- Email: [email protected]
Beam shaping in vertical-cavity surface-emitting laser cavities
15 December 1998
Optics Communications 158 Ž1998. 65–71
Beam shaping in vertical-cavity surface-emitting laser cavities S. Bischoff 1, S.W. Koch Department of Physics and Material Sciences Center, Philipps UniÕersitat ¨ Marburg, Renthof 5, D-35032 Marburg, Germany Received 21 April 1998; revised 19 September 1998; accepted 7 October 1998
Abstract The transverse mode dynamics of vertical-cavity surface-emitting lasers is analyzed studying the passive cavity beam shaping mechanisms. Diffraction in the Bragg mirrors is found to favor the fundamental transverse mode compared to the higher order modes. Mode dependent effective complex reflection coefficients are computed. The effect of diffraction is compared to the effects of thermal lensing due to heating of the Bragg mirrors and of the plasma and active layer lattice. q 1998 Elsevier Science B.V. All rights reserved. PACS: 42.25.Fx; 42.55.Px; 42.55.Sa; 94.20.Bb
1. Introduction Vertical-cavity surface-emitting lasers ŽVCSELs. have been proposed as devices used e.g. for two-dimensional optical switching arrays and optical data-links. For such applications and for further development and optimization purposes one needs experimental characterization and a good theoretical understanding of the device operational characteristics. A detailed numerical model for VCSELs requires long computing times, where typically the solution of Maxwell’s equation in the Bragg mirrors is computationally very demanding. Therefore, one often replaces the Bragg mirror geometry by an effective cavity with planar mirrors characterized by reflection coefficients w1x, or one solves the derived partial differential equations only in the steady state limit w2x. In this paper we investigate the beam shaping mechanisms of the Bragg mirrors and their influence on the transverse mode dynamics of VCSELs. The results lead to a very good understanding of the experimentally observed transverse mode dynamics. This general approach can be viewed as an application of the Fox and Li analysis of the transverse beam shape in optical resonators w3x. 1
Present address: Mikroelektronik Centret, Building 345 East, Technical University of Denmark, DK-2800 Lyngby, Denmark, E-mail: [email protected]
2. Beam propagation: theory The propagation of an optical field through the Bragg mirror structure is determined by Maxwell’s equations, which in cylindrical coordinates yields
E 2E Er 2
1 E 2E
1 EE q
r Er
q
r 2 Eu 2
E 2E q
E z2
1 E 2E y
c2 E t 2
s0 ,
Ž1.
where E is the complex electric field, c 0 and c are the speed of light in vacuum and in the material, respectively. In the slowly varying envelope approximation the solution of Eq. Ž1. can be written as Es
Ý Ez Ž t , z . Jm Ž k m, n r . e i mu m ,n
m,n
(
2 =exp yic k 2z q k m, n t " ik z z .
Ž2.
Here Jm is the Bessel function of order m and E z m , nŽ t, z . is the slowly varying electric field amplitude for the mode m,n. The electric field variation with respect to the radius r has been captured by choosing the Bessel functions basis. The value of k m, n is determined by defining rmax , so that JmŽ k m, n rmax . is the nth zero value. The constant k z is equal to v 0rc, where v 0 is the optical carrier frequency. The refractive index within each quarter wavelength layer, see Fig. 1, is assumed to be constant. The
0030-4018r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 5 4 8 - 3
66
S. Bischoff, S.W. Kochr Optics Communications 158 (1998) 65–71
Fig. 1. Schematic drawing of a VCSEL structure. The top and bottom Bragg mirrors consist of 17 and 20 pairs of quarter wavelength layers, respectively. The black shaded regions correspond to a refractive index of 3.66, while the white shaded regions have a refractive index of 2.97. The longitudinal mode structure is sketched at the right-hand side. The oxide layer has a refractive index of 2.904. ŽA. and ŽB. show schematically the two cases considered.
transmitted and reflected fields between two dielectric media are determined by Maxwell’s boundary conditions. Clearly, such a Bessel mode expansion is not necessary and can be replaced by other numerical schemes w4,5x. However, for situations where the longitudinal mode structure is almost equal to its steady state value w6x, the mode-expansion method has the advantage that one can calculate effective complex reflection coefficients for each Bessel mode Ž k m, n value. as will be further elucidated in the following.
