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Comparison of Bi3Fe5O12 film giant Faraday rotators grown on (111) and (001) Gd3Ga5O12 single crystals
Thin Solid Films 515 (2006) 477 – 480 www.elsevier.com/locate/tsf
Comparison of Bi3Fe5O12 film giant Faraday rotators grown on (111) and (001) Gd3Ga5O12 single crystals P. Johansson *, S.I. Khartsev 1, A.M. Grishin 1 Department of Condensed Matter Physics, Royal Institute of Technology, SE-164 40 Stockholm-Kista, Sweden Available online 27 January 2006
Abstract Bismuth iron garnet (Bi3Fe5O12, BIG) epitaxial thin films were grown on single crystal (Gd3Ga5O12, GGG) (111) and (001) substrates by rfmagnetron sputtering technique. Processing parameters have been optimized to obtain high deposition rate (2.74 Am/h) and the surface rms roughness less than 10 nm. X-ray diffraction reveals films epitaxial quality: exclusive (111) or (001) orientation with narrow rocking curves and strong in-plane texture. Films possess low optical loss and magneto-optical Faraday rotation (FR) as high as 5 deg/Am at 677 nm wavelength. Comparative analysis of films grown on (111) and (001) substrates clearly shows significant superiority of BIG/GGG(001) film. For this film, the coercive field ¨100 Oe appears to be 2.5 times lower while the optical transmission to be 10% higher than that for BIG/GGG(111) film. Enhanced magneto-optical performance of BIG/GGG(001) films relies upon better accommodation of the film-to-substrate mismatch strain through the tetragonal BIG lattice distortions compared to the rhombohedral one in BIG/GGG(111) films. D 2006 Elsevier B.V. All rights reserved. PACS: 78.20.Ls; 78.20.Bh; 78.20.Ci; 81.15.Cd; 85.70.Ge Keywords: Magnetooptical effects; Theory, models, and numerical simulation; Optical constants (refractive index, absorption); Deposition by sputtering; Ferrite and garnet devices
1. Introduction The main interest in BIG films stems from their large magneto-optical Faraday effect [1]. Since 1969 when it was realized that bismuth doping strongly enhances the Faraday rotation (FR) in iron garnets by about 2 deg/ı`m per Bi-atom per chemical formula unit [2], more researching has been done to increase Bi content. Completely substituted bismuth iron garnet Bi3Fe5O12 (BIG) shows the highest Faraday rotation per film thickness in visible light among all known garnet films [1]. Recently Bi3Fe5O12/Y3Fe5O12 and Bi3Fe5O12/Ga3Gd5O12 photonic crystals have been synthesized on a Gd3Ga5O12 substrate [3]. The latest year’s photonic crystals have an expanding niche of photonics. They have a potential for filtering, switching and light guiding [4]. There have been several techniques explored to grow BIG films on single crystal substrates. Okuda et al. [1] prepared BIG films by reactive ion beam sputtering, RIBS. Pulsed laser deposition, PLD, showed
higher deposition rate compared to RIBS [5]. BIG films can not be grown by liquid phase epitaxy, LPE, because Bi3Fe5O12 is thermodynamically nonequilibrium phase [6]. Rf-magnetron sputtering technique can be used in this case. There are two main advantages to use rf-magnetron sputtering instead of PLD. The first is the possibility to achieve a better transmittance in deposited films. This because sputtering is droplets free and therefore gives a better film surface compared with PLD. The second main advantage is that sputtering generally gives much higher thickness uniformity over a large substrate area. In this regard, it can be considered as an industrial technology. At the same time PLD and rf-sputtering are both reliable in stoichiometry preservation. This paper describes the growth and analysis of magnetooptical spectra of rf-magnetron sputtered BIG films with two different orientations (001) and (111) grown on the corresponding gadolinium gallium garnet Gd3Ga5O12 substrates. 2. Experimental
* Corresponding author. Tel.: +46 8 7904186. E-mail address: [email protected] (P. Johansson). 1 Tel.: +46 8 7904186. 0040-6090/$ - see front matter D 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2005.12.268
Bismuth iron garnet films were grown on GGG(111) and GGG(001) single crystal 5 5 0.5 mm3 substrates by rf-
