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Formation of photonic structures in glasses through phase separation
PERGAMON
Solid State Communications 117 (2001) 733±737
www.elsevier.com/locate/ssc
Formation of photonic structures in glasses through phase separation A.A. Lipovskii a,*, V.D. Petrikov a, V.G. Melehin b, D.K. Tagantsev c, B.V. Tatarintsev c a
St. Petersburg State Technical University, Polytechnicheskaja 29, St. Petersburg, 195251 Russian Federation b A.F. Ioffe Physical-Technical Institute, Polytechnicheskaja 26, St. Petersburg, 194021 Russian Federation c S.I. Vavilov State Optical Institute, Babushkina 36-1, St. Petersburg, 193171 Russian Federation Received 26 November 2000; accepted 25 December 2000 by E.I. Ivchenko
Abstract Uniformly size-distributed lithium-niobate microspheres were formed in a vitreous matrix through a liquid±liquid phase separation of glass under heat treatment. The size of the crystallites varied from 4 to 90 mm depending on the temperature and the duration of the anneal. The structures were studied by optical diffraction, microscopy as well as by Raman and X-ray techniques. q 2001 Published by Elsevier Science Ltd. Keywords: A. Insulators; D. Phase transitions PACS: 61.43.F; 64.75; 32.10.D; 81.70.P
1. Introduction Both natural [1] and arti®cial [2] structures consisting of particles of the size lying in the range of optical wavelengths possess exclusive optical properties, which depend on the type, size and mutual disposition of these particles. Being embedded into a solid optical medium and having a narrow size distribution (even without spatial ordering), the particles form such structures, which allow governing optical transmission of the medium via spectral dependence of light scattering [3], and they can be used for spectral ®ltering via optical or microwave resonance, especially in waveguides [4], or for optical feedback in resonator-less lasers [5]. If those particles are ordered, they form photonic crystals, which have been of interest for the last few years [6], particularly, due to the possibility of forming an arti®cial band structure [7] and to vary non-linear properties of the media [8]. The formation of randomly distributed particles of equal size is treated as the ®rst step toward photonic crystals, with spatial ordering of the particles being the second one [9]. In this paper, we present an approach to the formation of * Corresponding author. Fax: 17-812-552-7954. E-mail address: [email protected] (A.A. Lipovskii).
the structures consisting of the particles of equal size. The approach is based on the phase separation of glasses [10] resulting in the formation of either glassy or, after subsequent crystallisation, crystalline particles of the composition different from the medium one. The size dispersion of the particles depends on the type of phase decomposition (spinodal or binodal) and, therefore, on the composition of the glass used, while the average particle size can be controlled by the heat treatment conditions providing their formation. In addition, the variety of glass compositions allows forming particles of different composition and refractivity, and all this makes the approach prospective for producing photonic structures, e.g. microresonators. Here we describe the formation and the studies of microspheres containing lithium niobate crystallites formed in the especially designed glass. These microspheres are equally size distributed, and their index differs from one of the glassy matrix, which is embedded with them.
2. Formation of spherical lithium niobate particles The designed niobium±germanium±silica glass contains 26 mol% of lithium oxide and 20 mol% of niobium oxide. The exact composition of the glass is presented in Table 1.
