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Production of waveguides in LiF by MeV ion bombardment
Nuclear Instruments and Methods in Physics Research B 240 (2005) 234–238 www.elsevier.com/locate/nimb
Production of waveguides in LiF by MeV ion bombardment J.A.M. Pereira
a,*
, M. Cremona b, S. Pelli c, E.L.A. Macchione a, K. Koide a, S.S. Vasconcelos a, G.C. Righini c
a USP, Institute of Physics, CP 66318, 05315-970 Sa˜o Paulo, SP, Brazil PUC-Rio, Physics Department, CP 38071, 22452-970 Rio de Janeiro, RJ, Brazil Nello Carrara Institute of Applied Physics – CNR, Optoelectronics and Photonics Department, via Panciatichi 64, 50127 Firenze, Italy b
c
Available online 10 August 2005
Abstract Alkali fluorides containing color centers are promising systems for applications in new integrated optical devices like active waveguides and color center lasers. In this work, we report the development of a simple method, based on highenergy ion beam irradiation, to create active waveguides in alkali halide materials. MeV carbon and helium beams at normal incidence were used to irradiate lithium fluoride crystals, with different ion doses varying from 1014 up to 1016 cm2, producing thin colored strips. Irradiated waveguides were also characterized by means of optical absorption spectroscopy in order to obtain the distribution of the color centers induced by the ion beam. The results confirm the feasibility of integrated active devices based on color centers in LiF such as tunable light amplifiers, lasers and hybrid optoelectronic components. 2005 Elsevier B.V. All rights reserved. PACS: 42.82.Et; 42.79.Gn; 61.80.Jh; 61.82.Ms; 61.72.Ww Keywords: Ion implantation; Optoelectronics; Alkali halides
1. Introduction MeV particles impinging a solid deposit their energy within a very short time scale (1017 s), ˚ 2) at rates in small cross sectional areas (50 A ˚ the order of 100 eV/A. As a result, the material experiences very high energy and power densities *
Corresponding author. E-mail address: [email protected] (J.A.M. Pereira).
(TW/cm2) in a well localized solid volume. Electronically excited state recombination leads to a variety of physical phenomena such as sputtering from the surface and creation of defects in the bulk along the projectile path. In the case of ionic solids, ion implantation generates the creation of a region of optically active defects (color centers) with relatively small fluences. Such ion beam modified materials are promising systems for applications in new integrated optical devices.
0168-583X/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2005.06.122
J.A.M. Pereira et al. / Nucl. Instr. and Meth. in Phys. Res. B 240 (2005) 234–238
The production of color centers is accompanied by the local increase of the refractive index that creates a guiding structure. Within this class of materials, lithium fluoride (LiF) offers high photothermal stability of the defects and, for this reason it has been used for fabricating room temperature (RT) color center lasers [1–3]. In addition, due to the peculiar overlapping of the F2 and Fþ 3 color center absorption bands, colored LiF can produce simultaneous emission in the red and green spectral regions [4] when excited with the same pumping wavelength at 450 nm. Recently, the search regarding new technologies for the production of miniaturized devices, such as integrated tunable lasers, has driven several groups to investigate the production of waveguides using low energy electron beam irradiation [5–7]. However, little was done in obtaining the same effect through ion bombardment and only basic studies have been done for keV ions [8]. One of the advantages of the ion implantation procedure over other coloration methods, for instance high energy electrons, is that the path of an ion inside the target is virtually a straight line. Accordingly, a sharper edge between colored and non-colored regions can be produced leading to well defined boundary conditions for light propagation [9]. Unlike coloration by photons, for which the defect density is higher at the surface, irradiation with ions offers a controlled depth distribution of the defects by an appropriate choice of the incident energy. In this work, we report the results of the fabrication of planar waveguides in LiF crystals using ion beam irradiation. The aim of this investigation is to compare the optically active properties of LiF irradiated with different MeV ions at equivalent total deposited energy with different energy deposition densities.