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P. Johansson et al. / Thin Solid Films 515 (2006) 477 – 480
around (444) and u-scan were performed for (642) atomic plane. 3. Results and discussion XRD h – 2h scans in Fig. 1 clearly show the sintered BIG films are single phase and exclusively (001) or (111) oriented. Rocking-curves of (004) and (444) Bragg reflections are presented in Fig. 2. The full width at half maximum, FWHM, was calculated for the peaks. As showed in Fig. 2a, the rocking curves are narrower for the BIG(001) film compared to the BIG(111). BIG(111) has FWHM of 0.26(0.08- for GGG) and BIG(001) has FWHM of 0.21- (0.07for GGG). u-scans of (4 0 10) plane for BIG(1.39 Am)/ GGG(001) and of (642) plane for BIG(1.32 Am)/GGG(111) were recorded to examine in-plane texture in grown films relative to the substrate. As showed in inserts in Fig. 2, the reflections of the film replicate those of the substrate appearing at the same u-angle positions. Therefore, XRD patterns undoubtedly exhibit the epitaxial quality of BIG films on GGG substrates. Out-of plane lattice parameters for
Fig. 1. (a) X-ray diffraction (XRD) h – 2h scan in CuKa radiation for 1.39 Amthick Bi3Fe5O12 film grown by rf-magnetron sputtering technique onto single crystal Gd3Ga5O12(001). (b) XRD h – 2h scan for 1.32 Am-thick Bi3Fe5O12 film grown onto Gd3Ga5O12(111) crystal.
magnetron sputtering at 50 W of rf-power applied to 1VV composite (3 Bi2O3 + 5 Fe2O3) target. Details of the target preparation see elsewhere [5]. After reaching the ultimate pressure of 10 6 Torr Ar – O2 gas mixture (5 : 1) was introduced to build up the total pressure of 35 mTorr. The substrate temperature was 525 -C. BIG film deposition rate was ˚ /s. Deposition was finalized by in situ approximately 7.6 A post-annealing at 525 -C in 500 Torr of oxygen atmosphere for 15 min. Films thickness was controlled by the Tencor profilometer. It was 1.39 Am for BIG/GGG(001) and 1.32 Am for BIG/GGG(111) within of 5% accuracy. The X-ray diffractometer (XRD) Siemens D-5000 operating in CuKa ˚ ) was used to define the microradiation (k = 1.54056 A crystalline structure of sintered BIG films. h – 2h scans and rocking curves were recorded for the main Bragg reflections, and u-scans were traced for the oblique planes. For BIG/GGG(001) the rocking curve was scanned around (004) and u-scan was performed for (4 0 10) atomic plane. For BIG/GGG (111) the rocking curve was scanned
Fig. 2. (a) The rocking curves of the (004) Bragg reflections from the Bi3Fe5O12/Gd3Ga5O12 (001) structure. Inset: u-scans of the oblique (4010) planes in BIG/GGG(001). (b) The rocking curves of the (444) Bragg reflections from the Bi3Fe5O12/Gd3Ga5O12 (111) structure. Inset: u-scans of the oblique (642) planes in BIG/GGG(111).
P. Johansson et al. / Thin Solid Films 515 (2006) 477 – 480
479
the films of two different orientations were calculated using the Nelson –Riley [7] extrapolation: ðacosh a0 Þ 1 1 þ ¼ Ccos2 h : ð1Þ a0 sinh h Here a cosh is an interplane distance calculated from the apparent Bragg peak position at 2h and C is a fitting coefficient. The lattice parameter for BIG film on GGG(001) ˚ and for BIG on GGG(111) is was found to be 12.626 T 0.005 A ˚. 12.619 T 0.005 A The magneto-optical Faraday rotation (FR) H as a function of wavelength k was measured using a homemade magnetooptical spectrometer based on fibre-optic unit Ocean Optics PC 2000 (300 –1000 nm, 5.5 nm resolution) [8]. The FR is about the same for the two different orientations: about 5 deg/Am at 677 nm wavelength with the peak value of 24 deg/Am at k = 544 nm, see Fig. 3a. The optical dispersion of BIG films has been studied in transmitted light. The experimental spectra T(k) of the transmittance are presented in Fig. 3b. Experiments show that BIG/GGG(001) films have higher transmittance than BIG/ GGG(111). These data have been fitted to the theoretical expression [9,10]: T¼
Mx P Qx þ Rx2
ð2Þ
where: M = 16s (n 2 + k 2), P = [(n + 1)2 + k 2][(n + 1)(n + s 2) + k 2],
Fig. 4. The normalized absorption plotted versus photon energy. The absorption edge is about 2.2 eV for both orientations of the Bi3Fe5O12 films.
