Highly confined and continuous single-mode operation of self-align photonic crystal oxide VCSEL

Transcription

1 Appl Phys B (2010) 100: DOI /s Highly confined and continuous single-mode operation of self-align photonic crystal oxide VCSEL M.S. Alias S. Shaari P.O. Leisher K.D. Choquette Received: 21 November 2009 / Revised version: 8 February 2010 / Published online: 21 May 2010 Springer-Verlag 2010 Abstract In this letter, we report on single-mode operation of originally multi-mode oxide VCSEL by using etched photonic crystal air holes and unique trench structure. The device fabrication utilized conventional photolithography; with simplified lithography step of self-aligning the photonic crystal and trench structures to the laser aperture for efficient and vigorous device processing. The fabricated photonic crystal VCSEL with trench device exhibits a singlemode output power of 0.7 mw, threshold current of 3.5 ma, slope efficiency of 0.10 W/A, and continuous single-mode output spectra at wide operating current range. The results are compared with conventional multi-mode oxide VCSEL of similar device geometry. In addition, theoretical analysis is presented for developing further understanding of the photonic crystal VCSEL. M.S. Alias ( ) Microelectronics & Nanotechnology Program, Telekom Malaysia Research & Development (TMR&D), Lingkaran Teknokrat Timur, Cyberjaya, Selangor, Malaysia sharizal@tmrnd.com.my Fax: S. Shaari Institute of Microengineering & Nanoelectronics (IMEN), National University of Malaysia (UKM), Bangi, Selangor, Malaysia P.O. Leisher nlight Corporation, 5408 NE 88th St., Bldg. E, Vancouver, WA 98665, USA K.D. Choquette Electrical & Computer Engineering Department, University of Illinois at Urbana Champaign (UIUC), Urbana, IL 61801, USA 1 Introduction Vertical-Cavity Surface-Emitting (VCSEL) are becoming primary light source for short data communication with the applications evolving into long wavelength optical telecommunication, optical interconnect, automotive network, optical mouse, gas spectroscopy, and high definition display [1 3]. The promising growth is due to VCSEL device superiorities of low operating current and power consumption, high fiber coupling efficiency, high modulation bandwidth, two-dimensional (2-D) arrays capability and most importantly the low-cost manufacturing ability (on wafer test and similar microelectronics device processing). Single transverse mode operation of VCSEL is substantial for high coupling efficiency into a single-mode fiber which has small core diameter or for coupling into multimode fiber [4]. Standard multi-mode VCSEL at the moment used µm oxide aperture technique for indexand current-confinement scheme [5, 6]. In order to attain single transverse mode operation, the oxide aperture of VC- SEL must be smaller than 4 µm [7]. This rigorous requirement leads to low power output due to small oxide aperture, high optical losses due to scattering, and introduce high resistance, which reduces the VCSEL high speed ability. Lately, various methods have been reported for achieving single transverse mode VCSEL by introducing losses to the higher-order modes such as proton implanted structure [8], surface-relief etching [9 11], hybrid oxide-implant VCSEL [12], holey structure [13 15], and photonic crystal air holes [16 25]. In this letter, we report on single-mode operation of originally multi-mode oxide VCSEL by using etched photonic crystal air holes and trench structure. The photonic crystal VCSEL (PhC VCSEL) with trench device fabrication utilized conventional photolithography with simplified

2 454 M.S. Alias et al. Fig. 1 Schematic of a PhC VCSEL with trench cross section, and b PhC design top view self-align lithography step to ensure efficient and vigorous device processing. Also, we presented theoretical analysis (PhC model and mode profiles) based on the exact fabricated device properties. To the best of our knowledge, this is a first all-inclusive report on experimental work and theoretical analysis for this kind of PhC VCSEL (with trench and self-align photolithography process). Almost all of the previous reported PhC VCSEL is based on the air post mesa device structure [16 20] and used a complicated lithography process such as electron beam lithography [13, 17, 25, 26], focus ion beam lithography [19, 20], and chemically assisted ion beam lithography [18, 27]. 