235
diameter aperture placed at the chamber entrance and the irradiation was carried out at normal incidence to the sample crystal. A straight cyclic motion of the sample was made using a step motor mounted on a micrometer sample manipulator. Ion beam irradiation takes place during the back and forth movement of the crystal producing a colored strip (Fig. 1). The ion flux was j = 0.5 · 1013 particles/cm2 s corresponding to a 0.5 nA ion beam current, measured on the target holder. The 4.5 MeV energy for carbon and 1.1 MeV for helium were chosen in order to create about 3 lm deep defect layer, which in principle is suitable to achieve more than one guided mode, necessary for guiding and for a good optical characterization. Depending on the irradiation time, different fluences were deposited on LiF, varying from 4.5 · 1014 up to 2.4 · 1016 cm2. The irradiation time varied from 2 to 30 min. To obtain these results with electron beam irradiation, tenfold longer exposition times are necessary [7,10]. The irradiated samples were analysed by optical absorption spectroscopy. The results for helium irradiation were quite similar to those reported in a previous paper [9]. For carbon irradiation a more complex band structure was found especially in the region of the F2 absorption band around 3.0 eV (Fig. 2). The de-convolution of the spectra showed four different components due to the F2 centers and to aggregate centers F3 and F4. The fourth component, labelled as F2[C], is attributed to the presence of implanted carbon in the LiF lattice. One can see in Fig. 2, that the F2[C] band leads to a shift in the absorption energy of the colored region. The energies and widths of the components, together with their relative contributions
2. Experimental The irradiations were carried out with 1.1 MeV He+ and 4.5 MeV C3+ collimated beams from a TANDEM electrostatic accelerator available at Laborato´rio de Ana´lise de Materiais por Feixes Ioˆnicos (LAMFI) at University of Sa˜o Paulo, Brazil. The ion beam was collimated by a 0.30 mm
Fig. 1. Schematics of the irradiation experimental set-up.
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J.A.M. Pereira et al. / Nucl. Instr. and Meth. in Phys. Res. B 240 (2005) 234–238
Fig. 3. Experimental set-up for the coupling of light to optical waveguides by prism-coupling technique.
Fig. 2. Absorption band of LiF irradiated by carbon at doses of (a) 4.5 · 1014 cm2 and (b) 6.0 · 1015 cm2. The arrows locate the positions of the centres F4, F2, F2[C] and F3.
are presented in Table 1 for the two considered ion fluences. The characteristics of each component agrees with tabulated values for the F2, F3 and F4 bands in LiF [11]. The relative contributions of the bands are energy loss and dose dependent. The irradiated strips were tested to detect the existence of guided modes. The experimental setup is shown in Fig. 3. Prism-coupling with a tent-shaped prism was used to measure the effective indices of the modes in order to assess the
Table 1 Values of the energies and widths of each component, together with their relative contributions for the two considered carbon ion fluences Band
Energy (eV)
Width (eV)
/ = 4.5 · 1014 cm2 (%)
/ = 6.0 · 1015 cm2 (%)
F4 F2 F2[C] F3
2.47 2.79 3.00 3.18
0.30 0.12 0.075 0.27
14 22 41 23
18 31 11 23
refractive index change induced by irradiation [12]. At each resonance input angle, light is coupled into the waveguide and correspondingly part of the light does not appear in the reflected beam, originating a so called dark line, i.e. a narrow dark line cutting in half the reflected spot. Dark lines, which are a proof of the physical conditions for guided propagation, were observed for helium irradiated strips. No such propagation modes could be measured in the case of carbon irradiation due to the presence of the aggregate centers. The associated absorption bands lead to high attenuation of the input light signal. For the helium irradiation, the values of TE = 1.3934 and TM = 1.3937, characterizing the waveguide, could be calculated directly from the measurement of the coupling angle, refractive index and base angle of the prism for helium irradiation.
3. Discussion One important parameter governing such experiments is the ion energy loss that is depicted in Fig. 4(a) as a function of the ion impinging kinetic energy for carbon and helium projectiles [13]. At the MeV energy range, the major part of the kinetic energy of the projectile is transferred to target electrons by creating excitations and ionizations. Helium ions at energy E0 = 1.1 MeV
J.A.M. Pereira et al. / Nucl. Instr. and Meth. in Phys. Res. B 240 (2005) 234–238
Fig. 4. (a) Energy loss of He and C ions in LiF crystals as a function of the initial kinetic energy. The energy E0 = 1.1 MeV of the He beam and E0 = 4.5 MeV of C used for the LiF irradiation is shown. (b) Energy loss of He and C projectiles with 1.1 MeV and 4.5 MeV initial energy respectively, as function of the travelled distance inside the solid.