0038-1098/01/$ - see front matter q 2001 Published by Elsevier Science Ltd. PII: S 0038-109 8(01)00021-7
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A.A. Lipovskii et al. / Solid State Communications 117 (2001) 733±737
Table 1 Glass composition Component
Ta2O5
TiO2
ZrO2
BaO
Nb2O5
Li2O
Na2O
GeO2
SiO2
Concentration (mol%)
5
7
1
1
20
26
3
5
32
A batch of chemically pure and high-purity grade reagents was prepared and then melted at 14508C for 2 h whilst stirring. The melt was poured onto a massive brass plate and then put into a glass-oven for annealing. The glass was annealed at 5808C for 2 h with subsequent spontaneous cooling to the room temperature. All this allowed to obtain 180 g glass samples of high optical quality. The glass index was about 1.95 for the visible range of wavelengths. A choice of the conditions of the glass heat treatments was based on the temperature dependence of the glass viscosity (h ) measured by the penetration technique and on the results of differential thermal analysis. The latter additionally allowed the evaluation of the glass transition temperature region within 560±5908C. The derivatogram of the designed glass (heating up to 10008C with the rate 5 K/min) is presented in Fig. 1, where one can see a sharp exothermic peak at t7058C (h , 10 5 Pa s). The X-ray diffractometry of the powder resulting from the differential thermal analysis showed a set of X-ray diffraction peaks corresponding to trigonal lithium niobate. Upon a soft annealing (4±10 h) at temperatures falling within 640±6608C (h 10 7.5^0.5 Pa s), that is, below 7058C, the glass demonstrated phase separation resulting in the formation of equisized transparent spherical particles (drops) of high refractivity, those particles being inside the transparent glass matrix with some lower index. The microphotograph of those particles is shown in Fig. 2a, and b. X-ray diffractometry showed no crystalline phase had formed (see Fig. 3), and this appeared to be an evidence of the liquid±liquid phase separation taking place only. Increasing the heat treatment temperatures (up to 6808C, h 10 7 Pa s) and/or times (longer than 10 h) led to the crystallization of those liquid drops that was corroborated by
X-ray diffractometry and Raman scattering. The X-ray diffraction patterns in Fig. 3 illustrate the peaks of the lithium niobate phase increasing with a rise in the processing temperature and/ or time. The results of the Raman scattering studies are presented in Fig. 4. They show the formation of the phonon peaks typical for lithium niobate. These peaks grew under annealing, and the comparison of Raman signals from the glass matrix surrounding the afore-mentioned spherical drops and from a single microsphere, when completed with X-ray data, allowed us to conclude that the microsphere mainly consisted of microcrystals and demonstrated features typical for lithium niobate. Supposedly, ®rst, the glassy drops enriched with niobium and lithium are formed as a result of the liquid±liquid phase separation process, and then those drops tend to crystallize in the course of the heat treatment. All the processes result in the formation of spherical drops containing lithium niobate microcrystals, and these microspheres are embedded into the silica-rich glassy matrix. It should be noted that the microspheres have their own internal structure, which is illustrated by Fig. 2b. The structure of this kind is typical for crystallization which turns inwards from the spherical surface, i.e. from the interface of the drop [11].
3. Optical measurements The size of the microspheres was measured with an optical microscope and by diffraction techniques. The results of measuring are shown in Fig. 5. The size of the microspheres
Fig. 1. Results of differential thermal analysis of the designed glass.
A.A. Lipovskii et al. / Solid State Communications 117 (2001) 733±737
735
Fig. 2. (a) Microphotographs of multiple drops. (b) A single drop formed in the glass in the course of 16 h anneal at 6808C.
varies from several microns up to 90 mm depending on the thermal-temporal conditions of the heat treatment. While optical microscopy provides direct measurements of the microsphere diameters due to their high phase contrast in the glassy matrix, the light diffraction technique allows evaluation of their diameters on the base of a measured angle of Fraunhofer diffraction [3]. The high refractive microspheres embedded into the silica rich (i.e. lower index) matrix produce the cones of diffracted light, and the angle of the ®rst cone can be used to calculate microsphere diameters (see e.g. in Ref. [12]). It is necessary to point out that, due to the focusing properties of the microspheres, two phenomena are responsible for the formation of the diffraction pattern. The ®rst one is the diffraction of light focused by the microspheres, and the second is the diffrac-
tion of the plane wave by the set of non-transparent disks, where the disks correspond to the microspheres, which make strongly divergent beams. In our experiments samples of the glasses embedded with microspheres were illuminated with He±Ne laser, and the spatial distribution of intensity of the light passed through the sample was recorded with a camera. It is essential that the diffraction of this kind can be observed only in case of narrow size distribution of the objects, which diffract the light. The narrower the size distribution, the larger the amount of diffraction cones, and, correspondingly, the larger the amount of diffraction rings, which can be observed on the screen placed after the diffracting object. An unprocessed digital photograph of the screen illuminated by the diffraction pattern is shown in the insert in Fig. 5. Eight diffraction rings could be
Fig. 3. (a) X-ray (CuKa radiation) diffraction angular spectra for initial glass, (b) for the glass samples annealed at 6608C during 4 h, (c) 16 h, (d) At 6808C during 8 h, (e) during 16 h and (f) at 7008C during 8 h. The positions of vertical lines correspond to the peaks of crystalline lithium niobate (trigonal form), and the height of the lines is proportional to the height of corresponding peaks.