2 PhC VCSEL design and experiment Figure 1(a) shows the schematic device cross section of 850 nm PhC oxide VCSEL with trench structure. The VC- SEL device consists of an active region of three GaAs quantum wells and four Al 0.3 Ga 0.7 As barriers, embedded between 34.5 periods of bottom n-type and 25 periods of top p-type DBR mirrors. These DBRs composed of alternating high- and low-refractive index layers of Al 0.15 Ga 0.85 As/Al 0.9 Ga 0.1 As multilayer. The oxidation layer is formed by a high aluminum concentration layer of Al 0.98 Ga 0.02 As. The n- and p-contact are materialized by metal alloys of AuGe/Ni/Au and Ti/Au, respectively. Figure 1(b) illustrates the top view of the triangular lattice constant patterns with a single point defect at the center, used for the PhC design. Single-mode operation in PhC VCSEL can be realized similar to PhC optical fiber [28] by creating 2-D arrays of air holes onto the top DBR mirror. The PhC structure creates a step refractive index profile across the VCSEL device (perpendicular to the wafer plane). The PhC design parameters are connected by the triangular lattice constant, a, and air hole diameter, b. The transverse index guiding around the single defect point can be controlled by the air hole diameter-to-lattice constant ratio (b/a). We fixed the b/a ratio at 0.5 by defining a = 4µm and b = 2µm. Figure 2 summarizes the main stages during the PhC VC- SEL with trench device fabrication. The device fabrication started with metalization of the bottom n- and the top ring p- ohmic contacts by evaporating AuGe/Ni/Au (40/20/150 nm) and Ti/Au (15/150 nm), respectively. The top p-contact required photolithography patterning and lift-off process; however the bottom n-contact is a straightforward evaporation. The surface of the VCSEL epiwafer is proton implanted for device isolation. SiO 2 mask is deposited by the process of plasma-enhanced chemical vapor deposition (PECVD) followed by thick photoresist pillars coating on all VCSEL apertures for device protection prior to implantation. The photoresist is patterned by photolithography and SiO 2 mask is etched by reactive ion etching (RIE) for implantation. After implantation, the sample is cleaned by O 2 plasma and etched by RIE. Next, thick SiO 2 mask is redeposited by PECVD. PhC holes and trench patterning (selfaligned) are then transferred by single-step photolithography to SiO 2 mask followed by RIE mask etched. The leftover photoresist is removed and thicker photoresist pillars are re-coated on top ring contacts to protect the PhC holes mask patterns prior to trench etching by inductively coupled plasma RIE (ICP-RIE). After trench is partly etched, the thick photoresist pillars are removed and sample is cleaned by O 2 plasma. The epiwafer is then exposed to wet oxidation to form the oxide aperture. Next, PhC holes are etched onto the VCSEL aperture by second ICP-RIE process (trench will completely etched at the same time) followed by SiO 2 mask removal by RIE. The PhC VCSEL with trench device fab-

3 Highly confined and continuous single-mode operation of self-align photonic crystal oxide VCSEL 455 Fig. 2 PhC VCSEL with trench device fabrication process of a PhC and trench patterns are transferred in single-step photolithography (self-aligned) to the SiO 2 mask, b first stage ICP-RIE for partial trench etching, and c second stage ICP-RIE for PhC holes etching and complete trench etching (after oxide layer is formed by wet oxidation process) rication is completed by defining the fan-pad metal (formed by photolithography, Ti/Au of 25/1000 nm evaporation and metal lift-off). Typical oxide VCSEL with trench but without etched PhC holes is also fabricated (almost similar device processing but exclude PhC holes processing steps) to enable device comparison. A self-aligned technique is utilized for aligning the PhC holes and trench structure by single-step photolithography. This approach allowed the PhC holes to be aligned automatically to the laser aperture with high accuracy for simplicity and reproducibility device processing. Figure 3 demonstrates