A single oscillation-mode laser employing the asymmetric waveguide structure is designed and analyzed. The mode selection mechanism is realized by using an asymmetric Bragg reflection waveguide (BRW) and shown to be effective to achieve high side-mode suppression ratio (SMSR). As an example, a silicon-based quasi-one-dimensional BRW with Er-doped Si-nanocrystal in the silicon oxide core is considered and illustrated for the laser structure. Guidance properties and threshold conditions are examined to verify the design procedure and performance feasibility for the single oscillation mode laser.
© 2009 Optical Society of America
Lasers that are capable of operating in a single oscillation mode are desirable for a range of applications in communications, signal processing, and sensing in which narrow spectral width is required. The conventional Fabry-Perot (FP) lasers, though simple in design and fabrication, exhibit multiple oscillation modes and hence broad lasing spectra . So far, there have been several schemes for realizing the single-mode lasers such as the distributed feedback (DFB) or distributed Bragg reflector (DBR) lasers. Wavelength selective grating structures are placed along the axis of optical waveguides to achieve high side-mode suppression ratio (SMSR) .
In this paper, we investigate an alternative approach to realize single-mode laser based on an asymmetric transverse Bragg reflection waveguide (BRW) [2,5]. Different from the conventional DFB and DBR lasers, longitudinal structure of the laser is uniform similar to that of a standard FP laser, whereas gratings are utilized as an integral part of the transverse structure (Fig. 1). One of the unique features for the BRW [2–5] is that light guidance and amplification can be achieved in the low-index core region facilitated by constructive interference due to Bragg reflections from the relatively higher-index periodic dielectric claddings. Er-doped Si-nanocrystial (Si-NC) silica (or silicon-rich silicon oxide, SRSO) materials [6–8] used in silicon photonics can be well suited as the active core material for this BRW laser design. Also, the use of asymmetric Bragg gratings offers additional degree of freedom so that the trade-off between light confinement/loss and wavelength selectivity of the conventional symmetric BRW structure can be relaxed. By overlapping the stop-bands of two dissimilar wide-band Bragg reflectors, one can tailor the resultant waveguide to be selective in a very narrow wavelength range and in the meanwhile maintain high optical confinement by strong Bragg reflections from the claddings. For design of lasers, this means that we can achieve a high side-mode suppression ratio, while keeping the lasing threshold as low as possible. In Section 2, we will describe in detail the key design parameters and their impact on gain threshold of the laser. A sensitivity study of the structural parameters is also described for the case of finite-layer BRW lasers. In Section 3, a specific and representative design for silicon lasers using Er-doped SRSO as the active material is demonstrated. Section 4 presents the conclusions.
2. Operation principle and design guidelines
The schematic diagram for a typical BRW structure formed asymmetrically along the horizontal x-direction is shown in Fig. 1(a). Along the wave propagation direction (z-axis) the waveguide is uniform with total length of L. Reflectivities at the two facets of z=0 and z=L are assumed to be R 1 and R 2, respectively. The optical confinement along the vertical y-direction is neglected for the sake of simplicity without loss of generality. The refractive index profiles of the transverse waveguides are shown in Fig. 1(b), where parameters of the right and left Bragg reflectors are different so that their stop bands may be off-set  to produce wavelength selectivity of sub-nanometer resolution.
Physically, the lateral optical confinement along x is achieved by Bragg reflections from the wavelength-dependent reflectors as claddings of the waveguide. The optical field is confined in the low-index guiding layer, when the Bragg reflectors are working in the stop-band region [1,5]. Since characteristics of the band diagram are functions of the grating parameters, such as period, duty cycle, index difference, etc., we may control positions of the stop-bands of two different Bragg reflectors such that they overlap with each other by a small wavelength range (as observed in Fig 2). Only in the overlapped region, the optical field is confined and guided in the core; outside that range, light leaks from the pass-band of one of the Bragg reflectors and hence will suffer significant loss. Details of this asymmetric waveguide can also be found in Ref. , and here we will be more focused on the issues for laser designs. i.e., threshold analysis for asymmetric BRW structures with limited number (i.e., 10–20) of claddings.
