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The photonic density of states (PDOS) is one of the key physical quantities governing the lasing behavior for photonic band-edge lasers. The PDOS is conventionally altered by exploiting the high-Q band-edge mode within a device, which is typically achieved by increasing the contrast of periodic refractive index variation (Δn) or increasing the periodic number of the photonic crystals. In this paper, we propose a different approach to achieve a high-Q band edge mode within an active compound dielectric waveguide grating (CWDG). We demonstrate that the lasing threshold and intensity can be flexibly tuned by changing the filling factors of the CWDG. This design can effectively improve the performance of electrically pumped photonic band-edge lasers.
© 2016 Optical Society of America
1. Introduction
Due to the unique ability to realize nanoscale lasers, spasers (surface plasmon amplification by stimulated emission of radiation) [1–4] have aroused a great interest for applications in integrated photonic circuits (IPCs). Different nanoscale spasers have been reported in metal-coated nanopillars [5] and nanorings [6], semi-conductor nanowires on a silver film [7,8], and gold nanospheres [2]. Spasers inherit the unique properties of surface plasmons, in particular the compact (sub-wavelength) size, ultrafast dynamics and ultrasmall mode volumes [2,7,9]. However, they suffer from large metallic losses in the optical frequency range.
Except for the spasers, photonic crystal lasers have also attracted attention. They can be classified into two types regarding their working mechanisms. The first one is based on a resonant cavity and lasing actions arise from the resonant cavity modes [10–12]. The second, named photonic band-edge lasers, does not have any physically defined cavity. The operation of this laser is based on the enhancement of the optical density of states at the photonic band edges [13–18]. In principle, both ways can effectively prolong the interaction time between light and the gain medium though the slow-light effect, thereby forming a long time photon feedback. Therefore, the photonic crystal lasers are widely used in optical and biological applications. However, their large mode volumes and device volumes hamper the applications in IPCs.
For all photonic structures mentioned above, a high quality factor (Q-factor) is desired to enhance the lasing action. The Q-factor corresponds to the photon lifetime of the micro-cavity resonant modes and indicates the ability of the micro-cavity to confine light. A higher Q-factor indicates that the micro-cavity has a strong ability to restrict light. Generally speaking, a higher Q-factor yields a higher density of states, which, in turn, decreases the threshold of lasing action. Several strategies thus have been exploited to increase the Q-factor in periodic photonic structures, such as increasing the contrast of periodic refractive index variation (Δn) [19,20] or increasing the periodic number of the photonic crystals. Unfortunately, the value range of material refractive indices in nature is restricted, while the increased number of periods will certainly expand the device volume. Therefore, to some degree, these traditional modulating methods still have some limitations.
The micro-cavity can also be characterized by its mode volume, which corresponds to the energy density of the photons in the cavity [21, 22]. In the micro-cavity, the ratio Q/Vmode determines the strength of the interaction between light and matter. Therefore, researchers focus their attention on the realization of a micro-cavity with both high Q-factor and small mode volume V.
In this paper, a compound waveguide dielectric grating (CWDG) structure with both, high Q-factor and ultrasmall mode volume, is proposed to realize a controllable lasing behavior. The results show that there exists a flat dispersion region for the guided-mode resonance (GMR) with a low group velocity. Within this region, the interaction time between the light and the gain medium is extended, which provides a sufficiently long photon feedback time. The stronger light amplification and a more pronounced monochromatic response are clearly observed by introducing gain materials in the CWDG structure. Meanwhile, one can flexibly regulate the Q-factor and mode volume so as to modulate the lasing behavior (i.e. threshold and intensity of lasing) by controlling the geometric parameters of the CWDG structure. Besides the thin grating structure, our structure does not suffer from large intrinsic losses due to all dielectric configurations.
