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The strategies and approaches of designing chirped Distributed Bragg Reflector for group velocity compensation in metal-metal waveguide terahertz quantum cascade laser are investigated through 1D and 3D models. The results show the depth of the corrugation periods plays an important role on achieving broad-band group velocity compensation in terahertz range. However, the deep corrugation also brings distortion to the group delay behavior. A two-section chirped DBR is proposed to provide smoother group delay compensation while still maintain the broad frequency range (octave) operation within 2 THz to 4 THz.
© 2016 Optical Society of America
1. Introduction
The recent development of broad-band terahertz quantum cascade lasers (THz QCLs) paves the way for their applications in spectroscopy and metrology [1]. A next breakthrough in this field may come from combining a phase lock technique with a broadband gain medium to realize comb operation, which will lead to real practical applications of terahertz technology, such as high-precision THz frequency spectroscopy and THz optical clockwork [1–3]. Hugi et al. have achieved the broad-band frequency comb (60 cm−1) in mid-infrared (MIR) QCLs through an active mode-locking technique [4]. Achieving comb operation in THz QCLs is more challenging due to that their lasing frequency is much closer to the reststrahlen band of the laser material (GaAs/AlGaAs), where electro-magnetic (EM) waves strongly couple to semiconductor crystal lattice, resulting strong group velocity dispersion (GVD) in that frequency range. The strong GVD would induce unevenly-distributed Fabry-Perot modes in frequency domain, yielding unstable frequency comb operation, particularly over a broad frequency range. Rosch et al. presented a frequency comb THz QCL by employing a well-designed, ultra-broadband gain medium [5]. However, no special GVD compensation mechanisms were employed in their device, which limited the device to perform to its full capacity. A chirped distributed Bragg reflector (DBR) has been proposed to address the strong group velocity dispersion issue. The chirped DBR structure is fabricated along a section of the metal-metal waveguide side wall of THz QCLs to compensate the GVD resulted from the waveguide section without the chirped DBR structure [6]. This chirped DBR-based strategy has been implemented quite successfully in optical fiber lasers [7, 8]. Nevertheless, its application to THz QCLs is only at the beginning stage and technical challenges associated with the long wavelength and broad-band operation are yet to be explored.
In this paper, we present a theoretical study on designing and optimizing a chirped DBR to achieve GVD compensation over an ultra wide frequency range from 2 THz to 4 THz in metal-metal waveguides. A one-dimensional (1D) model based on transfer matrix method is firstly adopted to reveal the fundamental behaviors and characteristics the linearly chirped DBR structures. Then a more accurate three-dimensional (3D) model is deployed by using a COMSOL RF package to verify the 1D simulation results and finalize the DBR structure derived from the 1D model. The simulation results show that the design parameters of the DBR structures, such as the corrugation depth of the chirped periods and the distribution profile of the period length, play a critical role in determining the extent of compensation for GVD. Substantial modulations in calculated group delay curves will be provoked if the DBR structures are not properly designed. These group delay modulations are related to Gires-Tournois interferometer effect [9], which is caused by the abrupt interface between the waveguide sections with and without the chirped DBR. By including a buffer section between the two waveguide sections to achieve a smooth transition in effective refractive index, the group delay modulations can be significantly minimized, yielding a flat dispersion compensation curve over an octave frequency range of 2 to 4 THz in THz QCLs.