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A.A. Lipovskii et al. / Solid State Communications 117 (2001) 733±737
Fig. 4. (a) Raman spectrograms of initial glass, (b) the glass samples annealed at 6808C for 16 h, (c) annealed for 23 h and measuring point between the grains, (d) annealed for 23 h and measuring point corresponding to the position of a grain, (e) annealed for 32 h and (f) the spectrogram of lithium niobate crystal, YZ-orientation. Laser wavelength is 0.5145 mm.
observed by the naked eye in dark room, but the camera did not allow us to distinguish more then 4±5 rings without additional digital processing of the image, because the intensities of the highest diffraction orders were low. Direct optical measurements of the diameters of 79 microspheres showed that the size of 76 of these microspheres lay within 5% dispersion range. The number of observed rings also gives evidence that the width of the size distribution of the microspheres did not exceed 5% since modeling of the diffraction performed according to [12] showed that the seventh and eighth rings were hardly distinguishable for the size dispersion of non-transparent disks exceeding 3± 5%. As the highest rings were distinguishable, we concluded that the model of non- or weakly transparent microspheres
described the situation better. That is, in the course of our heat treatments the designed glass produced a very sharp distribution in the diameters of spherical particles precipitating. Therefore, this glass can be regarded as a prospective one for the formation of photonic structures.
4. Conclusions Finally, heating induced phase separation of the niobiumrich glass allowed us to form uniformly size-distributed microspheres containing lithium niobate crystallites in the glassy matrix with lower index. The size of the particles varied in the range from several microns to tens of microns
Fig. 5. (1) Diameter of the microspheres versus the duration of thermal treatment of glass samples at 6608C and (2) 6808C. Squares ± microscopy measurements, triangles ± diffraction measurements, lines ± least square approximation. Insert: the pattern of He±Ne laser light diffraction by the sample of glass annealed for 16 h at 6608C.
A.A. Lipovskii et al. / Solid State Communications 117 (2001) 733±737
depending on the temperature and duration of the thermal treatment inducing the phase separation. The variety and the ¯exibility of glass compositions provide chances to form microparticles of a different nature and in another range of sizes. The approach developed appears to be prospective to develop photonic structures, e.g. microresonators, and it may be prospective also for the photonic crystals since the long-range character of liquid±liquid phase separation gives hope to form spatially ordered structures. Acknowledgements The research has been supported by International Science and Technology Center (Grant #979) and Russian National Program `Physics of solid nanostructures'. References [1] A. van Blaaderen, Opals in a new light, Science 282 (1998) 887. [2] J.E.G.J. Wijnhoven, W.L. Vos, Preparation of photonic crystals made of air spheres in titania, Science 281 (1998) 802.
737
[3] C.F. Bohren, D.R. Huffman, Absorption and Scattering of Light by Small Particles, Wiley, New York, 1983, p. 536. [4] P. Affolter, B. Eliasson, Electromagnetic resonances and Qfactors of lossy dielectric spheres, IEEE Trans. Microwave Theory Technol. MTT-21 (1973) 573. [5] H. Cao, J.Y. Xu, E.W. Seelig, R.P.H. Chang, Microlaser made of disordered media, Appl. Phys. Lett. 76 (2000) 2997. [6] C.M. Soukulis, Photonic band gap materials: the semiconductors of the future, Phys. Scr. 66 (1996) 146. [7] M. Jacoby, Photonic crystals: whole lotta holes, Chem. Eng. News 76 (1998) 38. [8] S. John, T. Quang, Resonant nonlinear dielectric response in a photonic band gap material, Phys. Rev. Lett. 76 (1996) 2484. [9] A. VanBlaaderen, R. Ruel, P. Wiltzius, Template-directed colloidal crystallization, Nature 385 (1997) 321. [10] O.V. Mazurin, E.A. Porai-Koshits (Eds.), Phase Separation in Glass North-Holland, Amsterdam, 1984, p. 369. [11] M. Tomazawa, Phys. Chem. Glasses 13 (1972) 161 (see also Ohlber, S.H., Golob, H.R., Strichler, D.W., Symposium ion nucleation and crystallization on glass and melts, p.55, Ed. Reser, M.V., The American Ceramic Society, Ohio, 1972). [12] M. Born, E. Wolf, Principles of Optics, Cambridge University Press, London, 1999, p. 980.