the schematic top view of the PhC and trench singlestep photomask (single step) and top ring contact photomask (separate step) along with a schematic cross section of a completed PhC VCSEL with trench device. By subtracting the inner diameter of the trench pattern (43 µm) with outer diameter of the top ring contact pattern (39 µm), it will give 4 µm alignment tolerance (2 µm at each half-side) which is sufficient for photolithography alignment. The device experimental measurement is performed on wafer probing system at room temperature under continuous wave operation. The light versus current (L I ) and current versus voltage (I V ) characteristics are measured with precise variation of input current (using semiconductor parameter analyzer). The light output is detected by a silicon photodetector which also is connected to the semiconductor parameter analyzer. The output spectrum of VCSEL is obtained by coupling the lasing beam into an optical fiber connected to an optical spectrum analyzer. The theoretical measurement is analyzed using the Plane Wave Expansion (PWE) method [29] and Finite Difference Frequency Domain (FDFD) method [30]. 3 Results and discussion Figure 4(a) shows the scanning electron microscope (SEM) image for the fabricated PhC VCSEL with trench of b/a = Fig. 3 Schematic of self-align PhC & trench photomask along the fabricated PhC VCSEL with trench 0.5 (a = 4 µm and b = 2 µm). The PhC holes are perfectly etched onto the VCSEL top DBR with the unetched defect point at the center. The circular trench structure surrounding the VCSEL mesa is also well etched. It can be noticed that the PhC and trench pattern (single-step photomask) is slightly misaligned to the right of the top ring contact (initial step photomask). However, the PhC holes area is still within the VCSEL aperture (inner area of the top ring contact) due to sufficient alignment tolerance. This result shows the significance of the self-align technique used in order to ensure simplified, reproducible and vigorous PhC VCSEL with trench fabrication. Figures 4(b) and 4(c) shows the mi-

4 456 M.S. Alias et al. Fig. 4 Top view microscopy images of a tilted SEM, b PhC with trench single-step patterning, and c fabricated device (prior to fan-pad metalization) Fig. 5 L I V characteristic of a PhC VCSEL with trench, and b typical VCSEL with trench croscope image of PhC and trench single-step pattern, and fabricated PhC VCSEL device with trench, respectively. Figure 5(a) shows the L I V characteristic for PhC VC- SEL with trench of b/a = 0.5 (a = 4 µm and b = 2µm) and oxide aperture diameter of 9 µm. The fabricated device achieved threshold current, I th of 3.5 ma, slope efficiency of 0.10 W/A, and output power, P max of 0.7 mw. The L I V characteristic for the control device of oxide trench VCSEL (without PhC holes) with similar oxide aperture diameter of 9 µm is shown in Fig. 5(b) for comparison. The device exhibited I th of 0.8 ma, slope efficiency of 0.29 W/A, and P max of 3.0 mw. The high I th and low slope efficiency in PhC VCSEL with trench is because higher optical threshold gain is required to compensate high-order mode loss [18]. In addition, the etched PhC holes increased scattering loss [26], which also results in higher threshold current is required to ensure lasing. Also, PhC holes might lead to non-radiative recombination of the injected current [31]. Output power for PhC VCSEL with trench is low compared to control device since only fundamental mode is allowed for lasing and the laser emitting area is reduced due to the existence of PhC holes structure. However, the power composed of genuine single-mode operation. The output power can be improved by optimizing device parameters such as oxide aperture, etch depth and PhC design. From the slope of I V characteristic, the differential series resistance, R s of PhC VCSEL with trench is approximately 124. The control Fig. 6 Output spectrum of PhC VCSEL with trench and typical VC- SEL with trench device R s is approximately 87. The high series resistance of PhC VCSEL with trench is due to the current flow resistance induced by PhC holes and the etched trench structure. The etching of PhC holes partially removed the conductive p-doped material of the top DBR mirror. Figure 6 exhibits output spectrum for both PhC VCSEL with trench and control device above threshold. By combining PhC holes with trench structure in VCSEL, highly confined single-mode condition is obtained. Single-mode condition is defined if more than 30 db difference in side mode suppression ratio (SMSR) is achieved from threshold to rollover lasing operation. The PhC VCSEL with trench demonstrates approximately 35 db of SMSR. Also, it is ob-

