- Email: [email protected]
Influence of the humidity conditions on the reflectivity spectrum of a porous silicon microcavity
ARTICLE IN PRESS
Physica E 38 (2007) 172–175 www.elsevier.com/locate/physe
Influence of the humidity conditions on the reflectivity spectrum of a porous silicon microcavity E. Xifre´ Pe´reza, L.F. Marsala, J. Ferre´-Borrulla, T. Trifonova,b, J. Pallare`sa, a
Departament d’Enginyeria Electro`nica, Ele`ctrica i Automa`tica, Universitat Rovira i Virgili, Avda. Paı¨sos Catalans, 26, 43007 Tarragona, Spain b Departament d’Enginyeria Electro`nica, Universitat Polite`cnica de Catalunya, C/ Gran Capita` s/n, Campus Nord, 08074 Barcelona, Spain Available online 20 December 2006
Abstract We have designed and fabricated a porous silicon (PS) microcavity that shows a reflectivity resonance around 3 mm in between two spectral regions with high reflectivity values. The microcavity has been simulated following the photonic crystals formalism, which results in good agreement with the measured spectrum. The reflectivity spectrum of the microcavity has been analyzed under different humidity conditions. We demonstrate that the reflectivity resonance shifts to higher wavelengths and that the reflectivity decreases when the humidity increases. In addition, the reflectivity spectrum of the as-prepared device is recovered when the humidity returns to the initial laboratory conditions. Finally, the effect on the reflectivity spectrum of a condensed water layer at the surface of the microcavity is also analyzed. r 2007 Elsevier B.V. All rights reserved. PACS: 42.70.Qs; 68.65.Ac; 78.55.Mb Keywords: Porous silicon; Humidity sensor; Reflectivity spectrum
1. Introduction Porous silicon (PS) is a dielectric material obtained by HF electrochemical etching of silicon. PS refractive index is modulated by changing the current density during the etch and its thickness depends on the time for which the current is applied. It has excellent mechanical and thermal properties and is compatible with silicon-based microelectronics technology. PS multilayers have been used for many applications, such as waveguides [1,2], LEDs [3] and omnidirectional mirrors [4,5]. In particular, microcavity structures have been used for hydrocarbon sensors [6,7] or biosensors [8,9], where changes of the optical properties of the PS microcavity were analyzed. On the other hand, PS monolayers have been used in humidity sensors, designed to detect humidity through changes of its electrical properties [10,11]. Despite optical measurements may be more robust for some applications, only a few authors Corresponding author. Tel.: +34 977 559625; fax: +34 977 559605.
E-mail address: [email protected] (J. Pallare`s). 1386-9477/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2006.12.033
proposed the analysis of the optical properties of PS microcavities for the detection of humidity changes [12]. In this paper, we study the changes of the reflectivity spectrum of a PS microcavity under different humidity conditions. In Section 2, the fabrication, simulation and optical characterization of the microcavity are presented. In Section 3, the influence of the humidity on the resonance position and the reflectivity is discussed. Finally, in Section 4, the effect of a condensed water thin film at the surface of the microcavity on the reflectivity spectrum is studied. 2. Fabrication and characterization The PS microcavity has been obtained by the electrochemical etching of p+-type silicon wafers with a resistivity of 0.01 O cm in an ethanoic HF electrolyte with concentration of 15.4% (volumetric ratio). Fig. 1 shows the fabricated microcavity, which consists of a spacer layer inserted in between two symmetric Distributed Bragg Reflectors (DBR) with three periods. Each period consists of two layers with different refractive indices (n1, n2) and thicknesses (d1, d2). The current densities used have been
ARTICLE IN PRESS E. Xifre´ Pe´rez et al. / Physica E 38 (2007) 172–175
173
1.0
Reflectivity
0.8
0.6
0.4
0.2 Measured Simulated 0.0 1.5
2.0
2.5 3.0 Wavelength (µm)
3.5
4.0
Fig. 2. Measured (solid line) and simulated (dashed line) reflectivity spectrum of the fabricated microcavity for incidence angle 201.
Fig. 1. SEM image of the fabricated porous silicon microcavity on the silicon wafer. The spacer layer is located in between two DBR. Each DBR is formed by the periodic repetition (N ¼ 3) of two layers with low refractive index (gray layer) and high refractive index (light gray layer).