3. Diffractive beam shaping The cavity induced beam shaping is investigated for the example of a typical VCSEL cavity with 20 pairs of
quarter wavelength layers in the bottom mirror and 17 pairs of quarter wavelength layers in the top mirror, see Fig. 1. The refractive index of the different quarter wavelength layers is 2.97 and 3.66, respectively. The VCSEL is a l cavity, where the active material is placed at the center. Typically, a VCSEL is grown on a substrate. Often a proton bombarded region is introduced to improve the current confinement. Simultaneously, however, this bombarded region also causes optical loss since the Bragg mirrors are damaged. Furthermore, the VCSEL may have an oxide layer, which leads to improved current confinement and to weak index guiding of the optical modes w7x. In Figs. 1A and 2b we schematically display the two cases studied in this paper. The beam shaping is investigated assuming a CW injection signal through the bottom mirror. The injected signal is turned on at time t s 0 ps. The cavity beam shaping is then analyzed by computing the transmitted beam profile at the top mirror. First we investigate the case where the injected CW beam profile is assumed to consist of a zero and first order field component, as shown in Fig. 2a. The field intensity in the zero and first order mode is plotted together with the total field intensity for u s pr4. The total field intensity depends on the angle u given by the relation in Eq. Ž2.. The angle u s pr4 corresponds in Fig. 2 to the direction of positive interference. In Fig. 2 we assume an injected beam diameter of about 100 mm. In this case the transmitted beam profile is unchanged upon transmission through the cavity, see Fig. 2b. Here the field intensity is plotted for u s pr4 3.46 ps after turning on the CW injection signal. This result confirms our expectations, since the effect of diffraction should be negligible in the case of a beam profile with slow radial variation. Thus, the Bragg mirrors of a large VCSEL aperture can to a good approximation be replaced with an effective cavity length and mode independent planar mirror reflection coefficients. However, in many situations of practical interest VCSELs are built with apertures in the range of 3–20 mm. Clearly, this is much narrower than the case studied in Fig. 2. The computed results for such a smaller aperture configuration are shown in Fig. 3a where we reduced the injected beam spot diameter to about 10 mm. Again, we have plotted the fields for u s pr4. The transmitted beam profile is displayed in Fig. 3b 1.16 ps after turning on the CW injection field. The results show that the effect of diffraction leads to a significant reshaping of the beam profile, even though the modifications for a single pass are relatively small. This is displayed in Fig. 4, where the total field is plotted as a function of time for u s pr4. For each time step the peak intensity has been normalized to one, so that reshaping of the beam profile with time is more clearly visible. The normalization is necessary, since the field is still performing relaxation oscillations due to the instantaneous switching-on of the CW injection field.
S. Bischoff, S.W. Kochr Optics Communications 158 (1998) 65–71
67
Fig. 2. Both figures display the total field intensity as well as the different components related to the zero order and first order Bessel mode-expansion fields in the case of a very broad aperture. The total field intensity is plotted for u s pr4. Ža. The injected CW beam profile. The field contribution fitted by the first order Bessel mode has been multiplied by a factor of 10. Žb. The emitted field intensity at the top mirror 3.46 ps after turning on the injection field.
The beam in Figs. 3 and 4 is strongly reshaped due to diffraction in the VCSEL cavity. Thus, the Bragg mirror cavity itself has a significant influence on the transverse mode characteristic of the VCSEL emission. The effect of diffraction tends to focus the main energy of each mode to a particular range of radii. While the zero order mode is
focused towards the center of the VCSEL cavity, with increasing mode number the focusing range is further and further away from the beam center. The single transverse mode operation of small aperture VCSELs can be understood solely by the effect of diffraction in the Bragg mirrors. In comparison to the zero order
Fig. 3. Both figures display the total field intensity as well the different components related to the zero order and first order Bessel mode-expansion fields. The total field intensity is plotted for u s pr4. Ža. The injected CW beam profile. The field contribution fitted by the first order Bessel functions has been multiplied by a factor of 10. Žb. The emitted field intensity at the top mirror 1.16 ps after turning on the CW injection field.
68
S. Bischoff, S.W. Kochr Optics Communications 158 (1998) 65–71
Fig. 4. Normalized emitted field intensity of the top mirror as a function of time for u sp r4.