loose in average 40 eV for each angstrom travelled in the near surface region of a LiF crystal. Correspondingly, carbon has a greater energy loss and ˚ to the electronic at 4.5 MeV can transfer 160 eV/A system of LiF. In both cases, this amount of energy is high enough to produce electron–hole pairs that are the precursors of point defects such as F and H centers. For the helium ion particular case, the maximum energy loss occurs at 700 keV which is less than the initial kinetic energy given by the accelerator. Hence, the energy loss increases as the projectile penetrates the solid and reaches its maximum after the projectile travels a distance of 1 lm, as indicated in Fig. 4(a) and (b). Likewise it occurs for carbon but with higher kinetic energy and energy loss values. The depth profile of electronically induced defects should follow the energy loss profile as shown in Fig. 4(b). For both cases, it is roughly constant over the first 2 lm and then it decays more or less linearly until 3.5 lm. In fact, both energy loss profiles have quite the same behavior. The difference is that carbon deposits 4 times more energy than helium leading to a higher density of energy deposition within the same irradiated volume. The irradiation was carried out in a way that the carbon ion fluence was always one-fourth of the helium ion fluence so the total deposited energy is nearly the same for both irradiations. It is well known that ion irradiation is very effective in producing color centers in solids. However, the coloration intensity is not the only
237
requirement for the production of a good quality waveguide. There is a compromise between the refraction index change, coloration depth and absorption coefficient that must be achieved all together if the irradiated material is to be used for a given application. In the case of the present issue, the depth penetration of the ions should be fixed and this determines the particle beam energy. The energy loss of the ions and so the absorption coefficient, must be varied at constant depth. The only way to do this is to vary the atomic number of the impinging projectiles. The main finding of the work is that the carbon energy loss, at the required energy, is too high to the purpose of fabricating guiding structures, despite the chemical shift occurring in the absorption spectrum due to its implantation in LiF.
4. Conclusions Optical planar waveguides in LiF crystals were measured for helium and carbon MeV ion implantation. The results obtained for helium confirms the feasibility of integrated active devices based on color centers in LiF as found in our previous work [9]. Carbon irradiation produces aggregate defects that disturb light propagation in the waveguide. Besides, carbon produces unwanted chemical reactions creating shifts in the position of the absorption bands. For the present purpose, we conclude that the ion implantation technique works fine for light ion irradiation for which a good balance between coloration depth, refractive index change and light absorption is achieved. On the other hand, heavy ion irradiation induces too much aggregate centers due to the higher density of energy deposition per impact. As a result, in heavy ion generated guiding structures, light absorption does not allow the production of good quality waveguides.
Acknowledgments The authors would like to acknowledge the financial support from FAPESP, CNPq (Brazil) and CNR (Italy).
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J.A.M. Pereira et al. / Nucl. Instr. and Meth. in Phys. Res. B 240 (2005) 234–238
References [1] H.E. Gu, L.F. Wan, H.S. Guo, Opt. Commun. 70 (1989) 141. [2] T.T. Basiev, S.B. Mirov, V.V. Osiko, IEEE J. Quantum Electron. 24 (6) (1988) 1052. [3] T. Tsuboi, H.E. Gu, Appl. Opt. 33 (6) (1994) 982. [4] L.X. Zheng, L.F. Wan, Opt. Commun. 55 (1985) 277. [5] G. Baldacchini et al., Opt. Commun. 94 (1992) 139. [6] R.M. Montereali et al., Nucl. Instr. and Meth. B 166–167 (2000) 764.
[7] R.M. Montereali, A. Mancini, G.C. Righini, S. Pelli, Opt. Commun. 153 (1998) 223. [8] Q. Yan et al., Nucl. Instr. and Meth. B 115 (1996) 415. [9] M. Cremona, J.A.M. Pereira, S. Pelli, G.C. Righini, Appl. Phys. Lett. 81 (2002) 4103. [10] R.M. Montereali, M. Piccini, E. Burattini, Appl. Phys. Lett. 78 (2001) 4082. [11] D.W. Fowler, Physics of Color Centers, Academic Press Inc., 1968. [12] R. Ulrich, R. Torge, Appl. Opt. 12 (1973) 2901. [13] J.F. Ziegler, J.P. Biersack, U. Littmark, TRIM code.
