Spatially inhomogeneously polarized transverse modes in vertical-cavity surface-emitting lasers

We demonstrate spatially nonuniform polarization of the fundamental transverse mode in a vertical-cavity surface-emitting laser. Using a vectorial electromagnetic model, we find orthogonal linearly polarized components in the field of a solitary mode. We give experimental evidence for our predictions by polarizationresolved near-field measurements: the fundamental transverse mode contains a four-lobed intensity distribution in one linear polarization direction that is weaker than the dominant orthogonally polarized Gaussian-shaped distribution by 35 dB in excellent agreement between experiment and model. The commonly used approach to determine eigenmodes of optical resonators is based on a scalar description of the electromagnetic field @1#. To obtain cavity solutions within the framework of the scalar paraxial approximation, it is necessary to assume a uniform linear polarization of the field vector in any cross section of the spatially extended mode. However, it has been shown that a homogeneous linear polarization is only compatible with infinitely extended plane waves, whereas beams of finite size or nonplanar wave fronts require components of the field vector along the beam’s main propagation axis ~‘‘longitudinal polarization’’ ! and orthogonal to the dominant linear polarization direction ~‘‘cross polarization’’ ! to maintain consistency with the transverse nature of electromagnetic waves as a basic result of Maxwell’s equations @2,3#. This phenomenon has been addressed for the case of radiating spherical particles @4#, illuminated pinholes @2#, and Gaussian-Maxwell-beams of gas lasers @5,6#, where nonuniform distributions of the electric-field vector have been predicted and observed. The power ratio between the weak cross-polarization component and the total power in the propagating TEM00-mode of an Ar-ion laser has been measured to be only 10 211 @6#. Thus, the effect is usually assumed as negligible and transverse modes are regarded as spatially homogeneously polarized. This assumption has also been adopted for transverse modes in vertical-cavity surfaceemitting lasers ~VCSELs ! @ 7#, a special type of semiconductor laser whose polarization properties have attracted broad interest @ 8‐1 0#. Numerous investigations of polarization related phenomena have been conducted @11‐15#, but thorough studies of spatial variations in the polarization of a solitary transverse mode in a VCSEL are up to now missing. Instead, it has been initially assumed that the polarization state of the emitted light is well described by a single set of Stokes parameters for each transverse mode @16,17#. In this work, we demonstrate the transverse modes’ vectorial character. In particular, we use a joint theoretical and experimental approach to analyze the spatial distribution of the polarization vector in the near field of a VCSEL’s solitary cavity mode. For the computation of the modes we apply the model of @18# based on mode expansion and coupled-mode theory. We expand the vectorial electromagnetic field in terms of the continuous basis of the TE and TM free space modes labeled by the transverse wave vector k @19#. The refractive index perturbation introduced by the device structure leads to a coupling between the expansion modes and via coupled-mode theory @20# it is possible to define a relation for the electric field at any two longitudinal positions z of the resonator. The boundary condition is given by the demand for self-consistency between backward and forward propagating waves at two arbitrary values of z and defines a simple eigenvalue problem: eigenvalues are related to the threshold gain and lasing frequency of the modes, while the corresponding eigenvectors give the distribution of the expansion coefficients overk and thus allow the reconstruction of the vectorial field. We apply this model to particular structures that are available to us for experimental verification. The devices are oxide-confined AlGaAs VCSELs provided by Honeywell, Inc. They have oxide apertures of 3-mm diameter and emit at wavelengths of l’845 nm. For these devices, we calculate the complex vectorial field of the cavity solutions and so obtain complete information on the polarization state. In the following, we depict the vectorial fields by plotting the three Cartesian field components (Ex ,Ey ,Ez) because their particular shapes allow a very sensitive experimental investigation of the polarization state. In Fig. 1, we present the calculated intensity distribution in the near-field of the fundamental transverse mode. We obtain two vectorial solutions, labeled as even and odd, which are degenerate in frequency and gain due to the circular symmetry and isotropy initially assumed in the model. They both possess a nonzero z component as a consequence of kfi0 contributions in the field expansion. Moreover, both solutions simultaneously contain unequal components x and y, indicating that the modes’ polarizations are spatially inhomogeneously distributed. The dominant components have a Gaussian-like intensity profile, while the weaker components have a characteristic four-lobed profile. In order to explain the origin of these spatial structures, we explicitly derive the expressions of the field expansion