## Abstract

Dual longitudinal mode distributed feedback lasers have been fabricated using surface gratings with and without apodization. Analytic formulas and simulations that have been used to derive design guidelines are presented. The fabricated device characteristics are in good agreement with the simulations. The grating apodization enables a lower threshold current density, a higher output power and a broader range of difference frequency tunability by bias, which can be extended beyond the measured 15–55 GHz by changing the device structure. The apodization and the complex coupling of the surface gratings reduce the effects of the uncontrollable phase of facet reflections, enabling the use of higher facet reflectivities, which leads to narrower intrinsic short time-scale linewidths.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Stable, efficient and low cost photonic generation of high frequency RF signals has been under intense research [1, 2]. The photonic solutions for the next generation wireless systems require spectral purity, low latency, low cost, power efficiency, and scalability [3]. Dual-wavelength semiconductor lasers have been investigated for millimeter wave generation [4], but they suffer from large intrinsic phase noise [2]. Coupled-cavity structures [5], Y-branch lasers [6], monolithically integrated amplified feedback lasers with direct modulation [7], and varying feedback conditions [8] have been used to decrease the linewidth, but they are more complex to fabricate and control and are less efficient. Apodization has been used for making the output power extraction from facets asymmetric without the need for asymmetric facet coatings [9] and for increasing the effifficiency of broad-area distributed feedback (DFB) lasers [10]. The apodization has also been used for reducing the spatial hole burning [11], but it is difficult to implement in the fabrication of semiconductor laser buried gratings.

Surface gratings eliminate the re-growth, simplifying the fabrication process, and can achieve a relatively high coupling coefficient without being placed in areas with high optical field intensity, because they have a high optical contrast in the grating region [12]. Being placed away from the areas with the highest temperature and optical field intensity and involving a negligible interaction between the defect-prone processed interfaces and the carriers, the surface gratings lead to more stable devices with better performances and increased reliability. Also, the gain coupling associated with surface gratings [13] increases the stability of the grating modes with respect to laser cavity facet feedback [14]. Supplementary, the apodization can easily be implemented for surface gratings [15].

The paper presents dual-longitudinal-mode distributed feedback (DM-DFB) lasers with periodic phase-shifts, gives guidelines for varying the difference frequency between the emitted modes, and discusses surface grating implementation including apodization and its effects. Linear apodization in DM-DFB lasers leads to reduced threshold current and a broader and more sensitive tunability of the difference frequency by bias variation. The apodization also enables balancing the output modes in bias configurations that give higher output power at the facet with the weaker grating strength.

## 2. Device structure and fabrication

The epilayer structure used in the fabrication of the DM-DFB lasers has four 7 nm In_{0.689}Al_{0.055}Ga_{0.256}As quantum wells interleaved with In_{0.456}Al_{0.174}Ga_{0.37}As barriers, embedded between 80 nm In_{0.521}Al_{0.373}Ga_{0.106}As waveguide layers, and In_{0.52}Al_{0.48}As barrier reduction layers between the waveguide layers and InP claddings. Cladding doping was increased, starting from waveguide layers, between 1 × 10^{17} and 1.5 × 10^{18} cm^{−3} on the p-side and between 8 × 10^{17} and 8 × 10^{18} cm^{−3} on the n-side. The effective index was solved by a finite differences mode solver [12].

The laterally-coupled ridge-waveguide (LC-RWG) surface gratings, illustrated in Fig. 1, have been processed using UV nanoimprint lithography [16]. The apodization, which can be easily achieved with any longitudinal profile by varying the ridge width (W) and/or the lateral extension of the protrusions (D) along the device, was accomplished by linearly changing W between 1.4 and 2.0 µm along the longitudinal direction, while keeping D constant at 2.5 µm. The values for grating etching depth, ridge width (W) and lateral extension of the protrusions (D) have been chosen so that they ensure a stable single transverse mode operation [12]. Supplementary, the D value has been chosen such that it leads to a coupling coefficient close to the maximum achievable for the ridge width range, while having a minimal influence on the local effective refractive index [17] and on the target etching profile of the LC-RWG gratings. This is possible since the optical field decreases rapidly in the grating area away from the ridge, which, for the given structure, leads to a saturation in the increase of the coupling coefficient and of the local effective refractive index with increasing D beyond 2.5 µm. The change in W also induces a change in the local effective refractive index of the grating, corresponding to a calculated 0.6 nm Bragg resonance chirp between the wide-W and the narrow-W ends of the grating. Because the longitudinal structure of the laser has three sections with different contacts, the chirp effects can be controlled by asymmetrically biasing the three sections of the laser.