J1 ¼ 100 mA/cm2 and J2 ¼ 20 mA/cm2 for the DBR and JS ¼ 70 mA/cm2 for the spacer layer. They have been applied during 10 s for J1, 20 s for J2 and 30 s for JS, respectively, which lead to estimated layer thicknesses of d1 ¼ 0.55 mm, d2 ¼ 0.41 mm and dS ¼ 1.50 mm and a period thickness L ¼ d1+d2 ¼ 0.96 mm. The refractive index for each current density has been calculated and are n1 ¼ 1.22, n2 ¼ 1.82 and nS ¼ 1.30 for the studied wavelength range using Ref. [13]. These values of refractive indices and thicknesses have been used to simulate the optical properties of the PS microcavity using the transfer matrix method [14–16]. Fig. 2 shows the simulated reflectivity spectrum assuming an incidence angle of 201, where we can observe the reflectivity resonance around 3 mm. Fig. 2 also shows the experimental reflectivity spectrum of the microcavity for an incidence angle 201 using a FTIR spectrometer Bruker Vertex 70. The spectrum is characterized by a reflectivity resonance at 2.97 mm in between two high reflectivity bands, in good agreement with the simulated results. Narrower resonance widths could be obtained either by increasing the number of periods of the DBR or by enhancing the fabrication conditions [17]. 3. Influence of the relative humidity on the reflectivity spectrum Fig. 3 shows the measured reflectivity spectrum of the fabricated microcavity for different humidity levels. The changes in the spectrum are small, in good agreement with previous results [12]. The inset in Fig. 3 shows the zoom of reflectivity resonance and its fitting by using a Gaussian
Fig. 3. Measured reflectivity spectrum for humidity levels of 40%, 50%, 60%, 70% and 80%. Inset: Zoom of the measured (symbol) reflectivity spectrum of the microcavity centered at the reflectivity resonance for humidity levels of 40%, 60% and 80%. The solid lines are the fitted values using a Gaussian function.
function. We can observe that when humidity increases, the reflectivity resonance shifts to higher wavelengths and its reflectivity decreases. The inset in Fig. 4 shows this resonance shift. The reflectivity spectrum of the asprepared device is recovered when the humidity returns to the initial laboratory conditions. These variations are probably due to water vapor infiltrated into the pores of the PS layers. Water vapor slightly increases the effective refractive index of the layers as the air in the pores is substituted by water vapor. As the increase of the refractive index is very slight, we can explain the slight shift of the reflectivity resonance to higher wavelengths and the decrease of the reflectivity when the humidity rises.
ARTICLE IN PRESS E. Xifre´ Pe´rez et al. / Physica E 38 (2007) 172–175
174
5
R (%)
4
1.0
0.6
32 % 0.4
0.8
0.2 0.0 40
3
50 60 70 Humidity (%)
80
Reflectivity
Resonance Shift (nm)
6
0.6
0.4
2 60%
0.2 1 0.0 0
1.5 40
45
50
55
60
65
70
75
Humidity (%) Fig. 4. Variation of the merit function, defined by the Eq. (1), for different humidity levels at l ¼ 3 mm. The linear dependence is also shown. Inset: Variation of the reflectivity resonance position for different humidity levels.
250
4. Water condensation on the surface of the microcavity
150
4.0
5 4 3 2 1 0 30
35
100
40 45 50 Humidity (%)
55
60
50
0 30
In this section, we study the changes in the reflectivity spectrum when water is not only present as vapor infiltrated into the pores but also as a thin film on the surface of the microcavity due to the water condensation. Fig. 5 shows the measured reflectivity spectrum of the microcavity for different humidity levels with condensed water on the surface. The initial reflectivity measurement was realized when the microcavity was in a 60% relative humidity with a condensed water film on its surface. This measurement presents very low reflectivity values for the whole wavelength range. After this measurement, a desiccant has been inserted into the FTIR sample compartment to reduce the humidity of the ambient and the condensation on the microcavity, and the rest of measurements have been realized at different times during this drying process. We observe that when the humidity decreases, the reflectivity increases for the whole range of wavelengths and the position of the reflectivity resonance is also affected. To quantify the relative reflectivity variation, we used the merit function defined by the Eq. (1). Fig. 6 shows the merit function R for l ¼ 3 mm, where RHmax is 0.167 for
3.5
6
200
(1)
where RHmax is the measured reflectivity for the maximum relative humidity and RH is the measured reflectivity for a given relative humidity. Fig. 4 shows the linear dependence of this merit function for l ¼ 3 mm in the humidity range from 35% to 80%, where RHmax is 0.324 for 80%.
