## Abstract

The vectorial model of two-dimensional photonic crystals based on coherently coupled arrays of Vertical Cavity Surface - Emitting Lasers (VCSELs) is proposed in non-Hermitian Hamiltonian eigenproblem formulation. The polarization modes of square-symmetry photonic lattices are investigated theoretically. Rich mode structure with complimentary patterns of intensity for orthogonal polarizations of electromagnetic Bloch wave is predicted. The predicted near-field patterns of the polarization modes are confirmed in measurements of InGaAs/AlGaAs VCSEL arrays emitting at 965nm wavelength.

©2004 Optical Society of America

## 1. Introduction

Photonic crystals are periodic dielectric structures in which the photon modes can be tailored so as to control the propagation of electromagnetic waves as well as light-matter interaction [1]. A particular sub-class of two-dimensional (2D) photonic crystals consists of 2D arrays of optical waveguides or optical microcavities that are optically coupled perpendicular to the main axis of light propagation. In these structures only a small part of the wave vector, transversal to the main propagation direction, undergoes Bragg reflections. One example of implementation of such photonic crystals are so-called photonic crystal fibers, in which void patterns are introduced into the fiber core [2]. Matrices of vertical cavity surface emitting lasers (VCSELs) are an example of 2D photonic crystal systems incorporating gain and loss distributions [3–5]. The effect of including optical loss or gain distribution in such crystals is interesting both because of the expected impact on the photonic band structure and in view of their applications in optical devices.

In the case of VCSEL-based structures, the 2D photonic crystals are typically realized by patterning the reflectivity of the upper distributed Bragg reflector (DBR) of the VCSEL wafer using metallic overlays [3]. Experimental investigation of the photon mode structure [3,4], the effect of optical disorder [6,7] and photonic crystal strain [8], and photonic heterostructure confinement [9] in such VCSEL-based photonic crystals have been reported. Scalar models of the mode structure of VCSEL arrays have also been developed and employed to explain many of their observed features[4,10–12]. In particular, the photonic band structure of different VCSEL-based photonic crystal configurations have been analyzed [13,14]. However, little attention has been devoted so far to the polarization mode structure of these active photonic crystals [15].

In this letter we consider the polarization modes of VCSEL-based photonic crystals both theoretically and experimentally. The vectorial model is formulated as a non-Hermitian Hamiltonian eigenproblem. Peculiar polarization Bloch waves are predicted and confirmed experimentally using reflectivity-patterned phase-locked arrays of VCSELs.

## 2. Model

Our model system consists of a Fabry-Perot cavity in which the reflectivity of the upper mirror is modulated in two directions parallel to its plan. This corresponds to a VCSEL-based
photonic crystal utilizing conventional VCSEL wafer in which a patterned metal overlay is added to the top DBR. The pattern consists of high-reflectivity metal squares surrounded by
low-reflectivity metal, and is characterized by a fill factor FF that is the ratio between the area of the high-reflectivity pixel and that of the unit cell. The resulting composite DBR is
of high average reflectivity and of low contrast of the reflectivity pattern. The underlining DBR structure is finally neglected in our model system, since one can separate fast
longitudinal oscillations (along the cavity axis) and slow lateral oscillations of the electromagnetic field by the method of P.L. Kapitza [25]. In
this particular case, it consists in replacing the DBRs with mirrors of equivalent reflectivities [Fig.1.a]. Since the periodicity in the
reflectivity patterning enters through the boundary conditions at the cavity mirrors, the theoretical modeling of these 2D photonic crystals is carried out here using an equivalent,
unfolded cavity presentation. This equivalent presentation is based on the fact that the multiple reflections at the cavity mirrors effectively translate the cavity into a structure that
is periodic along the cavity axis. The resulting unfolded photonic crystal is thus three-dimensional and it can be analyzed using an orthogonal plane wave (OPW) expansion method [16]. The photonic modes are then given by electromagnetic Bloch waves propagating in the 3D equivalent crystal. Since the reflection operator
*r*(*x*, *y*)*$\widehat{\sigma}$* at a cavity mirror includes coordinate rotation (*$\widehat{\sigma}$*=*ÎĈ*
_{2} where *Î* represents the coordinate inversion and *Ĉ*
_{2} represents rotation by *π* about the optical axis), the equivalent crystal is of periodically varying “noninertiality” [17] in the *x*-*y* plane (parallel to the cavity mirrors plane) that reads [18,19]

where *r*(*x*,*y*) is the spatially modulated (amplitude) reflectivity with period *Λ*, and *L* is the cavity
length (the *z*-period of the crystal is 2*L*). In typical VCSEL arrays, |**K**
_{⊥}|≪*K _{z}* (paraxial approximation) and the contrast of the reflectivity pattern is small. In this case, the gauge transformation

