1. Introduction

Index-patterned Fabry-Pérot lasers [1,2] offer a simple regrowth free path to single-mode laser operation, requiring only a gain source, a lithographically defined perturbative refractive index pattern and reflective facets to function. In addition, the lasers have unique properties including narrow linewidth [3] and are robust against feedback [4]. The concept has recently been applied for Silicon-photonics integrated lasers [5]. The refractive perturbation can be achieved with a shallow slot or pit inserted in the waveguide [4,6]. The spectral content of the index-patterned laser comes from sculpting the gain profile of the individual longitudinal modes, for which a wide range of index patterns have been used [3–5].

Although a useful and widely used device, the impact on the device lasing spectrum of changing the index-pattern has never been specifically analyzed, nor has there been any analysis from a theoretical perspective as to how the device performance is affected as the grating position is shifted with respect to the facets [7]. The grating-facet phase is a major factor in these types of devices as mode selectivity is obtained based on an interplay between reflections from the device grating and from the facet. Errors in facet position from cleaving will result in a random grating-facet phase. In this paper, we investigate this grating-facet phase issue in detail. Our analysis is then applied to propose routes to improve the performance of index-patterned devices, from the point of view of both single-mode selectivity and device yield.

The index-patterned laser provides a distinct path to single-mode lasing, where the grating structure interacts with and enhances the device facet reflections, in contrast to laser devices that use distributed feedback mirrors or Bragg mirrors, and that rely on Bragg scattering. The unique spectral properties of index-patterned devices, particularly in the active regime, are coming under more intense study, as more exotic applications for laser devices are being sought with a growing application space in sensing, metrology and microwave photonics [8–10]. For creating integrated devices, a simple laser with an extremely high degree of frequency control is a boon. Along with this, given the convergent goals of single-mode laser devices (e.g. low linewidth) and the increased availability of device components integrateable with Silicon-photonics, it would appear the separate laser design approaches will become less distinguishable [5,11,12]. Also in the field of nitride devices, where Bragg gratings are difficult and expensive to make, index-patterned devices have the ability to take the lead in terms of single-mode generation [13].

Index patterns have been designed by using a regular or geometric spacing, by using a genetic algorithm and by using an inverse-Fourier transform of the desired spectral function [14–16]. Our work highlights a particularly useful method, namely the Fourier design method [16]. As the inverse-Fourier design approach has previously been demonstrated where it was shown how a function could be selected in frequency-space and its inverse-Fourier transform could be used to position perturbations in a grating [16], our work extends this idea so that a function can be selected in cavity-space, resulting in its grating Fourier transform being imprinted on the threshold spectrum. This Fourier transform approach allows tailor-made spectra to be designed with strong selection of a single mode for thermally stable devices. We show a route to improving the spectral characteristics, chiefly side-mode suppression ratio based on analysis of a range of different Fourier transform functions. We also demonstrate a reasonable ability to control the device spectrum, with lasing primarily occurring on the selected mode, or one of its two nearest neighbor modes. Indirectly we can influence the electric field profile in the cavity, which is potentially a useful application of the index-patterned device, for soliton generation [17]. The impact of changing the index-pattern on device performance is calculated using a first-order Fourier calculation of reflection between the perturbations and the device facets. The accuracy of the Fourier analysis is confirmed by benchmarking against transmission transfer matrix method (TMM) calculation that includes all orders of reflection. The precise device length of an index-patterned Fabry- Pérot laser is usually defined by the cleaved facets. As the cleave positions cannot be accurately pre-determined, this results in the grating-facet phase being random with respect to the index pattern, leading to potential yield issues.

