∂α . (11)
This was the missing link, since a reflector's geometry, or some of its segments', is considered converging/diverging/collimating if the angular derivative of its source-target mapping, ∂β/∂α, is positive/negative/zero.

APPENDIX C. TRANSFORMATION OF STRAIGHT LINES

A line after the conformal mapping √ z is transformed into a rectangular hyperbola, whose asymptotes make an angle of π/2. This can be understood intuitively: given that the light-curve is parallel to the reflected rays under √ z, we know the asymptotic reflection from a line (flat surface) will be parallel to it, i.e.,-π/2 and π/2 with respect to its normal, giving a total angle of π in the real space, but only half in the light space. Being more rigorous, consider a line in the real space with parameters (d, β) and its transformation,

r line (θ) = d csc(θ -β), rhyp. ( θ) = √ d csc(2 θ -β). (12)
Then, consider the canonical hyperbola given by x2 a 2 -

ȳ2 b 2 = 1, (13)
where a and b stand for its semi axis lengths. In the case of a rectangular hyperbola, a = b, hence, changing to polar coordinates and solving for r, r( θ) = a

√ sec 2θ. ( 14
)
If we wanted it to have an arbitrary orientation ϕ, we just need to let the angle be θ -ϕ, and so the final expression becomes r( θ) = a csc(2 θ -(2ϕ -π/2)), (15) meaning that the orientation of the rectangular hyperbola will be given by ϕ = β/2 + π/4.

APPENDIX D. FORMULAE OF CONIC COLLIMATORS

Lastly, it will be of use to derive the polar equation of a conic translated so that it is acting as a collimator. For example, in the case of a circle, this occurs when the translation was by half its radius, R/2. In general, we need to remember that the radius of curvature at the vertex of any conic is given by the semi latus rectum, , and so the translation required is given by /2, so

x -( /2) a 2 ± y 2 b 2 = 1, (16)
where, a and b stand for the semi axis lengths of the conic. Then, using the fact that = b 2 /a, and using carefully the eccentricity definition, as well as keeping the possible sign changes from hyperbola to ellipse, we have that ±b 2 = a 2 -c 2 =⇒ ±b 2 /a 2 = (1 -e 2 ), the equation for any conic becomes

) 17
Notice that the ± symbol disappears of the expression giving a unique formula, and more importantly, the semi latus rectum, , as well as the eccentricity, e, are preferred to express a conic since the conic constants a, b, and c can get undetermined in certain scenarios, e.g. for the parabola. Lastly, we need to convert to polar coordinates, obtaining

If we wanted it to have an arbitrary orientation φ, we just need to let the angle be θ -φ. The respective light curve can be obtained by doing θ → 2 θ and r → r2 .

Figure 1 .

 1 Figure 1. (a) Segmented reflector consisting of three confocal parabolic segments, each depicted in a different color. The limiting rays are shown as dashed lines. Each segment redirects the same amount of power from the source. (b) The corresponding light curve is constituted by three straight line segments at angles from the x-axis equal to half the reflected light's angles from the x-axis. Although the straight line segments' lengths are different, they subtend equal angles from the origin, which are in turn half the angle subtended by the parabolic segments.

Figure 2 .

 2 Figure 2. (a) The reflector is obtained after applying the z = z2 map. (b) Light curve construction given a source emitting uniformly and a source range α ∈ [90 • , 270 • ], i.e. from 45 • to 135 • in the light space. We want to redirect 10%, 20%, 30%, and 40% of light towards β = 0 • , 20 • , 40 • , and -20 • respectively, so we draw the corresponding straight-line segments oriented at half those target angles.

Figure 3 .

 3 Figure 3. Two reflectors composed of six parabolic segments (a),(c) and their light curves (b),(d). The reflectors have convergent (a) and divergent (c) geometry, but have equal source and target ranges. The light curves are concave/convex towards the origin for convergent/divergent reflectors.

Figure 4 .

 4 Figure 4. Segmented Fresnel-type reflectors: (a) convergent reflector with seven segments and (b) its light curve. (c) collimator, (i.e. neither divergent or convergent) with five segments and (d) its light curve. The baselines for the reflector segments map onto limiting rectangular hyperbolas for the light curves.

Figure 5 .