5 Highly confined and continuous single-mode operation of self-align photonic crystal oxide VCSEL 457 Fig. 7 Output spectrum for a typical VCSEL, and b PhC VCSEL; at various injected current Fig. 9 PBG diagram of PhC structure with out-of-plane propagation for b/a = 0.5 (a = 4µm&b = 2µm) Fig. 8 Near-field comparison for a trench VCSEL, and b PhC VCSEL with trench served that the lasing wavelength blue-shifted when PhC holes and trench are incorporated into the VCSEL. This effect could be donated by the change of VCSEL cavity resonance due to modification of top DBR refractive index when trench and PhC holes are etched into the semiconductors. Figure 7 exhibits the output spectrum for both VCSEL devices at increasing injected currents. PhC VCSEL with trench achieved single-mode condition (SMSR > 30 db) and narrow linewidth throughout the operating current range (above threshold current). No distinguish side modes appear at even high operating current. The typical oxide VCSEL shows multimode spectrum from low to high injection current levels. Several transverse modes existed within the operating wavelength range. The spectrum also exhibits broadening linewidth as the injected current increased. The peak lasing wavelength is red-shifted for both devices as injected current increased due to VCSEL self-heating effect (due to high number of heterojunctions). The near-field optical image comparison for both VCSEL devices is shown in Fig. 8. The PhC VCSEL with trench structure demonstrates superior laser beam confinement at the center of the PhC structure (no defect point) throughout the operating current range. Higher-order modes are discriminated and only fundamental mode is allowed for lasing. As for the trench VCSEL without PhC holes, the laser beam is distributed across the entire VCSEL aperture. Due to this, the trench VCSEL exhibits a higher output power compared to PhC VCSEL. Figure 9 shows the out-of-plane photonic band diagram (PBG) computed using PWE method for PhC triangular lattice design of b/a = 0.5 (a = 4 µm and b = 2 µm), with assumption of infinite holes etched and single point defect. This is a simplified approximation to describe the optical phenomena inside the VCSEL top DBR that has been etched with PhC holes. The area under the linear curve is defined by a line which refers to the lowest order out-of-plane mode for the PhC holes design. The slope of this line is the effective index of the structure formed by the PhC holes which induced the index guiding effect. By purposely introducing a point defect at center of the index guiding region (unetched hole), lateral index waveguiding is realized. In case of PhC VCSEL, the effective index represents the refractive index of top DBR region etched with PhC structure, n clad.therefractive index for the unetched region, n core is obtained by averaging the index of DBR materials used. This approach is similar to step index waveguide in optical fiber. Thus, the normalized frequency or V eff parameter [32] can be used to determine the number of confined modes (single mode expected if V eff < 2.405). However in PhC fiber, the holes are etched completely (infinite). In PhC VCSEL, the top DBR mirror is only 50 80% etched with holes due to fabrication limitation and to avoid high optical losses. Due to this (finite etch depth), the V eff parameter is modified by introducing