In this paper, we utilize the Fourier transform design approach to analyze and mitigate these facet to pattern phase issues. The applicability of the Fourier design approach is confirmed by comparison with full transmission matrix calculations in Supplement 1 Sec. S1. We make a comparison of the performance of three exemplar grating patterns, including a mathematical analysis which confirms that increasing a weighted average of the perturbation positions can increase the selectivity of the chosen mode. We then utilize the Fourier transform design approach to analyze and mitigate the facet to pattern phase issues. We investigate how the device characteristics are modified when the gratings are not ideally positioned with respect to the facets, firstly for a small sub-wavelength deviation in position and then for deviations on the scale of microns. We conclude with a discussion of how the Fourier design approach can be used for device optimization. Our analysis highlights the ability to select design functions that can give a high yield, despite grating-facet phase errors introduced during device fabrication.

2. Theory of spectral selection in index-patterned lasers

Our aim is to analyze how changes in the distribution of index perturbations along a device cavity (Fig. 1(a))) influence the spectrum of the lasing device. We outline here, and describe in more detail in Supplement 1 Sec. 2, how the wavelength-dependent change in threshold gain of the laser can be determined using a weighted Fourier transform of the distribution of perturbations along the device. The background theory describing the route to preferentially select a single-mode using the Fourier transform approach has been described in several previous publications [5,16,18–21]. We describe here the key features of the method relevant to our later analysis.

 

Fig. 1. a) Perturbation patterns for chosen design functions; details of how the perturbation patterns are determined are given in Supplement 1, Secs. S2 and S3. Calculated threshold change, $-\Delta G_{m}$, using b) in-phase cosine and c) out-of-phase sine Fourier transformations.

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We consider a Fabry-Pérot laser cavity of constant refractive index, $n$, with facets with equal amplitude reflectivities, $r$, into which we introduce a small index perturbation of short length $d$ on the right-hand side of the cavity (see Fig. S1). This reduces the local refractive index by $\Delta n$ over the length $d$. To simplify the analysis, we assume here that the index perturbation is a quarter wavelength long; $d=\frac {\lambda _0}{4(n-\Delta n)}$, where $\lambda _0$ is the target wavelength, and $m_0$ is its mode number in the cavity. The impact of the perturbation is both wavelength and position dependent. The perturbation can reduce the threshold gain for the chosen mode if it is placed so that the left facet of the perturbation is an integer number of half wavelengths from the left cavity facet. The threshold gain is increased if the perturbation is an odd number of quarter wavelengths from the left cavity facet.

We show in Supplement 1 Sec. S2 that the reduction in threshold gain level, $\Delta G_{m_0}$, for mode $m_0$ due to a quarter wavelength perturbation at arbitrary position $\epsilon$ is given by [16,18]

If we now introduce a number of perturbations, we can take advantage of the fact that each perturbation is small to make a first-order approximation and to ignore all reflections between the perturbations. This allows us to treat the overall change in gain as a sum of the change in gain due to the individual perturbations. Choosing $m_{0}$ as our reference mode, we can generalize Eq. (1) to calculate the reduction in threshold gain, $\Delta G_{\Delta m}$, for mode $m = m_{0} + \Delta m$ due to the introduction of $N$ perturbations. We show in Supplement 1 Sec. S2 that Eq. (1) can be written for $N$ perturbations as [18]

It has generally been assumed in the design of index-patterned lasers that all of the perturbations introduced are perfectly positioned with respect to the facets to select mode $m_0$, with the left side of each perturbation a whole number of half wavelengths from the left-hand laser facet. Therefore only the first term in the square brackets on the right-hand side of Eq. (2) has been included in the design process and analysis, as $\cos {(\pi \epsilon _j m_0)} =0$ for ideally placed perturbations. In this case, the index pattern constructed represents a discrete approximation to a continuous Fourier cosine transform of the real-space design function, corrected to allow for the variation of the amplitude of the modulation function $w(\epsilon ) = \sinh {(\alpha _{1}\epsilon L)}$ along the cavity. In addition to considering perturbations that are ideally positioned, we also include in our analysis below the second term on the right-hand side of Eq. (2), which describes a Fourier sine transform of the object function when the perturbations are each an odd number of quarter wavelengths from the left hand facet for mode $m_0$.