2. Modulation of the guided mode resonance linewidth
Figure 1(a) depicts the proposed structure, which is enabled by an active CWDG. For this type of configuration, the GMR effect [23,24] plays a key role in the light amplification. To illustrate the resonance effect, the dispersion relations is calculated with the aid of a simple model. In this approximation, the periodic structure is considered as a planar dielectric waveguide slab. The dispersion equation for transverse-electric (TE)-polarized light wave can be written as [25],
While for transverse- magnetic (TM)-polarized light wave it is expressed bywherek0 is the wavevector in vacuum, n1, n2 and n3 are the refractive indices of the waveguide layer, substrate and the cover layer, respectively. β is the wavevector along the x direction in the waveguide layer, m is the mode number (m = 0, 1, 2...) and d is the thickness of the waveguide layer. Figure 1(b) presents the dispersion relations of the guided-mode for TM and TE-polarized light in a uniform dielectric slab. It can be seen that the dispersion relation of the guided modes is similar to that of surface plasmon polaritons (SPPs) in a metal-dielectric interface [26]. Due to the momentum mismatch, the mode supported by the proposed structure will not interact with the incident light. However, the structural periodicity introduces an additional wavevector and gives rise to strong electromagnetic field localization at the interface (i.e. SPPs) or within the structures (waveguide mode). Therefore, light amplification can be achieved in these structures when the gain medium is introduced. Recently, several novel lasers based on this principle have been demonstrated [4,17]. Different from plasmonic lasers, our structure keeps the advantages of SPPs, namely, subwavelength scale and ultrasmall mode volume, but gets rid of the metal dissipation loss, which potentially lowers the threshold of the lasing action.
Fig. 1 (a) Scheme of the single-layer four-part periodic CWDG structures, consisting of alternating Si and Polyurethane, fabricated on a SiO2 substrate. nH, nL and nS are the refractive indices of Si, Polyurethane and SiO2, respectively. The thickness of the grating and substrate are dl and dS, respectively. The grating period is denoted as Λ. fa, fb and fc are the filling factors, respectively. (b) Calculated dispersion relations of the guided-modes for the TM- and TE-polarized light in a uniform dielectric slab. The Si slab has a thickness of 0.43Λ. Insert shows the magnetic/electric mode profile of the TM0/TE0 waveguide-mode in such a uniform dielectric slab.
The single-layer four-part periodic CWDG structure is composed of two grating ridges with identical width, as shown in Fig. 1(a). The resonance linewidth of the grating could be modulated by adjusting the gap between the two grating ridges [27]. The first layer is the grating, consisting of the alternated Si and Polyurethane. The second lays is the substrate, composed of SiO2. The refractive indices of Si, Polyurethane and SiO2 are nH = 3.48(Si), nL = 1.3(Polyurethane) and nS = 1.48(SiO2), respectively. The thicknesses of grating and substrate are dl = 215nm and dS = 600nm, respectively. The grating filling factors are selected as fa = 0.2 and fb + fc = 0.6, while the grating period is Λ = 500nm. In the following, the spectral response of the configuration is exactly performed by finite-difference time-domain (FDTD) simulations [17,28,29], in which the periodic boundary conditions are utilized. In our calculation, we consider a TM-polarized wave (magnetic field H polarized along the y direction) incident from the top side of our structure.
The reflectance of the CWDG structure for fb = 0.25 varies with the incident wavelength and incident angle, as plotted and shown in Fig. 2(a). It can be understood from Fig. 2(a) that the loci of the reflection maxima matches closely with the folded dispersion curves of guided modes. Similar folded dispersion curves of guided modes can be obtained by folding the band curves of uniform dielectric medium into the first Brillouin zone [30]. From Fig. 2 (a) and (b), we can find that the dispersion relation at the proximity of the wavelength of λ = 870nm is relatively flat at a small incident angle, while it changes with an increased slope at a larger incident angle. As the incident angle increases from 0 to 6°, the resonant wavelength shifts from 870nm to 884nm. The results indicate that the CWDG structure possesses a low group velocity, which is calculated by the group velocity formula Vg = dω/dk‖ (ω = 2πc/λ, k| = 2πsinθ/λ). Hence, the proposed structure can efficiently prolong the interaction time between light and the gain medium, thus providing a sufficiently long photon feedback time, when dye molecules are doped into the Polyurethane (Ref.17, Supplementary Fig. S1).