2. Results and discussion
2.1 Material dispersion and chirped DBRs
The severe material dispersion at THz frequencies in GaAs is ascribed to the strong coupling between EM waves and crystal lattice because the frequency is close to the reststrahlen region. The frequency-dependent material dielectric constant can be simply approximated as [10]
where is the lattice temperature ( = 40 K is used in all the simulations in this work), ε∞ is the high-frequency (optical) dielectric constant, ε0 is the low-frequency (optical) dielectric constant, ω is EM wave frequency, ωTO is transverse optical (TO) phonon frequency (ωTO ≈8 THz for GaAs), and γp is damping coefficient (neglected here). Equation (1) shows that at a frequency moderately lower than ωTO, at which most published by THz QCLs are operated [11, 12], the material dielectric constant strongly depends on frequency. As a result, the group velocity dispersion (GVD) can be as high as 87,400 fs2/mm at 3.5 THz, which is more than two orders of magnitude greater than the GVD value at mid-infrared wavelengths [13]. In order to achieve zero dispersion, one strategy is to fabricate a chirped DBR structure along the ridge of the metal-metal waveguide of the quantum cascade lasers, which introduces non-zero waveguide dispersion to compensate the material dispersion from the flat ridge waveguide section [6]. The period length of such a DBR structure gradually increases from its starting period to its ending period and the corresponding corrugation depth linearly increases, as shown in Fig. 1(a). The reflection peak wavelength of a DBR is defined by its period length, therefore shorter-wavelength waves which suffer longer material group delays are reflected earlier at the starting (shorter) periods, while longer-wavelength waves would travel further towards the ending (longer) periods. By carefully designing the chirped and corrugated periods, the DBR structure synchronizes the round-trip traveling of the EM waves at different wavelengths, substantially compensating the very large material GVD at THz frequencies.
Fig. 1 (a) An example of corrugation shape of a double chirped waveguide structure. The period lengths of sinusoidal shape gradually increase from left to right, while the corrugation depth tapers linearly from the starting period to the ending period. Inset: A full waveguide consisting of a chirped DBR structure (red) and a flat ridge waveguide section (blue). (b) The reflectance spectra of chirped DBRs with different period lengths (8 – 28 μm). The DBRs have a sinusoidal shape, similar to the one shown in (a). The period length in each DBR remains constant. The 1D simulation results show that the peak reflectance frequency shifts from 1.7 THz to 5.6 THz while the corrugation period length decreases from 28 to 8 μm. (c) The group delay from a 6 mm-long ridge waveguide (~ + 4.5 ps, blue curve) within 2.5 to 3.5 THz is compensated by that from a carefully designed chirped DBR structure (~-4.5 ps, red curve), resulting in a close to zero overall group delay (green curve).
For a first order Bragg grating, the peak reflection wavelength (λi) is determined by the length of the corrugation period (Λi) as
where neff is the effective refractive index of the waveguide. As such, the upper and lower cutoff frequencies of the reflection spectrum of the chirped DBR is defined by the corrugation period length of the starting and ending periods, respectively. A one-dimensional model based on transfer matrix method [13] was firstly developed and implemented to design a chirped DBR structure and to explore its basic features. The first step is to determine the lengths of its starting and ending periods. Figure 1(b) shows a series of the reflections spectra of multiple DBR structures, which have a similar corrugation shape as that shown in Fig. 1(a) but different while fixed period lengths. The reflectance band is red shifted from 5.6 THz to 1.7 THz when the corrugation period length increases from 8 to 28 μm. To cover a frequency range from 2 to 4 THz, the period lengths of the starting and ending periods are therefore set as 10 and 26 μm, respectively.Figure 1(b) reveals the reflection spectrum shifts at an accelerated rate to the higher frequency side and grows broader as the period length of the DBR structure increases. The combination of the individual spectra would yield the overall reflection spectrum of a chirped DBR structure in which the period lengths increases from 10 to 28 μm. Due to the aforementioned accelerated blue shift, such a chirped DBR may exhibit dips in the overall reflectance curve, particularly at higher frequencies [as shown by grey arrows in Fig. 1(b)]. In order to generate a continuous and flat-top reflectance curve and to prevent the formation of any discontinuities/gaps between neighboring reflectance spectra, more corrugation periods should be inserted at the beginning section (shorter period side) of the DBR structure, which means a nonlinear distribution of the corrugation periods along the longitudinal direction. In this paper, a quadratic distribution of corrugation periods (with more periods at starting (shorter) period side and less periods at ending (longer) side) is deployed in the following DBR structures to be discussed below.
