## Abstract

We perform a theoretical study of the spectral and polarization threshold characteristics of Vertical-Cavity Surface-Emitting Lasers with Liquid Crystal overlay (LC-VCSEL) in three different configurations of the LC cell. Our model predicts the possibility of selecting between two orthogonal directions of linear polarization (LP) of the fundamental mode (x or y LP) by choosing appropriate LC length. It further predicts very strong polarization discrimination with LP mode threshold gain difference as large as several times the threshold gain of the lasing mode. We also numerically demonstrate an active control of light polarization by electro-optically tuning the LC director and show that either polarization switching between x and y LP modes or continuous change of the LP direction would be possible. Finally, we numerically demonstrate that LC-VCSEL would be capable of efficient wavelength tuning.

©2011 Optical Society of America

## 1. Introduction

Combining photonic components with liquid crystals (LC) extends their functionality by making use of the LC exceptionally strong electro-optical and nonlinear properties [1]. Examples can be found for one dimensional (1D) photonic crystals and cavities [2, 3], 2D photonics crystal slabs [4] and fibers [5, 6], 3D photonic crystals [7] and metamaterials [8]. Added by the LC values are the temperature and/or electric (or magnetic) field wavelength tunability as well as the strong optical anisotropy and nonlinearity. Wavelength tunability of external cavity semiconductor lasers have also been demonstrated by intracavity Lyot-type LC filter [9] and recently by LC-based tunable mirror [10]. LCs have also been implemented in a feedback-loop of Vertical-Cavity Surface-Emitting Lasers in order to achieve polarization control [11] and high contrast modulation [12]. In these experiments the external cavity containing the LC is several tens of cm long resulting in a bulky system. We have recently demonstrated polarization control of VCSEL [13, 14] and wavelength tuning of edge-emitting semiconductor laser [15] with optical feedback from piezo-electrically controlled isotropic (air) extremely short external cavity, of the order of a few *μm*. Filling in this cavity with a LC would form in the case of VCSEL a laser with a LC overlay (LC-VCSEL), in which the normal VCSEL cavity is optically coupled to a cavity containing LC. Such devices can be independently biased using separate electrical contacts for the VCSEL and the LC cell and potentially offer extended functionality than common VCSELs. Monolithic Coupled-Cavity VCSELs (CC-VCSELs) have been first investigated by Stanley et al. [16] 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 [17] or electrically injected [18, 19] CC-VCSELs can occur on a single (short or long) wavelength mode, as well as on the two of them simultaneously. The experimentally measured single and double wavelength threshold characteristics of CC-VCSELs showed good agreement with theory [20]. The goal of this paper is to provide insight in the operation and design of coupled-cavity LC-VCSELs. To this aim we make use of a simple transfer-matrix method recently applied for studying threshold characteristics of CC-VCSELs [21]. We investigate the spectral and polarization characteristics of the LC-VCSELs for the case of electrically modulated LC cavity and fundamental transverse-mode operation. We show that such a LC-VCSEL can act as a voltage-controlled polarization switching or wavelength tuning device that is decoupled from the laser design. When the reflectivity of the coupling mirror is reduced to zero, much enhanced wavelength tuning Δ*λ* is possible as recently suggested for optically pumped VCSEL containing nano polymer dispersed LC material [22]. Experimentally, Δ*λ* of 30 and 40 nm has been recently demonstrated for optically pumped 1.5 *μ*m InGaAs/InP quantum well VCSEL with a-*Si*/a-*SiN _{x}* and

*SiO*

_{2}/

*TiO*

_{2}Bragg mirrors [23, 24].

The paper is organized as follows: In Section 2 we present the LC-VCSEL device with 3 different configurations of the LC cell, as well as the way we model it. In Section 3 we investigate the LC-VCSEL modal and polarization resolved spectral and threshold characteristics for different cavity lengths and for the different LC cell configurations and mirrors. In Section 4 we demonstrate how polarization control and electro-optic polarization switching can be achieved in LC-VCSEL. Wavelength tuning is analyzed in section 5, and finally, conclusions are given in Section 6.

