As can be seen from Figs. 2 and 3, both approximate solutions are in good agreement with the exact solution. Conclusion: Poisson equation is solved analytically in a AlGaAs/AlGaAs/GaAs structure under reverse bias subject to ohmic contact boundary conditions. It is shown that the position-dependent dielectric constant in the graded region can be approximated by an average constant dielectric constant with good accuracy. This should facilitate the numerical modelling of such device structures. Acknowledgments: The author would like to thank E. Gardner and F. Barnes for their encouragement. S. C. KWOK Department of Electrical and Computer Engineering University of Colorado Boulder, C O803094425. U S A 4th April 1990 References CAPASSO,F.: ‘New multilayer and graded gap optoelectronic and high speed devices by graded gap engineering’, Surface Sci., 1984, 142, pp. 513-528 2 CASEY, H. c., and PANISH, M. B.: ‘Heterostructure lasers, part A: fundamentalprinciples’ (Academic Press, New York, 1978) 3 ANDREWS, M.H.,MARSHAK, A.H.,and SHRIVASTAVA, R.: ‘The effect of position-dependent dielectric constant on the electric field and charge density in a p-n junction’, J . Appl. Phys., 1981,52, (ll),pp. 1 67834787 ANALYSIS OF COUPLING COEFFICIENT BETWEEN TWO VERTICAL CAVITY SURFACE E M l l T l N G LASERS FOR TWO-DIMENSIONAL PHASE-LOCKED ARRAY In order to address this issue, we report the results of the analysis of the coupling coellicient of two evanescently coupled VCSE lasers and discuss the design of twodimensional phase-locked array of vertical cavity surface emitting lasers. If the radius of the laser is comparable to the carrier diffusion length of a few microns, and the laser is pumped high enough for the electron density within the active region to be uniform, the laser can be approximated as a step index profile cylindrical waveguide. A similar problem was dealt with in optical fibre coupler analysis, however their interest was focused on the optical fibre which can be approximated by the weakly guiding or degenerate fibre condition. For a large refractive index difference between the laser and the surrounding medium (or cladding), like the VCSE laser array, a more rigorous analysis is necessary. In this analysis, the InGaAs strained layer quantum well active layer and the GaAs cavity are taken into account.2 The emission wavelength of the laser is l p m and the refractive index of the cavity is 3.5. A computer program was written to obtain the single transverse conditions for various values of the laser radius with the refractive index of cladding as a parameter. If the surrounding medium is air, the laser radius has to be <0.2pm for single mode operation which is consistent with the observation of J. L. Jewell.’ For an AIo.,Ga,.,As cladding, the cut-off radius is increased to 0.55pm. It can be seen that the use of an AI, ,Ga, ,As cladding region can alleviate the design constraint so that the proportion of the peripheral length to area of the laser cross-section can be reduced. It also acts as a passivation layer that is capable of reducing the surface recombination currentof the laser which reduces the carrier ~ an increase of the refractive lifetime to about ~ O O P S .With index of the cladding and a decrease of radius of the laser the optical field spreads into the cladding region. If a second laser is brought into close proximity with the first, the optical field will have a finite amplitude in the second laser. Since the field has an overlap in the second laser it can couple power into the second laser if the modal propagation constants of the two lasers are identically matched. The eigenmode can be analysed by the coupled mode equations. The coupling coefficient, c, between two layers of radius a and centre to centre distanced is given byio I ndexing terms: Lasers and laser applications, Semiconductor lasers The coupling coefficient between two adjacent circular vertical cavity surface emitting lasers having an InGaAs strained quantum well active region (Ag = 1pm) and a GaAs cavity has been analysed. The analysis was performed for AI, ,Ga, ,As cladding ( n = 3.325)and air (n = 1). In order lo obtain a coupling coefinent of IO-’, which is known for the stable phase locking of array, the laser separation must be less than 0.25pm for the lasers of radius 0.4prn with air cladding and be less than 0.72pm for the lasers of radius 1.2pm with AI,,Ga,,As cladding. Various structures of two-dimensionalphase-locked arrays, such as periodic, circular and concentriccircular arrays, can be realised by bringing lasers into close proximity of which value can be obtained from the analysis. The successful fabrication of low threshold current vertical cavity surface emitting (VCSE) lasers has been reported.’