3. Results

Having reviewed the Fourier theory of spectral selection in index-patterned lasers, we now turn to investigate its application in more detail. We apply the method to three different exemplar slot distribution functions; $f_1(\epsilon )=1$, $f_2(\epsilon )=|\epsilon |$ and $f_3(\epsilon )=\epsilon \sin {(2 \pi \epsilon )}$, whose Fourier transforms are listed in Table 1. Our motivation for choosing these three functions was as follows. The first function, $f_1 (\epsilon )=1$, was selected as it was the function used in the original paper on this Fourier design approach [16] and provides a mathematically pure signal in frequency-space, with the Fourier transform in Table 1 being zero for all modes except $m_0$. The function $f_2 (\epsilon )=|\epsilon |$ was selected as it gives a slot distribution which has a close to uniform distribution along the cavity. This is another approach used previously when designing the gratings for these laser types [14], and makes a useful comparison. Finally, the function $f_3 (\epsilon )=\epsilon \sin {(2\pi \epsilon )}$ was chosen as it gives a relatively pure signal in frequency-space as well as having a favorable slot distribution in terms of the mode selectivity (see Supplement 1 Sec. S3).

Table 1. The cosine and sine transforms for the exemplar design functions, each normalized so that the cosine transform equals 1 for $\Delta m = 0$.

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As in previous work, $f_1(\epsilon )$ is set to zero for $0 < \epsilon < 0.0125$, to avoid the divergence in its perturbation distribution function, $\frac {1}{\sinh {(\alpha _1 \epsilon L)}}$ as $\epsilon \rightarrow 0$ [18]. We investigate for the three exemplar slot distribution functions their expected mode selectivity characteristics both for the selected mode $m_0$ and also for the neighboring modes $m_0 + \Delta m$. We consider first the ideal case, where all slots are placed to maximize the selectivity of $m_0$. We show that $f_2(\epsilon )$ and $f_3(\epsilon )$ both reduce the threshold gain for mode $m_0$ more strongly than $f_1(\epsilon )$ does, with the neighboring modes $m_0 \pm 1$ however also displaying a reduction in threshold gain for $f_2(\epsilon )$ and $f_3(\epsilon )$ when compared with $f_1(\epsilon )$. A quantitative mathematical analysis is also presented in Supplement 1 Sec. S3, confirming that having a perturbation distribution weighted away from the cavity centre can enhance the threshold reduction for mode $m_0$.

Finally, we investigate the impact of non-ideal perturbation position, considering firstly the case where the perturbations are all moved by the same fraction of a wavelength from their ideal positions with respect to the cavity facets. This is followed by analysis allowing for larger shifts (of order microns) in perturbation position, as could potentially occur if there is an error in where the laser facet is cleaved. We find for the sub-wavelength shifts in perturbation position that the calculated mode distribution is very well described by a linear combination of the cosine and sine Fourier distribution functions. The sine Fourier transform of these exemplar functions preferentially selects mode $m_0 \pm 1$. As the mirror phase is shifted and the relative contributions of the cosine and sine transforms change, threshold tends to shift between different modes, with mode $m_0$ or modes $m_0 \pm 1$ being most frequently selected. We also find that good mode selectivity is maintained with larger shifts in perturbation position, as expected based on a simple analysis using the function $f_2(\epsilon )$.

Improved selectivity can be obtained over a broader spectral range by increasing the number of perturbations in the cavity, $N$. However, as each perturbation introduces loss, irrespective of its position, there is an upper limit to the number of perturbations that can be introduced along the cavity [6]. We work here with $N = 20$, in line with previous work [16,18,22]. This and the other parameters used in this work are listed in Table 2.

Table 2. Modeling parameters used in this paper for the Fourier calculations, and TMM calculations in Supplement 1 Sec. S1.

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3.1 Analysis of design functions for perfectly in-phase grating index pattern

In this section, we analyze the Fourier transforms of each of our design function perturbation patterns. The validity of the Fourier approach is confirmed in Supplement 1 Sec. S1, where we show the difference between the mode selectivity predicted using the Fourier cosine transform compared to the results of a full transmission matrix method calculation. These results show that the Fourier method provides an excellent prediction of the modal distribution function for the values of index perturbation $\frac {\Delta n}{n}$ and number of perturbations $N$ used here and in previous studies [16,18,19].