Production of waveguides in LiF by MeV ion bombardment J.A.M. Pereira
a,*
, M. Cremona b, S. Pelli c, E.L.A. Macchione a, K. Koide a, S.S. Vasconcelos a, G.C. Righini c
a USP, Institute of Physics, CP 66318, 05315-970 Sa˜o Paulo, SP, Brazil PUC-Rio, Physics Department, CP 38071, 22452-970 Rio de Janeiro, RJ, Brazil Nello Carrara Institute of Applied Physics – CNR, Optoelectronics and Photonics Department, via Panciatichi 64, 50127 Firenze, Italy b
c
Available online 10 August 2005
Abstract Alkali fluorides containing color centers are promising systems for applications in new integrated optical devices like active waveguides and color center lasers. In this work, we report the development of a simple method, based on highenergy ion beam irradiation, to create active waveguides in alkali halide materials. MeV carbon and helium beams at normal incidence were used to irradiate lithium fluoride crystals, with different ion doses varying from 1014 up to 1016 cm2, producing thin colored strips. Irradiated waveguides were also characterized by means of optical absorption spectroscopy in order to obtain the distribution of the color centers induced by the ion beam. The results confirm the feasibility of integrated active devices based on color centers in LiF such as tunable light amplifiers, lasers and hybrid optoelectronic components. 2005 Elsevier B.V. All rights reserved. PACS: 42.82.Et; 42.79.Gn; 61.80.Jh; 61.82.Ms; 61.72.Ww Keywords: Ion implantation; Optoelectronics; Alkali halides
1. Introduction MeV particles impinging a solid deposit their energy within a very short time scale (1017 s), ˚ 2) at rates in small cross sectional areas (50 A ˚ the order of 100 eV/A. As a result, the material experiences very high energy and power densities *
Corresponding author. E-mail address: [email protected] (J.A.M. Pereira).
(TW/cm2) in a well localized solid volume. Electronically excited state recombination leads to a variety of physical phenomena such as sputtering from the surface and creation of defects in the bulk along the projectile path. In the case of ionic solids, ion implantation generates the creation of a region of optically active defects (color centers) with relatively small fluences. Such ion beam modified materials are promising systems for applications in new integrated optical devices.
0168-583X/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2005.06.122
J.A.M. Pereira et al. / Nucl. Instr. and Meth. in Phys. Res. B 240 (2005) 234–238
The production of color centers is accompanied by the local increase of the refractive index that creates a guiding structure. Within this class of materials, lithium fluoride (LiF) offers high photothermal stability of the defects and, for this reason it has been used for fabricating room temperature (RT) color center lasers [1–3]. In addition, due to the peculiar overlapping of the F2 and Fþ 3 color center absorption bands, colored LiF can produce simultaneous emission in the red and green spectral regions [4] when excited with the same pumping wavelength at 450 nm. Recently, the search regarding new technologies for the production of miniaturized devices, such as integrated tunable lasers, has driven several groups to investigate the production of waveguides using low energy electron beam irradiation [5–7]. However, little was done in obtaining the same effect through ion bombardment and only basic studies have been done for keV ions [8]. One of the advantages of the ion implantation procedure over other coloration methods, for instance high energy electrons, is that the path of an ion inside the target is virtually a straight line. Accordingly, a sharper edge between colored and non-colored regions can be produced leading to well defined boundary conditions for light propagation [9]. Unlike coloration by photons, for which the defect density is higher at the surface, irradiation with ions offers a controlled depth distribution of the defects by an appropriate choice of the incident energy. In this work, we report the results of the fabrication of planar waveguides in LiF crystals using ion beam irradiation. The aim of this investigation is to compare the optically active properties of LiF irradiated with different MeV ions at equivalent total deposited energy with different energy deposition densities.