6 458 M.S. Alias et al. Fig. 10 Comparison of LP mode profiles for single and multi-mode PhC VCSEL with trench etch depth factor, γ as shown in (1)[33], where D is the defect diameter (2a b), λ is the operating wavelength, n clad and n core as described above. V eff = πd n λ 2 core [ ] 2 (1 γ)n core + γn clad (1) In order to compute the V eff parameter for the given PhC design, D is calculated from the given a and b dimension, λ is at 850 nm, n core is , n clad is the calculated effective index, and γ is assumed as 0.1 (19 DBR pairs etched) [33]. The calculated V eff parameter for the PhC design of b/a = 0.5 (a = 4 µm and b = 2 µm) is which indicate a single-mode behavior (V eff < 2.405). By incorporating this PhC design into VCSEL, single-mode PhC VCSEL operation is expected. In contrast, by calculation of V eff parameter for multi-mode PhC design of b/a = 0.5 (a = 5.5µm and b = 2.8 µm) we obtain 2.517, which shows us to operate at multi-mode condition (V eff > 2.405). Figure 10 exhibits the computed PhC guided modes based on the fabricated PhC VCSEL with trench structure (b/a ratio = 0.5) by using FDFD method. Comparison is made between a device that achieved single-mode operation (a = 4 µm and b = 2 µm) and device that shows multi-mode operation (a = 5.5 µm and b = 2.8 µm). The computation shows the existence of four guided modes, which are classified using the linearly polarized (LP) modes approximation since PhC VCSEL is a weak guiding structure. In PhC fiber, only fundamental guided mode is allowed because of high refractive index contrast (since the PhC hole is completely etched) and large PhC holes size. In case of PhC VCSEL, it has a weak guiding structure due to the smaller refractive index contrast (PhC hole is partially etched) and smaller PhC holes size. In LP 01 mode of Fig. 10, the single-mode device shows a well confined fundamental mode within the PhC holes structure and the oxide aperture. Most of the PhC holes structure for single-mode PhC design is overlapping inside the oxide aperture (only slight edges outside the oxide aperture), thus inducing strong PhC guiding effect. The PhC guiding effect is very minimal for the multi-mode device since only a small part of the PhC holes overlap within inside the oxide aperture. The mode distribution area of multi-mode PhC design is around 54.1 µm 2 (considering only 0.35 µm out of 2.8 µm diameter for each PhC hole structure is overlapping inside the oxide aperture). In single-mode PhC design, the mode distribution area is much smaller, around 29.6 µm 2 (considering 1.43 µm out of 2.0 µm diameter for each PhC hole structure is overlapping inside the oxide aperture). Thus, only a central area of 46.5% from original oxide aperture area of 63.6 µm 2 is permitted for laser emission in single-mode PhC device due to the strong PhC guiding effect. By this, only the fundamental mode is supported for lasing while other high-order modes (side modes) are suppressed. As for the multi-mode PhC design, the area of mode distribution is larger consisting 85.5% of the oxide aperture area. The larger area permitted for laser emission reflect a weak PhC guiding effect in multi-mode device, which enables multiple high-order modes to exist in the laser emission. Although several higher-order modes are existed for the single-mode device (as shown in the computed LP 11,LP 21, and LP 02 modes in Fig. 10), it is only fractionally exist due to the device weakly guiding feature in PhC VCSEL. These modes are weakly formed and do not strongly reflect actual LP patterns for high-order modes. The device of multi-mode shows more dominant higher-order modes patterns (similar to LP modes pattern of multi-mode fiber). 4 Conclusions In this letter, we presented the output characteristics of PhC VCSEL with unique trench structure. The device is fab-

7 Highly confined and continuous single-mode operation of self-align photonic crystal oxide VCSEL 459 ricated by conventional photolithography, with simplified lithography step of self-aligning the photonic crystal and trench structure to the laser aperture. By this, we proposed an efficient, simplified and reproducible device processing for the intended device. The fabricated PhC VCSEL with trench device exhibits single-mode output power of 0.7 mw, threshold current of 3.5 ma, slope efficiency of 0.10 W/A, and continuous single-mode output spectra at wide operating current range. These device characteristics are compared with a control device (oxide multi-mode VCSEL with trench) for further analysis. Although the PhC VCSEL with trench exhibits lessen electrical characteristic (due to optical loss