Figure 1(a)) shows the perturbation distribution patterns for the three different design functions. Figure 1(b)) shows the corresponding Fourier transforms of the three patterns for the ideal case, where all perturbations are aligned to maximize the selectivity of $m_0$. We see that $f_2$ and $f_3$ both reduce the threshold gain for mode $m_0$ more strongly than $f_1$ does, with the neighboring modes $\Delta m =\pm 1$ however, also displaying a reduction in threshold gain, when compared with $f_1$. These reductions are close to 41% and 26% of mode $m_0$, as expected from the Fourier transforms listed in Table 1 ($\frac {4}{\pi ^2}$ and $\frac {1}{4}$ respectively). Figure 1(b)) shows some of the key features associated with threshold mode selection in index-patterned lasers. The threshold modulation for the modes neighboring $m_0$ follows that expected from the Fourier transform of the chosen object function, $f(\epsilon )$. However, the calculated threshold gain starts to deviate from that predicted by the Fourier method, once $\Delta m \sim N$. Deviations from the Fourier function in these areas can involve fluctuations in the threshold level ($f_1$), or there can be a large change in threshold for a single mode ($f_2$), due to constructive interference across the perturbation distribution between waves scattered at that wavelength.

It can be observed that mode $m_0$ shows an approximately equal reduction in threshold gain for functions $f_2$ and $f_3$, and a smaller reduction for $f_1$. This difference in selectivity can be understood using an integral expression derived in Supplement 1 Sec. 3 to estimate the maximum reduction in threshold gain, $\Delta G_{m_0}$, that can be obtained for a given object function, $f(\epsilon )$. We derive there that

To simplify the calculation of $\Delta G_{m_0}$, we can treat the $\sinh {(\alpha _1 \epsilon L)}$ function as close to linear over most of the length of the cavity and therefore approximate it by $\alpha _1 \epsilon L$. This makes the calculation of Eq. (3) analytical for our three exemplar functions. We calculate an identical reduction in threshold gain for $f_2$ and for $f_3$, namely

This approximation gives a value of $2.02$ $cm^{-1}$, which is the maximum threshold selectivity obtainable for a grating pattern with a relatively even covering of half of the length of the laser cavity. For $f_1$ we place the perturbations between $\epsilon =a$ ($a$ being the starting point for the density function calculation) and $\epsilon =\frac {1}{2}$, for which the ratio of the two integrals in Eq. (3) is $\frac {\frac {1}{2}-a}{-\ln {(2a)}}$. This gives an estimated $m_0$ threshold gain modulation of $0.83$ $cm^{-1}$, lower than the fully calculated value of $1.08$ $cm^{-1}$. The continuous approximation of Eq. (3) is less accurate in this case due to the divergence in the perturbation distribution function, $\frac {f(\epsilon )}{\sinh {(\alpha _1 \epsilon L})}$, as $\epsilon \rightarrow 0$. Nevertheless, the model provides a clear explanation for the smaller threshold reduction observed for $f_1$ compared to the other two functions.

Table 3 compares for the three design functions considered here, the calculated reduction in threshold gain for mode $m_0$ with respect to the background and with respect to modes within 2 to 20 ($\Delta \lambda \sim 19$ $nm$) and 2 to 50 ($\Delta \lambda \sim 47$ $nm$) of the selected mode, with the final column showing the reduction with respect to the background calculated analytically using Eq. (3). It can be seen that the best modal selectivity in an ideal cavity is obtained using the function $f_3$.

Table 3. Comparison of the spectral performance of each function when the grating is perfectly aligned to the facet, showing the threshold reduction for $m_0$ with respect to the background ($\Delta G$), the neighboring $\pm 20$ and $\pm 50$ modes, and as calculated using Eq. (4).