mode the higher order modes typically experience a higher loss in the Bragg mirrors due to the more pronounced influence of the proton bombarded regions, see Fig. 1. Furthermore, due to the diffraction effects also the zero order mode can take advantage of the rather high gain at high radii due to beam reshaping. Thus the zero order mode can be very stable even in the presence of spatial hole burning. From the analysis presented so far, one can already conclude that for the purpose of studying the transverse mode dynamics the Bragg mirror structure of small aperture VCSELs cannot be modelled accurately by considering only an effective cavity length with mode independent mirror reflection coefficients. Even in the case of calculations of the threshold current w7,8x it is important to include the effect of diffraction, since the effective gain depends on beam reshaping. However, a large signal model with a detailed Bragg mirror model is computationally very demanding, since the maximum allowed numerical time step will be determined by the time it takes to propagate the distance of a quarter-wavelength. In order to avoid this computer-time consuming procedure, it is of advantage to calculate effective reflectivity coefficients, which contain the information of diffraction. In the case of Bessel function expansion this can be done, since the longitudinal mode structure is almost equal to its steady state during relaxation oscillations w6x. In Fig. 5 we plot the computed modal reflection and transmission coefficients for the top mirror of the VCSEL cavity shown in Fig. 1. The solid lines in Figs. 5a and 5b show the power reflection and transmission coefficients as a function of k m, n . The dashed line in Fig. 5a is the sum of the power reflection and transmission coefficients, which is independent of k m, n . The effect of internal loss can be
included in the reflection and transmission coefficients, in that case the sum is lower than 1. The power reflection and transmission coefficients are almost constant as a function of k m, n . Hence, looking at the power reflectivity alone it appears to be somewhat surprising that we obtained such a strong change in the transmitted beam profile. However, looking at the computed complex reflection and transmission coefficients for the field in Figs. 5c and 5d one sees the origin of strong reshaping of the beam profile. Basically, upon propagation the different field components experience different phase changes, which are responsible for the reshaping of the beam. In our analysis, where the electric field is expressed by different Bessel functions a spatially slowly varying field typically carries its main energy part in the contributions characterized by k m, n values, which are lower than 1–2 mmy1. In this case the transmission and reflection coefficients are almost constant and the effect of diffraction is negligible. Narrow beams on the other hand contain much more energy in components characterized by high k m, n values which leads to strong propagation induced reshaping. The calculated complex reflection and transmission coefficients can be used in large signal simulations of circular cylindrical VCSELs. The reflection coefficients can replace the distributed Bragg reflector ŽDBR. mirrors as long as Ži. the field is changing slowly compared to the time it takes to pass through the DBR mirrors, see Fig. 4, and Žii. the longitudinal mode structure is close to steady state. Our analysis showed that the longitudinal mode structure in VCSELs is very close to steady state even during relaxation oscillations w6x, since the gain per pass through the active region is very small. Thus, each pass through the active region only results in a small perturbation of the longitudinal mode structure. However, when steady state is reached a lot of energy is stored within the VCSEL cavity, and the transmitted beam profile is dominated by the VCSEL cavity eigenmodes and not by the injected beam profile. Accordingly, the calculated reflection coefficients can replace the detailed reflectivity of the DBR mirrors. In this case the propagation problem in the Bragg mirrors only requires the multiplication of each Bessel mode function by the appropriate reflection coefficient. This significantly reduces the computer resources when modelling VCSELs. The proton bombarded region should in this case be modelled as a radius dependent loss, which is acting on the reflected field before one calculates the induced polarization in the active medium. However, the transverse mode dynamics of VCSELs does not only depend on diffraction in the Bragg mirrors. The effect of self-focusing due to heating of the Bragg mirrors or other wave-guiding effects may also influence the transverse beam shape. To check the relevance of these effects, we investigate the influence of self-focusing or
S. Bischoff, S.W. Kochr Optics Communications 158 (1998) 65–71
69
Fig. 5. Calculated reflection and transmission coefficients as a function of k m, n for the top mirror shown in Fig. 1.
weak index guiding using the same method used for the ideal Bragg mirror structure. When the refractive index is a function of the in-plane coordinate Ž r , u ., the electric field cannot be expanded in terms of Bessel functions since they are not a solution to Maxwell’s equations in this case. Therefore, we used a beam propagation method w4,5x to compute the transmitted light field through the VCSEL cavity when we study the effect of self-focusing or weak index guiding. The results are shown in Fig. 6. Here, the solid line has been obtained using the beam propagation method for a pure VCSEL cavity. The beam shape of the injected field intensity for the zero, first and second order field components is shown in Fig. 6d. The line with the open circles in Figs. 6a–6c represents the computed results for the case of a VCSEL cavity with an oxide layer as schematically shown in Fig. 1B. The oxide layer is placed next to the l cavity and has a diameter of 8 mm. The refractive index for the oxide layer has an effectiÕe refractive index step compared to the quarter-wavelength material of y0.066 w7x. The oxide layer results in better current confinement and thus reduces the threshold current of the VCSEL considerably w7,8x. Furthermore, the optical modes are slightly index guided by the oxide layer, which also enhances the effect of self-focusing of the fundamental mode. Broadening of the higher order modes towards high radii is reduced significantly showing that the oxide layer
has a more significant effect on these higher order modes. This could explain the experimental observation w9x that large aperture Ž) 4 mm. oxide VCSELs tend to operate in a multi-transverse mode regime. The oxide layer simply reduces the higher order mode threshold current more than the zero order threshold current. Another important effect when discussing the transverse mode dynamics is self-focusing due to plasmarlattice heating in the active medium w2x, or self-focusing due to heating of the Bragg mirror structure. We investigate these phenomena and their relevance in comparison to the effect of diffraction by assuming a radial index variation in the entire VCSEL structure or the active layer. The lines with triangles in Figs. 6a–6c show the effect of thermal lensing, where the temperature induced index variation in the VCSEL is assumed to yield D n s nd exp Ž y0.4 r 2 . ,
Ž3.
