## Abstract

We discuss the design, modelling, fabrication and characterisation of an integrated tuneable birefringent waveguide for quantum cascade lasers. We have fabricated quantum cascade lasers operating at wavelengths around 4450 nm that include polarisation mode converters and a differential phase shift section. We employed below laser threshold electroluminescence to investigate the single pass operation of the integrated device. We use a theory based on the electro-optic properties of birefringence in quantum cascade laser waveguides combined with a Jones matrix based description to gain an understanding of the electroluminescence results. With the quantum cascade lasers operating above threshold we demonstrated polarisation control of the output.

© 2013 OSA

## 1. Introduction

For many applications including laser diode spectroscopy and ellipsometry the control of wavelength and polarisation of a laser is required. Presently, bulk optics elements are routinely used for polarisation control [1] and wavelength tuning [2, 3]. In this paper we describe the design, fabrication and characterisation of an integrated electro-optically tuneable birefringent waveguide for polarisation control and wavelength tuning of a semiconductor laser, the quantum cascade laser (QCL) [4].

QCLs are based on intersubband transitions in quantum wells but we can expect that QCL material will exhibit electro-optic properties based on intersubband transitions and interband transitions [5, 6]. The selection rules for intersubband transitions dictate that only light polarised with its electric vector perpendicular to the plane of the quantum well can couple to the intersubband transitions. In the typical waveguide arrangement used in a QCL this means that there is only optical gain for Transverse Magnetic (TM) polarised modes. In addition it means that, at the operating wavelengths, QCL waveguides exhibit birefringence and indeed it has been previously reported [7] that intersubband transitions can lead to birefringence. In the QCL waveguides the strength of the intersubband transition can be altered by injecting current into the waveguide and therefore the QCL waveguides have birefringence that can be adjusted with injection current. Also associated with the quantum wells that make up the QCL material are interband transitions and these transitions are strong and highly dispersive at photon energies close to the band gap resonance [6]; at the photon energies well away from the band-edge where the QCL operates the effect of the interband transitions is weaker and much less dispersive. The interband transitions are also a function of applied electric field and injected current and so at the QCL operating wavelength we can expect an interband electro-optic effect which is not dispersive. Altering the phase relationship between the polarisations is the basis of the wavelength tuning and polarisation control of birefringent waveguides – but wavelength and polarisation tuning requires that both TM and TE modes are present and that is usually achieved by incorporating waveplates into the laser.

The feasibility of an electro-optically altering the phase relationship between the polarisations has been recently demonstrated for quantum well semiconductor laser diodes by monolithically integrating the laser, polarisation mode convertor (PMC) section and differential phase shift (DPS) section [8].

The integrated equivalents of waveplates are polarisation mode convertors [8, 9]. We have previously report how PMCs are fabricated for QCLs [10]. The PMCs are made by etching trenches on the waveguides to make the waveguide cross-section asymmetric. With PMCs it is possible to make the integrated equivalent of a bulk quarter waveplate and to produce a hybrid mode that is 50% TE and 50% TM. The other component required for a tuneable waveguide is a DPS section; this is a section of the QCLs waveguide with electrical contacts that allow injection of current that alters the strength of the intersubband transitions.

In this paper we discuss the design of the integrated tuneable birefringent waveguide (ITBW) and we report results on experiments that employ the sub-threshold electroluminescence (EL) to probe the operation of the ITBW. We develop a theory of the operation based on a Jones matrix [11] description of the ITBW and a description of the electro-optic properties of the birefringence of the QCLs waveguide that splits the electro-optic birfringence into a current dependent but wavelength independent effect that arises from interband transitions and current and wavelength dependent birfringence that arises from the intersubband transitions. Finally we demonstrate wavelength and polarisation tuning of the quantum cascade laser operating above threshold.