3.0
Fig. 5. Measured reflectivity spectrum for different humidity levels with the presence of condensed water on the surface of the microcavity.
R (%)
RH RHmax 100ð%Þ RHmax
2.5
Wavelength (µm)
To quantify the sensor sensitivity at a given l, we define the merit function R as R¼
2.0
80
Resonance Shift (nm)
35
35
40
45
50
55
60
Humidity (%) Fig. 6. Variation of the merit function for different humidity levels at l ¼ 3 mm. The exponential dependence is also shown. Inset: Variation of the reflectivity resonance position for different humidity levels.
60%. The obtained values follow an exponential law instead of the linear law obtained in Section 3. According to these results, we could say that the condensation on the surface could mask the humidity measurements of this type of sensor. The inset in Fig. 6 shows the resonance shift. We can observe that the thin film of condensed water on the microcavity surface leads to larger shifts than when there is only water vapor infiltrated into the pores. 5. Conclusions A PS microcavity has been fabricated, characterized and simulated. The FTIR reflectivity spectrum of this device depends not only on the humidity level of the ambient medium but also on the presence of a water thin film on its surface due to the condensation of water. In both cases, the
ARTICLE IN PRESS E. Xifre´ Pe´rez et al. / Physica E 38 (2007) 172–175
reflectivity resonance shifts to higher wavelengths and the reflectivity decreases when the humidity increases, but the relative changes in the reflectivity spectrum are much more significant with the presence of condensed water. Acknowledgements This work was supported by the Spanish Commission of Science and Technology (CICYT) under Grant number TEC2006-06531. J. Ferre´ acknowledges the Ramon y Cajal fellowship from the Spanish Ministerio de Ciencia y Tecnologı´ a. References [1] A. Bruyant, I. Stefanon, G. Lerondel, S. Blaize, S. Aubert, R. Bachelot, P. Royer, P. Pirasteh, J. Charrier, P. Joubert, Phys. Status Solidi A 202 (2005) 1417. [2] A.M. Rossi, G. Amato, V. Camarchia, L. Boarino, S. Borini, Appl. Phys. Lett. 78 (2001) 3003. [3] S. Chan, P.M. Fauchet, Opt. Mater. 17 (2001) 31. [4] A. Bruyant, G. Le´rondel, P.J. Reece, M. Gal, Appl. Phys. Lett. 82 (2003) 3227.
175
[5] E. Xifre´-Pe´rez, L.F. Marsal, J. Pallare`s, J. Ferre´-Borrull, J. Appl. Phys. 97 (2005) 064503. [6] L. De Stefano, L. Moretti, I. Rendina, A.M. Rossi, Sens. Actuators A 104 (2004) 179. [7] J. Dorvee, M.J. Sailor, Phys. Status Solidi A 202 (2005) 1619. [8] C. Pacholski, M. Sartor, M.J. Sailor, F. Cunin, G.M. Miskelly, J. Am. Chem. Soc. 127 (2005) 11636. [9] H. Ouyang, C.C. Striemer, P.M. Fauchet, Appl. Phys. Lett. 88 (2006) 163108. [10] Z.M. Rittersma, A. Splinter, A. Bodecker, W. Benecke, Sens. Actuators B 68 (2000) 210. [11] M. Bjo¨rkqvist, J. Salonen, J. Paski, E. Laine, Sens. Actuators A 112 (2004) 244. [12] C.J. Oton, L. Pancheri, Z. Gaburro, L. Pavesi, C. Baratto, G. Faglia, G. Sberveglieri, Phys. Stat. Sol. A 197 (2003) 523. [13] E. Xifre´-Pe´rez, J. Pallare`s, J. Ferre´-Borrull, T. Trifonov, L.F. Marsal, Phys. Status Solidi C, in press. [14] P. Yeh, Optical Waves in Layered Media, Wiley, New York, 1988, p. 102. [15] E. Hecht, Optics, Addison-Wesley, San Francisco, 1987, p. 373. [16] Y. Fink, J.N. Winn, S. Fan, C. Chen, J. Michel, J.D. Joannopoulos, E.L. Thomas, Science 282 (1998) 1679. [17] P.J. Reece, G. Lerondel, W.H. Zheng, M. Gal, Appl. Phys. Lett. 81 (2002) 4895.