Q R u
[(n 2 1 + k 2)(n 2 s 2 + k 2) 2k 2(s 2 + 1)] 2 cos u k [2(n 2 s 2 + k 2) + (s 2 + 1)(n 2 1 + k 2)] 2 sin u, = [(n 1)2 + k 2][(n 1)(n s 2) + k 2], = 4pnd / k, x = exp( ad), a = 4 pk / k.
=
Here d is the film thickness, n is the refractive index of film, k is the extinction coefficient. The dispersion of refractive index of the substrate s(k) was taken from Ref.[11]. Film thickness d was measured with the profilometer. Surface roughness was taken in account as a Gaussian distribution of film thickness centred at d with the standard deviation r d [10]. Transmittance of the rough film T is expressed as: " 2 # Z 1 1 dV d pffiffiffiffiffiffi exp T¯ ðk; d; rd ; n; k; sÞ ¼ 2 rd rd 2p T ðk; dV; n; k; sÞddV
ð3Þ
The theoretical model compared with experimental data is rather good for wavelength 650– 1000 nm. The refractive index n for BIG films was found to follow the Sellmeier formula: n = A + (k o / k)2 [10] with the coefficients A = 2.315 T 0.1 and 2.41 T 0.1, k o = 423 T 2 and 425 T 2 nm for BIG/GGG(001) and
Fig. 3. (a) Faraday rotation spectra for Bi3Fe5O12 (001) and (111) films grown onto Gd3Ga5O12 crystals. (b) Normalized transmission spectra for BIG films, the theoretical curves were calculated from Eq. (2). The transmittance is higher for BIG(1.39 Am)/GGG(001) than for BIG(1.32Am)/GGG(111).
Fig. 5. Faraday rotation at wavelength 677 nm versus magnetic field in rfmagnetron sputtered BIG/GGG(001) and BIG/GGG(111) films.
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BIG/GGG(111), correspondingly. The normalized absorption calculated and plotted versus photon energy in Fig. 4 revealed the absorption edge about 2.2 eV. The magneto-optical Faraday rotation at k = 677 nm was also measured as a function of applied magnetic field (Fig. 5). The homemade pulsed magnetic field tracer enables recording of magneto-optical hysteresis loops up to H = 2 T. The coercive field H c is 101.7 T 2 Oe for BIG/GGG(001) compared to 267.0 T 2 Oe in BIG/GGG(111). 4. Conclusions Bismuth iron garnet (Bi3Fe5O12, BIG) epitaxial films were grown on single crystal (Gd3Ga5O12, GGG) (111) and (001) substrates by rf-magnetron sputtering technique. BIG/ GGG(001) film compared to BIG/GGG(111) shows better magneto-optical performance: lower coercive force and higher optical transmittance.
References [1] T. Okuda, T. Katayama, H. Kobayashi, N. Kobayashi, K. Satoh, H. Yamamoto, J. Appl. Phys. 67 (1990) 4944. [2] C.F. Buhrer, J. Appl. Phys. 40 (1969) 4500. [3] S. Kahl, A.M. Grishin, Appl. Phys. Lett. 84 (2004) 1438; S.I. Khartsev, A.M. Grishin, Appl. Phys. Lett. 87 (2005) 122504. [4] M. Levy, IEEE J. Sel. Top. Quantum Electron. 8 (2002) 1300. [5] N. Adachi, V.P. Denysenkov, S.I. Khartsev, A.M. Grishin, T. Okuda, J. Appl. Phys. 88 (2000) 2734. [6] S. Kahl, A.M. Grishin, S.I. Khartsev, K. Kawano, J.S. Abell, IEEE Trans. Magn. 37 (2001) 2457. [7] B.D Cullity, Elements of X-ray Diffraction, Addison-Wesley, Reading, MA, 1988. [8] S. Kahl, S.I. Khartsev, A.M. Grishin, K. Kawano, G. Kong, R.A. Chakalov, J.S. Abell, J. Appl. Phys. 91 (2002) 9556. [9] R. Swanepoel, J. Phys. E 16 (1983) 1214. [10] S. Kahl, V. Popov, A. Grishin, J. Appl. Phys 94 (2003) 5688. [11] Marvin J. Weber, Handbook of Optical Materials, CRC Press, 2002.