The dual-mode emission is derived from the superposition of two different gratings, which, in the simplified case of sinusoidal effective refractive index variation, is given by *n(x)* = *n*_{0} + ∆*n* · sin *a +* ∆*n* · sin *b*, where *a* and *b* are related to the Bragg resonance frequencies *ν*_{1}* _{B}* and $a=\frac{2\pi \cdot x}{{\Lambda}_{1}}=\frac{4\pi}{m\cdot c}\cdot {\nu}_{1B}\cdot {n}_{\text{eff}\_1}$ and $b=\frac{2\pi \cdot x}{{\Lambda}_{2}}=\frac{4\pi}{m\cdot c}\cdot {\nu}_{2B}\cdot {n}_{\text{eff}\_2}$, with Λ

*the periods,*

_{i}*n*

_{eff_i}the effective refractive indexes and

*m*the grating order of the two gratings. Under the assumption that

*n*

_{eff_0}=

*n*

_{eff_1}≈

*n*

_{eff_2}, which is a good approximation for gratings with the same profile, contrast, and filling factor, the resulting superposition $n(x)={n}_{0}+2\Delta n\cdot \mathrm{sin}\left(\frac{a+b}{2}\right)\cdot \mathrm{cos}\left(\frac{a-b}{2}\right)$ corresponds to a grating with a period ${\mathrm{\Lambda}}_{0B}=m\cdot \frac{c}{2\cdot {n}_{\text{eff}\_0}({\nu}_{1B}+{\nu}_{2B})/2}=m\cdot \frac{c}{2\cdot {n}_{\text{eff}\_0}\cdot {\nu}_{0B}}$ modulated with a period ${\mathrm{\Lambda}}_{M}=m\cdot \frac{c}{2\cdot {n}_{\text{eff}\_0}({\nu}_{2B}-{\nu}_{1B})/2}=m\cdot \frac{c}{2\cdot {n}_{\text{eff}\_0}\cdot {\nu}_{M}}$. If a 1

^{st}-order modulation is implemented (in order to have the shortest modulation period) by introducing a corresponding phase shift after every

*M*periods of the grating, i.e. Λ

*= 2 ·*

_{M}*M*· Λ

_{0}

*, (which for gratings having a rectangular profile of the effective index variation and a 0.5 filling factor, corresponds to introducing*

_{B}*λ*

_{0B}/4 phase-shifts after every

*M*periods), then two stopbands are created with their Bragg resonances spaced by:

*λ*

_{Bragg}≈

*λ*

_{0B}/(

*m · M*). When the two stopbands are placed around the peak gain wavelength, the modulated grating supports two modes placed close to the reflectivity nodes next to the inner (i.e. between the stopbands) edges of the stopbands. In such a case, a good approximation for the frequency difference between the two emitted modes is obtained by subtracting the stopband frequency width between the encompassing nodes (which is approximately the same for the two stopbands) from the difference between the two Bragg resonance frequencies:

*ν*

_{sb}is the approximate frequency difference between the first reflectivity nodes encompassing the stopband, adapted from [18];

*S*is a factor related to the grating strength, which was fitted as

*S*≈ 1.8/

*m*· (1 − 0.1 ·

*κ*·

*L*) for the studied structures; 2 · ∆

*n*is the (effective) refractive index difference between two longitudinal grating slices;

*n*

_{eff_0}is the longitudinally averaged effective refractive index; and P is the number of discrete phase shifts. The carrier grating order

*m*is included in the formula when the modulation is of 1

^{st}-order.

*κ*is the grating coupling coefficient and

*L*the total grating length. The approximation works well when the grating filling factor is not close to the values leading to minima in the coupling coefficient variation with filling factor [12] and when

*κ*·

*L*is relatively high, both conditions being required for grating-induced mode selection. It should be noted that the lasing modes’ frequencies can differ slightly from the frequencies of the inner nodes next to the reflectivity stopbands, depending on the complete resonance condition for the cavity.