throughout the entire VCSEL with nd s 0.001. A full calculation of the temperature profile and resulting index variation is beyond the scope of this paper. Therefore, we have assumed a Gaussian shaped index profile and refractive index changes, which are close to values found in the literature w2x. The effect of thermal lensing is seen to have a strong effect on the fundamental mode, while the effect on the higher order modes is less pronounced. However,
70
S. Bischoff, S.W. Kochr Optics Communications 158 (1998) 65–71
Fig. 6. Ža. – Žc. Emitted beam out of the top mirror of a VCSEL 1.75 ps after turning on the injection field at the bottom mirror. The injected field is shown in Žd.. Ža., Žb. and Žc. show the zero, first and second order transverse modes, respectively. The solid line in Ža. – Žc. corresponds to a pure Bragg mirror structure ŽFig. 1A.. The line with triangles and the dotted curve correspond to self-focusing throughout the entire structure or only the active region Ž14 nm., respectively. The line with open circles corresponds to a weak index guided structure ŽFig. 1B..
the effect of thermal lensing on the higher order modes depends critically on the actual transverse variations of the refractive index. In the case of defocusing Ž nd - 0. the VCSEL operation may generate a doughnut mode, which is made up of the zero order mode components w2x. As can be seen in Fig. 6 the radial refractive index profile in the Bragg mirrors resulting from heating of the device has an effect comparable to diffraction on the transverse beam dynamics, and should therefore be taken into consideration
when calculating the transverse mode dynamics of VCSELs. However, our analysis shows that the carrier induced refractive index changes in the active region have a negligible effect on the transverse beam profile. The dotted line in Figs. 6a–6c display the self-focusing effect by assuming an active region of 14 nm in the center of the l cavity with an index profile given by Eq. Ž3. with nd s 0.002. The self-focusing of the beam in the active material is insignifi-
S. Bischoff, S.W. Kochr Optics Communications 158 (1998) 65–71
cant due to the short width of the active layer. The solution of Maxwell’s equations in the active layer can thus be done by neglecting diffraction, so that the method used in Ref. w10x can be used to model the gain of the active material.
71
tract F49620-97-1-0002. We thank J. Moloney, D. Burak, W. Chow, and C.Z. Ning for stimulating discussions.
References 4. Summary We have presented data discussing the transversal beam shaping of Bragg mirrors of VCSEL cavities. Diffraction has been shown to be an important effect when calculating the transverse beam profile. Self-focusing in the Bragg mirrors due to the radial dependence of the refractive index due to heating results in a beam reshaping comparable in strength to the effect of diffraction. The effect of self-focusing due to plasmarlattice heating of the active material is found to be of less importance. Furthermore, we have presented a numerical method, which can be used to calculate final reflection coefficients for gain guided VCSEL simulators, where index guiding and thermal lensing are neglegible.
Acknowledgements This work was supported by the DFG, the Leibniz prize, NATO travel grant CRG971186 and AFOSR con-
w1x C.Z. Ning, R.A. Indik, J.V. Moloney, J. Opt. Soc. Am. B 12 Ž1995. 1993. w2x Y.-G. Zhao, J.G. McInerney, IEEE J. Quantum Electron. 32 Ž1996. 1950. w3x A.E. Siegman, Lasers, University Science Book, Stanford University, 1986. w4x F. Gonthier, A. Henault, S. Lacroix, R.J. Black, J. Bures, J. Opt. Soc. Am. B 8 Ž1991. 416. w5x J.-P. Zhang, K. Petermann, IEEE J. Quantum Electron. 30 Ž1994. 1529. w6x C.Z. Ning, S. Bischoff, S.W. Koch, G.K. Harkness, J.V. Moloney, W.W. Chow, Optical Engineering – Bellingham 37 Ž1998. 1175. w7x G.R. Hadley, K.L. Lear, M.E. Warren, K.D. Choquette, J.W. Scott, S.W. Corzine, IEEE J. Quantum Electron. 32 Ž1996. 607. w8x H.K. Bissessur, F. Koyama, K. Iga, IEEE J. Selected Topics Quantum Electron. 3 Ž1997. 344. w9x D.L. Huffaker, J. Shin, D.G. Deppe, Appl. Phys. Lett. 66 Ž1995. 1723. w10x T. Strouken, A. Knorr, P. Thomas, S.W. Koch, Phys. Rev. B 53 Ž1996. 2026.