Comparison of Bi3Fe5O12 film giant Faraday rotators grown on (111) and (001) Gd3Ga5O12 single crystals P. Johansson *, S.I. Khartsev 1, A.M. Grishin 1 Department of Condensed Matter Physics, Royal Institute of Technology, SE-164 40 Stockholm-Kista, Sweden Available online 27 January 2006
Abstract Bismuth iron garnet (Bi3Fe5O12, BIG) epitaxial thin films were grown on single crystal (Gd3Ga5O12, GGG) (111) and (001) substrates by rfmagnetron sputtering technique. Processing parameters have been optimized to obtain high deposition rate (2.74 Am/h) and the surface rms roughness less than 10 nm. X-ray diffraction reveals films epitaxial quality: exclusive (111) or (001) orientation with narrow rocking curves and strong in-plane texture. Films possess low optical loss and magneto-optical Faraday rotation (FR) as high as 5 deg/Am at 677 nm wavelength. Comparative analysis of films grown on (111) and (001) substrates clearly shows significant superiority of BIG/GGG(001) film. For this film, the coercive field ¨100 Oe appears to be 2.5 times lower while the optical transmission to be 10% higher than that for BIG/GGG(111) film. Enhanced magneto-optical performance of BIG/GGG(001) films relies upon better accommodation of the film-to-substrate mismatch strain through the tetragonal BIG lattice distortions compared to the rhombohedral one in BIG/GGG(111) films. D 2006 Elsevier B.V. All rights reserved. PACS: 78.20.Ls; 78.20.Bh; 78.20.Ci; 81.15.Cd; 85.70.Ge Keywords: Magnetooptical effects; Theory, models, and numerical simulation; Optical constants (refractive index, absorption); Deposition by sputtering; Ferrite and garnet devices
1. Introduction The main interest in BIG films stems from their large magneto-optical Faraday effect [1]. Since 1969 when it was realized that bismuth doping strongly enhances the Faraday rotation (FR) in iron garnets by about 2 deg/ı`m per Bi-atom per chemical formula unit [2], more researching has been done to increase Bi content. Completely substituted bismuth iron garnet Bi3Fe5O12 (BIG) shows the highest Faraday rotation per film thickness in visible light among all known garnet films [1]. Recently Bi3Fe5O12/Y3Fe5O12 and Bi3Fe5O12/Ga3Gd5O12 photonic crystals have been synthesized on a Gd3Ga5O12 substrate [3]. The latest year’s photonic crystals have an expanding niche of photonics. They have a potential for filtering, switching and light guiding [4]. There have been several techniques explored to grow BIG films on single crystal substrates. Okuda et al. [1] prepared BIG films by reactive ion beam sputtering, RIBS. Pulsed laser deposition, PLD, showed
higher deposition rate compared to RIBS [5]. BIG films can not be grown by liquid phase epitaxy, LPE, because Bi3Fe5O12 is thermodynamically nonequilibrium phase [6]. Rf-magnetron sputtering technique can be used in this case. There are two main advantages to use rf-magnetron sputtering instead of PLD. The first is the possibility to achieve a better transmittance in deposited films. This because sputtering is droplets free and therefore gives a better film surface compared with PLD. The second main advantage is that sputtering generally gives much higher thickness uniformity over a large substrate area. In this regard, it can be considered as an industrial technology. At the same time PLD and rf-sputtering are both reliable in stoichiometry preservation. This paper describes the growth and analysis of magnetooptical spectra of rf-magnetron sputtered BIG films with two different orientations (001) and (111) grown on the corresponding gadolinium gallium garnet Gd3Ga5O12 substrates. 2. Experimental
* Corresponding author. Tel.: +46 8 7904186. E-mail address: [email protected] (P. Johansson). 1 Tel.: +46 8 7904186. 0040-6090/$ - see front matter D 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2005.12.268
Bismuth iron garnet films were grown on GGG(111) and GGG(001) single crystal 5 5 0.5 mm3 substrates by rf-
478
P. Johansson et al. / Thin Solid Films 515 (2006) 477 – 480
around (444) and u-scan were performed for (642) atomic plane. 3. Results and discussion XRD h – 2h scans in Fig. 1 clearly show the sintered BIG films are single phase and exclusively (001) or (111) oriented. Rocking-curves of (004) and (444) Bragg reflections are presented in Fig. 2. The full width at half maximum, FWHM, was calculated for the peaks. As showed in Fig. 2a, the rocking curves are narrower for the BIG(001) film compared to the BIG(111). BIG(111) has FWHM of 0.26(0.08- for GGG) and BIG(001) has FWHM of 0.21- (0.07for GGG). u-scans of (4 0 10) plane for BIG(1.39 Am)/ GGG(001) and of (642) plane for BIG(1.32 Am)/GGG(111) were recorded to examine in-plane texture in grown films relative to the substrate. As showed in inserts in Fig. 2, the reflections of the film replicate those of the substrate appearing at the same u-angle positions. Therefore, XRD patterns undoubtedly exhibit the epitaxial quality of BIG films on GGG substrates. Out-of plane lattice parameters for
Fig. 1. (a) X-ray diffraction (XRD) h – 2h scan in CuKa radiation for 1.39 Amthick Bi3Fe5O12 film grown by rf-magnetron sputtering technique onto single crystal Gd3Ga5O12(001). (b) XRD h – 2h scan for 1.32 Am-thick Bi3Fe5O12 film grown onto Gd3Ga5O12(111) crystal.
