## Abstract

We perform a theoretical linear study of the spectral and polarization threshold characteristics of coupled-cavity Vertical-Surface-Emitting Lasers (CC-VCSEL) on the base of a simple matrix approach. We show that strong wavelength discrimination can be achieved in CC-VCSELs by slightly detuning the cavities. However, polarization discrimination is not provided by the coupled-cavity design. We also consider the case of reverse-biasing one of the cavities, i.e. using it as a modulator via linear and/or quadratic electrooptic effect. Such a CC-VCSEL can act as a voltage-controlled polarization or wavelength switching device that is decoupled from the laser design. We also show that using QD stack instead of quantum wells in the top cavity would lead to significant reduction of the driving electrical field.

© 2010 Optical Society of America

## 1. Introduction

Coupled-cavity Vertical-Cavity Surface-Emitting Lasers (CC-VCSELs), i.e. monolithic devices with two optical cavities separated by a coupling mirror have attracted considerable interest recently [1–11]. Such devices can be independently biased using three electrical contacts and therefore offer much extended functionality than common VCSELs. Coupled microcavities are first investigated by Stanley et al. [1], demonstrating pronounced wavelength anticrossing effect: two different resonant wavelengths appear in the reflectance spectrum with separation that depends on the coupling mirror transmission. Light emission in optically pumped [2,3] or electrically injected [4, 5] CC-VCSELs can occur on a single (short or long) wavelength mode, as well as on the two of them simultaneously depending on the detuning between the two cavities. The experimentally measured single and double wavelength threshold characteristics of CC-VCSELs showed good agreement with theory [6]. Lasing can also occur on multiple transverse mode in two distinct combs belonging to the short and long cavity modes [7]. CC-VCSEL can also extend the fundamental transverse-mode operation achieving output power as high as 6.1 mW [8]. CC-VCSELs are easily operated in a Q-switch mode by optical loss modulation by reverse biasing one of the cavities [9]. Such reverse bias can also lead to voltage-controlled polarization switching in CC-VCSELs [10, 11]. This extended functionality of CC-VCSELs asks for a simple modeling approach that could provide basic understanding of the various phenomena that can occur and also provide some design guidelines. In this paper we implement such modeling based on simple transfer-matrix (TM) approach. Previously, a second order Taylor expansion of characteristic TM has been developed to provide analytical results for wavelength selectivity of CC-VCSELs with the goal of achieving two-wavelength emission for THz generation [12]. Recently, TM has been also used to demonstrate the principle of push-pull modulation of CC-VCSEL namely, that the output optical field can be modulated by dynamically detuning the two cavities using carrier-induced refractive index change [13]. Much higher intrinsic modulation bandwidth, not limited by the relaxation oscillation frequency is predicted. In this paper we complement these previous studies by showing that the TM method can be also used to calculate the threshold gain of CC-VCSELs. We then investigate the spectral and polarization characteristics of CC-VCSELs for the case of reverse-biasing one of the cavities, i.e. using it as a modulator via linear and/or quadratic electrooptic effect. We show that such a CC-VCSEL can act as a voltage-controlled polarization or wavelength switching device that is decoupled from the laser design.

The paper is organized as follows: In Section 2 we describe our transfer-matrix procedure for obtaining the resonant wavelengths and threshold gains of planar multilayer structure. In Section 3 we investigate CC-VCSEL spectral and threshold characteristics for different cavity lengths and cavity coupling strengths. In Section 4 and 5 we analyze, respectively, the CC-VCSEL polarization characteristics and the impact of electrooptic effect introduced by reverse biasing one of the cavities. Conclusions are given in Section 6.