Figure 1(c) shows an example to demonstrate how GVD dispersion compensation is achieved in a chirped DBR. For an EM wave with a specific angular frequency (ω) travelling a round trip in a waveguide, the change rate of its phase shift with respect to angular frequency represents its group delay,
where r(ω) is the reflectivity calculated by using the one-dimensional transfer matrix method, and arg[r(ω)] is the phase shift at the corresponding frequency. As shown in Fig. 1(c), the material group delay (one round trip) in a 6 mm-long rectangular ridge waveguide increases monotonically with frequency (the blue curve with ~ + 4.5 ps from 2.5 THz to 3.5 THz). The monotonically-decreasing waveguide group delay (one round trip) of the chirped DBR (the red curve with ~-4.5 ps from 2.5 THz to 3.5 THz in this chirped DBR structure) can therefore compensate the material group delay partially or even completely, depending on the group delay differences between the lower and higher frequencies from the material contribution and from the chirped DBR waveguide contribution are exactly matching but with opposite signs. The green curve in Fig. 1(c) is the total group delay dispersion that combines the red and blue curves, showing ~0 ps group delay from 2.5 THz to 3.5 THz.2.2 Effects of period number and corrugation depth
For a DBR mirror structure, the strength of reflection at a corresponding frequency becomes stronger either by increasing the repeating number of the corrugation periods or etching a deeper corrugation for the periods. To examine the extent of the compensation for group delay dispersion with an extended broad operation band from 2 THz to 4 THz, a series of chirped DBR structures (SA, SB, SC, SD, SE and SF) with ridge widths of 20 μm are proposed and simulated. The key dimensional parameters of the structures are listed in Table 1. SA, SB, and SC are three chirped DBR structures with the same parameters except the number of the corrugation periods (thus the total length). SA, SD, SE, and SF are almost identical except the starting corrugation depth. The starting period length in these six DBR designs is 10 μm, which corresponds to an upper reflection cutoff frequency of ~4 THz. Nevertheless, Fig. 2(b) shows that the calculated reflectance of SA (red) drops to ~70% at ~3.2 THz, leading to a much-narrower-than-expected reflection band. This is because the corrugation of the first few periods at the beginning section of the chirped DBR structure is too shallow (starting from 0 μm), and the resultant reflectance at higher frequencies is therefore very weak or even negligible. To overcome this deficiency, two different design strategies-increasing the number of the periods and implementing deeper corrugation- are investigated independently and their outcomes are presented in Fig. 2(b)-2(e)
Table 1. Dimensional Parameters of SA, SB, SC, SD, SE, and SF
Fig. 2 (a) Six 20 μm wide chirped DBR structures are simulated by using the one dimensional model. Structure A (SA in red) is set as a baseline sample with following parameters: starting period length (10 μm), ending period length (26 μm), number of periods (30), ending period corrugation depth (8.5 μm), and starting period corrugation depth(0 μm). Structure B and C (SB in orange and SC in yellow) have the same parameters as those in SA, except the number of periods (35 periods in SB and 40 periods in SC). Structure D, E, and F (SD in green, SE in blue, and SF in purple) have the same parameters as those in SA, except the starting period corrugation depth(1, 2, 3 μm for SD, SE, SF, respectively). (b) The calculated reflectance spectra of SA (red), SB (orange) and SC (yellow). (c) The calculated group delay of SA (red), SB (orange) and SC (yellow). A difference of ~3.9 ps in group delay compensation between SA and SC is shown. (d) The calculated reflectance spectra of SA (red), SD (green), SE (blue) and SF (purple). (e) The calculated group delay of SA (red), SD (green), SE (blue) and SF (purple).
