1. Introduction

Ultra-compact dielectric broadband reflectors are essential elements in the design of laser cavities, light trapping microcavities, cavity QEDs, nonlinear optics, and quantum computing systems [1–4]. Traditionally, they can be realized by using the metal films or the stacked dielectric thin films (i.e., distributed Bragg reflectors, DBRs). Metal films can offer larger reflection bandwidth but are limited by the intrinsic absorption losses. Stacked dielectric thin films can achieve very low losses. But they typically require many individual layers with the stringent refractive index and thickness tolerances for each layer. Two dimensional photonic crystal slab (2D PCS) broadband reflectors can be realized for in-plane directions based on the photonic bandgap principle [5]. Recently more attention has been paid to the single layer patterned broadband reflectors for surface-normal incident direction operation, based on the guide mode resonance, or Fano resonance [6–8] principles, for single layer one-dimensional sub-wavelength grating (1D SWG) structures, or 2D PCS structures [8–17].

Under surface-normal incidence, DBR, 1D SWG, and 2D PCS reflectors can all exhibit similar reflection properties with extremely high reflection and broad reflection spectral band. However, reflection mechanisms are different. For 1D SWG and 2D PC mirrors, the incident wave couples to the in-plane guided-mode based on phase matching conditions. The wave then reradiates at one edge with a zero phase difference and at another edge with a π phase difference. Consequently, these constructive and destructive interferences result in high reflection and low transmission, respectively [16]. While for DBR, the high reflectivity arises from the multiple reflections with constructive interference among these reflected waves. Due to the large index difference, Bragg mirrors possess a broad reflection spectral band [18]. For 1D SWG and 2D PCS mirrors, the broad reflection spectral band most likely originates from the cooperating of the several adjacent guided mode resonances [16].

In addition to the reflector spectral amplitude properties, such as high reflectivity R and broad reflection band, it is equally important to understand the phase discontinuity and dispersion behavior (reflection phase shift, Φ_r), the field/energy penetration depths (L_p, L_e), and the field/cavity modal characteristics in cavities formed by these types of patterned single layer dielectric reflectors. However, most attentions so far have been paid to the spectral reflection amplitude properties. Very little work has been reported on phase/dispersion and mode/energy characteristics. In Ref [19], the authors reported excellent work on the phase discontinuity Φ_r and the energy penetration depth L_e, which were estimated from the mode spacing in the Fabry-Perot (FP) cavity. In this work, we investigate the phase discontinuity and the dispersion properties in 2D PCS Fano reflectors, for applications in multi-wavelength cavity design. We will also discuss the distinctively different behavior for phase and energy penetration depths. The phase penetration depth L_p is related to the reflection delay. A large phase penetration depth will lead to longer photon lifetime and longer cavity resonance. The energy penetration depth L_e is related to the energy decay. A smaller energy penetration depth will lead to more modal confinement [18,20,21]. In addition, the different reflection mechanisms, guided mode resonance in PC mirrors and constructive interferences in DBRs, result in distinctively different field distribution profiles inside the cavity, which is another important factor to be considered in the laser cavity design. In what follows, we first introduce the cavity configurations of three types of surface-normal dielectric reflectors. Secondly, we investigate and compare the two penetration depths, L_p and L_e according to the calculated Φ_r and R values, based on the 3D finite difference time domain (FDTD) technique. We then compare the field distributions of the resonant modes in FP cavity formed by these three types of dielectric reflectors. Finally, a conclusion is given.

2. Dielectric mirrors configuration and corresponding FP cavities

Here, we consider three types of dielectric mirrors and their corresponding FP cavities, as shown in Fig. 1 . They all consist of two kinds of materials, Si and SiO₂. Here Si and SiO₂ are assumed to be lossless and dispersion-free, with the refractive indexes are of 3.48 and 1.48, respectively, over the spectral range of interest around 1550 nm. In Fig. 1(a), the top and the bottom mirrors of the FP cavity (denoted as “Cavity I”) are 1D Si SWG structures with the same lattice parameters, with the Si layer thickness h of 0.46μm, the grating period Λ of 0.7μm, and the air slit width w of 0.25Λ. The bottom SWG mirror is formed on a silicon-on-insulator (SOI) substrate, with the buffered oxide (BOX) layer thickness of 0.83μm. These two 1D SWG reflectors exhibit a high reflection over 1.1-2.06μm wavelength band for TM polarization only (H field is parallel to the air slit, y direction) [16].