Optics Communications 158 Ž1998. 65–71
Beam shaping in vertical-cavity surface-emitting laser cavities S. Bischoff 1, S.W. Koch Department of Physics and Material Sciences Center, Philipps UniÕersitat ¨ Marburg, Renthof 5, D-35032 Marburg, Germany Received 21 April 1998; revised 19 September 1998; accepted 7 October 1998
Abstract The transverse mode dynamics of vertical-cavity surface-emitting lasers is analyzed studying the passive cavity beam shaping mechanisms. Diffraction in the Bragg mirrors is found to favor the fundamental transverse mode compared to the higher order modes. Mode dependent effective complex reflection coefficients are computed. The effect of diffraction is compared to the effects of thermal lensing due to heating of the Bragg mirrors and of the plasma and active layer lattice. q 1998 Elsevier Science B.V. All rights reserved. PACS: 42.25.Fx; 42.55.Px; 42.55.Sa; 94.20.Bb
1. Introduction Vertical-cavity surface-emitting lasers ŽVCSELs. have been proposed as devices used e.g. for two-dimensional optical switching arrays and optical data-links. For such applications and for further development and optimization purposes one needs experimental characterization and a good theoretical understanding of the device operational characteristics. A detailed numerical model for VCSELs requires long computing times, where typically the solution of Maxwell’s equation in the Bragg mirrors is computationally very demanding. Therefore, one often replaces the Bragg mirror geometry by an effective cavity with planar mirrors characterized by reflection coefficients w1x, or one solves the derived partial differential equations only in the steady state limit w2x. In this paper we investigate the beam shaping mechanisms of the Bragg mirrors and their influence on the transverse mode dynamics of VCSELs. The results lead to a very good understanding of the experimentally observed transverse mode dynamics. This general approach can be viewed as an application of the Fox and Li analysis of the transverse beam shape in optical resonators w3x. 1
Present address: Mikroelektronik Centret, Building 345 East, Technical University of Denmark, DK-2800 Lyngby, Denmark, E-mail: [email protected]
2. Beam propagation: theory The propagation of an optical field through the Bragg mirror structure is determined by Maxwell’s equations, which in cylindrical coordinates yields
E 2E Er 2
1 E 2E
1 EE q
r Er
q
r 2 Eu 2
E 2E q
E z2
1 E 2E y
c2 E t 2
s0 ,
Ž1.
where E is the complex electric field, c 0 and c are the speed of light in vacuum and in the material, respectively. In the slowly varying envelope approximation the solution of Eq. Ž1. can be written as Es
Ý Ez Ž t , z . Jm Ž k m, n r . e i mu m ,n
m,n
(
2 =exp yic k 2z q k m, n t " ik z z .
Ž2.
Here Jm is the Bessel function of order m and E z m , nŽ t, z . is the slowly varying electric field amplitude for the mode m,n. The electric field variation with respect to the radius r has been captured by choosing the Bessel functions basis. The value of k m, n is determined by defining rmax , so that JmŽ k m, n rmax . is the nth zero value. The constant k z is equal to v 0rc, where v 0 is the optical carrier frequency. The refractive index within each quarter wavelength layer, see Fig. 1, is assumed to be constant. The
0030-4018r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 5 4 8 - 3
66
S. Bischoff, S.W. Kochr Optics Communications 158 (1998) 65–71
Fig. 1. Schematic drawing of a VCSEL structure. The top and bottom Bragg mirrors consist of 17 and 20 pairs of quarter wavelength layers, respectively. The black shaded regions correspond to a refractive index of 3.66, while the white shaded regions have a refractive index of 2.97. The longitudinal mode structure is sketched at the right-hand side. The oxide layer has a refractive index of 2.904. ŽA. and ŽB. show schematically the two cases considered.
transmitted and reflected fields between two dielectric media are determined by Maxwell’s boundary conditions. Clearly, such a Bessel mode expansion is not necessary and can be replaced by other numerical schemes w4,5x. However, for situations where the longitudinal mode structure is almost equal to its steady state value w6x, the mode-expansion method has the advantage that one can calculate effective complex reflection coefficients for each Bessel mode Ž k m, n value. as will be further elucidated in the following.