Band diagrams for the left and right infinite-layer Bragg reflectors in Fig. 1(a) are calculated from the transfer matrix method  and shown in Fig. 2(a) as blue and left-shaded bands, respectively. By properly positioning the two stop bands, we find that their overlap can be designed sufficiently small to produce desirable band shape as shown in Fig. 2(b). For the ideal infinite-layer case, no wave leaks out of the waveguide and a selected bandwidth of the guidance can be narrowed down to 0.4 nm. For the finite-layer (50-period) BRW designs, leakage through the claddings will lower the confinement and increase the total loss. Nevertheless, a narrow wavelength range with high confinement and low loss can still be guaranteed as illustrated by the green-triangle lines in Fig. 2(c). As a comparison, symmetric BRWs with high (Δn=|n 1−n 2|=0.5) and low (Δn=0.05) index-contrast are included as the red (dotted) and black (squared) lines in the same figure. The high index-contrast symmetric BRW, as seen, has weak wavelength selectivity, while the low contrast one suffers high wave-penetration into the passive waveguide claddings, which cause low confinement and high loss in the finite-layer case. Therefore, neither of them can be used as the asymmetric one to balance the trade-off between the confinement/loss and the wavelength selectivity, which are the main factors determining a desirable threshold gain pattern for a laser. In Fig. 2(d), we also compared the effect of number of claddings on the same properties of the asymmetric BRWs. As we see, confinement and loss profiles of the waveguide are smoothed by the reducing number of claddings. Consequently, sharpness of the band-edges can be ameliorated. This leads to the desirable results that the structural complexity, threshold level, and side-mode suppression are all balanced for a stable and realizable BRW laser, as we will further explain later in this section.
The numerical calculation of the single guided mode electric field is given in Fig. 3(a) (with power normalized to unity), and the confinement factor with corresponding leakage loss for the 20-period-cladding case is shown in Fig. 3(b). From the figures, we see that the confinement factor of the 20-period case maximizes to 59.5%, while the loss profile minimizes to 48.3 cm−1 around 1558 nm. By properly adjusting these parameters, we can design the asymmetric BRW that has desirable threshold gain (gth) pattern to select proper oscillation mode and to ensure single-mode lasing with high SMSR. Eq. (1) is the formula we use to carry out threshold gain analysis of the BRW lasers:
where parameter γ (λ) is the leakage loss due to wave penetrations through the finite-period Bragg reflectors as shown by the red line in Fig. 3(b) and L is the FP cavity length. The longitudinal resonance condition can be given as
where neff(λ) is the core effective index and k is the wave number in vacuum. The FP mode spacing can be obtained from the following formula  as
For L=1000µm and FP mirror reflectivities R 1=0.3 and R 2=0.3, we could plot the threshold gain for the BRW laser in Fig. 4(a) with 10, 20 and 100 periods of the Bragg gratings, respectively. It is observed that, by reducing the number of layers in the cladding, the threshold gain increases due to overall higher leakage loss. In the meanwhile, the opening angle of the curve, which qualitatively represents sharpness of the stop-band edges, becomes wider. This latter effect will help to ameliorate the band-edge sensitivities and reduce mode-hoping due to shift of the band-edge. Also, with reducing number of cladding layers, the structural complexity is reduced. Therefore, for the design of a stable and optimized BRW laser, the threshold level, the structural complexity and the side-mode suppressions have to be all balanced.
To quantitatively study the sensitivity of threshold gain to the core index changes, we plot gth with ncore=1, 1.001 and 1.002 for the 20-period case in Fig. 4(b), from which we could see that the wavelength shift in the extreme situation (ncore=1.002) is still less than one mode spacing (i.e., dλshift=0.8nm<2.2nm) for this design. This shows that the system is to remain in single-mode operation condition when core index is changed during the lasing process.