 5 Figure 5. (a) Circular reflector of radius R at five distances d from the source. (b) Their corresponding light curves.Within ±45 • of the x-axis, (green dashed lines), this profile at d = R/2 has good performance paraxially as a collimator. Collimation degrades at higher angles, e.g. at ±90 • (red dashed lines). For a small translation, the collimation range can be extended to ≈ 67.5 • . For closer positions there is clear variation from divergent to convergent behavior, the crosses indicating the inflection points (far-field caustics). (c) Logarithmic radiant intensity diagrams. These become multivalued for non-purely convergent/divergent reflectors due to the appearance of far-field caustics (cusps on the plot).

2

 2

Figure 6 .

 6 Figure 6. (a) A CPC reflector (θc = π/3, f = 1) that collimates light from the edge source point along π -θc, and six points along the extended source. (b) Light curves for these six source points. The gray-dashed lines , included for reference, indicate the directions for the acceptance angles θc and π -θc.

Figure 7

 7 Figure 7. Schematic construction of a curve given a correspondence between its tangent angle and its polar coordinate: (left) for a reflector with source-target map β(α), the tangent angle is equal to (α + β)/2; (right) for the light curve with polar angle ᾱ, the tangent angle is β. Notice that for the reflector, β is negative, i.e., light is going downwards.

( 1 - 2 4 ( 3 +

 123 e 2 )x 2 + y 2 -(1 + e 2 )x -e 2 ) = 0. (

  r(θ) = (1 + e 2 ) cos(θ) + (1 -e 2 ) cos 2 (θ) + (3 + e 2 ) 2[1 -e 2 cos 2 (θ)].

ACKNOWLEDGEMENTS

We thank Mark Dennis, Jannick Rolland, Bill Cassarly and Cristina Canavesi for separate discussions whose intersection led to this idea.

FUNDING

University of Rochester's PumpPrimer II Award (OP212647), CONACYT fellowship (LAC), Excellence Initiative of Aix-Marseille University-A*MIDEX, a French "Investissements d'Avenir" programme (MAA).

APPENDIX A. EXTENSION TO SMOOTH AND CONTINUOUS REFLECTORS

Our approach is rigorous for segmented reflectors, and to extend it to the continuum case, one could argue, heuristically, that in the limit of an infinite number of small segments it will hold. Nevertheless, we would like to show its validity more strictly, and we will do so for the z 2 inverse map, although the proof for √ z is also straight forward. Following Elmer's construction, [START_REF] Elmer | The Optical Design of Reflectors[END_REF][START_REF] Winston | Nonimaging Optics[END_REF] we can obtain a 2D curve given the map between its polar coordinate, α, and its angle of direction, β, see Fig. 7. It can be seen that the curve would follow the relation

from which if we do the change of variable ω = π -ᾱ and use that γ = π/2 -β we would end up with

Integrating from θ0 to θ leads to the final expression

where r0 is the initial point distance that fixes the solution.

We would like to apply z 2 , i.e. r → r2 = r and θ → 2 θ = θ, to show that the light curve is transformed into the corresponding reflector. So, squaring expression (6), renaming r = r2 along with β = 2 β, and doing 2 ᾱ = α, we get

where we dropped the explicit dependency of β(α) (to ease writing). This equation is almost what we are looking for, we just have to recognize that given the doubling of the angle, now θ = 2 θ, so

which is exactly the equation derived by Elmer [START_REF] Elmer | The Optical Design of Reflectors[END_REF] for a reflector that sends the light coming from the source at an angle α to an angle β. As shown in Fig. 7, the relation given in Eq. ( 4) changes for the reflector to

from which, similarly, after using that ω = π -α and integrating between θ 0 and θ, Eq. ( 8) can be obtained.

Additionally, it can be shown that the angle of a curve segment in the real space, ν, under the map √ z is transformed into as ν = ν -α 2 , meaning that the angular change is given by the shrinking or expansion of the polar coordinate α, and does not depend on the original angle. Since the light curve segment is only at half of the angle in which light is reflected after hitting the reflector, the transformation leads ν = (β + α)/2, that is the same geometrical relation between the angle of incidence, the local angle of the reflector, and the angle in which light gets reflected, described in the literature [START_REF] Elmer | The Optical Design of Reflectors[END_REF] and discussed previously (see Fig. 7).