A rectangular-step effective refractive index variation (e.g. with *n _{high}* = 3.1977 and

*n*= 3.1966 in the alternating slices of the un-apodized gratings) was calculated for the studied gratings, and lasers having modulated gratings with varying M,

_{low}*κ*, and P have been simulated. Lasers with two phase-shifts (P=2) separating three sections of M=818 3

^{rd}-order grating periods (Λ

_{0B}=733 nm) (resulting in a total length L≈1.8 mm) with linearly-apodized and with un-apodized gratings have been fabricated and characterized. The structural parameters of the fabricated devices have been chosen to achieve difference frequencies measurable with the bandwidth of the available photodetectors as well as to have a

*κ*·

*L*product in the range of 1.5, in order to avoid spatial hole burning and the associated modal instability. The fabricated devices have three independent contacts over the three grating sections separated by the phase shifts. The measured difference frequencies included the 26–28 GHz range for bias combinations that compensated the uncontrollable phase of the facet reflections, in good agreement with the analytic approximation of Eq. (2) (which gives a difference frequency of 27.17 GHz for

*S*≈ 0.5) and with the numeric simulations (giving 27.3 GHz), both of which do not take into account the effects of facet reflections. The measured difference frequency between the emitted modes is tunable (in a range up to 15–55 GHz for the lasers with apodized gratings, which are less sensitive to facet reflection phase variation) by changing the bias of these three sections.

#### 2.1 Simulation

The variation of the real part of *κ* and of the effective refractive index with the width of the ridge (W), calculated as described in [12], are shown in Fig. 2.

Transfer matrix method (TMM) and time-domain traveling wave (TDTW) [19] simulations were used for the design of the apodized structures. TMM was used to determine the effects of different structural variations on the stop bands, mode positions and mirror losses, while TDTW was used to determine the time-dependent longitudinal photon and carrier densities initiated by spontaneous emission noise sources under different bias conditions. The effects of variations in *M*, *κ*, and *P* on the stop band and mode positions simulated with TMM are shown in Fig. 3. The top panel shows that the mode spacing reduces significantly with increasing *M*. It also indicates that the mode selection is weaker when the grating has a small number of sections (P+1) with a relatively small number of grating periods (M) and a low coupling coefficient (*κ*), leading to a small *κ* · *L*. This can be mitigated by increasing the number of grating sections. The middle panel of Fig. 3 reveals that the variation of *κ* has a much smaller effect on mode spacing than the variation of M; while the bottom panel shows that mode spacing increases and *P* − 1 reflectivity lobes appear between the two stopbands with increasing *P*.

Figure 4 shows the calculated dependencies of the difference frequency between the emitted modes (∆*ν*_{modes}) on structural parameter variations. The top-left panel of Fig. 4 shows the variation of ∆*ν*_{modes} with *κ* and M when the 3^{rd}-order grating has three equal sections (i.e. for *P* + 1 = 3). Constant *κ* · *L* lines have been overlaid on top of the ∆*ν*_{modes} variation map. The panel shows that *d*∆*ν*_{modes}/*dκ* decreases for higher M, which indicates that structures with a higher number of periods are more tolerant to etching profile variations. The horizontal solid lines from the top-right panel of Fig. 4, corresponding to Eq. (1) calculated at 1562 nm for *m* = 3 and different values of *M*, point out that larger frequency differences, entailing a smaller *M* value, would require an increased number of sections (*P* + 1) in order to achieve a reasonably high *κ* · *L* when *κ* is relatively low. The top-right panel of Fig. 4 also shows the simulated values of the difference frequency and of the side-mode suppression ratio (SMSR) for *m* = 3, *M* = 150 and different values of *P*, indicating that, by increasing the number of phase sections while keeping *M* constant, the difference frequency can be smoothly increased with only a moderate penalty to the SMSR. The circles showing simulated difference frequency values coincide well with the line calculated using the analytic approximation of Eq. (2). The bottom-left panel of Fig. 4 shows the effects of changing *M* in the end sections (between facets and the outermost phase shifts), while keeping *M* = 150 constant for the inner sections of the grating, illustrating the effect of the variable position of cleaving planes. The upper lines show the variation of the difference frequency for different values of *P*, while the lower lines show that for *P* > 2 the SMSR is reduced when the number of periods in the end sections is substantially reduced or increased with respect to the number of periods in the inner grating sections. The bottom-right panel of Fig. 4 shows the variation of ∆*ν*_{modes} with the number of periods (*M*) in the grating sections for different *P*. The values obtained with the analytic approximation coincide with TMM numerical simulation results, validating Eq. (2).