 

Fig. 1 Sketches of different dielectric reflectors and the corresponding Fabry-Perot cavities: (a) cross section (xz plane) of Cavity I consisting of top and bottom reflectors based on 1D single Si-layer sub-wavelength grating (SWG) with the same pattern parameters; (b) overview of Cavity II consisting of top and bottom mirrors based on 2D PCS patterns with a square air holes lattice; (c) cross section (xz plane) of Cavity III consisting of top and bottom mirrors based on 4-pairs of Si/SiO₂ (0.11/0.27μm) DBR stacked layers.

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Shown in Fig. 1(b) is the second FP cavity (“Cavity II”) formed with top and bottom 2D PCS Fano reflectors, where the Si slab layers with a thickness h = 0.34um are patterned with 2D square lattice air hole arrays. The PC lattice constant Λ is equal to 0.98μm. Again, the bottom reflector is processed on a SOI wafer, with the BOX layer thickness of 1μm. To enable the reflection bands of the top and bottom mirrors with large spectral overlap, their radius of air hole are set to r_t = 0.26Λ and r _b = 0.28Λ, respectively. The resulting overlapping reflection range of these two 2D PC mirrors is over 1.49-1.58μm wavelength band for both TE and TM polarizations (Broader and more flat band could be found through carefully optimizing the structure parameters). For comparison, we also consider the third FP cavity (“Cavity III”), based on two classical high index contract Si/SiO₂ DBRs, as shown in Fig. 1(c). The DBRs consist of 4-pairs of Si/SiO₂ stacked layers, where the thickness of Si and SiO₂ are chosen to be 0.11 and 0.27μm, respectively. Such a DBR with a very large index difference possesses a wide reflection band over 1.22-2.07μm. Note all the parameters chosen here are optimized for broadband reflectors with peak reflection around 1550nm.

3. Phase penetration depth and energy penetration depth

In dielectric mirrors, the reflection is not an instantaneous process. It includes a reflection time delay (τ) and energy storage in the mirrors. The reflection delay increases the laser cavity round-trip time or the photon lifetime. The energy storage in mirrors decreases the modal volume and the confinement factor in small cavity in which the cavity round-trip time and the cavity volume have a comparable magnitude as the reflection delay and the mirror storage [20,21]. The reflection delay is directly related to the slope of the reflection phase shift Φ_r. The relation can be expressed as$\tau =\partial {\mathrm{\Phi }}_r/\partial \omega$. The phase penetration depth, L_p, is defined as the half-distance that light propagates in the incident medium during this delay time,

The energy storage is always associated to the parameter of energy penetration depth, L_e. It is the length that the field intensity decays to 1/e of its maximum from the edge of cavity into the mirrors. However, this method is not suitable to calculate L_e of the PC mirrors we discuss here, because the guided modes are excited inside the mirrors. But L_e can be estimated from the mirror transmission or reflection based on the following equation,

3.1 Reflection and the phase shift

First, we utilize FDTD simulation method to get R and Φ_r of these dielectric mirrors. A Gaussian temporal pulse excitation is used to simulate the reflectivity R. In order to calculate Φ_r, a continuous plane wave of a single wavelength (λ) is vertically incident on the dielectric mirrors and Φ_r is extracted from the stable reflected field. Here the calculated Φ_r is set in the range of [0, 2π]. In order to validate our numerical simulation method, we compare the simulated R and Φ_r of the top DBR based on FDTD with the theoretical calculated values according to the multiple thin film matrix theory [22]. The calculated results completely overlap with the theoretical ones.

Plotted in Fig. 2(a) are the simulated R and Φ_r values for both top and bottom DBRs. In the high reflection (R>0.95) spectral band, covering from 1.2 to 2.1μm, Φ_r slowly increases from 0.815π to 1.19π. Shown in Fig. 2(b) are the simulated R and Φ_r spectra of top and bottom 1D SWG reflectors. The high reflection TM (R>0.95) spectral band spans from 1.1 to 2.06μm. Although both DBR and 1D SWG reflector have similar broad high reflection spectra bands, their Φ_r changes are very different. Φ_r of 1D SWG reflector rapidly varies in the range of [0, 2π], much faster than those of DBRs. The calculated R and Φ_r spectra of for the top and bottom 2D PCS reflectors are plotted in Fig. 2(c), where the overlapping high reflection spectra band covers from 1.49 to 1.58μm. Owing to the different air fill factors (r_t = 0.26Λ < r _b = 0.28Λ), high reflection spectral band for the bottom mirror is narrower than that of the top mirror. In this high reflection band range, the phase shift of top mirror Φ_rt varies in range of (0.91π, 1.17π), and the phase shift of bottom mirror Φ_rb changes in range of (0.7π, 1.14π). For all these three types of mirrors, Φ_r varies over the high reflection spectral band at drastically different change rates, which can be found by comparing the phase penetration depth L_p.