## 2. Transfer-matrix method applied to CC-VCSELs

Our procedure of finding the resonant wavelength and threshold gains of CC-VCSEL is based on the transfer matrix method [14], which provides an efficient plane-wave solution of Maxwell equations in planar multilayer structures. Each layer “i”, homogenous in the transverse plane, is described by a characteristic 2 × 2 matrix [14]:

*β*= (2

_{i}*π/λ*)

*n*(normal incidence is considered) with

_{i}d_{i}*n*being the (generally complex) refractive index of the

_{i}*i*layer and

^{th}*d*its thickness. The electric and magnetic fields in the

_{i}*i*-layer are obtained as matrix multiplication, i.e.:

^{th}*N*is the number of the last layer in the structure and

*N*+ 1 denotes the semiinfinite medium on the right side of the structure. Considering light coming from the left side of the multilayer, i.e. no incoming field in the last, semiinfinite medium provides [

*E*

_{N}_{+1},

*H*

_{N}_{+1}]

*= [1,*

^{T}*n*

_{N}_{+1}]

*. The electric and magnetic field components in the*

^{T}*i*layer can be decomposed in traveling waves: to the right (superscript “+”) and to the left (superscript “−”) with electric fields given by:

^{th}To find the resonant wavelengths and threshold gains of the CC-VCSEL we impose the condition that there is no in-coming (traveling to the right) field to the whole CC-VCSEL multilayer structure, i.e.

with*n*

_{0}on the left side of the multilayer. Here,

*M*

_{0}is the matrix of the whole multilayer as given by Eq. (2), i.e. ${M}_{0}={\prod}_{k=1}^{N}{m}_{k}$.

The real and imaginary parts of Eq. (4) provide two implicit equations for two variables, namely the resonant wavelength *λ _{res}* and the imaginary part of the quantum well refractive index
${n}_{QW}^{im}$. The numerical solution of these two equations then provides the threshold gain as
${G}_{th}=4\pi {n}_{QW}^{im}/{\lambda}_{\mathit{res}}$.

## 3. Spectral and threshold characteristics of CC-VCSELs

The CC-VCSEL structure is shown schematically in Fig. 1 together with layer refractive index and thickness composition. We consider InGaAs QW basic structure lasing around 960nm, however the obtained results are generic and would be similar for more sophisticated structures and/or other VCSEL material compositions providing lasing on different wavelengths. Spectral dependencies of the complex refractive indices of different layers can be taken into account in the TM by recalculating all refractive indices at each wavelength prior to characteristics matrices calculation. We do not implement such procedure here to keep the study simple and easily reproducible; considering material dispersion does not change the basic results of our study. Our procedure of finding the resonant wavelength and threshold gains of CC-VCSEL is based on the transfer matrix method as described in the previous section.

In Fig. 2 we present the CC-VCSEL resonant wavelengths and threshold gains as a function of the relative difference of the two cavity thicknesses, Δ = 2(*d _{tC}* −

*d*)

_{bC}*/*(

*d*+

_{tC}*d*) where

_{bC}*d*(

_{tC}*d*) is the thicknesses of the top (bottom) cavity (cladding plus active layer thickness). Gain is provided by either the two cavities [Fig. 2(b)] or the bottom cavity only [Fig. 2(c)]. The wavelength anticrossing for the symmetric case (Δ = 0) is clearly visible in Fig. 2(a). The optical power distributions for the short and long wavelength modes are shown in Fig. 2(d) and 2(e), respectively, for the symmetric case Δ = 0 and in Fig. 2(f) and 2(g) for the asymmetric case Δ = 0.02 [at the positions of the dots in Fig. 2(a)]. The longitudinal profile of the refractive index of the CC-VCSEL structure is shown by black lines. As can be seen from Fig. 2(d)–2(g), the values of the optical power within the QW active regions that determine the modal confinement factors strongly depend on the symmetry of the structure. Therefore, for the case of positive (negative) Δ in Fig. 2(c), the threshold gain for the long (short) wavelength mode drastically increases as the mode shifts to the top cavity without a gain [see also Fig. 2(f) for Δ = 0.02]. However, if both cavities provide gain the change of the threshold current is much reduced [see Fig. 2(b)]. Nevertheless, strong wavelength discrimination can be achieved in CC-VCSELs by slightly detuning the cavities. For example, detuning the cavities by Δ = 0.02 increase the threshold gain of the short wavelength mode as much as 1.5 (5.8) times the one of the long wavelength mode for the case of Fig. 2(b) and 2(c).