Grating structures with two phase shifts were chosen for the experiments since they are the shortest that can achieve a given ∆*ν*_{modes} with the best SMSR. A high SMSR is helpful when a high speed photodetector is employed for detecting the mode-beating difference frequency. The difference frequency tuning by bias variations is modeled in TMM by changing the effective indexes of the three sections independent of each other, with the magnitude of change derived from carrier density variations [21]. An example of carrier and photon density distributions along the laser cavity, simulated by the TDTW method, is shown in Fig. 5. The distributions have been plotted for the nonuniform bias conditions which lead to balanced powers of the two emitted modes from the output facet at 0 cavity position. The apodization can be used to direct the emission toward the lower *κ* end of the device at 0 cavity position, as illustrated in Fig. 5. Besides this, other goals of employing apodization, with respect to the dual-mode emission, were to decrease the mode-beating RF spectrum linewidth and to increase the sensitivity and range of difference frequency tuning by bias.

The DM-DFB lasers with apodized gratings were characterized before and after applying AlO_{x} anti-reflection (AR) coating with atomic layer deposition. The reflectivity achieved with single layer AR coating is between 2 and 3 %.

## 3. Device performance

The DM-DFB lasers were biased with three DC drivers, two Thorlabs ITC510s and one Thorlabs LDC340. The output beam was collimated and coupled to a single mode fiber after a Thorlabs IO-2.5-1550-VLP free space Faraday isolator. The spectrum of the fiber-coupled light was recorded with an optical spectrum analyzer (OSA). For the mode-beating linewidth measurements the light was transmitted to a Finisar XPDV2320R broadband photodiode, whose output was amplified with a Centellax UA0L65VM RF amplifier before being measured with a 26.5 GHz electrical spectrum analyzer (ESA).

The light-current (LI) characteristics of the three-contact lasers with apodized and un-apodized gratings were obtained by shining the collimated beam into an integrating sphere with an InGaAs photodiode. Both devices were similarly biased, using the three independent drivers to achieve uniform currents through all sections. The measured LI characteristics, given in Fig. 6, show that the apodized DM-DFB lasers have a lower threshold current and a higher maximum power than the un-apodized DM-DFB lasers.

#### 3.1 Optical domain

Figure 7 shows measured optical spectra from an AR-coated un-apodized DM-DFB laser and from apodized DM-DFB lasers with cleaved facet and with AR-coated facets. The spectra, which have been overlaid in frequency for easier comparison, show that the apodization does not induce detrimental effects on the spectral characteristics, and that the AR coating suppresses the Fabry-Pérot modes well. The narrow side-modes present next to the main two modes before and after AR coating are attributed to four-wave mixing.

The measured difference frequency variations with grating sections’ bias levels for the un-apodized and apodized DM-DFB lasers are shown in Fig. 8. The apodized structure shows a larger bias-dependent difference frequency variation, with a maximum range from 15 to 55 GHz while the difference frequency in the un-apodized lasers varies between 25 and 44 GHz. It should be noted that the range of difference frequency variation with bias depends on structural adjustments (e.g. varying *M*, *P*, *n*_{eff}, *κ*). The difference frequency derivative with respect to the front section bias current is also higher for the apodized structure although the average ridge width (1.5 µm) and the current density variation are the same in the front sections of both structures. While the complex coupling coefficient of LC-RWG surface gratings enables grating-defined behavior with relatively high facet reflectivities irrespective of facet reflection phases [14], AR facet coating is still beneficial for achieving stable dual-mode operation with DM-DFB lasers under a broader range of variable bias. For the DM-DFB laser with apodized gratings the AR coating extends the range of balanced dual-mode operation, increases the difference frequency tuning range and reduces the influence of the middle section bias.