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The integral expression Eq. (3) provides an excellent estimate of the mode selectivity obtained when approximating $f(\epsilon )$ by a discrete distribution of $N$ perturbations. More importantly, the calculation confirms the benefits for mode selectivity of choosing a perturbation distribution function that avoids placing perturbations near to the cavity centre due to the $w(\epsilon )=\sinh {(\alpha _{1}\epsilon L)}$ weighting for a laser with equal reflectivity mirrors.

3.2 Impact of non-ideal perturbation positions

Errors in facet position from cleaving can result in a random grating-facet phase. We therefore now investigate the effect of having the perturbations in the incorrect position, each shifted from the ideal designed position by a constant amount. Figure 1(c)) shows the change in the threshold mode selection function when each perturbation is moved by $\frac {\lambda _0}{4n}$ from their ideal position relative to the end facets. With this positioning, the grating threshold gain spectrum is determined by the sine Fourier transform of the design function. It can be seen in each case that shifting the perturbations by $\pm \frac {\lambda _0}{4n}$ leads to selection of mode $m_0 \pm 1$, with some reduction in threshold also for mode $m_0 \pm 2$, for functions $f_2$ and $f_3$. As can be observed, the overall behavior of the sine transform shows similar behavior to that in the ideal, cosine transform case, with a breakdown of the pattern once $\Delta m \sim N$.

Any intermediate phase shift can be described using Eq. (2). This allows for a design framework to improve the yield and performance of index-pattered lasers. As an example of this process and the potential yield improvements offered by the function $f_3$, we took each of the three functions; $f_1(\epsilon )=1$, $f_2(\epsilon )=|\epsilon |$ and $f_3(\epsilon )= \epsilon \sin {(2\pi \epsilon )}$ and cycled them through a phase variation of $\pi$ (position variation of $\frac {\lambda _0}{2n}$) between the grating and the facet. Considering a mode bandwidth of $\Delta m = \pm 20$ and setting a base-line mode selectivity of $0.35$ cm$^{-1}$, $f_3$ gave a $>300\%$ improvement on yield over $f_1$ and $>100\%$ improvement over $f_2$, as shown in Fig. 2(a)), with Fig. 2(b)) showing the mode selected as the perturbations are shifted relative to the facet. It can be expected from Figs. 1(b)) and c) that functions $f_2$ and $f_3$ should both allow strong selection of modes $m_0$ and $m_0 \pm 1$ for a wide range of facet positions, with the function $f_3$ then potentially shifting to mode $m_0 \pm 2$ over the remaining range of positions. It is clear from Fig. 1(c)) and Fig. 2) that selecting a design function with clear single-mode selectivity in the sine transform is much more likely to yield a lasing device within the range of $m_0\pm 1$ than a function that does not.

 

Fig. 2. Calculated change in a) mode selectivity and b) lowest threshold mode number as the perturbations are shifted with respect to the facet position.

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3.3 Impact of larger changes in the relative grating-facet position

While we have looked at the impact of sub-wavelength shifts on the grating-facet position, what can be as important in terms of manufactured devices is the impact of grating-facet position changes on the order of microns, due to errors in cleaving. To investigate the impact of such larger changes in relative grating-facet position, we have undertaken exemplar mode selectivity calculations, with a $+10$ $\mu m$ and a $-10$ $\mu m$ shift in perturbation position. The results shown in Fig. 3) remain very similar to those in Fig. 2) for $f_2$ and $f_3$, with $f_1$ showing stronger variation, due to the large density of features close to $\epsilon =0$. We note also that the $f_3$ function is more robust to facet cleaving errors when taking into account the safety-margin between the last perturbation in the device and the device facet. The $f_2$ function loses a perturbation when the grating is shifted by $+10$ $\mu m$. The $f_1$ function selectivity improves significantly for the $+10$ $\mu m$ shift but degrades significantly for the $-10$ $\mu m$ shift, in agreement with the importance of weighted average perturbation position highlighted by Eq. (3).