As shown in Fig. 2(b), by increasing the period number to 35 (in SB) and 40 (in SC), the reflection band is extended moderately to higher frequencies (up to ~3.4 THz at ~70% of the reflectance), confirming that a larger number of corrugation periods helps strengthen the reflection and push the upper cutoff frequency to higher frequencies. However, the period number has to be much larger than 40 to further push the upper cutoff frequency to 4 THz. This would inevitably lead to a very long DBR cavity and introduce excess group delay dispersion within the desired frequency range.
The calculated group delay curves for SA, SB, and SC are plotted in Fig. 2(c), which show a monotonic decrease in the corresponding reflection band. Note that longer DBR structures bring in larger group delay differences between the lower and higher frequencies. The longest structure (SC) introduces 3.9 picoseconds extra group delay than the shortest structure (SA) does. As revealed earlier in Fig. 1(c), the material group delay over the frequency range is predetermined by material properties and is fixed, while the waveguide group delay increases with the length of the DBR structure. As such, increasing the period number may partially address the insufficient reflection issue at higher frequencies, but it is unlikely to yield exact dispersion compensation.
An alternative approach is to implement chirped DBRs with deeper corrugation depth, especially to increase the corrugation depth at the starting periods. Figure 2(d) compares the calculated reflectance curves of the four chirped DBR structures with identical parameters except the starting period corrugation depths, which are 0 μm for SA (red), 1 μm for SD (green), 2 μm for SE (blue), and 3 μm for SF (purple). The upper-limit cutoff frequency (at ~70% of the reflectance) of the reflectance band is effectively extended from ~3.2 THz to ~3.5 THz, ~3.8 THz, and ~4.1 THz as the corrugation depth of the starting period increases from 0 to 3 μm. Clearly, with a corrugation depth of 3 μm to get started in a chirped DBR, the reflection band is now extended to a wide range from 2 to 4 THz. It seems the implementation of deeper corrugation is a good solution. Nevertheless, deeper corrugation of the DBRs also generates a substantial and detrimental side effect. Figure 2(e) shows the calculated group delay curves of SA, SD, SE, and SF. Over the frequency range within its reflection band, the group delay curve of SA is monotonically decreasing and almost smooth. Starting with SD, periodic ripples can be observed in the group delay curves in the frequency range of interest even though the overall decreasing trend is retained. The modulation of the periodic ripples becomes stronger as the corrugation depth of the DBRs increases. These periodic ripples clearly disrupt the group velocity dispersion compensation as the material group delay can be cancelled out only at some discrete frequencies.
2.3 Two-section chirped DBRs
The ripples observed in Fig. 2(e) can be explained by Gires-Tournois interferometer effect. At the interface between the flat rectangular waveguide section and the chirped DBR section [see the SG curve in Fig. 3(a)], there is an abrupt change in effective refractive index. This discontinuity yields the formation of an internal cavity, leading to the emerging ripples (the fast modulations in the group delay curves). To minimize the ripples, one should eliminate the refractive index discontinuity. One effective approach is to insert a buffer structure between the flat rectangular waveguide section and the chirped DBR section. The buffer structure is a linearly-chirped DBR in which the corrugation depth of the starting period is zero [see the SH curve in Fig. 3(a)].
Fig. 3 (a) Comparison of the corrugation shape of two 20 μm wide chirped DBR Structure G (SG) and Structure H (SH). These two structures share the same parameters of starting period length (10 μm), ending period length (26 μm), number of periods (40), starting period corrugation depth (4 μm), and ending period corrugation depth (8.5 μm). However, different from SG, SH has an additional 10 periods of corrugations as a transition buffer from the flat rectangular ridge waveguide to the chirped DBR section. (b) Calculated group delay of SG (blue) and SH (red). The buffer region provides SH with a smoother group delay curve between 2 THz and 4 THz with less modulation, but still retains the same decaying trend (~7 ps) as that in SG. (c) Calculated modulations of the group delays within the frequency band for DBR structures with different number of periods in the buffer region. All other dimensional parameters are the same as those in SH.