_{bC}The coupling strength between the two cavities, determined by the transmissibility of the middle DBR strongly impacts the wavelength splitting and the difference of gains between the short and long wavelength CC-VCSEL modes. This is illustrated in Fig. 3, in which we show the CC-VCSEL resonant wavelengths (a) and threshold gains (b) as a function of the relative difference of the two cavity thicknesses Δ for two different numbers of layers of the middle DBR. As can be seen, increasing the number of middle DBR pairs results in decreasing the wavelength splitting and simultaneously in increasing the threshold gain difference due to the fact that the modes are better localized in each of the cavities.

## 4. Polarization characteristics of CC-VCSELs

In order to describe CC-VCSEL polarization properties we consider that small birefringence is always present in VCSELs and that it results in emission of two orthogonal linearly polarized (LP) modes along [110] (taken hereafter as x LP) and [11̄0] (taken hereafter as y LP) crystallographic directions [15, 16]. Therefore, we add for one (say y) LP a small contribution *n _{brf}* to the refractive indices of all the layers in the structure. This lifts the polarization degeneracy and results in 4 LP modes, with pairwise orthogonal polarization as shown in Fig. 4. For the chosen

*n*= 4 × 10

_{brf}^{−4}the wavelength splitting between the LP modes is very small, of the order of 1

*Å*. It results in a very small gain anisotropy, of the order of 0.5

*cm*

^{−1}. We therefore conclude that the coupled-cavity design by itself does not provide efficient polarization discrimination.

## 5. Electrooptic modulation of CC-VCSELs

In this section, we consider electro-optic modulation in one of the cavities while the other one provides the necessary for lasing gain. To this aim, we suppose that the top-cavity is subject to a linear electro-optical effect, i.e. its refractive index is given by:

*r*= 1.7 × 10

^{12}

*m/V*- the linear electro-optic coefficient and

*E*- the modulation electric field. In addition, quadratic electro-optic effect can take place in appropriately designed QWs that can replace the gain-providing InGaAs QWs in the top-cavity, i.e.

_{tCav}*s*= 4 × 10

^{−18}

*m*/

^{2}*V*

^{2}[17, 18].

We consider that the top-cavity works as a p-i-n modulator when reverse biased, i.e. only the refractive index changes according to Eqs. (6) and (7) while the width of the *i ^{th}*-region (top-cavity) remains unchanged. Furthermore, the top-cavity QW design is modified in a way that the QWs do not absorb light at the emission wavelengths and posses enhanced quadratic electrooptic effect as in [17, 18] or they are completely removed. In Fig. 5 we show the CC-VCSEL polarization resolved resonant wavelengths (a,c) and threshold gains (b,d) as a function of the modulation electric field

*E*for the case of linear electro-optic effect (LEO) only (a,b) and both linear and quadratic electro-optic effect (QEO) (c,d). Comparing Fig. 5(a) and 5(c), one can conclude that QEO effect produces much stronger wavelength shift than the LEO effect solely. Similarly, comparing Fig. 5(b) and 5(d) reveals that QEO effect produces much larger gain difference between the short and the long wavelength modes. However, it does not influence the CC-VCSEL polarization properties since it impacts the pairs of orthogonal LP modes in the same way. On the contrary, the LEO effect increases the refractive index for one LP mode and decreases it for the orthogonal one. Therefore, using QEO effect seems not to bring advantage for voltage induced polarization switching in CC-VCSELs. For example, an electric field in the top cavity of

_{tCav}*E*= −1×10

_{tCav}^{7}

*V/m*results in gain difference between x and y LP short wavelength modes of Δ

*G*≈ −7

_{th}*cm*

^{−1}for both the case of LEO and combined LEO and QEO effects.