235
diameter aperture placed at the chamber entrance and the irradiation was carried out at normal incidence to the sample crystal. A straight cyclic motion of the sample was made using a step motor mounted on a micrometer sample manipulator. Ion beam irradiation takes place during the back and forth movement of the crystal producing a colored strip (Fig. 1). The ion flux was j = 0.5 · 1013 particles/cm2 s corresponding to a 0.5 nA ion beam current, measured on the target holder. The 4.5 MeV energy for carbon and 1.1 MeV for helium were chosen in order to create about 3 lm deep defect layer, which in principle is suitable to achieve more than one guided mode, necessary for guiding and for a good optical characterization. Depending on the irradiation time, different fluences were deposited on LiF, varying from 4.5 · 1014 up to 2.4 · 1016 cm2. The irradiation time varied from 2 to 30 min. To obtain these results with electron beam irradiation, tenfold longer exposition times are necessary [7,10]. The irradiated samples were analysed by optical absorption spectroscopy. The results for helium irradiation were quite similar to those reported in a previous paper [9]. For carbon irradiation a more complex band structure was found especially in the region of the F2 absorption band around 3.0 eV (Fig. 2). The de-convolution of the spectra showed four different components due to the F2 centers and to aggregate centers F3 and F4. The fourth component, labelled as F2[C], is attributed to the presence of implanted carbon in the LiF lattice. One can see in Fig. 2, that the F2[C] band leads to a shift in the absorption energy of the colored region. The energies and widths of the components, together with their relative contributions
2. Experimental The irradiations were carried out with 1.1 MeV He+ and 4.5 MeV C3+ collimated beams from a TANDEM electrostatic accelerator available at Laborato´rio de Ana´lise de Materiais por Feixes Ioˆnicos (LAMFI) at University of Sa˜o Paulo, Brazil. The ion beam was collimated by a 0.30 mm
Fig. 1. Schematics of the irradiation experimental set-up.
236
J.A.M. Pereira et al. / Nucl. Instr. and Meth. in Phys. Res. B 240 (2005) 234–238
Fig. 3. Experimental set-up for the coupling of light to optical waveguides by prism-coupling technique.
Fig. 2. Absorption band of LiF irradiated by carbon at doses of (a) 4.5 · 1014 cm2 and (b) 6.0 · 1015 cm2. The arrows locate the positions of the centres F4, F2, F2[C] and F3.
are presented in Table 1 for the two considered ion fluences. The characteristics of each component agrees with tabulated values for the F2, F3 and F4 bands in LiF [11]. The relative contributions of the bands are energy loss and dose dependent. The irradiated strips were tested to detect the existence of guided modes. The experimental setup is shown in Fig. 3. Prism-coupling with a tent-shaped prism was used to measure the effective indices of the modes in order to assess the
Table 1 Values of the energies and widths of each component, together with their relative contributions for the two considered carbon ion fluences Band
Energy (eV)
Width (eV)
/ = 4.5 · 1014 cm2 (%)
/ = 6.0 · 1015 cm2 (%)
F4 F2 F2[C] F3
2.47 2.79 3.00 3.18
0.30 0.12 0.075 0.27
14 22 41 23
18 31 11 23
refractive index change induced by irradiation [12]. At each resonance input angle, light is coupled into the waveguide and correspondingly part of the light does not appear in the reflected beam, originating a so called dark line, i.e. a narrow dark line cutting in half the reflected spot. Dark lines, which are a proof of the physical conditions for guided propagation, were observed for helium irradiated strips. No such propagation modes could be measured in the case of carbon irradiation due to the presence of the aggregate centers. The associated absorption bands lead to high attenuation of the input light signal. For the helium irradiation, the values of TE = 1.3934 and TM = 1.3937, characterizing the waveguide, could be calculated directly from the measurement of the coupling angle, refractive index and base angle of the prism for helium irradiation.
3. Discussion One important parameter governing such experiments is the ion energy loss that is depicted in Fig. 4(a) as a function of the ion impinging kinetic energy for carbon and helium projectiles [13]. At the MeV energy range, the major part of the kinetic energy of the projectile is transferred to target electrons by creating excitations and ionizations. Helium ions at energy E0 = 1.1 MeV
J.A.M. Pereira et al. / Nucl. Instr. and Meth. in Phys. Res. B 240 (2005) 234–238
Fig. 4. (a) Energy loss of He and C ions in LiF crystals as a function of the initial kinetic energy. The energy E0 = 1.1 MeV of the He beam and E0 = 4.5 MeV of C used for the LiF irradiation is shown. (b) Energy loss of He and C projectiles with 1.1 MeV and 4.5 MeV initial energy respectively, as function of the travelled distance inside the solid.