Physica E 38 (2007) 172–175 www.elsevier.com/locate/physe
Influence of the humidity conditions on the reflectivity spectrum of a porous silicon microcavity E. Xifre´ Pe´reza, L.F. Marsala, J. Ferre´-Borrulla, T. Trifonova,b, J. Pallare`sa, a
Departament d’Enginyeria Electro`nica, Ele`ctrica i Automa`tica, Universitat Rovira i Virgili, Avda. Paı¨sos Catalans, 26, 43007 Tarragona, Spain b Departament d’Enginyeria Electro`nica, Universitat Polite`cnica de Catalunya, C/ Gran Capita` s/n, Campus Nord, 08074 Barcelona, Spain Available online 20 December 2006
Abstract We have designed and fabricated a porous silicon (PS) microcavity that shows a reflectivity resonance around 3 mm in between two spectral regions with high reflectivity values. The microcavity has been simulated following the photonic crystals formalism, which results in good agreement with the measured spectrum. The reflectivity spectrum of the microcavity has been analyzed under different humidity conditions. We demonstrate that the reflectivity resonance shifts to higher wavelengths and that the reflectivity decreases when the humidity increases. In addition, the reflectivity spectrum of the as-prepared device is recovered when the humidity returns to the initial laboratory conditions. Finally, the effect on the reflectivity spectrum of a condensed water layer at the surface of the microcavity is also analyzed. r 2007 Elsevier B.V. All rights reserved. PACS: 42.70.Qs; 68.65.Ac; 78.55.Mb Keywords: Porous silicon; Humidity sensor; Reflectivity spectrum
1. Introduction Porous silicon (PS) is a dielectric material obtained by HF electrochemical etching of silicon. PS refractive index is modulated by changing the current density during the etch and its thickness depends on the time for which the current is applied. It has excellent mechanical and thermal properties and is compatible with silicon-based microelectronics technology. PS multilayers have been used for many applications, such as waveguides [1,2], LEDs [3] and omnidirectional mirrors [4,5]. In particular, microcavity structures have been used for hydrocarbon sensors [6,7] or biosensors [8,9], where changes of the optical properties of the PS microcavity were analyzed. On the other hand, PS monolayers have been used in humidity sensors, designed to detect humidity through changes of its electrical properties [10,11]. Despite optical measurements may be more robust for some applications, only a few authors Corresponding author. Tel.: +34 977 559625; fax: +34 977 559605.
E-mail address: [email protected] (J. Pallare`s). 1386-9477/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2006.12.033
proposed the analysis of the optical properties of PS microcavities for the detection of humidity changes [12]. In this paper, we study the changes of the reflectivity spectrum of a PS microcavity under different humidity conditions. In Section 2, the fabrication, simulation and optical characterization of the microcavity are presented. In Section 3, the influence of the humidity on the resonance position and the reflectivity is discussed. Finally, in Section 4, the effect of a condensed water thin film at the surface of the microcavity on the reflectivity spectrum is studied. 2. Fabrication and characterization The PS microcavity has been obtained by the electrochemical etching of p+-type silicon wafers with a resistivity of 0.01 O cm in an ethanoic HF electrolyte with concentration of 15.4% (volumetric ratio). Fig. 1 shows the fabricated microcavity, which consists of a spacer layer inserted in between two symmetric Distributed Bragg Reflectors (DBR) with three periods. Each period consists of two layers with different refractive indices (n1, n2) and thicknesses (d1, d2). The current densities used have been
ARTICLE IN PRESS E. Xifre´ Pe´rez et al. / Physica E 38 (2007) 172–175
173
1.0
Reflectivity
0.8
0.6
0.4
0.2 Measured Simulated 0.0 1.5
2.0
2.5 3.0 Wavelength (µm)
3.5
4.0
Fig. 2. Measured (solid line) and simulated (dashed line) reflectivity spectrum of the fabricated microcavity for incidence angle 201.
Fig. 1. SEM image of the fabricated porous silicon microcavity on the silicon wafer. The spacer layer is located in between two DBR. Each DBR is formed by the periodic repetition (N ¼ 3) of two layers with low refractive index (gray layer) and high refractive index (light gray layer).