3. Diffractive beam shaping The cavity induced beam shaping is investigated for the example of a typical VCSEL cavity with 20 pairs of
quarter wavelength layers in the bottom mirror and 17 pairs of quarter wavelength layers in the top mirror, see Fig. 1. The refractive index of the different quarter wavelength layers is 2.97 and 3.66, respectively. The VCSEL is a l cavity, where the active material is placed at the center. Typically, a VCSEL is grown on a substrate. Often a proton bombarded region is introduced to improve the current confinement. Simultaneously, however, this bombarded region also causes optical loss since the Bragg mirrors are damaged. Furthermore, the VCSEL may have an oxide layer, which leads to improved current confinement and to weak index guiding of the optical modes w7x. In Figs. 1A and 2b we schematically display the two cases studied in this paper. The beam shaping is investigated assuming a CW injection signal through the bottom mirror. The injected signal is turned on at time t s 0 ps. The cavity beam shaping is then analyzed by computing the transmitted beam profile at the top mirror. First we investigate the case where the injected CW beam profile is assumed to consist of a zero and first order field component, as shown in Fig. 2a. The field intensity in the zero and first order mode is plotted together with the total field intensity for u s pr4. The total field intensity depends on the angle u given by the relation in Eq. Ž2.. The angle u s pr4 corresponds in Fig. 2 to the direction of positive interference. In Fig. 2 we assume an injected beam diameter of about 100 mm. In this case the transmitted beam profile is unchanged upon transmission through the cavity, see Fig. 2b. Here the field intensity is plotted for u s pr4 3.46 ps after turning on the CW injection signal. This result confirms our expectations, since the effect of diffraction should be negligible in the case of a beam profile with slow radial variation. Thus, the Bragg mirrors of a large VCSEL aperture can to a good approximation be replaced with an effective cavity length and mode independent planar mirror reflection coefficients. However, in many situations of practical interest VCSELs are built with apertures in the range of 3–20 mm. Clearly, this is much narrower than the case studied in Fig. 2. The computed results for such a smaller aperture configuration are shown in Fig. 3a where we reduced the injected beam spot diameter to about 10 mm. Again, we have plotted the fields for u s pr4. The transmitted beam profile is displayed in Fig. 3b 1.16 ps after turning on the CW injection field. The results show that the effect of diffraction leads to a significant reshaping of the beam profile, even though the modifications for a single pass are relatively small. This is displayed in Fig. 4, where the total field is plotted as a function of time for u s pr4. For each time step the peak intensity has been normalized to one, so that reshaping of the beam profile with time is more clearly visible. The normalization is necessary, since the field is still performing relaxation oscillations due to the instantaneous switching-on of the CW injection field.
S. Bischoff, S.W. Kochr Optics Communications 158 (1998) 65–71
67
Fig. 2. Both figures display the total field intensity as well as the different components related to the zero order and first order Bessel mode-expansion fields in the case of a very broad aperture. The total field intensity is plotted for u s pr4. Ža. The injected CW beam profile. The field contribution fitted by the first order Bessel mode has been multiplied by a factor of 10. Žb. The emitted field intensity at the top mirror 3.46 ps after turning on the injection field.
The beam in Figs. 3 and 4 is strongly reshaped due to diffraction in the VCSEL cavity. Thus, the Bragg mirror cavity itself has a significant influence on the transverse mode characteristic of the VCSEL emission. The effect of diffraction tends to focus the main energy of each mode to a particular range of radii. While the zero order mode is
focused towards the center of the VCSEL cavity, with increasing mode number the focusing range is further and further away from the beam center. The single transverse mode operation of small aperture VCSELs can be understood solely by the effect of diffraction in the Bragg mirrors. In comparison to the zero order
Fig. 3. Both figures display the total field intensity as well the different components related to the zero order and first order Bessel mode-expansion fields. The total field intensity is plotted for u s pr4. Ža. The injected CW beam profile. The field contribution fitted by the first order Bessel functions has been multiplied by a factor of 10. Žb. The emitted field intensity at the top mirror 1.16 ps after turning on the CW injection field.
68
S. Bischoff, S.W. Kochr Optics Communications 158 (1998) 65–71
Fig. 4. Normalized emitted field intensity of the top mirror as a function of time for u sp r4.