## 2. VCSEL with liquid crystal overlay: longitudinal and transversal LC cells

Three LC-VCSEL structures are considered as shown schematically in Fig. 1: one longitudinal and two transversal LC cells with electric field applied to the LC, respectively, along (*E _{LC}* ||

*Oz*), Fig. 1(a) and transverse (

*E*||

_{LC}*Oy*), Figs. 1(b) and 1(c) to the light propagation - or VCSEL cavity. We call hereafter these three types of LC-VCSEL cells

*L*

_{1},

*T*

_{1}and

*T*

_{2}, respectively. Nematic LC is considered with planar alignment of the LC molecules close to the glass boundaries (the longer axis of the LC molecules along

*Oy*,

*Oz*and

*Ox*axes in Figs. 1(a)–1(c), respectively). Such alignment could be achieved by depositing and rubbing thin polyimide layers on top of the transparent ITO (indium tin oxide) electrodes of the glass plates forming the LC cell. The length of the LC cell for the case of Fig. 1(a) is determined by the spacer used to separate the glass plate from the plate on which the VCSEL wafer is fixed and can vary from

*μ*m-s to hundreds of

*μ*m-s, while the transverse sizes can be very large (cm-s). An electric field

*E*applied to the LC cell will turn the LC director (the averaged direction of the longer axis of the LC molecules) at an angle

_{LC}*θ*, which depends on the distance to the anchoring boundary, as schematically illustrated in Fig. 1 [1]. To simplify the calculations, we consider that this angle is constant in the area above the VCSEL aperture where light is propagating. This assumption is quite reasonable for transversal LC cells as the VCSEL aperture of several micrometers is much less the size of the LC cell in x- (and y-) directions: hundred(s) (thousands) of

*μm*. However, it is questionable for the case of thin longitudinal LC cell. Calculations of the distribution of the LC director orientation

*θ*(

*z*) for this case will be presented in section 5 assuming hard-boundary conditions.

For longitudinal LC cell (Fig. 1(a)) and transversal LC cell with planar alignment along the *Oz*-axis (Fig. 1(b)) the LC director (along the *Oz*′ axis) is turned in the *yOz* plane at an angle *θ* with respect to the *Oz* axis; the coordinate transformation matrix being:

*Oz*axis and linearly polarized along the

*Ox*and

*Oy*axes, respectively:

*n*and

_{o}*n*are the LC ordinary and extraordinary refractive indices.

_{e}For transversal LC cell with planar alignment along the *Ox*-axis (Fig. 1(c)) the LC director along *Ox*′ is turned in the *xOy* plane at an angle *ϕ* with respect to the *Ox* axis, i.e.:

*Oz*axis will experience the highest birefringence if it is linearly polarized along

*Ox*′ and

*Oy*′, i.e.:

*N*of the VCSEL top distributed Bragg mirror (DBR) and the LC length

_{top}*L*.

_{LC}Our procedure of finding the resonant wavelengths and threshold gains of the LC-VCSEL is based on the transfer matrix method [25], by imposing the condition that there is no in-coming field to the whole LC-VCSEL multilayer structure [21]. The real and imaginary parts of the so obtained implicit characteristic equation are solved for two variables, namely the resonant wavelength *λ _{res}* and the imaginary part of the quantum well refractive index
${n}_{QW}^{im}$ [21]. The threshold gain is then obtained as
${G}_{th}\hspace{0.17em}=\hspace{0.17em}4\pi {n}_{QW}^{im}/{\lambda}_{res}$.

## 3. LC-VCSEL: optical field distribution and threshold gain

In Fig. 2 we present the optical power distributions for two modes of LC-VCSEL with *N _{top}* = 27 in Figs. 2(a)–2(d), which corresponds to a commercial off-shelf VCSEL and with

*N*= 0 in Figs. 2(e)–2(f), i.e. half a VCSEL without the top-DBR. The LC cell length and mirror are: Figs. 2(a) and 2(b)

_{top}*L*= 5.05

_{LC}*μm*and ITO/Au mirror; Figs. 2(c) and 2(d):

*L*= 5.08

_{LC}*μm*and ITO/dielectric DBR mirror and Figs. 2(e) and 2(f):

*L*= 5.1

_{LC}*μm*and ITO/dielectric DBR mirror. Refractive index profile of the LC-VCSEL structure is shown by black lines. LC ordinary refractive index is considered, which corresponds to x-linearly polarized (LP) mode in LC-VCSEL type

*L*

_{1}or

*T*

_{1}.

For LC cell with ITO/Au mirror, the calculated resonant wavelengths and threshold gains of the two modes are: Fig. 2(a)
*λ* = 0.80516 *μm*, *G _{th}* = 324869

*cm*

^{−1}and Fig. 2(b)

*λ*= 0.85156

*μm*,

*G*= 2699

_{th}*cm*

^{−1}. As can be seen from Figs. 2(a) and 2(b), the values of the optical power within the QW active regions that determine the modal confinement factors are very different and therefore the threshold gains for the two modes differ by more than two orders of magnitude. This situation resembles the case of monolithic CC-VCSEL with strongly detuned cavities [21]. We will further consider only modes with realistic threshold gain (

*G*

_{th}*<*5000

*cm*

^{−1}).