## 2. ITBW design

Our design for an ITBW for QCLs is shown in Fig. 1. It consists of 2 PMCs configured as quarter waveplates with a DPS section between them. The PMCs are passive optical waveguides that are made asymmetric by, for example, etching trenches on one side on the waveguide. Full theoretical and design description of PMCs on QCLs can be found in [10]. The DPS section adjusts the relative phase of the TM and TE components of the circularly polarised light from the first PMCs. This is achieved because of the selection rules of the intersubband transitions and in a QCL structure the strength of the intersubband transitions can be altered by the injection of current so the birefringence of DPS section can be adjusted by injection current. The relative phase of the TE and TM components at the input into the second birefringent waveguide determines the output polarisation of the second PMC. In a QCL, only TM light will be amplified and the wavelengths that are rotated back to TM polarisation are determined by the current in the DPS section thus we have a QCL that has its operating wavelength selected by an electro-optic tuneable birefringent waveguide.

## 3. Electro-optic birefringence of the QCL and the Jones matrix description

In the DPS section of the ITBW we utilize the electro-optic birefringent properties of the quantum cascade laser structure to tune the polarisation and wavelength. We regard the electro-optic birefringence of the DPS section as made up three parts, a natural birefringence (i.e birefringence with zero injection current into the DPS section), $\Delta n$, an interband birefringence, $\Delta {n}_{IB}({i}_{DPS})$, and an intersubband birefringence, $\Delta {n}_{IS}(\lambda ,{i}_{DPS})$. The electro-optic interband birefringence comes from interband transitions and arises from the effect of current on interband transitions and the electro-optic intersubband birefringence arises from the effect of the current on the intersubband transitions. For$\Delta {n}_{IB}({i}_{DPS})$, we ignore the wavelength dependence because a QCL operates at wavelengths far from the band-gap resonance of our quantum well material and so is almost constant for wavelengths close to the laser wavelength. For$\Delta {n}_{IS}(\lambda ,{i}_{DPS})$, we explicitly include the wavelength dependence because it is much more dispersive for wavelengths close to the laser wavelength [7]. Note that the$\Delta {n}_{IS}(\lambda ,{i}_{DPS})$, only gives rise to a change to the TM mode. So we have:-

Jones matrices are routinely used to describe the polarisation behaviour of bulk optics systems [11] and have been used to describe the operation of a laser incorporating a birefringent crystal and gain medium [12]. Here we use Jones matrices to describe the operation of the ITBW. The coordinate system we use is as follows; y is the direction of growth of the epilayer (so TM light is y-polarised) and x is perpendicular to the direction of growth (TE light is x-polarised).

In what follows we assume that the PMCs can be described by a Jones matrix description and that their operation is wavelength independent [9]. For the co-ordinate system illustrated in Fig. 1 the generic Jones matrix is:-

*θ*is the angle the optic axis makes with the x direction and

*δ*is the phase retardation.

From our previous work on PMCs [10] we have designed a PMC that converts a TM mode into a hybrid mode that is 50% TM and 50% TE. Our experiments do not allow us to measure the relative phase between the TM and TE modes and so we can adapt the generic Jones matrix to describe this behaviour in various ways; for example an optics axis rotation of θ = π/4 and phase retardation of *δ* = π/2, or we could set θ = π/8 and *δ* = π. As explained in [9] it appears that PMCs that conversion to 50% TM and 50% TE in a half beat length are best described by setting θ = π/8 and *δ* = π:- Using these values we obtain the Jones matrix for the PMC.

*θ*= 0, because the plane of the quantum wells is in the x- direction and$\delta =\frac{2\pi \Delta n(\lambda ,{i}_{DPS})L}{\lambda}$, where

*L*is the length of the DPS,

*is the free space wavelength. Thus the Jones matrix for DPS section is,*

_{λ}

*T*_{D}for the ITBW is given by:-We assume the input into the ITBW section is TM polarised because the QCL gain section selects for TM polarisation, $\left[\begin{array}{c}0\\ 1\end{array}\right]$, then the polarisation vector representing the ITBW response is given by:-

In the dynamic polarisation control operation of ITBW we measure the angle of polarisation, *φ*, of the output from the QCL as a function of input current into the DPS and if we assume a single pass through the ITBW then that is given by

## 4. Fabrication of QCL-ITBW devices

The QCLs used in this work are based upon a double phonon resonance QCL wafer structure reported previously in [13] that emitted at a wavelength of around 4600 nm.