Fig. 2 (a) Reflectivity (represented by the color ramp) as a function of wavelength and incident angle for fb = 0.25. (b) The right shows a magnified view of the resonance response at the proximity of the critical wavelength of λ = 870nm for lasing effects.
According to the rigorous coupled-wave theory [27, 31], the grating Fourier harmonics εn are related to the coupling between evanescent diffraction fields and the leaky guided modes. In addition, |εn| are highly dependent on the filling factor fb, which signifies that the coupling can be dominated by adjusting the filling factor fb. With a proper design of the waveguide gratings, the leaky guided modes can be excited solely through the first evanescent diffracted order of the grating. In this situation, the coupling is mainly decided by the first Fourier harmonic ε1. Normally, a smaller ε1 manifests a weaker coupling, it brings a decreased spectral width and vice versa. To illustrate the theoretical analysis, the reflection spectrum of the CWDG structure with various filling factors fb for the normal incident TM-polarized light (with magnetic-field parallel to the y direction) is shown in Fig. 3.
Fig. 3 Reflectance of the CWDG structure with various filling factor fb for the normal incident TM-polarized light.
One can see that the linewidth of the reflectance could be effectively controlled by changing the filling factor fb. In the range of 0 to 0.29, the linewidth decreases with increasing fb. On the contrary, it increases with increasing fb in the range of 0.31 to 0.60. The result agrees with our theoretical analysis. According to the theoretical analysis, with the increase of fb in the range of 0 to 0.29, the coupling strength gradually weakens and the reflectance linewidth decreases, leading to an increase of Q-factor and vice versa. When |ε1| equals to zero, at fb = fc = 0.3, the coupling strength between ± 1 order evanescent diffraction waves and the leaky guided modes is zero, which corresponds to a zero linewidth of the resonance. Meanwhile, the magnetic field distribution of the CWDG structure at the resonant wavelength of 870nm is plotted in Fig. 4 for various filling factors fb. When the filling factor fb = 0, the two grating ridges overlap, the magnetic field at the resonant wavelength mainly is concentrated in the center of the grating ridge (not shown in the figure). While fb≠0, the two grating ridges begin to separate from the center. In this case, the magnetic field, localized in the middle of the grating ridge, is exposed to the Polyurethane. Furthermore, with the increase of fb, the gap between the two gratings ridge increases. Meanwhile, the coupling strength gradually weakens, resulting in an enhancement of the magnetic field intensity exposed to the Polyurethane.
Fig. 4 Magnetic field distribution of the CWDG structures at the resonant wavelength of 870nm for filling factor (a) fb = 0.15, (b) fb = 0.20 and (c) fb = 0.25, respectively. Black dotted lines indicate the Si gratings and white solid lines represent the Polyurethane.