For LC cell with ITO/dielectric DBR mirror and off-shelf VCSEL, the calculated resonant wavelengths and threshold gains of such two realistic modes are: Fig. 2(c)
*λ* = 0.85131 *μm*, *G _{th}* = 1357

*cm*

^{−1}and Fig. 2(d)

*λ*= 0.85222

*μm*,

*G*= 4814

_{th}*cm*

^{−1}. Now, as can be seen from Figs. 2(c) and 2(d), the optical field distributions of the two modes are quite similar, resulting in much smaller difference of the threshold gains. We should mention that modes with very small confinement factors and huge threshold gains, like the one shown in Fig. 2(a) also exist for this LC cell but will not be considered hereafter.

For LC cell with ITO/dielectric DBR mirror and half-VCSEL, the calculated resonant wavelengths and threshold gains are: Fig. 2(e)
*λ* = 0.84092 *μm*, *G _{th}* = 2755

*cm*

^{−1}and Fig. 2(f)

*λ*= 0.86839

*μm*,

*G*= 3371

_{th}*cm*

^{−1}. The optical field now occupies largely the LC region leading to increased threshold gain in comparison to VCSEL with full top-DBR.

## 4. LC-VCSEL: polarization control and electro-optic polarization switching

Stand-alone VCSELs emit linearly polarized (LP) fundamental mode oriented along either [110] or [11̄0] crystallographic direction due to small residual-stress birefringence (Δ*n* = |*n*
_{[110]} – *n*
_{[11̄0]}| ∼ 10^{−4}) [26, 27]. The LC birefringence, (Δ*n* = 0.2 for E7 LC considered here), would easily overcome this small inherent VCSEL birefringence and would therefore determine light polarization orientation, i.e. the LC overlay will make it possible to control the polarization of the emitted light by the VCSEL. We demonstrate this polarization control in Fig. 3, where the resonant wavelengths *λ* and threshold gains *G _{th}* for two LP modes oriented along x and y axes are shown as a function of the LC cell length

*L*. The LC cell is with either a metal (Figs. 3(a), 3(b) and 3(e), 3(f)) or a dielectric (Figs. 3(c), 3(d)) LC mirror. The LC refractive index is given by Eq. (3) and we take

_{LC}*θ*=

*π*/2, corresponding to

*L*

_{1}cell without electric field

*E*applied to it and to

_{LC}*T*

_{1}cell with

*E*such that the LC molecules are completely turned along it. As can be seen from Fig. 3, the wavelength splitting between the two orthogonal LP modes and their threshold gain difference, $\Delta {G}_{th}\hspace{0.17em}=\hspace{0.17em}|{G}_{th}^{x}\hspace{0.17em}-\hspace{0.17em}{G}_{th}^{y}|$, strongly increase at certain LC cell lengths where the two cavities, the VCSEL and the LC one, strongly interact with each other. At these resonances the two cavities share the optical field, i.e. much larger than out-of-resonance part of the optical field resides in the LC cavity (see e.g. Fig. 2(d)). This leads to a strong decrease of the QW confinement factor and, therefore, to a huge increase of the threshold gain. Simultaneously, the wavelength splitting between the orthogonal LP modes increases as the LC confinement factor has increased. Quite importantly, the large LC birefringence makes the resonances for the two orthogonal LP modes happen at different LC lengths. Therefore, the threshold gain difference could become very large; for example, at LC length of

_{LC}*L*= 5.2(5.23)

_{LC}*μm*the threshold gain difference is as large as Δ

*G*= 2477(1346)

_{th}*cm*

^{−1}for the case of metal (dielectric) LC mirror, i.e. several times larger than the threshold gain itself. As can be deduced from Figs. 3(b), 3(d), and 3(f), a continuous change of the LC length would lead to consecutive polarization switching between the x and y LP modes (and eventually mode competition (hoping) at the points where the

*G*curves cross). By choosing an appropriate LC length one can therefore control the polarization of the light emitted by the LC-VCSEL. Moreover, a range of LC lengths exists where the threshold gain difference is larger than the threshold gain itself, i.e. where a strong polarization discrimination is possible. Scaling the length of the LC cell does not impact the polarization selection mechanism: it remains nearly the same as can be seen by comparing (a,b) and (e,f) graphs of Fig. 3. The wavelength splitting between successive longitudinal modes of the LC-VCSELs in Figs. 3(a) and 3(b) is about 40nm and 4.5nm, respectively.