To study the sensitivity of threshold gain to the small change in cladding index, we perturbed the original configuration of (n1=3.5, n2=2.6, n3=2.9658, n4=2.0658) to (n1=3.497, n2=2.603, n3=2.9628, n4=2.0648) as shown in Fig. 5(a) for the infinite layer case. Threshold gain for the corresponding 20-period case is calculated and shown in Fig. 5(b). The results also show stability of the structure when operated in the single-mode condition. Mode-hopping is not likely to happen under small perturbations of the cladding or core indices.
From the above, we see that the 20-period asymmetric BRW laser structure exhibits a good balance between the structural complexity, threshold level, and mode selectivity. And a threshold-gain margin of more than 3 cm−1 can be achieved. This gain difference may be directly linked to SMSR from the following formula as 
where P 0 is the main-mode power, Δα is the threshold-gain margin, vg is the group velocity, nsp is the spontaneous emission factor, γtot and γm are the total loss and mirror loss for the mode with lowest loss, respectively. By assuming nsp=1 for the purpose of illustration, we obtain the SMSR as a function of the main-mode power as follows:
3. Practical design of the Si-based BRW laser with Er-doped SRSO core
Following the above discussions of the asymmetric BRW for laser applications, we further simplify the design by using only one material for the waveguide substrate and etching different widths of air-slots on the surface to make the claddings as shown in Fig. 7. For the sake of simplicity, we assume that the slots are etched deep enough to ensure the validity and accuracy for the one-dimensional (1D) approximation in our modeling and simulation.
As the first step, we calculate the band-gap dispersion curve for the ideal infinite-period structure  and the field pattern for the finite-period case, as shown in Figs. 8(a) and 8(b). It is noted that, in the extreme case, the bandwidth of guidance can be as narrow as 0.6 nm. We observe, in the finite-layer case, that the optical field leaks more into the right-hand-side cladding due to its weaker reflective nature. From the band-gap point of view, this means that the stop band of the right-side cladding is narrower than that on the left, due to the fact that air slots are of smaller size on the right-side of the Si substrate. To equalize the reflections from both Bragg reflectors, we may use more periods on the right in the following example (e.g., 30 on the right and 10 on the left), whose parameters are ncore=1.5 (for Er-doped Si-NC in silica), n1=3.45 (silicon substrate), n2=1.0 (air), tc=1934 nm, a=108 nm, b=196 nm, c=249 nm and d=51 nm.
To examine the performance of this BRW design as a single-mode laser, we calculate the confinement factor, the leakage loss, and the threshold gain by assuming the cavity length L=500µm and the facet reflectivity R 1=R 2=0.3 , as shown in Fig. 9.
From Fig. 9(b), we observe that the threshold gain difference is 0.5 cm−1, which is sufficient to guarantee a 20 dB side-mode suppression ratio (SMSR) as predicted from Eq. 4 for the single-mode CW operation. To further validate this point, variation of the SMSR with the central mode power is plotted in Fig. 10, where nsp=1.34 is assumed .
We have analyzed the asymmetric BRW structure for realizing single-mode lasers with high side-mode suppression ratio. Design procedure and guidelines to enhance the selection of longitudinal FP cavity modes by using the stop-band overlap of two different Bragg reflectors are described and discussed. In particular, we have considered a Si-based BRW laser with Er-doped SRSO core as an illustrative and practical example, and investigated its guidance properties and lasing threshold conditions. The feasibility for achieving single-mode oscillation based on these design parameters is demonstrated by way of simulation.
Special thanks go to Yuchun Lu, Mehdi Ranjbaran, Qingyi Guo, Lin Han and Jianwei Mu for their helpful advises. We also thank the referee for the helpful comments.
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