It is known that the strong EM field confinement is critical for the interaction between light and the gain medium. As a result, an efficient enhancement of the spontaneous emission, i.e. the Purcell factor, and a lower threshold are expected for the proposed CWDG structure. According to the Fermi’s golden rule [32], the Purcell factor is given by
where Q is the cavity Q-factor, Vmode is the mode volume, λ is the resonant wavelength and n is the refractive index of the medium, respectively. The Q-factor can be obtained by Q = λ/Δλ, where Δλ is the spectral linewidth. From Fig. 3, we find Q-factors of 54.2, 131.3 and 546.4 for filling factors of 0.15, 0.20 and 0.25, respectively.The effective mode volume can be obtained according to the formula
where ε (r) is the dielectric constant at the position r, and |E(r)|2 the corresponding field intensity. With filling factors 0.15, 0.20 and 0.25, one obtains values of Vmode≈1.84 × 10−3(λ/n)3, Vmode≈1.33 × 10−3(λ/n)3 and Vmode≈1.28 × 10−3(λ/n)3, respectively [33,34]. In combination with Eq. (4) and Eq. (5), the Purcell factors are estimated as 280, 939 and 4058 for filling factors take 0.15, 0.20 and 0.25, respectively. With an effectively enhanced Purcell factor, an optimized lasing action with low threshold can be expected [35–38].3. Simulation on the enhanced lasing action
As a next step, a commercial gain medium is introduced into the CWDG structure, with both high Q-factor and ultrasmall mode volume, to verify the enhanced lasing action. When the polyurethane layer is mixed with infrared fluorescence dye molecules (IR-140, Sigma-Aldrich [17]) and optically pumped, the emission spectra can be calculated as is shown in Fig. 5. Here, a four-level two-electron atomic system is used to describe the time evolution of the population density of the gain material [39], with the following rate equations:
whereand represent the net macroscopic polarizations that are caused by the transition from state 1 to state 2 and state 0 to state 3, respectively. ωa and ωb are the frequencies of the transitions between state 1and 2, and that between state 0 and 3, respectively. is the total electric field. N0, N1, N2 and N3 are the population densities of the molecules corresponding to the energy levels 0, 1, 2 and 3, respectively.τxy is the lifetime of the transition between energy levels x and y. The other gain medium parameters used in the simulations are chosen as: (1) ωa = 2.165 × 1015 Hz and ωb = 2.512 × 1015 Hz, (2) lifetimes: τ21 = τ30 = 1ns, τ32 = τ10 = 10fs; (3) The initial population density: N = 2.0 × 1024 m−3.Fig. 5 Lasing actions in the CWDG structures. (a) Emission spectra for fb = 0.25 as a function of input pump amplitude and wavelength. (b) Emission spectra linewidth as a function of pump amplitude.
Figure 5(a) depicts the emission spectra of the CWDG structure for fb = 0.25 as a function of the input pump amplitude and wavelength. Steep and narrow emission spectra with a well-defined threshold behavior can be observed, indicating the lasing behavior in the vicinity of 870nm. To present the enhanced lasing action more clearly, the emission spectra linewidth as a function of the pump amplitude is also plotted and shown in Fig. 5(b). We find that the linewidth of the emission spectra is narrowed by almost one order of magnitude, from about 4.5nm to 0.8nm. This phenomenon clearly demonstrates a coherent lasing behavior [34].
By extracting the peak values of the spectra in Fig. 5(a), we can also determine the maximum emission intensity as a function of the pump amplitude, which is shown in Fig. 6. Here, the red solid, blue dashed and black dotted lines represent the results for fb = 0.25, fb = 0.20 and fb = 0.15, respectively. It is found that the curves for these three cases have the same shape. The emission is nearly zero below a certain threshold value, and then quickly rises as the pump amplitude increases. However, the threshold and maximum intensity are obviously different. The estimated threshold for fb = 0.25, fb = 0.20 and fb = 0.15 are about 0.60 × 107V/m, 0.73 × 107V/m and 0.84 × 107V/m, respectively. The results demonstrate that the threshold of the lasing behavior can be conveniently tuned by adjusting the grating profile, i.e. the filling factor.
Fig. 6 Maximum emission intensity for various filling factors as a function of the input pump amplitude. The red solid, blue dashed and black dotted lines represent the results for fb = 0.25, fb = 0.20 and fb = 0.15, respectively.
4. Conclusions
In conclusion, a compound waveguide dielectric grating structure with both, high Q-factor and ultrasmall mode volume, is demonstrated to realize a controllable lasing behavior. By adjusting the grating profile one can regulate the Q-factor, magnetic field intensity and mode volume. With gain materials mixed into the polyurethane layer, a stronger light amplification and a more pronounced monochromatic response are observed. In the meantime, the threshold and intensity of lasing can be further tuned by alternating the filling factor. Besides the thin grating structure, our structure does not suffer from large intrinsic losses due to all dielectric configurations.
Funding
National Key Research Program of China (No. 2016YFA0301101); National Natural Science Foundation of China (NSFC) (Nos. 51377003, 61137003, 11234010, and 11404102); Natural Science Foundation of Jiangxi Province (No. 150167), and the Fundamental Research Funds for the Central Universities.
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