_{th}We will now consider electro-optic modulation of LC-VCSEL with LC cells type *L*
_{1} and *T*
_{1} by applying an external electric field that turns the LC director at an angle *θ* with respect to the *Oz* axis (cf. Eq. (3)). In Fig. 4 we show the LC-VCSEL polarization resolved resonant wavelength (a,c) and threshold gain (b,d) as a function of the angle *θ* for the case of LC cell with a metal mirror and for two LC lengths: (a,b) *L _{LC}* = 5.1

*μm*and (c,d)

*L*= 50.2

_{LC}*μm*. As can be seen from Figs. 4(b), 4(d), the threshold gain for the y LP mode (blue line) is modulated so that in certain regions of

*θ*it is smaller than the threshold gain for the x LP mode (green line). This is similar to the modulation by the LC length discussed in the previous paragraph - ${G}_{th}^{y}$ would strongly increase when the VCSEL and LC cavities become in resonance and would drop otherwise. Therefore, we can electro-optically switch the polarization state of the light emitted by the LC-VCSEL between two orthogonal LP states, as well as chose the amount of the LP mode suppression (threshold gain difference). We should mention that it is straightforward to consider LC losses in our model; a realistic LC losses of 23

*cm*

^{−1}[1] produce only a slight shift of the threshold gain curves without noticeable change of the resonance wavelength (the dotted lines in Figs. 4(a), 4(b)). The LC length plays a crucial role electro-optic modulation of LC-VCSEL - it determines how many windows of y LP emission would open for one complete modulation of

*θ*between 0 and

*π*/2 radians (compare Figs. 4(b) and 4(d)). This is easily understandable as it is the LC phase thickness

*n*(

_{y}*θ*)

*L*that tunes the LC cavity in and out of resonance with the VCSEL cavity, i.e. the longer the LC-cell the larger the number of polarization switchings.

_{LC}According to Figs. 3(b), 3(d), 3(f), a precise control of the LC length (within 0.1*μm*) seems necessary in order to achieve
${g}_{th}^{x}\hspace{0.17em}>\hspace{0.17em}{g}_{th}^{y}$, i.e. a condition that ensures electro-optic polarization control and modulation. We would like to stress that such precise control of the LC length is actually not necessary when considering a practical LC-VCSEL device. A very simple and practical way to easily match the above condition would be to exploit the temperature dependence of the LC refractive index. As an example, let us consider a LC cell length of *L _{LC}* = 5.2

*μm*such that the green curve is bellow the blue curve (c.f. Fig. 3(b)). Then, taking LC thermooptic coefficient of

*dn/dT*≈ 1 × 10

^{−3}, a temperature change of 40 degrees would bring the green curve above the blue curve, thus making it possible to electro-optically control the LC-VCSEL polarization. Less temperature change would be required for longer LC cells.

We now discuss electro-optical modulation for the case of *T*
_{2} LC cell (cf. Fig. 1(c)). As shown in Fig. 3, we can chose appropriate EC length such, that the threshold gain for the y LP mode at *E _{LC}* = 0 is smaller that the one for the x LP (
${G}_{th}^{x}\hspace{0.17em}<\hspace{0.17em}{G}_{th}^{y}$). Now, with applying an external electric field

*E*the LC director will turn in the xOy plane at an angle

_{LC}*ϕ*as given by Eq. (3). As a result, the threshold gains for the LP modes oriented along Ox’ and Oy’ axis will not change: ${G}_{th}^{x\prime}\hspace{0.17em}=\hspace{0.17em}{G}_{th}^{x}\hspace{0.17em}<\hspace{0.17em}{G}_{th}^{y}\hspace{0.17em}=\hspace{0.17em}{G}_{th}^{y\prime}$. Therefore, we expect that LC cell type

*T*

_{2}would allow to electro-optically turn the direction of the LP fundamental mode of the LC-VCSEL at any angle in the xOy plane.

We conclude this section by mentioning that the speed of electro-optical polarization switching depends on the LC dynamical response and the length of the LC cell and is in the millisecond range.