#### 3.2 RF domain

The mode-beat RF spectra from un-apodized and apodized DM-DFB lasers, measured around the mode separation frequency, are shown in Fig. 9. The measured lineshapes have been fitted in the least squares sense with unconstrained pseudo-Voigt line shapes, in which the widths of the Gaussian and Lorentzian components are not fixed. This has been done since the Gaussian linewidth induced by technical noise during the beat signal spectrum acquisition time varies and is much larger than the Lorentzian linewidth. The technical noise was mainly produced by thermal fluctuations and by fluctuations in the drive currents of the three independent sources. In contrast to typical heterodyne linewidth measurement setup, where the beat signal frequency is derived using a stable RF oscillator, in our measurement scenario the frequencies of both modes vary, contributing to the beat signal width and shape. This is the main reason why the conventional Voigt profile does not fit well to the measured RF spectra.

A longer photon lifetime inside the laser cavity induces linewidth narrowing. The AR-coated un-apodized structure has a larger overall *κ*L product and thus has a narrower linewidth than the AR-coated apodized device, because a higher *κ*L leads to a longer photon lifetime in the cavity. However, a high *κ*L has certain drawbacks, since it also leads to spatial hole burning, which affects the range and stability of grating-based operation [19]. The apodized structure has a lower overall *κ*L product, but the complex-coupled apodized surface gratings allow higher facet reflectivities without affecting the dual-mode operation significantly. Thus the DM-DFB lasers with apodized LC-RWG gratings achieve a dual-mode operation range that is both broader and more sensitive to bias changes, and a narrower linewidth when higher reflectivity facets are employed to increase the photon lifetime in the laser cavity, as shown in Fig. 9.

In Fig. 10 the full-width-at-half-maximum (FWHM) of the Lorentzian component of the unconstrained pseudo-Voigt fit is shown as a function of integration time per bandwidth, for DM-DFB lasers with un-apodized gratings and AR-coated facets and for DM-DFB lasers with apodized gratings and as-cleaved facets. Figure 10 shows that the DM-DFB laser with apodized gratings has a narrower intrinsic Lorentzian linewidth and a smaller linewidth variance on the short time scale. The Lorentzian linewidth broadening with increasing integration times (1.82 × 10^{12} Hz^{2} s^{−1} and 2.95 × 10^{12} Hz^{2} s^{−1} for lasers with un-apodized and with apodized gratings, respectively) are derived from increased noise contribution to the power spectral density as the integration time increases. The higher slope in the linewidth broadening with increasing integration time for the lasers with apodized gratings is related to the higher sensitivity of the emitted mode frequencies and of the difference frequency to the fluctuations in the cavity, which are induced by spontaneous-emission events as well as by thermal and current variations. The difference frequency jitter contributes to the linewidth broadening with longer integration times, but it is not significantly influencing the linewidth for short integration times. The smaller linewidth variance indicates better dual-mode operation stability under random variations at those short integration times. These observations imply that if the difference frequency were locked, the long term linewidth would also stay in the range observed for short integration times.

## 4. Conclusions

The theory and guidelines for designing dual-mode DFB lasers with LC-RWG surface gratings have been outlined. The surface gratings have been studied since they enable re-growth free fabrication and easy implementation of grating apodizations with arbitrary profiles. The effects of structural parameter variations on the difference frequency between the emitted modes have been analyzed and an analytic approximation formula for the difference frequency dependence on the main structural parameters was derived. The effects of linear grating apodization have been analyzed in simulation studies and have been experimentally investigated. DM-DFB lasers with linearly apodized LC-RWG surface gratings have a lower threshold current density and a higher maximum output power. They also have a more stable dual-mode operation, an increased sensitivity of the difference frequency on bias currents and a broader difference frequency tuning range by bias variations. The measured bias-controlled difference frequency tuning range was increased from 25–44 GHz for DM-DFB lasers with un-apodized LC-RWG gratings to 15–55 GHz by linear apodization of the gratings. The apodized surface gratings have reduced the influence of the un-controllable phase of the facet reflections, enabling the use of higher facet reflectivities, which, combined with the grating reflectivity, increase the photon lifetime in the cavity, narrowing the intrinsic Lorentzian linewidth of the emitted modes.

The improved characteristics of DM-DFB lasers with apodized gratings can be exploited for the generation of high-frequency RF signals in different frequency bands by using a reduced number of laser types (with tunable difference frequency) and a reduced number of photonic RF transceiver components. The exploitation of tunable DM-DFB lasers can thus reduce the complexity, footprint, power consumption, and cost of photonic RF transceivers as well as reduce the required laser inventory.

## Acknowledgments

The authors wish to thank Kimmo Lahtonen from Tampere University of Technology for making the antireflection coatings to the devices.

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