-, The strong advantage of the VCSE lasers over other surface emitting laser structures, such as grating coupled or 45” beam deflector, comes from the fact that a high density twodimensional array of microlasers can be achieved.’ In the case of a compact two-dimensional array the coupling or crosstalk between adjacent lasers limits the distance between lasers, i.e., the integration density if independent operation is desired. Alternatively if the lasers are in close proximity, one can use its coupling property to achieve a phase locked array for highpower, narrow beam divergence4 The use of a twodimensional phase-locked array leads to more freedom in the design of the arrays suggesting that the shape of the output beam can be better controlled to conform to the pattern which a specific application requires.’ However, the coupling between VCSE lasers has been neglected in spite of its importan~e.~.’ 896 The quantities U, w and ki are given as a (k: - j3’)”’, Q 2 - k:)‘/’ and k,? = nA2n/A))’, respectively, where n , is the refractive index of the laser and n2 the refractive index of cladding. The coupling coellicients against d / a for various values of a ; for n2 = 1 and 3.325 (refractive index of AI, &a, ,AS) are shown in Fig. 1. When the radius, a, is small the field amplitude is spread out far into the cladding and a fairly insensitive coupling coefficient is obtained. The coupling coefficient is very small for large a because the field is essentially confined to the laser region and there is little evanescent field penetrating the cladding region. At the same time, the coupling coefficient decreases steeply with increasing d / a since the evanescent field, which penetrates the cladding, falls off exponentially. In addition, for the same radius and distance, the AIo.,Gao ,As cladding gives a higher coupling coefficients and less critical dependence of the coupling coellicient with distance. This knowledge can be applied to the design of twodimensional phase-locked arrays. Phase-locked laser arrays use optical coupling between lasers to bring a spatial coherency across the array. It has been known“.” that the coupling coellicent between lasers for stable phase-locked operation of the array is in the range of 10-3-10-4. If we choose a coupling coellicient of IO-, for the design criteria for the array, the separation between lasers of radius 1 pm should be less than 0.15pm and lasers of radius ELECTRONICS LETERS 21st June 1990 Vol. 26 No. 13 0.4pm be less than 0.25pm with air cladding. With Al,.,Ga,.,As cladding, the separation between lasers of 1.2pm radius should be less than 0 . 7 2 ~ These . values of 1 F i 1 Coupling coef?ient against dla Md al.4 a Air cladding b AI,.,Ga,.,As cladding separation can be easily obtained using a standard e-beam lithography. An example is shown in Fig. 2 where vertical cavity surface emitting lasers of 0.6pm radius are in close proximity of less than 0.1p n separation. This array shows a laser action and its far-field beam width is as narrow as 7’ which is evidence of phase l ~ k i n g In . ~ addition, use of twodimensional array makes it possible to achieve various geometry of the phase locked array, such as rectangular, triangular, circular and concentric circular (centred polygonal) arrays (Fig. 3). The separation and cladding material for the various twodimensional phase locked array should be chosen based on the analysis of the coupling coefficient. In conclusion, we have analysed the use of different materials for the cladding layer of circular vertical surface emitting laser diodes. Based on this analysis, the coupling coefficient between two identical VCSE lasers was calculated for various values of separation and radius