 

Fig. 3. Calculated changes in mode selectivity and mode with the lowest threshold mode number with facet position when perturbation pattern is shifted by a), b) $+10$ $\mu$m and c), d) $-10$ $\mu$m with respect to the centre of the laser cavity.

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The very small change in the selectivity for functions $f_2$ and $f_3$, when the facets are shifted by as much as $10$ $\mu m$, can be understood by considering how the Fourier transforms in Table 1 are modified if the pattern is shifted and the limits of integration are therefore changed. As an example, if function $f_2$ is shifted to start at $\epsilon = a$ rather than 0, then the Fourier transform in Table 1 is modified by the addition of an extra term $\dfrac {1 - a\pi \Delta m \sin \left (a \pi \Delta m \right )-\cos \left (a \pi \Delta m \right )}{(\pi \Delta m)^2}$, introducing a correction of order $a^2$ to the calculated Fourier transform for modes close to the selected mode $m_0$, so long as the number of perturbations $N$ is unchanged.

4. Conclusions and further perspectives

In conclusion, we have used a Fourier design approach to investigate mode selectivity and robustness in index-patterned semiconductor lasers. We considered three exemplar perturbation patterns for application to as-cleaved laser devices. We showed how mode selectivity can be enhanced for ideally positioned perturbations by choosing a pattern with a larger weighted average perturbation position. We addressed for the first time the issue of yield from facet cleaving for index-patterned devices by investigating how the mode selectivity evolved as the perturbations shifted from being ideally positioned with respect to the facets. Our results demonstrate a good ability to control the device spectrum, with lasing primarily occurring on the selected mode, or its nearest neighbors.

Because we are primarily interested in how a given distribution of perturbations changes the threshold gain value for mode $m_0$ and its neighboring modes $m_0 + \Delta m$, we treated the threshold gain level as being independent of wavelength throughout this work. In practice, this is never the case, with the gain spectrum showing a peak at a particular wavelength, which is temperature dependent. Device optimization then requires to optimize several aspects of the laser design, with the specific optimization requirements dependent on the intended application. If a laser is intended to operate at a well defined wavelength over a limited temperature range, then the design method presented here can give a good first pass at the required perturbation distribution function. We note however, that it is not obvious a-priori that the highest yield is obtained when all perturbations have the same phase with respect to the left-hand facet for mode $m_0$; more work is required to identify whether further optimization can be achieved beyond that shown here.

If operation is required over a wider temperature range, then it is necessary to maximize the threshold reduction $\Delta G_{m_0}$ of mode $m_0$. In this case Eqs. (2) and (3) can give a very useful guide as to the maximum threshold reduction obtainable for a given perturbation pattern, whether the pattern is generated using the Fourier method or another approach, such as the use of a genetic algorithm. Fig. S3 in Supplement 1 Sec. S4 shows how a typical gain spectrum would be modified due to each of the three functions that we have considered. If we assume that the gain peak wavelength shifts with respect to the wavelength of mode $m_0$ by $0.4$ nm $^\circ$C$^{-1}$ [23], then we estimate for the gain spectrum in Fig. S3 that we can achieve single-mode lasing on mode $m_0$ over a temperature range of order $38 ^{\circ }$C for function $f_1$, $45 ^{\circ }$C for function $f_2$ and $46^{\circ }$C for function $f_3$.

The value of $\Delta G_{m_0}$ can be increased by allowing different reflectivities on the two facets, with e.g. a high reflectivity coating on the left-hand facet and an uncoated right-hand facet. Eqs. (2) and (3) must then be modified to take account of the different reflectivities, as described in Ref. [16]. The modified Fourier expressions could then be used either directly for the design of increased temperature stability, or else used in tandem with other approaches to achieve optimum designs.

Overall, our analysis highlights the ability of the Fourier method to select design functions for index-patterned lasers which offer high mode selectivity and yield. Further work would be valuable both to understand the physics behind optimized solutions, based on use of genetic algorithms; and to compare the capabilities of the different approaches and to identify the potential of hybrid approaches for further optimization.