Solid State Communications 117 (2001) 733±737
www.elsevier.com/locate/ssc
Formation of photonic structures in glasses through phase separation A.A. Lipovskii a,*, V.D. Petrikov a, V.G. Melehin b, D.K. Tagantsev c, B.V. Tatarintsev c a
St. Petersburg State Technical University, Polytechnicheskaja 29, St. Petersburg, 195251 Russian Federation b A.F. Ioffe Physical-Technical Institute, Polytechnicheskaja 26, St. Petersburg, 194021 Russian Federation c S.I. Vavilov State Optical Institute, Babushkina 36-1, St. Petersburg, 193171 Russian Federation Received 26 November 2000; accepted 25 December 2000 by E.I. Ivchenko
Abstract Uniformly size-distributed lithium-niobate microspheres were formed in a vitreous matrix through a liquid±liquid phase separation of glass under heat treatment. The size of the crystallites varied from 4 to 90 mm depending on the temperature and the duration of the anneal. The structures were studied by optical diffraction, microscopy as well as by Raman and X-ray techniques. q 2001 Published by Elsevier Science Ltd. Keywords: A. Insulators; D. Phase transitions PACS: 61.43.F; 64.75; 32.10.D; 81.70.P
1. Introduction Both natural [1] and arti®cial [2] structures consisting of particles of the size lying in the range of optical wavelengths possess exclusive optical properties, which depend on the type, size and mutual disposition of these particles. Being embedded into a solid optical medium and having a narrow size distribution (even without spatial ordering), the particles form such structures, which allow governing optical transmission of the medium via spectral dependence of light scattering [3], and they can be used for spectral ®ltering via optical or microwave resonance, especially in waveguides [4], or for optical feedback in resonator-less lasers [5]. If those particles are ordered, they form photonic crystals, which have been of interest for the last few years [6], particularly, due to the possibility of forming an arti®cial band structure [7] and to vary non-linear properties of the media [8]. The formation of randomly distributed particles of equal size is treated as the ®rst step toward photonic crystals, with spatial ordering of the particles being the second one [9]. In this paper, we present an approach to the formation of * Corresponding author. Fax: 17-812-552-7954. E-mail address: [email protected] (A.A. Lipovskii).
the structures consisting of the particles of equal size. The approach is based on the phase separation of glasses [10] resulting in the formation of either glassy or, after subsequent crystallisation, crystalline particles of the composition different from the medium one. The size dispersion of the particles depends on the type of phase decomposition (spinodal or binodal) and, therefore, on the composition of the glass used, while the average particle size can be controlled by the heat treatment conditions providing their formation. In addition, the variety of glass compositions allows forming particles of different composition and refractivity, and all this makes the approach prospective for producing photonic structures, e.g. microresonators. Here we describe the formation and the studies of microspheres containing lithium niobate crystallites formed in the especially designed glass. These microspheres are equally size distributed, and their index differs from one of the glassy matrix, which is embedded with them.
2. Formation of spherical lithium niobate particles The designed niobium±germanium±silica glass contains 26 mol% of lithium oxide and 20 mol% of niobium oxide. The exact composition of the glass is presented in Table 1.