by PhC holes), the device achieved continuous singlemode operation throughout wide operating current and emitting genuine single-mode power. We also reported the theoretical analysis of the PhC model and guided mode profiles based on the exact fabricated device properties. Acknowledgements This work is sponsored by Telekom Malaysia Berhad through project number R , R and RDTC/ We would like to acknowledge the Photonics Device Research Group of University of Illinois at Urbana-Champaign for their scientific collaboration. Also, we are grateful for the technical support by RSoft Design Group Inc. and Lumerical Solutions Inc. References 1. K. Iga, IEEE J. Quantum Electron. 6, 1201 (2000) 2. F. Koyama, IEEE J. Lightwave Technol. 24, 4502 (2006) 3. C.W. Wilmsen, H. Temkin, L.A. Coldren, Vertical-Cavity Surface- Emitting Lasers: Design, Fabrication, Characterization, and Applications (Cambridge University Press, Cambridge, 1999) 4. P. Pepeljugoski, D. Kuchta, Y. Kwark, P. Pleunis, G. Kuyt, IEEE Photon. Technol. Lett. 14, 717 (2002) 5. K.D. Choquette, K.L. Lear, R.P. Schneider, K.M. Geib, J.J. Figiel, R. Hull, IEEE Photon. Technol. Lett. 7, 1237 (1995) 6. B. Weigl, M. Grabherr, C. Jung, R. Jager, G. Reiner, R. Michalzik, D. Sowada, K.J. Ebeling, IEEE J. Sel. Top. Quantum Electron. 3, 409 (1997) 7. C. Jung, R. Jager, M. Grabherr, P. Schnitzer, R. Michalzik, B. Weigl, S. Muller, K.J. Ebeling, Electron. Lett. 33, 1790 (1997) 8. R.A. Morgan, G.D. Guth, M.W. Focht, M.T. Asom, K. Kojima, L.E. Rogers, S.E. Callis, IEEE Photon. Technol. Lett. 4, 374 (1993) 9. H. Martinsson, J.A. Vukusic, M. Grabherr, R. Michalzik, R. Jager, K.J. Ebeling, A. Larsson, IEEE Photon. Technol. Lett. 11, 1536 (1999) 10. A. Haglund, J.S. Gustavsson, J. Vukusic, P. Modh, A. Larsson, IEEE Photon. Technol. Lett. 16, 368 (2004) 11. E. Soderberg, P. Modh, J.S. Gustavsson, A. Larsson, Z.Z. Zhang, J. Berggren, M. Hammar, IEEE Photon. Technol. Lett. 19, 327 (2007) 12. E.W. Young, K.D. Choquette, S.L. Chuang, K.M. Geib, A.J. Fischer, A.A. Allerman, IEEE Photon. Technol. Lett. 13, 927 (2001) 13. A. Furukawa, S. Sasaki, M. Hoshi, A. Matsuzono, K. Moritoh, T. Baba, Appl. Phys. Lett. 85, 5161 (2004) 14. P.O. Leisher, A.J. Danner, J.J. Raftery Jr., D.F. Siriani, K.D. Choquette, IEEE J. Quantum Electron. 42, 1091 (2006) 15. H.P.D. Yang, I.-C. Hsu, F.-I. Lai, H.C. Kuo, J.Y. Chi, Jpn. J. Appl. Phys. 45, L871 (2006) 16. D.S. Song, S.H. Kim, H.G. Park, C.K. Kim, Y.H. Lee, Appl. Phys. Lett. 80, 3901 (2002) 17. N. Yokouchi, A.J. Danner, K.D. Choquette, IEEE J. Sel. Top. Quantum Electron. 9, 1439 (2003) 18. J.H. Baek, D.S. Song, I.K. Hwang, K.H. Lee, Y.H. Lee, Y. Ju, T. Kondo, T. Miyamoto, F. Koyama, Opt. Express 12, 859 (2004) 19. A.J. Danner, J.J. Raftery Jr., P.O. Leisher, K.D. Choquette, Appl. Phys. Lett. 88, (2006) 20. P.O. Leisher, A.J. Danner, K.D. Choquette, IEEE Photon. Technol. Lett. 18, 2156 (2006) 21. A.M. Kasten, M.P. Tan, J.D. Sulkin, P.O. Leisher, K.D. Choquette, IEEE Photon. Technol. Lett. 20, 2010 (2008) 22. A.M. Kasten, J.D. Sulkin, P.O. Leisher, D.K. McElfresh, D. Vacar, K.D. Choquett, IEEE J. Sel. Top. Quantum Electron. 14, 1123 (2008) 23. T. Czyszanowski, R.P. Sarzala, M. Dems, H. Thienpont, W. Nakwaski, K. Panajotov, Phys. Status Solidi (a) 206, 1396 (2009) 24. T. Czyszanowski, R.P. Sarzala, M. Dems, W. Nakwaski, H. Thienpont, K. Panajotov, J. Appl. Phys. 105, (2009) 25. D.F. Siriani, P.O. Leisher, K.D. Choquette, IEEE J. Quantum Electron. 45, 762 (2009) 26. Paul O. Leisher, J.D. Sulkin, K.D. Choquette, IEEE J. Sel. Top. Quantum Electron. 13, 1290 (2007) 27. K.H. Lee, J.-H. Baek, I.-K. Hwang, Y.-H. Lee, G.-H. Lee, J.-H. Ser, H.-D. Kim, H.-E. Shin, Opt. Express 12, 1301 (2007) 28. T. Birks, J. Knight, P. Russell, Opt. Lett. 22, 961 (1997) 29. A. Ferrando, E. Silvestre, J.J. Miret, P. Andres, M.V. Andres, Opt. Lett. 24, 276 (1999) 30. Z. Zhu, T.G. Brown, Opt. Express 10, 853 (2002) 31. H.-P.D. Yang, I.-C. Hsu, Y.-H. Chang, F.-I. Lai, H.-C. Yu, G. Lin, R.-S. Hsiao, N.A. Maleev, S.A. Blokhin, H.-C. Kuo, J.Y. Chi, IEEE J. Sel. Top. Quantum Electron. 26, 1387 (2008) 32. G.P. Agrawal, Fiber-Optic Communication Systems (Wiley, New York, 1997) 33. N. Yokouchi, A.J. Danner, K.D. Choquette, Appl. Phys. Lett. 82, 1344 (2003)