*E*=

_{α}**Ê**

*v*

_{αβ}*(*

_{β}*x*,

*y*),

*H*=

_{γ}*e*

_{zαβ}**Ê**

*v*

_{γβ}*(*

_{α}*x*,

*y*), where, e.g., $\hat{\mathbf{E}}\mathbf{v}=\mathbf{v}+i{K}_{z}^{-1}\hat{\mathbf{z}}+({\nabla}_{\perp}\xb7\mathbf{v})+\frac{1}{2}{K}_{z}^{-2}{\nabla}_{\perp}({\nabla}_{\perp}\xb7\mathbf{v})-\frac{1}{4}{K}_{z}^{-2}{\Delta}_{\perp}\mathbf{v}$ [21], converts Maxwell’ equations for the curl of

**E**and

**H**into the same form of a 2D non-Hermitian Hamiltonian eigenproblem with respect to the photonic state wave function |

**v**

_{mK}〉 [19]

where *n* is the refractive index in the cavity and *ω _{m}*

_{K}assume complex values.

The analytic solution of Eq.(2) at the high symmetry points Δ, *Z* and *T* of the first Brillouin zone (BZ) was
obtained for a square lattice by means of group theoretical analysis [see Fig.1.b; note that ${K}_{z}=\frac{\pi}{L}N$ for cavity modes while the first BZ boundaries are at ${K}_{z}=\pm \frac{\pi}{2L}$]. The photonic band structure for such lattice, obtained from the empty lattice test, is presented in Fig.1.c indicating the
symmetry notation used for the wave functions. The squared modulus of the wave functions |**v**
_{mK}〉 shown in Fig. 2 for the high symmetry points defines the intensity patterns of the main polarization components of the
respective photonic modes (all states are doubly degenerate by polarization).

Figure 3 shows the calculated losses (per cavity roundtrip) for the modes of Fig. 2. The *T*
point $\mathbf{K}=(\frac{\pi}{\Lambda},\frac{\pi}{\Lambda},{K}_{z})$ is of practical interest, since the photonic state *T*
_{5} has the lowest cavity loss and is the main lasing mode of the coupled laser array. This state is doubly degenerate by polarization, and the two degenerate modes,
|*T*
_{5}, **x̂**〉 and |*T*
_{5},**ŷ**〉 have the major polarization component along the *x*- and *y*-directions, respectively. Their electric fields are given by

$${\hat{\mathbf{E}}\mid T}_{5},\hat{\mathbf{y}}\u3009\propto \hat{\mathbf{y}}\mathrm{cos}\left(\frac{\pi}{\Lambda}x\right)\mathrm{cos}\left(\frac{\pi}{\Lambda}y\right)-\hat{\mathbf{z}}\frac{i\pi}{\Lambda {K}_{z}}\mathrm{cos}\left(\frac{\pi}{\Lambda}x\right)\mathrm{sin}\left(\frac{\pi}{\Lambda}y\right)+\hat{\mathbf{x}}\frac{{\pi}^{2}}{2{\Lambda}^{2}{K}_{z}^{2}}\mathrm{sin}\left(\frac{\pi}{\Lambda}x\right)\mathrm{sin}\left(\frac{\pi}{\Lambda}y\right)$$

The structure of the electromagnetic field is shown in Fig.4 for the case of the |*T*
_{5},**x̂**〉 mode. Each mode exhibits the anti-phase modulation at adjacent lattice sites and has the well known intensity pattern (*z*-component of the
Poynting vector) with maxima located at the high-reflectivity pixels. We will denote this intensity pattern “pixel mode”. This pattern is set by the main polarization component of the mode
(*x* and *y* polarizations for |*T*
_{5},**x̂**〉 and |*T*
_{5},**ŷ**〉 modes, respectively). Besides the main polarization component, each mode also has a minor component polarized at the orthogonal direction
(*y*-direction and *x*-direction for |*T*
_{5},**x̂**〉 and |*T*
_{5},**x̂**〉 respectively). This minor component has an amplitude smaller by a factor $\frac{{\pi}^{2}}{2{\Lambda}^{2}{K}^{2}}\ll 1$ as the wave vector is almost parallel to *z*. (Note that the *z*-polarized components do not contribute to the energy flow.) Our model
predicts that the weak polarization component has intensity maxima located at the cross points of the array grid; we will refer to this pattern as “grid mode”.