0038-1098/01/$ - see front matter q 2001 Published by Elsevier Science Ltd. PII: S 0038-109 8(01)00021-7
734
A.A. Lipovskii et al. / Solid State Communications 117 (2001) 733±737
Table 1 Glass composition Component
Ta2O5
TiO2
ZrO2
BaO
Nb2O5
Li2O
Na2O
GeO2
SiO2
Concentration (mol%)
5
7
1
1
20
26
3
5
32
A batch of chemically pure and high-purity grade reagents was prepared and then melted at 14508C for 2 h whilst stirring. The melt was poured onto a massive brass plate and then put into a glass-oven for annealing. The glass was annealed at 5808C for 2 h with subsequent spontaneous cooling to the room temperature. All this allowed to obtain 180 g glass samples of high optical quality. The glass index was about 1.95 for the visible range of wavelengths. A choice of the conditions of the glass heat treatments was based on the temperature dependence of the glass viscosity (h ) measured by the penetration technique and on the results of differential thermal analysis. The latter additionally allowed the evaluation of the glass transition temperature region within 560±5908C. The derivatogram of the designed glass (heating up to 10008C with the rate 5 K/min) is presented in Fig. 1, where one can see a sharp exothermic peak at t7058C (h , 10 5 Pa s). The X-ray diffractometry of the powder resulting from the differential thermal analysis showed a set of X-ray diffraction peaks corresponding to trigonal lithium niobate. Upon a soft annealing (4±10 h) at temperatures falling within 640±6608C (h 10 7.5^0.5 Pa s), that is, below 7058C, the glass demonstrated phase separation resulting in the formation of equisized transparent spherical particles (drops) of high refractivity, those particles being inside the transparent glass matrix with some lower index. The microphotograph of those particles is shown in Fig. 2a, and b. X-ray diffractometry showed no crystalline phase had formed (see Fig. 3), and this appeared to be an evidence of the liquid±liquid phase separation taking place only. Increasing the heat treatment temperatures (up to 6808C, h 10 7 Pa s) and/or times (longer than 10 h) led to the crystallization of those liquid drops that was corroborated by
X-ray diffractometry and Raman scattering. The X-ray diffraction patterns in Fig. 3 illustrate the peaks of the lithium niobate phase increasing with a rise in the processing temperature and/ or time. The results of the Raman scattering studies are presented in Fig. 4. They show the formation of the phonon peaks typical for lithium niobate. These peaks grew under annealing, and the comparison of Raman signals from the glass matrix surrounding the afore-mentioned spherical drops and from a single microsphere, when completed with X-ray data, allowed us to conclude that the microsphere mainly consisted of microcrystals and demonstrated features typical for lithium niobate. Supposedly, ®rst, the glassy drops enriched with niobium and lithium are formed as a result of the liquid±liquid phase separation process, and then those drops tend to crystallize in the course of the heat treatment. All the processes result in the formation of spherical drops containing lithium niobate microcrystals, and these microspheres are embedded into the silica-rich glassy matrix. It should be noted that the microspheres have their own internal structure, which is illustrated by Fig. 2b. The structure of this kind is typical for crystallization which turns inwards from the spherical surface, i.e. from the interface of the drop [11].
3. Optical measurements The size of the microspheres was measured with an optical microscope and by diffraction techniques. The results of measuring are shown in Fig. 5. The size of the microspheres
Fig. 1. Results of differential thermal analysis of the designed glass.
A.A. Lipovskii et al. / Solid State Communications 117 (2001) 733±737
735
Fig. 2. (a) Microphotographs of multiple drops. (b) A single drop formed in the glass in the course of 16 h anneal at 6808C.
varies from several microns up to 90 mm depending on the thermal-temporal conditions of the heat treatment. While optical microscopy provides direct measurements of the microsphere diameters due to their high phase contrast in the glassy matrix, the light diffraction technique allows evaluation of their diameters on the base of a measured angle of Fraunhofer diffraction [3]. The high refractive microspheres embedded into the silica rich (i.e. lower index) matrix produce the cones of diffracted light, and the angle of the ®rst cone can be used to calculate microsphere diameters (see e.g. in Ref. [12]). It is necessary to point out that, due to the focusing properties of the microspheres, two phenomena are responsible for the formation of the diffraction pattern. The ®rst one is the diffraction of light focused by the microspheres, and the second is the diffrac-
tion of the plane wave by the set of non-transparent disks, where the disks correspond to the microspheres, which make strongly divergent beams. In our experiments samples of the glasses embedded with microspheres were illuminated with He±Ne laser, and the spatial distribution of intensity of the light passed through the sample was recorded with a camera. It is essential that the diffraction of this kind can be observed only in case of narrow size distribution of the objects, which diffract the light. The narrower the size distribution, the larger the amount of diffraction cones, and, correspondingly, the larger the amount of diffraction rings, which can be observed on the screen placed after the diffracting object. An unprocessed digital photograph of the screen illuminated by the diffraction pattern is shown in the insert in Fig. 5. Eight diffraction rings could be
Fig. 3. (a) X-ray (CuKa radiation) diffraction angular spectra for initial glass, (b) for the glass samples annealed at 6608C during 4 h, (c) 16 h, (d) At 6808C during 8 h, (e) during 16 h and (f) at 7008C during 8 h. The positions of vertical lines correspond to the peaks of crystalline lithium niobate (trigonal form), and the height of the lines is proportional to the height of corresponding peaks.