for air and Alo.,Ga,.,As cladding region. As the refractive index of the cladding material increases, the distance between the lasers and the size of the laser decreases, the optical field penetrates further into the cladding material increasing the coupling coefficient. This increase of the coupling coefficient can Emit the design of the close packed two dimensional micro laser array because of increased coupling loss. In contrast, we can fabricate two-dimensional phase locked array using the coupling properties. In order to obtain a coupling coefficient of lo-,, which is known to give a stable phase locking of array, the laser separation must be less than 0.25pm for the lasers of radius 0.4pm with air cladding and be less than 0.72pm for the lasers of radius 1.2pm with Al,.,Ga,.,As cladding. Twodimensional phase-locked arrays lead to a great freedom in the design of array structures. We have suggested three different two-dimensional phase-locked array structures, i.e., periodic (rectangular, triangular), circular and concentric circular arrays. Authors would like to acknowledge a helpful discussion with Dr. Tae-Kyung Yo0 of Gold Star Central Research Institute, Dr. Jung Woong of Kukje Co., and Drs. A. Scherer and E. Kapon of Bellcore. H.-J. YOO‘ J. U. HAYES Bellcore, Red Bank, N J 07701-7040, U S A 12th March 1990 Y . 3 . KWON Department of Electrical Engineering Korea Advanced Institute ofscience and Technology Cheongryang PO Box 150, Seoul, Korea Hoi-Jun Yo0 is on leave from the Department of Electrical Engineering, Kaist, Seoul, Korea References 1 IGA, K., KOVOMA, F., and KINOSHITA, s.: ‘Surface emitting semiconductor lasers’, IEEE J . QuMtwn Electron., 1988, QE24,pp. 1 8 4 5 1855 2 JEWELL, I. L., SCHERER, A., MCCALL, S. L., HARBINSON, 1. P., and FLOREZ, L. I.: ‘Low Y. H., WALKER, S. I., threshold electricallypumped vertical cavity surface emitting micro-lasers’, Electron. Lett., 1989,25, pp. 1123-1124 3 HARBISON, I. P., and a b lj@ Fig. 2 SEM photograph of two-dimensional phase-locked array of verti- cal cavity surface emitting lasers a Whole structure b Detail periodic array 0000 0000 0000 0000 0000 circular array 0 0 0 0 0 0 centered polygon array 1950191 Fig. 3 Two-dimensional phase locked arrays ELECTRONICS LETTERS 21st June 1990 Vol. 26 No. 13 E. G., CHANG, 0. K., FLOREZ, L. T.: ‘Low series resistance vertical VOO, H.-I., HAVE& I. R., ANDREAKIDAS, N., PAEK, cavity frong surface emitting laser diode (FSELD)’,to be published in Appl. Phys. Lett. 4 YOO,H.-J., SCHERwl A., HARBISON, I., FLOREZ,L., PAEK, E. G., VAN DER GAAG, B. P., HAYFS, 1. R., VON LEHMEN, A., KAPON, E., and KWON, Y. s.: ‘Fabricationof two-dimensional phased array of vertical cavity surface emitting lasers’, Appl. Phys. Lett., 1990 5 YOO, H.-I., HAYES, 1. R., PAEK, E. G., XHERUI, A., and KWON, Y. s.: ‘Arraymode analysis of twodimensional phased arrays of vertical cavity surface emitting lasers’, to be published in June issue of IEEE J . Quantum. Electron., 1990 6 IEWELL, I. L., SCHERER, A., MCCALL, s. L., COSSARD, A. c., and ENGLISH, J. H.: ‘GaAs-AIAs monolithic microresonator arrays’, Appl. Phys. Lett., 1987,51, pp. 94-96 7 SCHERER, A., JEWELL, I. L., HARBINSON,J., and FLOREZ, L. T.: ‘Fabrication of surface emitting lasers and microresonators’, to be published 8 W C U S E , D.: ‘The coupling of degenerate modes in two parallel dielectric waveguides’, Bell Syst. Tech. J., 1971, So, pp. 1791-1816 9 IEWELL, 1. L., MCCALL, S. L., SCHERER, A., HOUH, H. H., WHITAKER, N. A., GOSSARD, A. c., and ENGLISH, I. H . : ‘Transverse modes, waveguide dispersion and 30ps recovery in submicron GaAs/AIAs microresonators’, Appl. Phys. Lett., 1989,55, pp. 22-24 10 V A N C L OR., O and S ~ PHARISFAU, , P.: The coupling of two parallel dielectric fibers, Part 11’, Physica, 1970,47, pp. 501-514 1 1 BUTLER, 1. K., ACKLEY, D. E., and ETTENBERG, M. E.: ‘COUpkd-mode analysis of gain and wavelength oscillation characteristics of diode laser phased arrays’, IEEE J. Quantum Electron., 1985, QE21, pp. 458463 12 ACKLEY, D. E., BUTLFR, 1. K., and ETTENBFRG, M. E.: ‘Phase-locked injection laser arrays with variable stripe spacing’, IEEE J. Quantum Electron., 1986, QE2f pp. 2204-221 1 897

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