## 3. Experimental results and discussion

The results of the model were tested experimentally using bottom emitting, electrically pumped InGaAs/AlGaAs VCSEL arrays operating at 965 nm wavelength [8]. The patterned top mirror consists of square Au-pixels embedded in a Cr-background, with a typical square lattice periodicity of Λ=5.8 µm and fill factor of FF=0.74.

The measured polarization-resolved near field (NF) pattern of such 4×4 VCSEL array (see Fig. 5) confirms the field distribution predicted by Eq.(3) and shown in Fig. 4. It corresponds to the state |*T*
_{5},**x̂**〉 with the main polarization component directed along the *x*- axis and with intensity maxima located at the VCSEL pixels (“pixel mode”). In this
state, the far-field pattern (not shown in the figure) has the familiar four-lobe pattern and indicates that the adjacent near- filed pixels are out of phase. The previously unknown, weak
*y*- polarized component of the field with intensity maxima located at the cross points of the array grid arises from the last term in Eq.(3). Similar polarization-resolved patterns were observed also for arrays of different sizes and the array size was shown to impact significantly the spectral and
dynamic behavior of the two orthogonal modes of Eq.(3) as well as the power ratio of polarization components [20].

Our simple model was developed for 2D photonic crystals of infinite extent (the details of the model will be given elsewhere [19]). Nevertheless, it
well predicts polarization features that are observed in VCSEL arrays of finite size. This is because the polarization features discussed are related to the Bloch term rather than the
envelope function part of the photonic wavefunction. Indeed, in arrays of finite size, the main polarization term in Eq.(3) should be
multiplied by a slowly varying envelope function *f*(*x*,*y*),e.g.,${\mathbf{E}}^{\left(0\right)}=\hat{\mathbf{x}}f(x,y)\mathrm{cos}\frac{\pi}{\Lambda}x\mathrm{cos}\frac{\pi}{\Lambda}y$ for an *x*-polarized mode. Respectively the weak cross-polarized term ${\mathbf{E}}^{\left(2\right)}=\frac{1}{2}\hat{\mathbf{y}}{K}_{z}^{-2}{\partial}^{2}(\hat{\mathbf{x}}\xb7{\mathbf{E}}^{\left(0\right)})\u2044\partial x\partial y$ assumes the form [21]

$$+\hat{\mathbf{y}}\frac{1}{2{K}_{z}^{2}}\mathrm{cos}\left(\frac{\pi}{\Lambda}x\right)\mathrm{cos}\left(\frac{\pi}{\Lambda}y\right)\frac{{\partial}^{2}f(x,y)}{\partial x\partial y}$$

where terms are arranged in descending order of their magnitudes (*∂f*/*∂x*, *∂f*/*∂y*≪*π*/Λ). For a nearly
constant envelope function as in our Fig.5, the main contribution to the weak cross-polarized term arises from the periodic part of the Bloch wave
[first term in Eq.(4)]. Thus the observed polarization inhomogeneity within the unit cell of the array is not influenced by the finite size
of array. Within the accuracy of the slowly varying envelope function, the polarization mode pattern coincides with the polarization Bloch wave of Eq.(3).

The contribution of the envelope function variation to the weak cross-polarized term is of minor importance [last term in Eq.(4)]. This term has intensity pattern similar to the “pixel-mode” pattern of the main polarization component and thus does not affect the polarization modulation within the unit cell of the array. It was observed and reproduced by the model in Ref.[15]. Different from our bottom emitting VCSEL arrays, the structures examined there were top emitting with VCSEL pixels definition via openings in the metallic grid deposited on the top DBR. In these structures, the major cross-polarized term with “grid mode” intensity pattern is blocked by the metallic grid of the top DBR. The polarization features reported there thus have only a large scale inhomogeneity. The numerical simulations were also performed for the field pattern outside the cavity. Therefore they reproduce well the slow polarization modulation across the entire array but do not reveal the polarization Bloch term of the intra-cavity mode.

In our arrays, deviations from the predicted “grid mode” pattern are observed near the boundaries of the array where the envelope function varies more rapidly. This variation results from depolarization and scattering near the array boundaries [22,23]. Analysis of the modification in the mode pattern near the photonic crystal boundaries should take into account band mixing caused by detuning from the high symmetry point of BZ [10,24] and is left for future studies.

## 4. Conclusion

In summary, we have studied the polarization mode structure of VCSEL-based photonic crystals. A photonic band structure model was developed and predicted the structure of a polarization Bloch wave revealing complimentary intensity patterns of main and weak orthogonal anti-phase mode components. The predicted polarization-resolved patterns were experimentally observed in mirror-patterned VCSEL arrays.

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