Figure 2(a) shows the SEM image of a PMC after ICP etching and Fig. 2(b) a SEM image of the cross section of a PMC. Figure 2(c) shows a photograph of the wire bonded QCL-ITBW device with different sections (GS, PMC1, DPS and PMC2) indicated on the micrograph. Fabrication processes were similar to those used for the PMCs described in [10]. In the ITBW device, only the GS and DPS section contact windows were opened on the top of the waveguides with the rest of the structure protected by Si_{3}N_{4} and SiO_{2} insulating layers. GS and DPS section metal contacts were electrically isolated using a metal lift-off technique. Detailed step-by-step information of the fabrication process can be found in [10].

Using the design methodology outlined in [10] the PMCs were designed to convert 50% TM and 50% TE in a half beat length; so that in the first PMC (PMC1) in Fig. 1 light from the GS, (polarised TM), originating from intersubband transitions is converted into to 50% TM and 50% TE light at the output of the PMC and input to the DPS. This required trenches to be etched into the PMC waveguides, the trenches were 0.5 μm wide and 3.518 μm deep and were fabricated using a reactive ion etching technique [14]. PMC1 and PMC2 are designed to be identical.

## 5. Characterisation and results

The QCL-ITBW devices were soldered epilayer-up onto aluminium nitride tiles and then mounted onto the cold finger of closed cycle helium cryostat. Spectral characteristics were measured using a Bruker Vertex 70 Fourier transform infrared (FTIR) spectrometer. The spectrometer was equipped with a liquid nitrogen cooled indium antimonide (InSb) detector, with a lock-in amplifier used for detector signal recovery. For polarisation measurements, a grid polariser was mounted in front of the detector, in a rotatable mounting. Characterisation was carried out at a laser heat sink temperature of 250K. The results that we report here were for devices that had the gain section length of 2500 μm then DPS section length 1125 μm and the PMC sections were designed to be half waveplates [9] with length 151 μm to give 50%, 50% TE, TM mode intensity split and cross-section of 3.6 μm wide and 4.9 μm high waveguides.

The GS current density was pulse injection with values around 4.2 kA/cm^{2} and the DPS had direct current (DC) injection at around 1/10 of the GS current density – to avoid excessive TM gain in the DPS section.

#### 5.1 Sub-threshold single pass measurements

To gain an insight into the operation of the ITBW we carried out single pass measurement using the sub-threshold EL from the GS. This allows as to estimate the values of$\Delta n$, natural birefringence without DPS current and$\Delta n(\lambda ,{i}_{DPS})$of the DPS. The intersubband birefringence, $\Delta {n}_{IS}(\lambda ,{i}_{DPS})$, was estimated by measuring the small amount of current dependent gain as a function of wavelength in the DPS and taking a Kramers-Kronig (KK) transformation [15] of the gain spectra.

Figure 3 shows the experimental setup used to characterise the ITBW. The GS was driven with 100 ns duration current pulses (${i}_{G}$ = 4.2 kA/cm^{2}) with a 60 kHz repetition rate and the DPS section driven with DC current. The device was driven below the lasing threshold current to ensure single-pass operation and avoid multimode beating effects on PMCs [9, 16]. The TM fraction of the EL spectra was measured by placing a grid polariser between the ITBW device and FTIR (with the optical axis of the grid polariser parallel to the QCL waveguide).

Figure 4(a) shows the experimental TM response curve of the sub-threshold EL spectra emitted from the ITBW device with and without application of DC current into the DPS section. When DC current (0.33 kA/cm^{2} and 0.69 kA/cm^{2}) is applied into the DPS section, increased transmission is observed through electrically modulating the gain [17].