The SH structure consists of two sections – a short chirped DBR structure with its corrugation depth linearly increasing from 0 to 4 μm (buffer section) and a much longer chirped DBR, attached right behind the buffer section, with a starting period corrugation depth of 4 μm (compensation section). The compensation section ensures that strong reflectance is achieved over a wide frequency range, while the buffer section provides a transition to eliminate the abrupt change in refractive index and thus minimizes the ripples in the group delay curve. Simulation results indeed confirm it. Ripples in the group delay curve is almost completely eliminated in SH [Fig. 3(b)]. A group of two-section DBRs similar to SH has been simulated. The number of periods (N) in the buffer section of these DBRs increases from 3 to 15, while the structures of their compensation sections remain unchanged. With the increasing of the period number, the ripples in the calculated group delay curves quickly diminish and the modulations of group delays (which is defined by the standard deviation of the calculated group delay data that deviate from a fitted smooth group delay curve) drops from 0.33 ps (for N = 3) to ~0.07 ps (for N = 15) [Fig. 3 (c)]. Simulation results clearly reveal that such a two-section DBR with a period number of N = 10 in the buffer section warrants satisfactorily smooth group delay curve (with a deviation of 0.1 ps) in the octave frequency range of 2 to 4 THz.
Although the above results obtained from the transfer matrix method present a basic understanding of chirped DBRs in terms of group delay compensation in a metal-metal THz waveguide, the one-dimensional (1D) model is certainly oversimplified for simulating the EM wave transmission processes. In particular, the one-dimensional model can’t predict at all the impact of less-than-unity confinement factor on DBR’s reflection and dispersion performance. It simply assumes a uniform mode distribution inside the waveguide. As a result, the effective refractive index contrast in the corrugated section of the chirped DBR is certainly overrated in the 1D model, which leads to overestimation of calculated reflectance. In this regard, a three-dimensional model should be employed to improve simulation accuracy and finalize device design parameters.
2.4 Simulation with a 3D model
A three-dimensional (3D) model based on COMSOL RF package is used to simulate the chirped DBR structures discussed above. In the 3D model, two ports are deployed – one at each side of the chirped DBR waveguide. The EM wave is injected from the starting (shorter) period side, while the port at the ending (longer) period side is set to be off. The S-parameters of the chirped DBR waveguides are calculated as a function of frequency. Due to the narrow ridge width (20 μm) of the waveguide, only TEM00 lateral mode is considered, and the reflectivity is determined by S11 parameter. Then, the group delay as a function of frequency is calculated the same as the 1D model. Three chirped DBRs (SI, SJ, SK), with a thickness of 10 μm and a ridge width of 20 μm, are designed and simulated using the 3D model. The dimensional parameters are listed in Table 2.