736
A.A. Lipovskii et al. / Solid State Communications 117 (2001) 733±737
Fig. 4. (a) Raman spectrograms of initial glass, (b) the glass samples annealed at 6808C for 16 h, (c) annealed for 23 h and measuring point between the grains, (d) annealed for 23 h and measuring point corresponding to the position of a grain, (e) annealed for 32 h and (f) the spectrogram of lithium niobate crystal, YZ-orientation. Laser wavelength is 0.5145 mm.
observed by the naked eye in dark room, but the camera did not allow us to distinguish more then 4±5 rings without additional digital processing of the image, because the intensities of the highest diffraction orders were low. Direct optical measurements of the diameters of 79 microspheres showed that the size of 76 of these microspheres lay within 5% dispersion range. The number of observed rings also gives evidence that the width of the size distribution of the microspheres did not exceed 5% since modeling of the diffraction performed according to [12] showed that the seventh and eighth rings were hardly distinguishable for the size dispersion of non-transparent disks exceeding 3± 5%. As the highest rings were distinguishable, we concluded that the model of non- or weakly transparent microspheres
described the situation better. That is, in the course of our heat treatments the designed glass produced a very sharp distribution in the diameters of spherical particles precipitating. Therefore, this glass can be regarded as a prospective one for the formation of photonic structures.
4. Conclusions Finally, heating induced phase separation of the niobiumrich glass allowed us to form uniformly size-distributed microspheres containing lithium niobate crystallites in the glassy matrix with lower index. The size of the particles varied in the range from several microns to tens of microns
Fig. 5. (1) Diameter of the microspheres versus the duration of thermal treatment of glass samples at 6608C and (2) 6808C. Squares ± microscopy measurements, triangles ± diffraction measurements, lines ± least square approximation. Insert: the pattern of He±Ne laser light diffraction by the sample of glass annealed for 16 h at 6608C.
A.A. Lipovskii et al. / Solid State Communications 117 (2001) 733±737
depending on the temperature and duration of the thermal treatment inducing the phase separation. The variety and the ¯exibility of glass compositions provide chances to form microparticles of a different nature and in another range of sizes. The approach developed appears to be prospective to develop photonic structures, e.g. microresonators, and it may be prospective also for the photonic crystals since the long-range character of liquid±liquid phase separation gives hope to form spatially ordered structures. Acknowledgements The research has been supported by International Science and Technology Center (Grant #979) and Russian National Program `Physics of solid nanostructures'. References [1] A. van Blaaderen, Opals in a new light, Science 282 (1998) 887. [2] J.E.G.J. Wijnhoven, W.L. Vos, Preparation of photonic crystals made of air spheres in titania, Science 281 (1998) 802.
737
[3] C.F. Bohren, D.R. Huffman, Absorption and Scattering of Light by Small Particles, Wiley, New York, 1983, p. 536. [4] P. Affolter, B. Eliasson, Electromagnetic resonances and Qfactors of lossy dielectric spheres, IEEE Trans. Microwave Theory Technol. MTT-21 (1973) 573. [5] H. Cao, J.Y. Xu, E.W. Seelig, R.P.H. Chang, Microlaser made of disordered media, Appl. Phys. Lett. 76 (2000) 2997. [6] C.M. Soukulis, Photonic band gap materials: the semiconductors of the future, Phys. Scr. 66 (1996) 146. [7] M. Jacoby, Photonic crystals: whole lotta holes, Chem. Eng. News 76 (1998) 38. [8] S. John, T. Quang, Resonant nonlinear dielectric response in a photonic band gap material, Phys. Rev. Lett. 76 (1996) 2484. [9] A. VanBlaaderen, R. Ruel, P. Wiltzius, Template-directed colloidal crystallization, Nature 385 (1997) 321. [10] O.V. Mazurin, E.A. Porai-Koshits (Eds.), Phase Separation in Glass North-Holland, Amsterdam, 1984, p. 369. [11] M. Tomazawa, Phys. Chem. Glasses 13 (1972) 161 (see also Ohlber, S.H., Golob, H.R., Strichler, D.W., Symposium ion nucleation and crystallization on glass and melts, p.55, Ed. Reser, M.V., The American Ceramic Society, Ohio, 1972). [12] M. Born, E. Wolf, Principles of Optics, Cambridge University Press, London, 1999, p. 980.