To gain an insight into the operation of the QCL-ITBW device we compare the theory of the electro-optic birefringence and the Jones matrix description of the ITBW against the experimental results. To make the comparison we calculate the TM intensity fraction of the ITBW response function using Eq. (7) and multiply the result with the EL spectra measured from the GS facets, shown in Fig. 4(b). We assume that when no current is injected that the birefringence is entirely dominated by the material birefringence,$\Delta n$,thus in Eq. (1) $\Delta n(\lambda ,{i}_{DPS})$replace with$\Delta n$, and we regard $\Delta n$, as an adjustable parameter. We take $\Delta n$as positive when the TE refractive index is greater than the TM refractive index. To fit the Jones matrix model to the experimental result, we adjust the value of$\Delta n=0.005$, with $\Delta n(\lambda ,{i}_{DPS}=0)$ with these values we obtain an approximate fit with the experimental result as is shown in Fig. 5(a).

In order to fit the Jones matrix model with the experimental results, when DC current is applied into the DPS section, we require to estimate the value of $\Delta {n}_{IS}(\lambda ,{i}_{DPS})$. To estimate the $\Delta {n}_{IS}(\lambda ,{i}_{DPS})$, we required to measure the gain using $I={I}_{\text{o}}{e}^{-\alpha L}$, where ${I}_{\text{o}}$is TM EL intensity without the DPS current, *I* is the TM EL intensity with the DPS current, *L* is the length of the DPS section and -α is the gain [18]. Figure 5(b) shows the measured gain at DPS current of 0.33 kA/cm^{2} and 0.69 kA/cm^{2}, the modulation observed with a period of ~130 nm in the gain spectra fits with a Fabry-Perot resonance associated with an additional ~40 μm long waveguide section between the end of PMC2 and the end facet (present due to imprecise cleaving). Using the KK relation [15] the gain measurement is transformed into $\Delta {n}_{IS}(\lambda ,{i}_{DPS})$ and shown in Fig. 6. In our model we include the electro-optic interband birefringence contribution, $\Delta {n}_{IB}({i}_{DPS})$, −0.0014 and −0.0022 were the values used, for 0.33 kA/cm^{2} and 0.69 kA/cm^{2} respectively that gave the best fit to the results.

So using the Jones matrix description, regarding $\Delta n$and $\Delta {n}_{IB}({i}_{DPS})$as an adjustable parameters and calculating $\Delta {n}_{IS}(\lambda ,{i}_{DPS})$from the gain measurements we obtain an approximate fit of the theory to the experimental results - see Fig. 7. Clearly, from Fig. 7(b), for high current densities and at the longer wavelengths there is substantial deviation of the theory from the measured response and some more refinements to the theory may be required, for example taking account of relative TE and TM losses [19], for a more complete understanding of the QCL-ITBW device.

#### 5.2 Theory of laser operation with birefringent components

As is discussed in [12], the operation of a laser with polarisation dependent gain and including birefringent components is a more complex situation than the single pass case we have analysed in the previous section.

In general the resonance requirement for a resonator with birefringent components imposes the following conditions, first the phase change in a round trip must be a multiple of 2π and second the polarisation must repeat after a round trip. These two conditions were expressed in the following equation:-

Where ** M** is the round trip Jones matrix,

**is the two-component vector representing polarisation and**

*X**β*is a positive real number.

We can some insight into the operation of the laser by analysing the QCL that includes the ITBW optical circuit using Jones matrix method as described in [20]. In this analysis we will derive the phase conditions for laser action and so we ignore the amplitude of the modes and assume an amplitude normalised to one for all the components.