Table 2. Dimensional Parameters of SI, SJ, and SK for the 3D Models in COMSOL
The structure SK is a chirped DBR waveguide with a buffer section of 10 periods of corrugations, which is 75 μm long in total. The structures SI and SJ are control designs, which both include a 75 μm long flat ridge waveguide section instead of a chirped DBR buffer section. The starting period corrugation depth of SJ is 6 μm, same as that of SK, while SI has a starting period with corrugation depth of 0 μm. Figure 4(a) and 4(b) show the calculated reflectance and group delay curves of the chirped DBR structures using the 3D model, respectively. Similar to the result from the 1D model, SI cannot provide adequate reflection at frequencies above ~2.8 THz due to the shallow corrugation depth of its starting periods. In contrast, both SJ and SK have a close-to-unity reflectance even at a frequency of 4 THz or beyond. This is more clearly revealed in the mode distribution diagram shown in Fig. 4(c) and (d). In Fig. 4(c), at a frequency of 3 THz, only part of the EM wave is bounced back by the corrugated periods of SI, which occurs at the middle of the chirped DBR structure. A substantial portion of the injected EM waves leaks to the right side. However, as shown in Fig. 4(d), EM waves at 4 THz are strongly reflected back to the left at the first quarter of the waveguide and the mode is not present on the right side, indicating close-to-unity reflectance. Although both SK and SJ yield broadband reflectance spectra, the calculated group delay curves shows multiple ripples are observed in SJ, but not in SK within the frequency range of 2 to 4 THz [see Fig. 4(b)]. As discussed earlier, this distortion comes from the undesired resonance of a Gires-Tournois interferometer like cavity. To further reveal this phenomenon, the mode distribution at frequencies of 2.30 THz (ripple peak), 2.37 THz (ripple valley), 2.44 THz (next ripple peak) in SJ are calculated, as shown in Fig. 4(e). Due to the abrupt discontinuity in effective mode index, the interface between the flat ridge waveguide section and the chirped DBR section acts as a reflection mirror that can reflect the backward EM waves bounced from the right side. At resonance frequencies, such as 2.30 and 2.44 THz, the EM waves can be trapped in the chirped DBR structure, which is confirmed by the higher intensity in the corresponding section (Fig. 4(e)). At off-resonance frequencies, such as 2.37 THz, mode trapping is much relieved and the mode intensity is thus much lower. As such, the group delay at the resonance frequencies is greater than that at the off-resonance frequencies. By inserting a buffer section (such as the one in SK), the abrupt mode transition from the flat ridge waveguide section to the chirped DBR section is pretty much removed and the Gires-Tournois interferometer effect is largely eliminated. A much smoother group delay curve is obtained in SK.
Fig. 4 (a) Calculated reflectance of three chirped DBR structures (Structure I (SI in blue), Structure J (SJ in green), and Structure K (SK in red)) within a frequency range of 2 - 4 THz from simulations based on a 3D model. The dimensional parameters of the structures are listed in Table 2. SI only provides a sufficient reflectance frequency band from 2 THz to 3 THz, while SJ and SK still have close-to-unity reflectance at 4 THz. (b) Calculated group delay of SI (blue), SJ (green) and SK (red). For SI, the modulation on its group delay curve appears with the frequency increasing beyond its cutoff frequency (~2.8 THz at ~70% of reflectance) from its reflectance band. (c) Calculated mode distribution in SI at 3 THz. (d) Calculated mode distribution in SK at 4 THz. (e) Calculated mode distribution in SJ at 2.30 THz, 2.37 THz, and 2.44 THz corresponding to the resonance and off-resonance frequencies, labeled by vertical dashed lines in (b).
3. Conclusion
This paper presents a systematic study of chirped DBR structures in terahertz metal-metal waveguides for the purpose of group velocity dispersion compensation over an octave-spanning frequency range (2 to 4 THz). A one dimensional model based on transfer matrix method is firstly deployed to simulate the DBR structures, revealing the basic features and properties of the structures. It is discovered that the implementation of deeper corrugations in the DBRs is more effective to achieve broadband reflectance than increasing the period number. Nevertheless, it can strongly distort the group delay behavior of the DBRs. To eliminate the strong ripples observed in the group delay curves, which are resulted from the abrupt transition from flat ridge waveguide to chirped DBR, a two-section chirped DBR structure is proposed. The new two-section DBR consists of a buffer section between the flat ridge waveguide section and the compensation section. Both one-dimensional and three-dimensional modeling reveal that the new two-section chirped DBR provide a much smoother group delay compensation in a very broad frequency range, which may significantly improve device performance of ultra broad band frequency comb THz QCLs.
Acknowledgment
The authors would like to thank Seyed Ghasem Razavipour for technical assistance. This work has been supported by the Natural Sciences and Engineering Research Council (NSERC), the Canada Foundation for Innovation (CFI), Ontario Research Foundation (ORF), Canadian Microelectronics Corporation (CMC), and the University of Waterloo.
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