J1 ¼ 100 mA/cm2 and J2 ¼ 20 mA/cm2 for the DBR and JS ¼ 70 mA/cm2 for the spacer layer. They have been applied during 10 s for J1, 20 s for J2 and 30 s for JS, respectively, which lead to estimated layer thicknesses of d1 ¼ 0.55 mm, d2 ¼ 0.41 mm and dS ¼ 1.50 mm and a period thickness L ¼ d1+d2 ¼ 0.96 mm. The refractive index for each current density has been calculated and are n1 ¼ 1.22, n2 ¼ 1.82 and nS ¼ 1.30 for the studied wavelength range using Ref. [13]. These values of refractive indices and thicknesses have been used to simulate the optical properties of the PS microcavity using the transfer matrix method [14–16]. Fig. 2 shows the simulated reflectivity spectrum assuming an incidence angle of 201, where we can observe the reflectivity resonance around 3 mm. Fig. 2 also shows the experimental reflectivity spectrum of the microcavity for an incidence angle 201 using a FTIR spectrometer Bruker Vertex 70. The spectrum is characterized by a reflectivity resonance at 2.97 mm in between two high reflectivity bands, in good agreement with the simulated results. Narrower resonance widths could be obtained either by increasing the number of periods of the DBR or by enhancing the fabrication conditions [17]. 3. Influence of the relative humidity on the reflectivity spectrum Fig. 3 shows the measured reflectivity spectrum of the fabricated microcavity for different humidity levels. The changes in the spectrum are small, in good agreement with previous results [12]. The inset in Fig. 3 shows the zoom of reflectivity resonance and its fitting by using a Gaussian
Fig. 3. Measured reflectivity spectrum for humidity levels of 40%, 50%, 60%, 70% and 80%. Inset: Zoom of the measured (symbol) reflectivity spectrum of the microcavity centered at the reflectivity resonance for humidity levels of 40%, 60% and 80%. The solid lines are the fitted values using a Gaussian function.
function. We can observe that when humidity increases, the reflectivity resonance shifts to higher wavelengths and its reflectivity decreases. The inset in Fig. 4 shows this resonance shift. The reflectivity spectrum of the asprepared device is recovered when the humidity returns to the initial laboratory conditions. These variations are probably due to water vapor infiltrated into the pores of the PS layers. Water vapor slightly increases the effective refractive index of the layers as the air in the pores is substituted by water vapor. As the increase of the refractive index is very slight, we can explain the slight shift of the reflectivity resonance to higher wavelengths and the decrease of the reflectivity when the humidity rises.
ARTICLE IN PRESS E. Xifre´ Pe´rez et al. / Physica E 38 (2007) 172–175
174
5
R (%)
4
1.0
0.6
32 % 0.4
0.8
0.2 0.0 40
3
50 60 70 Humidity (%)
80
Reflectivity
Resonance Shift (nm)
6
0.6
0.4
2 60%
0.2 1 0.0 0
1.5 40
45
50
55
60
65
70
75
Humidity (%) Fig. 4. Variation of the merit function, defined by the Eq. (1), for different humidity levels at l ¼ 3 mm. The linear dependence is also shown. Inset: Variation of the reflectivity resonance position for different humidity levels.
250
4. Water condensation on the surface of the microcavity
150
4.0
5 4 3 2 1 0 30
35
100
40 45 50 Humidity (%)
55
60
50
0 30
In this section, we study the changes in the reflectivity spectrum when water is not only present as vapor infiltrated into the pores but also as a thin film on the surface of the microcavity due to the water condensation. Fig. 5 shows the measured reflectivity spectrum of the microcavity for different humidity levels with condensed water on the surface. The initial reflectivity measurement was realized when the microcavity was in a 60% relative humidity with a condensed water film on its surface. This measurement presents very low reflectivity values for the whole wavelength range. After this measurement, a desiccant has been inserted into the FTIR sample compartment to reduce the humidity of the ambient and the condensation on the microcavity, and the rest of measurements have been realized at different times during this drying process. We observe that when the humidity decreases, the reflectivity increases for the whole range of wavelengths and the position of the reflectivity resonance is also affected. To quantify the relative reflectivity variation, we used the merit function defined by the Eq. (1). Fig. 6 shows the merit function R for l ¼ 3 mm, where RHmax is 0.167 for
3.5
6
200
(1)
where RHmax is the measured reflectivity for the maximum relative humidity and RH is the measured reflectivity for a given relative humidity. Fig. 4 shows the linear dependence of this merit function for l ¼ 3 mm in the humidity range from 35% to 80%, where RHmax is 0.324 for 80%.
