IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 9, NO. 5, MAY 1997 557

Polarization Behavior of Birefringent Multitransverse Mode Vertical-Cavity Surface-Emitting Lasers

A. Valle, L. Pesquera, and K. A. Shore

Abstract-An analysis has been undertaken of the effect of bire- frigence on the selection of polarization states of a weakly index guided vertical-cavity surface-emitting laser (VCSEL) supporting both a fundamental and a first-order transverse mode. It is shown that for small index steps polarization switching due to spatial- hole burning effects can occur. For larger index steps it is found that higher order modes can emerge which are orthogonally polarized to the dominant polarization of the fundamental mode.

Index Terms-Birefringence, charge carrier processes, optical polarization, semiconductor lasers, surface-emitting lasers.

OTENTIAL applications of vertical-cavity surface- P emitting lasers (VCSEL' s) such as magneto-optic discs and coherent detection require a control of the polarization characteristics of the device. On the other hand, the polarization of the fundamental transverse mode in VCSEL's is randomly oriented in the plane of the active layer [l]. Polarization switching of the fundamental transverse mode when increasing the injection current also occurs and several mechanisms have been proposed to explain it [2], [3]. It has also been found that the steady-state distribution of lasing power between polarization states changes markedly as higher order transverse modes are excited [4], [SI. It has been noted [6] that strong birefringence can arise in nominally isotropic VCSEL's due to strain induced during the fabrication process. Also, as shown in earlier work [7], birefringence consequent to refractive index anisotropies of order 0.005% are adequate for ensuring preferential excitation of a specific polarization state. It has also been suggested that a polarization switching behavior in some devices may arise due to spatial-hole burning (SHB) between the orthogonal polarization modes [8].

The main aim of this letter is to examine this possibility using numerical simulations. It is to be expected that the polarization behavior in a weakly index-guided VCSEL will depend on the magnitude of the index step, An, between the core and cladding region. It is found here that when An is small, polarization switching due to SHB effects can occur. It is also shown that when An is increased it is possible for higher order modes to arise in a state orthogonally polarized to the dominant polarization of the fundamental mode.

Manuscript received November 5, 1996; revised January 14, 1997. The work of A. Valle and L. Pesquera was supported by CICYT Project TIC95 0563-CO5-01 and by EU Project CHRX-CT94-0594. The work of K. A. Shore was supported in part by the U.K. EPSRC under Grant GWJ50149.

A. Valle and I.. Pesquera are with the Instituto de Fisica de Cantabria (CSIC-UC), Facultad de Ciencias, E-39005 Santander, Spain.

K. A. Shore is with the School of Electronic Engineering and Computer Systems, University of Wales, Bangor LL 57 1 UT, Wales, U.K.

Publisher Item Identifier S 1041-1 135(97)03263-1.

0 1 2 3 4 Radial Coordinate, r (pm)

Fig. I . (a) Normalized intensity profiles of LPol and LPll modes in both directions. The normalized intensity difference versus radial coordinate for index steps of 0.01 and 0.1 is plotted in parts (b) and (c), respectively.

The cylindrically symmetric weak index-guided VCSEL structure considered in the following analysis is illustrated schematically in [7, Fig. I]. Subscripts i = 1, 2 will be used to denote the polarization direction. Birefringence is taken into account by assuming that the core refraction in- dex (n?'") in direction 1 is greater than in direction 2 (nYe), while the cladding refractive indices (n?ladd, n;ladd)

are the same for both directions. The modes supported by the assumed structure are the LP,, modes [9]. Subscript j = f , h will be used to denote the fundamental (LPol) and the azimuthaly independent higher order mode (LP11) respectively. The transverse intensity profiles of these modes in both polarization directions are plotted in Fig. l(a). The intensity profile is defined as &(r) = pz,(r)/ p%, ( r )dr where pz3 ( r ) is the profile obtained by solving the Helmholtz equation [7]. In this figure, we have taken nyore = 3.5002, nyre = 3.5, a cavity radius, a = 3pm, and an index step, an = - ncladd - - 0.01, obtaining in this way a spectral

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IEEE PHOTONICS TECHNOLOGY LETTERS, VOL 9, NO 5 , MAY 1997

I , , I I I 4 1 , , , , , I I , , , , I 1 \ , , , I

splitting between orthogonal polarizations of 0.5 8, for both transverse modes which is of the order of typical experimental values [SI. The intensity profiles of different polarizations for a given transverse mode are very similar. The relative differences are smaller than 0.4%. Another way of looking at this difference is by considering the normalized difference, S$,, defined as S$, = r(ti1& - t i 2 $ z 3 ) , where U, is the group velocity in the %-direction. This definition is useful since the modal gain difference between orthogonal polarizations for the j-transverse mode is given by the overlap integral of the local gain and ( s i / j j . As n y r e > nyre the polarization with index 1 is better confined than the orthogonal one and hence the difference is positive near the center and negative near the edge of the device as is seen from Fig. l(b) and l(c) for An = 0 01 and 0 1, respectively The function ~ ( $ 1 ~ - $ z J ) is symmetric since the modulus of the maximum and minimum is similar and tends to zero when An increases because the ratio of the birefringence to An decreases. Therefore, SG, becomes asymmetric and mainly negative when An increases since 64, = 711 [ T ( i / j l g - I j i ~ ~ ) -~(112/211 - 1)$2,] and 212 > U I ,

as is seen from Fig. l(c). The numerical model used here to study polarization selec-

tion dynamics is described in detail in [7]. An extension of this model to treat the case of competition between polarization states of the fundamental and first higher order transverse modes is applied here The principal components of the model are a carrier continuity equation and appropriate photon rate equations. Current is injected by using a circular disc contact. We include the current spreading effect by using the following profile for the injected current [IO]: j o for T < s and j o exp[-(r - S ) / T O ] otherwise, where s is the radius of the contact and TO is the effective diffusion length of the carriers.