The Jones matrix of an ideal mirror (~100% reflection) is given by

Using the convention described in [20], when the direction of light propagation through a reciprocal element is reversed, the Jones matrix of the element is transformed as follows:**, for the ITBW is given by the following multiplication of matrices:-**

*M*In the above we have followed the usual convention for the Jones matrices and omitted the phase factor that describes the common phase change for TM and TE modes as the wave travels through the component. In order to completely take account of the phase change in the round trip matrix we follow [12] and explicitly include the phase factor that is common for TM and TE modes. If we assign the following variables ${L}_{d}$is the average path for a single pass and the average refractive index for the round trip of the whole device ${n}_{d}$is as follows -then Eq. (15) becomes:-

If we compare Eq. (17) with Eq. (10) then it is apparent that the exponential factor in Eq. (17) can only be a real number, that is 1, if the following phase condition is satisfied:-

The laser modes in the QCL-ITBW device have to satisfy this phase condition.In general ${n}_{d}$and ${L}_{d}$(${L}_{d}\approx $total length of device) will be a weighted mixture of TM and TE refractive indices and paths as the laser mode swaps between TM and TE propagation. In an approximate approach we can expect that ${n}_{d}\approx \frac{({n}_{TM}+{n}_{TE})}{2}$ and ${n}_{TM}$effective index for the TM mode and ${n}_{TE}$effective index for the TE mode.

According to this simple analysis the mechanism for tuning the QCL-ITBW will be the injection current induced change of the phase condition given by Eq. (18) and although this gives some insight it does not include the effect of gain. For lasing to occur, in addition to the phase condition the optical gain experienced by the TM mode must equal the optical loss after a round trip. But it can be seen from Eq. (18) that the phase condition for positive feedback in the QCL-ITBW device can be tuned by injecting current in the DPS section and altering the $\Delta n(\lambda ,{i}_{DPS})$factor. However, there will also be some injection current dependence of the ${n}_{d}$factor.

#### 5.3 Measurement of laser operation with birefringent components

Figure 9 shows the observed emission spectra and tuning at different DC current densities from 0 to 1.11 kA/cm^{2} applied to the DPS while a fixed pulsed peak current of 1.3**I*_{th} (6.8 kA/cm^{2}) is applied to the GS. No single mode emission was observed and 24 nm discontinuous tuning with tuning rate of 25 nm/(kA/cm^{2}) was recorded.

The spacing between the modes shown in Figs. 9(b) and 9(e) is 11 nm. According to the phase condition theory in section 5.2, Eq. (18), and using the values $\Delta n(\lambda ,{i}_{DPS})L\approx {10}^{3}nm$and ${n}_{d}{L}_{d}\approx {10}^{6}nm$the mode spacing is dominated by the second factor and we would have expected mode spacing more like 0.7 nm. The 11 nm spacing that we do observe is consistent with there being sub-cavities present in the device.

The PMC, employed in the ITBW, is an asymmetric waveguide and is formed by etching trenches into one side of the symmetric waveguide. The index contrast between symmetric and asymmetric waveguide can lead to reflections and we have from a finite- difference time-domain analysis estimated that the mirror reflectivity could to be as large as 20%. This reflectivity may lead to the formation of sub-cavities. The 11 nm spacing that we observe is consistent with the gain section and DPS section acting as coupled sub-cavities. This is similar to the Vernier mode selection described in [21]. To eliminate this effect the interfaces between waveguide elements will need to be redesigned with optimized etch depths and tapering, ensuring smooth mode transitions without reflections.

## 6. Active polarisation control

For polarisation control measurements, the GS pulsed current was fixed at 6.8 kA/cm^{2} (1.3* *I*_{th}) and the current to the DPS section was varied between 0 kA/cm^{2} and 0.97 kA/cm^{2} in steps of 0.13 kA/cm^{2}. We measured the maximum transmission as a function of wire-grid polariser rotation angle (*φ* = 0° to 360°), where *φ* = 0° is defined as the angle of the first maxima in transmission for the TM light through the wire grid polariser. This experiment does not completely determine the output polarisation; for example, it cannot discriminate between elliptical and linear polarisation but it does indicate that the polarisation state can be tuned by the current injected into the DPS.

Figure 10 shows the change in polarisation angle for maximum transmission as a function of the current injection into the DPS section. At 0 kA/cm^{2} a polarisation angle of 37° (TM fraction > TE fraction) was recorded and the polarisation angle increases upto 82° (TM fraction < TE fraction) with an applied DC current of 0.97 kA/cm^{2} with no further change in polarisation angle recorded above 0.97 kA/cm^{2} current.