mode the higher order modes typically experience a higher loss in the Bragg mirrors due to the more pronounced influence of the proton bombarded regions, see Fig. 1. Furthermore, due to the diffraction effects also the zero order mode can take advantage of the rather high gain at high radii due to beam reshaping. Thus the zero order mode can be very stable even in the presence of spatial hole burning. From the analysis presented so far, one can already conclude that for the purpose of studying the transverse mode dynamics the Bragg mirror structure of small aperture VCSELs cannot be modelled accurately by considering only an effective cavity length with mode independent mirror reflection coefficients. Even in the case of calculations of the threshold current w7,8x it is important to include the effect of diffraction, since the effective gain depends on beam reshaping. However, a large signal model with a detailed Bragg mirror model is computationally very demanding, since the maximum allowed numerical time step will be determined by the time it takes to propagate the distance of a quarter-wavelength. In order to avoid this computer-time consuming procedure, it is of advantage to calculate effective reflectivity coefficients, which contain the information of diffraction. In the case of Bessel function expansion this can be done, since the longitudinal mode structure is almost equal to its steady state during relaxation oscillations w6x. In Fig. 5 we plot the computed modal reflection and transmission coefficients for the top mirror of the VCSEL cavity shown in Fig. 1. The solid lines in Figs. 5a and 5b show the power reflection and transmission coefficients as a function of k m, n . The dashed line in Fig. 5a is the sum of the power reflection and transmission coefficients, which is independent of k m, n . The effect of internal loss can be
included in the reflection and transmission coefficients, in that case the sum is lower than 1. The power reflection and transmission coefficients are almost constant as a function of k m, n . Hence, looking at the power reflectivity alone it appears to be somewhat surprising that we obtained such a strong change in the transmitted beam profile. However, looking at the computed complex reflection and transmission coefficients for the field in Figs. 5c and 5d one sees the origin of strong reshaping of the beam profile. Basically, upon propagation the different field components experience different phase changes, which are responsible for the reshaping of the beam. In our analysis, where the electric field is expressed by different Bessel functions a spatially slowly varying field typically carries its main energy part in the contributions characterized by k m, n values, which are lower than 1–2 mmy1. In this case the transmission and reflection coefficients are almost constant and the effect of diffraction is negligible. Narrow beams on the other hand contain much more energy in components characterized by high k m, n values which leads to strong propagation induced reshaping. The calculated complex reflection and transmission coefficients can be used in large signal simulations of circular cylindrical VCSELs. The reflection coefficients can replace the distributed Bragg reflector ŽDBR. mirrors as long as Ži. the field is changing slowly compared to the time it takes to pass through the DBR mirrors, see Fig. 4, and Žii. the longitudinal mode structure is close to steady state. Our analysis showed that the longitudinal mode structure in VCSELs is very close to steady state even during relaxation oscillations w6x, since the gain per pass through the active region is very small. Thus, each pass through the active region only results in a small perturbation of the longitudinal mode structure. However, when steady state is reached a lot of energy is stored within the VCSEL cavity, and the transmitted beam profile is dominated by the VCSEL cavity eigenmodes and not by the injected beam profile. Accordingly, the calculated reflection coefficients can replace the detailed reflectivity of the DBR mirrors. In this case the propagation problem in the Bragg mirrors only requires the multiplication of each Bessel mode function by the appropriate reflection coefficient. This significantly reduces the computer resources when modelling VCSELs. The proton bombarded region should in this case be modelled as a radius dependent loss, which is acting on the reflected field before one calculates the induced polarization in the active medium. However, the transverse mode dynamics of VCSELs does not only depend on diffraction in the Bragg mirrors. The effect of self-focusing due to heating of the Bragg mirrors or other wave-guiding effects may also influence the transverse beam shape. To check the relevance of these effects, we investigate the influence of self-focusing or
S. Bischoff, S.W. Kochr Optics Communications 158 (1998) 65–71
69
Fig. 5. Calculated reflection and transmission coefficients as a function of k m, n for the top mirror shown in Fig. 1.
weak index guiding using the same method used for the ideal Bragg mirror structure. When the refractive index is a function of the in-plane coordinate Ž r , u ., the electric field cannot be expanded in terms of Bessel functions since they are not a solution to Maxwell’s equations in this case. Therefore, we used a beam propagation method w4,5x to compute the transmitted light field through the VCSEL cavity when we study the effect of self-focusing or weak index guiding. The results are shown in Fig. 6. Here, the solid line has been obtained using the beam propagation method for a pure VCSEL cavity. The beam shape of the injected field intensity for the zero, first and second order field components is shown in Fig. 6d. The line with the open circles in Figs. 6a–6c represents the computed results for the case of a VCSEL cavity with an oxide layer as schematically shown in Fig. 1B. The oxide layer is placed next to the l cavity and has a diameter of 8 mm. The refractive index for the oxide layer has an effectiÕe refractive index step compared to the quarter-wavelength material of y0.066 w7x. The oxide layer results in better current confinement and thus reduces the threshold current of the VCSEL considerably w7,8x. Furthermore, the optical modes are slightly index guided by the oxide layer, which also enhances the effect of self-focusing of the fundamental mode. Broadening of the higher order modes towards high radii is reduced significantly showing that the oxide layer
has a more significant effect on these higher order modes. This could explain the experimental observation w9x that large aperture Ž) 4 mm. oxide VCSELs tend to operate in a multi-transverse mode regime. The oxide layer simply reduces the higher order mode threshold current more than the zero order threshold current. Another important effect when discussing the transverse mode dynamics is self-focusing due to plasmarlattice heating in the active medium w2x, or self-focusing due to heating of the Bragg mirror structure. We investigate these phenomena and their relevance in comparison to the effect of diffraction by assuming a radial index variation in the entire VCSEL structure or the active layer. The lines with triangles in Figs. 6a–6c show the effect of thermal lensing, where the temperature induced index variation in the VCSEL is assumed to yield D n s nd exp Ž y0.4 r 2 . ,
Ž3.