## 5. LC-VCSELs for wavelength tuning

Finally, we consider LC-VCSEL for electro-optical wavelength tuning. In order to obtain maximum efficiency and widest wavelength region of tuning, the light confinement factor in the LC should be as large as possible; to this aim we implement a half-VCSEL (without the top-DBR) in either *L*
_{1} or *T*
_{1} LC cell with a dielectric mirror (cf. Figs. 2(e), 2(f)). Furthermore, as discussed in the previous section (cf. Fig. 4), the LC length should be as small as possible if one wish to avoid multiple polarization switching when electro-optically changing the LC director orientation *θ*. In Fig. 5 we show the calculated resonant wavelengths *λ* (a,c) and threshold gains *G _{th}* (b, d) for the two LP modes oriented along x (green lines) and y (blue lines) as a function of the LC director angle

*θ*. For very small LC length,

*L*= 0.64

_{LC}*μm*in Figs. 5(a), 5(b), a single y LP polarization is emitted. Wavelength tuning of 16.5 nm, 840

*nm*<

*λ*< 856.5

*nm*, is achieved when electro-optically tuning the LC director between 0 <

*θ*<

*π*/2

*rad*. Simultaneously, the threshold gain is decreased reaching a minimum of

*G*= 2400

_{th}*cm*

^{−1}at

*θ*= 0.9

*rad*. As expected, increasing the LC length -

*L*= 5.11

_{LC}*μm*in Figs. 5(c), 5(d) - the threshold gain curves of the two orthogonal LP modes cross multiple times, dividing the tuning region in several windows of either x or y LP mode lasing as denoted in the figure. In this way, one can more efficiently tune the wavelength of laser emission. For example, in the first window of y LP mode emission its wavelength can be tuned over 25 nm range between 842

*nm*<

*λ*< 867

*nm*when electro-optically tuning the LC director between 0 <

*θ*< 0.72

*rad*.

So far, we have considered that the LC director angle *θ* is constant in the area above the VCSEL aperture where light is propagating. As mentioned above, this assumption is questionable for a very thin longitudinal LC cell (c.f. Fig. 1(a)). In order to calculate the longitudinal distribution of the LC director *θ* (*z;E*) for a given electric field *E*, we assume hard-boundary conditions *θ* (*z* = 0) = *θ* (*z* = *L _{LC}*) = 0 and solve the boundary-value problem in one-elastic-constant (K) approximation, namely [1]:

*ɛ*is the dielectric anisotropy and

*ɛ*

_{0}is the vacuum permittivity.

The results for a longitudinal cell of length *L _{LC}* = 1.2

*μm*filled in with E7 LC (

*K*= 11 × 10

^{−12}

*N*,Δ

*ɛ*= 13.8 [28]) are presented in Fig. 6(a) for three LC voltages of

*U*= 2, 3 and 10

*V*(dotted,dashed and solid lines, respectively). Once the distribution

*θ*(

*z;E*) is known, we approximate it by a piecewise-constant function, i.e. we divide the LC layer in a number

*N*of homogenous layers with fixed but different director angle and calculate the LC-VCSEL resonant wavelength and threshold gain by the transfer matrix method (we take

_{LC}*N*= 40). The results are shown in Figs. 6(b) and 6(c) for two LC lengths:

_{LC}*L*= 0.64

_{LC}*μm*(blue line) and

*L*= 1.2

_{LC}*μm*(red line) and demonstrate that indeed, electrical wavelength tuning is possible for such very thin LC cells. However, as evident by comparing Fig. 5(a) and Fig. 6(b) (blue curve) the assumption of constant along the LC length

*θ*somehow overestimates the induced wavelength shift. Nevertheless, comparable wavelength span can still be achieved by increasing the length of the LC cell - c.f. Fig. 6(b) (red curve). Figure 6(b) provides further information for the LC voltage necessary for a certain wavelength shift, and also reveals that the LC voltage has to exceed a certain voltage, the so called Freedericksz threshold voltage, before the LC director starts reorienting according to the applied electric field [1].

## 6. Conclusions

We theoretically study Vertical-Cavity Surface-Emitting Laser with Liquid Crystal overlay in three different configurations of the LC cell and with either metallic or dielectric mirror. We calculate the modal and polarization resolved spectral and threshold gain characteristics of such LC-VCSEL and demonstrate the possibility of selecting between two orthogonal directions of linear polarization of the fundamental mode (x or y LP) by choosing appropriate LC length. The LP mode threshold gain difference can be as large as several times the threshold gain of the lasing mode, i.e. very strong polarization discrimination is possible. Scaling the length of the LC cell does not impact the polarization selection mechanism. We also consider the case of active control of light polarization by electro-optically tuning the LC director and show that either polarization switching between x and y LP modes or continuous change of the LP direction is possible. Finally, we numerically show that LC-VCSEL is capable of very efficient wavelength tuning. The experimental development of such electro-optically driven LC-VCSEL is now under way.

## Acknowledgments

The authors acknowledge the financial support of FWO-Vlaanderen project G.0656.09N and of OZR-VUB. KP acknowledges fruitful discussions with Profs. Kristiaan Neyts, Jeroen Beeckman and Minko Petrov.

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