magnetron sputtering at 50 W of rf-power applied to 1VV composite (3 Bi2O3 + 5 Fe2O3) target. Details of the target preparation see elsewhere [5]. After reaching the ultimate pressure of 10 6 Torr Ar – O2 gas mixture (5 : 1) was introduced to build up the total pressure of 35 mTorr. The substrate temperature was 525 -C. BIG film deposition rate was ˚ /s. Deposition was finalized by in situ approximately 7.6 A post-annealing at 525 -C in 500 Torr of oxygen atmosphere for 15 min. Films thickness was controlled by the Tencor profilometer. It was 1.39 Am for BIG/GGG(001) and 1.32 Am for BIG/GGG(111) within of 5% accuracy. The X-ray diffractometer (XRD) Siemens D-5000 operating in CuKa ˚ ) was used to define the microradiation (k = 1.54056 A crystalline structure of sintered BIG films. h – 2h scans and rocking curves were recorded for the main Bragg reflections, and u-scans were traced for the oblique planes. For BIG/GGG(001) the rocking curve was scanned around (004) and u-scan was performed for (4 0 10) atomic plane. For BIG/GGG (111) the rocking curve was scanned
Fig. 2. (a) The rocking curves of the (004) Bragg reflections from the Bi3Fe5O12/Gd3Ga5O12 (001) structure. Inset: u-scans of the oblique (4010) planes in BIG/GGG(001). (b) The rocking curves of the (444) Bragg reflections from the Bi3Fe5O12/Gd3Ga5O12 (111) structure. Inset: u-scans of the oblique (642) planes in BIG/GGG(111).
P. Johansson et al. / Thin Solid Films 515 (2006) 477 – 480
479
the films of two different orientations were calculated using the Nelson –Riley [7] extrapolation: ðacosh a0 Þ 1 1 þ ¼ Ccos2 h : ð1Þ a0 sinh h Here a cosh is an interplane distance calculated from the apparent Bragg peak position at 2h and C is a fitting coefficient. The lattice parameter for BIG film on GGG(001) ˚ and for BIG on GGG(111) is was found to be 12.626 T 0.005 A ˚. 12.619 T 0.005 A The magneto-optical Faraday rotation (FR) H as a function of wavelength k was measured using a homemade magnetooptical spectrometer based on fibre-optic unit Ocean Optics PC 2000 (300 –1000 nm, 5.5 nm resolution) [8]. The FR is about the same for the two different orientations: about 5 deg/Am at 677 nm wavelength with the peak value of 24 deg/Am at k = 544 nm, see Fig. 3a. The optical dispersion of BIG films has been studied in transmitted light. The experimental spectra T(k) of the transmittance are presented in Fig. 3b. Experiments show that BIG/GGG(001) films have higher transmittance than BIG/ GGG(111). These data have been fitted to the theoretical expression [9,10]: T¼
Mx P Qx þ Rx2
ð2Þ
where: M = 16s (n 2 + k 2), P = [(n + 1)2 + k 2][(n + 1)(n + s 2) + k 2],
Fig. 4. The normalized absorption plotted versus photon energy. The absorption edge is about 2.2 eV for both orientations of the Bi3Fe5O12 films.