loose in average 40 eV for each angstrom travelled in the near surface region of a LiF crystal. Correspondingly, carbon has a greater energy loss and ˚ to the electronic at 4.5 MeV can transfer 160 eV/A system of LiF. In both cases, this amount of energy is high enough to produce electron–hole pairs that are the precursors of point defects such as F and H centers. For the helium ion particular case, the maximum energy loss occurs at 700 keV which is less than the initial kinetic energy given by the accelerator. Hence, the energy loss increases as the projectile penetrates the solid and reaches its maximum after the projectile travels a distance of 1 lm, as indicated in Fig. 4(a) and (b). Likewise it occurs for carbon but with higher kinetic energy and energy loss values. The depth profile of electronically induced defects should follow the energy loss profile as shown in Fig. 4(b). For both cases, it is roughly constant over the first 2 lm and then it decays more or less linearly until 3.5 lm. In fact, both energy loss profiles have quite the same behavior. The difference is that carbon deposits 4 times more energy than helium leading to a higher density of energy deposition within the same irradiated volume. The irradiation was carried out in a way that the carbon ion fluence was always one-fourth of the helium ion fluence so the total deposited energy is nearly the same for both irradiations. It is well known that ion irradiation is very effective in producing color centers in solids. However, the coloration intensity is not the only
237
requirement for the production of a good quality waveguide. There is a compromise between the refraction index change, coloration depth and absorption coefficient that must be achieved all together if the irradiated material is to be used for a given application. In the case of the present issue, the depth penetration of the ions should be fixed and this determines the particle beam energy. The energy loss of the ions and so the absorption coefficient, must be varied at constant depth. The only way to do this is to vary the atomic number of the impinging projectiles. The main finding of the work is that the carbon energy loss, at the required energy, is too high to the purpose of fabricating guiding structures, despite the chemical shift occurring in the absorption spectrum due to its implantation in LiF.
4. Conclusions Optical planar waveguides in LiF crystals were measured for helium and carbon MeV ion implantation. The results obtained for helium confirms the feasibility of integrated active devices based on color centers in LiF as found in our previous work [9]. Carbon irradiation produces aggregate defects that disturb light propagation in the waveguide. Besides, carbon produces unwanted chemical reactions creating shifts in the position of the absorption bands. For the present purpose, we conclude that the ion implantation technique works fine for light ion irradiation for which a good balance between coloration depth, refractive index change and light absorption is achieved. On the other hand, heavy ion irradiation induces too much aggregate centers due to the higher density of energy deposition per impact. As a result, in heavy ion generated guiding structures, light absorption does not allow the production of good quality waveguides.
Acknowledgments The authors would like to acknowledge the financial support from FAPESP, CNPq (Brazil) and CNR (Italy).
238
J.A.M. Pereira et al. / Nucl. Instr. and Meth. in Phys. Res. B 240 (2005) 234–238
References [1] H.E. Gu, L.F. Wan, H.S. Guo, Opt. Commun. 70 (1989) 141. [2] T.T. Basiev, S.B. Mirov, V.V. Osiko, IEEE J. Quantum Electron. 24 (6) (1988) 1052. [3] T. Tsuboi, H.E. Gu, Appl. Opt. 33 (6) (1994) 982. [4] L.X. Zheng, L.F. Wan, Opt. Commun. 55 (1985) 277. [5] G. Baldacchini et al., Opt. Commun. 94 (1992) 139. [6] R.M. Montereali et al., Nucl. Instr. and Meth. B 166–167 (2000) 764.
[7] R.M. Montereali, A. Mancini, G.C. Righini, S. Pelli, Opt. Commun. 153 (1998) 223. [8] Q. Yan et al., Nucl. Instr. and Meth. B 115 (1996) 415. [9] M. Cremona, J.A.M. Pereira, S. Pelli, G.C. Righini, Appl. Phys. Lett. 81 (2002) 4103. [10] R.M. Montereali, M. Piccini, E. Burattini, Appl. Phys. Lett. 78 (2001) 4082. [11] D.W. Fowler, Physics of Color Centers, Academic Press Inc., 1968. [12] R. Ulrich, R. Torge, Appl. Opt. 12 (1973) 2901. [13] J.F. Ziegler, J.P. Biersack, U. Littmark, TRIM code.