3.0
Fig. 5. Measured reflectivity spectrum for different humidity levels with the presence of condensed water on the surface of the microcavity.
R (%)
RH RHmax 100ð%Þ RHmax
2.5
Wavelength (µm)
To quantify the sensor sensitivity at a given l, we define the merit function R as R¼
2.0
80
Resonance Shift (nm)
35
35
40
45
50
55
60
Humidity (%) Fig. 6. Variation of the merit function for different humidity levels at l ¼ 3 mm. The exponential dependence is also shown. Inset: Variation of the reflectivity resonance position for different humidity levels.
60%. The obtained values follow an exponential law instead of the linear law obtained in Section 3. According to these results, we could say that the condensation on the surface could mask the humidity measurements of this type of sensor. The inset in Fig. 6 shows the resonance shift. We can observe that the thin film of condensed water on the microcavity surface leads to larger shifts than when there is only water vapor infiltrated into the pores. 5. Conclusions A PS microcavity has been fabricated, characterized and simulated. The FTIR reflectivity spectrum of this device depends not only on the humidity level of the ambient medium but also on the presence of a water thin film on its surface due to the condensation of water. In both cases, the
ARTICLE IN PRESS E. Xifre´ Pe´rez et al. / Physica E 38 (2007) 172–175
reflectivity resonance shifts to higher wavelengths and the reflectivity decreases when the humidity increases, but the relative changes in the reflectivity spectrum are much more significant with the presence of condensed water. Acknowledgements This work was supported by the Spanish Commission of Science and Technology (CICYT) under Grant number TEC2006-06531. J. Ferre´ acknowledges the Ramon y Cajal fellowship from the Spanish Ministerio de Ciencia y Tecnologı´ a. References [1] A. Bruyant, I. Stefanon, G. Lerondel, S. Blaize, S. Aubert, R. Bachelot, P. Royer, P. Pirasteh, J. Charrier, P. Joubert, Phys. Status Solidi A 202 (2005) 1417. [2] A.M. Rossi, G. Amato, V. Camarchia, L. Boarino, S. Borini, Appl. Phys. Lett. 78 (2001) 3003. [3] S. Chan, P.M. Fauchet, Opt. Mater. 17 (2001) 31. [4] A. Bruyant, G. Le´rondel, P.J. Reece, M. Gal, Appl. Phys. Lett. 82 (2003) 3227.
175
[5] E. Xifre´-Pe´rez, L.F. Marsal, J. Pallare`s, J. Ferre´-Borrull, J. Appl. Phys. 97 (2005) 064503. [6] L. De Stefano, L. Moretti, I. Rendina, A.M. Rossi, Sens. Actuators A 104 (2004) 179. [7] J. Dorvee, M.J. Sailor, Phys. Status Solidi A 202 (2005) 1619. [8] C. Pacholski, M. Sartor, M.J. Sailor, F. Cunin, G.M. Miskelly, J. Am. Chem. Soc. 127 (2005) 11636. [9] H. Ouyang, C.C. Striemer, P.M. Fauchet, Appl. Phys. Lett. 88 (2006) 163108. [10] Z.M. Rittersma, A. Splinter, A. Bodecker, W. Benecke, Sens. Actuators B 68 (2000) 210. [11] M. Bjo¨rkqvist, J. Salonen, J. Paski, E. Laine, Sens. Actuators A 112 (2004) 244. [12] C.J. Oton, L. Pancheri, Z. Gaburro, L. Pavesi, C. Baratto, G. Faglia, G. Sberveglieri, Phys. Stat. Sol. A 197 (2003) 523. [13] E. Xifre´-Pe´rez, J. Pallare`s, J. Ferre´-Borrull, T. Trifonov, L.F. Marsal, Phys. Status Solidi C, in press. [14] P. Yeh, Optical Waves in Layered Media, Wiley, New York, 1988, p. 102. [15] E. Hecht, Optics, Addison-Wesley, San Francisco, 1987, p. 373. [16] Y. Fink, J.N. Winn, S. Fan, C. Chen, J. Michel, J.D. Joannopoulos, E.L. Thomas, Science 282 (1998) 1679. [17] P.J. Reece, G. Lerondel, W.H. Zheng, M. Gal, Appl. Phys. Lett. 81 (2002) 4895.