throughout the entire VCSEL with nd s 0.001. A full calculation of the temperature profile and resulting index variation is beyond the scope of this paper. Therefore, we have assumed a Gaussian shaped index profile and refractive index changes, which are close to values found in the literature w2x. The effect of thermal lensing is seen to have a strong effect on the fundamental mode, while the effect on the higher order modes is less pronounced. However,
70
S. Bischoff, S.W. Kochr Optics Communications 158 (1998) 65–71
Fig. 6. Ža. – Žc. Emitted beam out of the top mirror of a VCSEL 1.75 ps after turning on the injection field at the bottom mirror. The injected field is shown in Žd.. Ža., Žb. and Žc. show the zero, first and second order transverse modes, respectively. The solid line in Ža. – Žc. corresponds to a pure Bragg mirror structure ŽFig. 1A.. The line with triangles and the dotted curve correspond to self-focusing throughout the entire structure or only the active region Ž14 nm., respectively. The line with open circles corresponds to a weak index guided structure ŽFig. 1B..
the effect of thermal lensing on the higher order modes depends critically on the actual transverse variations of the refractive index. In the case of defocusing Ž nd - 0. the VCSEL operation may generate a doughnut mode, which is made up of the zero order mode components w2x. As can be seen in Fig. 6 the radial refractive index profile in the Bragg mirrors resulting from heating of the device has an effect comparable to diffraction on the transverse beam dynamics, and should therefore be taken into consideration
when calculating the transverse mode dynamics of VCSELs. However, our analysis shows that the carrier induced refractive index changes in the active region have a negligible effect on the transverse beam profile. The dotted line in Figs. 6a–6c display the self-focusing effect by assuming an active region of 14 nm in the center of the l cavity with an index profile given by Eq. Ž3. with nd s 0.002. The self-focusing of the beam in the active material is insignifi-
S. Bischoff, S.W. Kochr Optics Communications 158 (1998) 65–71
cant due to the short width of the active layer. The solution of Maxwell’s equations in the active layer can thus be done by neglecting diffraction, so that the method used in Ref. w10x can be used to model the gain of the active material.
71
tract F49620-97-1-0002. We thank J. Moloney, D. Burak, W. Chow, and C.Z. Ning for stimulating discussions.
References 4. Summary We have presented data discussing the transversal beam shaping of Bragg mirrors of VCSEL cavities. Diffraction has been shown to be an important effect when calculating the transverse beam profile. Self-focusing in the Bragg mirrors due to the radial dependence of the refractive index due to heating results in a beam reshaping comparable in strength to the effect of diffraction. The effect of self-focusing due to plasmarlattice heating of the active material is found to be of less importance. Furthermore, we have presented a numerical method, which can be used to calculate final reflection coefficients for gain guided VCSEL simulators, where index guiding and thermal lensing are neglegible.
Acknowledgements This work was supported by the DFG, the Leibniz prize, NATO travel grant CRG971186 and AFOSR con-
w1x C.Z. Ning, R.A. Indik, J.V. Moloney, J. Opt. Soc. Am. B 12 Ž1995. 1993. w2x Y.-G. Zhao, J.G. McInerney, IEEE J. Quantum Electron. 32 Ž1996. 1950. w3x A.E. Siegman, Lasers, University Science Book, Stanford University, 1986. w4x F. Gonthier, A. Henault, S. Lacroix, R.J. Black, J. Bures, J. Opt. Soc. Am. B 8 Ž1991. 416. w5x J.-P. Zhang, K. Petermann, IEEE J. Quantum Electron. 30 Ž1994. 1529. w6x C.Z. Ning, S. Bischoff, S.W. Koch, G.K. Harkness, J.V. Moloney, W.W. Chow, Optical Engineering – Bellingham 37 Ž1998. 1175. w7x G.R. Hadley, K.L. Lear, M.E. Warren, K.D. Choquette, J.W. Scott, S.W. Corzine, IEEE J. Quantum Electron. 32 Ž1996. 607. w8x H.K. Bissessur, F. Koyama, K. Iga, IEEE J. Selected Topics Quantum Electron. 3 Ž1997. 344. w9x D.L. Huffaker, J. Shin, D.G. Deppe, Appl. Phys. Lett. 66 Ž1995. 1723. w10x T. Strouken, A. Knorr, P. Thomas, S.W. Koch, Phys. Rev. B 53 Ž1996. 2026.