Figure 5(b) shows that strong polarization dichroism between the x and y LP modes (the solid and dashed lines) is introduced in the CC-VCSEL when a reversed voltage is applied to the top cavity. Threshold gain difference of about
${g}_{th}^{x}-{g}_{th}^{y}\sim -30c{m}^{-1}$ is achievable at electric field of −5×10^{7}*V/m*. However, for such symmetric design the long wavelength LP modes still posses threshold gains that are only about 13*cm*^{−1} higher than the one of the short-wavelength LP modes. It is desirable to further increase the threshold gain for the long wavelength mode which, as shown in section III, is possible by slightly increasing the top-cavity length. An example of such a CC-VCSEL with Δ = 5.5 ×10^{−3} leading to stronger wavelength discrimination and similar voltage controlled LP mode selectivity is given in Fig. 6. In practice, the VCSEL birefringence is caused by residual in-plane anisotropic strain [15, 16]. As shown in [19, 20], QW uniaxial strain leads to different gains for the two orthogonal LP modes polarized along the [110] and [11̄0] crystallographic directions. Furthermore, both the birefringence and the dichroism can be easily manipulated by applying external uniaxial strain as demonstrated in [20]. Alternatively, one can introduce mirror-loss dichroism by etching subwavelength diffraction grating on top of the VCSEL cavity [21]. It is therefore possible to shift relatively the threshold gain curves for the two orthogonal LP modes in such a way that they cross at a certain value of the applied reversed voltage to the CC-VCSEL top cavity [see Fig. 6(c)]. By doing so, a voltage controlled polarization switching in suitably designed CC-VCSEL is possible due to the linear electrooptic effect.

Recently, a very large linear electrooptical effect with *r* ∼ 3 × 10^{−11}*m/V*, similar to the one of Lithium Niobate, has been demonstrated in quantum dots (QD) [22]. Using such QD stack instead of quantum wells in the top cavity would lead to significant reduction of the driving electrical field. For example, taking the same CC-VCSEL parameters as in Fig. 6(a) and 6(b) and an effective thickness of the QD stack of *d _{QD}* = 10

*nm*to account for the smaller QD filling factor, we obtain about 2.5 times larger threshold gain difference [see Fig. 6(d)].

We conclude this section by considering wavelength electrooptical switching in CC-VCSELs. As already shown in Fig. 5(d), quadratic electrooptic effect causes much stronger threshold gain modulation than the linear one. Therefore, better controlled, larger threshold gain difference wavelength switching can be realized by suitably detuning the two coupled cavities. An example of such electrooptic switching is shown in Fig. 6(e), 6(f) for Δ = 1.5 × 10^{−3}. As can be seen, threshold gain difference of 230(200)*cm*^{−1} can be achieved for electric field of −3(−2) × 10^{7}*V/m* for the case of Fig. 6(e), 6(f). The drastic decrease of the strength of the electric field necessary for wavelength switching in Fig. 6(f) is due to the pre-biasing of the top cavity with a constant electric field of *E _{BS}* = −1 × 10

^{7}

*V/m*.

## 6. Conclusions

In conclusion, by utilizing a simple matrix approach we study the spectral and polarization threshold characteristics of coupled-cavity Vertical-Surface-Emitting Lasers. We show that while strong wavelength discrimination can be achieved in CC-VCSELs by slightly detuning the cavities, polarization discrimination is not provided by the coupled-cavity design. We also consider the case of using one of the cavities as a modulator via linear and/or quadratic electrooptic effect. We demonstrate that such a CC-VCSEL can act as a voltage-controlled polarization or wavelength switching device that is decoupled from the laser design and can be separately optimized for high modulation speed. We also show that using QD stack instead of quantum wells in the top cavity would lead to significant reduction of the modulation voltage. The study of dynamical characteristics of electro-optically driven CC-VCSEL is beyond the scope of this paper and is now under developing.

## Acknowledgments

The authors acknowledge the financial support of FWO-Vlaanderen project G.0657.09N and of OZR-VUB.

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