The calculations presented here are aimed at demonstrating the influence of the index step of the guide on the polarization properties of the laser. The parameters used in the calculation are those used in [7 ] . In Fig. 2(a), the light output power in both modes and in both polarizations, PZj, is given as a function of j o for s a, TO = 0.01 pm, and a small An = 0.01. The observed behavior can be explained by noting that the modal gain difference between orthogonal polarizations

A s r ( N ( r ) -N0)6$~(~)dr where N ( r ) is the carrier density (A = 3 ' 10-16cm2,No = 1.33 . 10'' ~ m - ~ ) . At low currents SHB is not significant and the carriers accumulate near the center of the device where S$, is positive and, therefore, the gain of the better confined polarization (1) is larger for both transverse modes as is seen from Fig 2(b). The situation will change when increasing the current because SHB in the local gain is deeper which leads to an accumulation of carriers toward the boundary of the active layer and hence to an increase in the modal gain of the less confined polarization (2) for both transverse modes (see Fig. 2(b) for j , less than 9 kA/cm2). A polarization switching can then occur when both transverse modes in polarization 2 reach the threshold gain. This polarization switching is made clearer in the inset of Fig. 2(a) where the total power in both polarizations has been plotted. The polarization switching occurs at higher values of the current when the radius of the contact decreases since

for the j-transverse mode is given by vlgl, - 2129~~ - -

I -- 11.110 m a v

c. m U ...

11.105 -0

2

t

SHB mechanisms are less effective [9]. The effect of current spreading can be studied by increasing the effective diffusion length of the carriers, r0. This causes an extra injection of carriers in the cladding region that results in a broader carrier density. Polarization switching is then maintained but with a smaller switching current since current spreading helps SHB in concentrating the carrier profile in the near cladding region.

The polarization behavior of the device changes when An is increased. In Fig. 3(a), the light output power is shown for An = 0.1, s = 2.1 pm, and T O = 0.01 pm. When the current density is smaller than 9 W c m 2 light is emitted in the fundamental transverse mode and is mainly polarized in the direction of smaller refraction index (2). Above that current the first higher order mode is excited with a polarization orthogonal to that of the fundamental mode. This behavior can be explained by reference to Fig. 1(c) where the normalized difference, 6$f, is plotted when An = 0.1. As was previously discussed, 61,!1f is mainly negative in such a way that vlglf - vzgzf = A J o m ( N ( ~ ) - No)S$f(r)dr is also negative in spite of the fact that carriers accumulate near the center of the device. In this way, the carrier density overlaps better with the less confined polarization of the fundamental transverse mode and, therefore, P2f > P l f . When the higher order mode appears the SHB in the gain by the fundamental mode is such that the carrier density has a maximum at T = 1.7 pm. This carrier profile has a larger overlap with the better confined polarization of the higher order transverse mode and, therefore, Plh > P2h. The previous behavior appears in a range of contact radius of 1.95 < s < 2.3 pm. When the contact radius increases the less confined polarization of both transverse modes overlaps better with the carrier density profile and

VALLE et U/.: POLARIZATION BEHAVIOR OF BIREFRINGENT MULTITRANSVERSE MODE VCSEL’S 559

1

’m 1

‘3 1

h “

a c U

v

3

< l a

1

1

0.4

a 0.2

0.0

1.1110

1.1108

1.1106

1.1104

1.1102

1.1100 0

i

Current Density (kA/cm‘)

Fig. 3. (a) Steady-state power of the polarized transverse modes for an index step of 0.1. The total output power in each polarization appears in the inset. (b) Steady-state modal gain versus injection current density.

that is the polarization that dominates in the light-current characteristic.

In summary, the effect of birefrigence on the selection of polarization states for a weakly index guided VCSEL sup- porting both a fundamental and a first-order transverse mode has been studied. Several polarization behaviors, including polarization switching due to spatial hole buming effects, can appear depending upon the refractive index step of the VCSEL waveguide. The polarization switching results were obtained for a weakly index guided structure with an index

step An = 0.01. This laser has an intermediate An with respect to VCSEL’s having polarization switchings that could be attributed to SHB [81 (for instance the one observed in cruciform VCSEL’s 181 and the one observed at 8 mA in [2, Fig. l(b)]. To the best of our knowledge, relevant experimental results are not available for An M 0.01, but it is expected that this work can be useful in understanding the role played by spatial hole buming on polarization switching.

ACKNOWLEDGMENT

The authors would like to thank Dr. S. Balle and Dr. K. Choquette for helpful discussions.

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