Q R u
[(n 2 1 + k 2)(n 2 s 2 + k 2) 2k 2(s 2 + 1)] 2 cos u k [2(n 2 s 2 + k 2) + (s 2 + 1)(n 2 1 + k 2)] 2 sin u, = [(n 1)2 + k 2][(n 1)(n s 2) + k 2], = 4pnd / k, x = exp( ad), a = 4 pk / k.
=
Here d is the film thickness, n is the refractive index of film, k is the extinction coefficient. The dispersion of refractive index of the substrate s(k) was taken from Ref.[11]. Film thickness d was measured with the profilometer. Surface roughness was taken in account as a Gaussian distribution of film thickness centred at d with the standard deviation r d [10]. Transmittance of the rough film T is expressed as: " 2 # Z 1 1 dV d pffiffiffiffiffiffi exp T¯ ðk; d; rd ; n; k; sÞ ¼ 2 rd rd 2p T ðk; dV; n; k; sÞddV
ð3Þ
The theoretical model compared with experimental data is rather good for wavelength 650– 1000 nm. The refractive index n for BIG films was found to follow the Sellmeier formula: n = A + (k o / k)2 [10] with the coefficients A = 2.315 T 0.1 and 2.41 T 0.1, k o = 423 T 2 and 425 T 2 nm for BIG/GGG(001) and
Fig. 3. (a) Faraday rotation spectra for Bi3Fe5O12 (001) and (111) films grown onto Gd3Ga5O12 crystals. (b) Normalized transmission spectra for BIG films, the theoretical curves were calculated from Eq. (2). The transmittance is higher for BIG(1.39 Am)/GGG(001) than for BIG(1.32Am)/GGG(111).
Fig. 5. Faraday rotation at wavelength 677 nm versus magnetic field in rfmagnetron sputtered BIG/GGG(001) and BIG/GGG(111) films.
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P. Johansson et al. / Thin Solid Films 515 (2006) 477 – 480
BIG/GGG(111), correspondingly. The normalized absorption calculated and plotted versus photon energy in Fig. 4 revealed the absorption edge about 2.2 eV. The magneto-optical Faraday rotation at k = 677 nm was also measured as a function of applied magnetic field (Fig. 5). The homemade pulsed magnetic field tracer enables recording of magneto-optical hysteresis loops up to H = 2 T. The coercive field H c is 101.7 T 2 Oe for BIG/GGG(001) compared to 267.0 T 2 Oe in BIG/GGG(111). 4. Conclusions Bismuth iron garnet (Bi3Fe5O12, BIG) epitaxial films were grown on single crystal (Gd3Ga5O12, GGG) (111) and (001) substrates by rf-magnetron sputtering technique. BIG/ GGG(001) film compared to BIG/GGG(111) shows better magneto-optical performance: lower coercive force and higher optical transmittance.
References [1] T. Okuda, T. Katayama, H. Kobayashi, N. Kobayashi, K. Satoh, H. Yamamoto, J. Appl. Phys. 67 (1990) 4944. [2] C.F. Buhrer, J. Appl. Phys. 40 (1969) 4500. [3] S. Kahl, A.M. Grishin, Appl. Phys. Lett. 84 (2004) 1438; S.I. Khartsev, A.M. Grishin, Appl. Phys. Lett. 87 (2005) 122504. [4] M. Levy, IEEE J. Sel. Top. Quantum Electron. 8 (2002) 1300. [5] N. Adachi, V.P. Denysenkov, S.I. Khartsev, A.M. Grishin, T. Okuda, J. Appl. Phys. 88 (2000) 2734. [6] S. Kahl, A.M. Grishin, S.I. Khartsev, K. Kawano, J.S. Abell, IEEE Trans. Magn. 37 (2001) 2457. [7] B.D Cullity, Elements of X-ray Diffraction, Addison-Wesley, Reading, MA, 1988. [8] S. Kahl, S.I. Khartsev, A.M. Grishin, K. Kawano, G. Kong, R.A. Chakalov, J.S. Abell, J. Appl. Phys. 91 (2002) 9556. [9] R. Swanepoel, J. Phys. E 16 (1983) 1214. [10] S. Kahl, V. Popov, A. Grishin, J. Appl. Phys 94 (2003) 5688. [11] Marvin J. Weber, Handbook of Optical Materials, CRC Press, 2002.