In order to fit the polarisation angle measurement result with the Jones matrix model, we are required to estimate the phase retardation with and without the DPS current. Phase retardation of DPS section can be written as:

where,and$\Delta \delta \left({i}_{DPS}=0\right)$and $\Delta \delta \left(\lambda ,{i}_{DPS}\right)$are the phase retardation without and with DPS current, $\Delta n$is the birefringence before current is applied, $\Delta {n}_{{i}_{DPS}}$is the rate of change of birefringence with respect to current density, ${i}_{DPS}$, we assume a simple linear dependence. Equation (19) is substituted with the single pass Jones matrix Eqs. (7) and (8), the polarisation control operation is than analysed using Eq. (9). To fit the Jones matrix model to the experimental result, we adjust the value of $\Delta n=0.005$, and, $\Delta {n}_{{i}_{DPS}}$ = – 1 x 10^{−3}cm

^{2}/kA and with these values we obtain a good fit with the experimental result for lower current densities as is shown in Fig. 10. The discrepancy between the theory and experiment at higher current densities may be due the fact that the theory is incomplete and it does not take account of gain and loss for the TE and TM mode in the DPS section. At the higher DPS injection currents there will be some gain for TM light and this prevents the output from being completely TE polarised.

## 7. Conclusions

We have described the design, modelling, fabrication and characterisation of integrated tunable birefringence waveguide (ITBW) for QCLs and demonstrated that an electro-optically tuneable birefringent property can be used in quantum cascade laser waveguides to achieve active control of polarisation and wavelength. We have used a simple Jones matrix description of the polarisation components in the QCL-ITBW device to construct a Jones matrix model that gives a good fit to the results obtained by measuring the output polarisation rotation as a function of DPS injection current. The model also helps explain the single pass experiments obtained by operating the device below the laser threshold. Further, from the Jones matrix description, we developed a simple phase only theory for the phase condition for modes in a laser with birefringent components, however applying our theory to the observed wavelength output spectra and DPS injection current tuning was complicated by sub-cavity effects.

By comparing the active polarisation angle measurement result with the Jones matrix model, we estimate the natural birefringence (no DPS current), $\Delta n$around 0.005 for the QCL employed in this work. This compares with the value that we obtain from the single pass measurements which is also 0.005. Also we can fit the dynamical polarisation angle control results if we assume a value of $\Delta {n}_{{i}_{DPS}}$ = –1 x 10^{−3} cm^{2}/kA – the value indicated by the fit to the single pass measurements is – 3.1 x 10^{−3} cm^{2}/kA. The results in Fig. 6 indicate that at the laser wavelengths, the intersubband contribution, $\Delta {n}_{IS}$, to birefringence is small at around 1.0 x 10^{−5} compared to the values given above. This value of $\Delta {n}_{IS}$is in approximate agreement for the values reported in [22, 23] for the linewidth enhancement factor for QCL albeit for different structures and wavelengths. Therefore we conclude that for laser operation, the electro-optic effect– the current dependent birefringence, is dominated by the interband transition and is not highly dispersive. This limits the wavelength tuning. To obtain significant wavelength tuning requires a redesign of the QCL active region.

Further experimental and theory work is required to obtain a more complete understanding of the operation of the QCL-ITBW device. Experimentally a QCL active region designed specifically to enhance the electro-optic properties is required and a variety of device architectures should be investigated. The theory of birefringent effects in lasers needs to be extended to include both amplitude and phase effects.

The polarisation angle control range is 45° and we are not aware of any other QCL device that is capable of electronically controlling the output polarisation. Active control of the polarisation state of a QCL output could find applications in polarisation dependent spectroscopy [24] or in infrared ellipsometry [25].

## Acknowledgments

The authors acknowledge helpful conversations with D. C. Hutchings. First Author, D. Dhirhe acknowledges the financial support from Ministry of Social Justice and Empowerment, Government of India, New Delhi, India.

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