 

Fig. 2 Calculated reflection R (blue solid and black dash lines) and phase shift Φ_r (red dash-dot and green dot lines) spectra of top and bottom (a) DBR reflectors, (b) 1D SWG reflectors and (c) 2D PCS reflectors.

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3.2 Phase penetration depth and energy penetration depth

The phase penetration depth L_p can then be calculated based on Eq. (1) and the simulated Φ_r shown in Fig. 2 The results are plotted in Fig. 3(a) , for the bottom DBR, 1D SWG, and 2D PCS reflectors in the wavelength range of 1.5-1.6μm, denoted as L_{p, DBR}, L_{p, 1DSWG}, L_{p, 2DPCS}, respectively. It can be seen that L_{p, 2DPCS} (~2.1μm) and L_{p, 1DSWG} (~3.2μm) are one order larger than L_{p, DBR} (~0.22μm). Additionally, while L_{p, DBR} and L_{p, 1DSWG} remains largely unchanged over the spectral range, L_{p, 2DPCS} does change significantly over different spectral locations. This not only results in a much longer reflection delay time in PC mirrors, it also leads to different resonance cavity locations. Such a long phase delay may be due to the guided mode excitation, even if 2D PCS reflectors are very thin. To verify this point, we record the dynamic process of the reflected field with λ = 1.540μm at one monitor above the 2P PCS reflector and the DBR, as shown in Fig. 3(b). One can find, for the PCS reflector based on guided mode Fano resonance, the reflected field reaches stable condition only after a long 200fs period, while it only takes about 60fs for DBR to reach the stable condition. So, it is very clear these two different reflection mechanisms result in very different L_p values in PC reflectors and DBRs.

 

Fig. 3 (a) The phase penetration depths for three types of bottom reflectors; (b) Reflected field at λ = 1.540μm changes as function of time, for 1D SWG reflectors (top), 2D PCS reflectors (middle), and DBR reflectors (bottom).

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The energy penetration depth L_e can be obtained based on T = 1-R, and Eqs. (2) and (3). The results are sown in Fig. 4 for the bottom DBR, 1D SWG, and 2D PCS reflectors, respectively. Different from the phase penetration characteristics discussed earlier, both L_{e, 2DPCS} (~100 nm) and L_{e, 1DSWG} (~60 nm) are much lower than L_{e, DBR} (~220 nm). And all energy penetration depths have much less wavelength dependence, as compared to that of phase penetration depths. It is worth noting that much higher energy penetration depths are expected in DBRs with smaller index contrast (e.g. GaAs/AlGaAs, InGaAsP DBRs). Such a small energy penetration length in PCS mirrors can attribute from the tight mode confinement. This is also very favorable in achieving better modal confinement inside the cavity. For classical DBRs, the phase penetration depth is typically smaller or similar to the energy penetration depth, due to smaller index contract and larger energy penetration depth [18,22]. However, for 1D SWG and 2D PCS reflectors discussed here, L_p is much larger than L_e, consistent with in-plane 2D PCS reflectors, as reported in Ref. 18. This could be due to the large phase discontinuities originated from the modal interaction between in-plane guided modes and vertical radiation modes. On the other hand, the energy penetration depth can be very small and localized with the thin PCS layer.

 

Fig. 4 The energy penetration depths for three types of bottom reflectors.

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4. FP cavities and field distributions

Finally, for the resonant FP cavity modes, we can investigate the field distribution properties in these FP cavities shown in Fig. 1. To have resonance cavity modes with similar spectral locations, the cavity lengths are chosen as L_c1 = 5.4μm, L_c2 = 5.2 and 5.4μm, L_c3 = 5.4μm in three different cavities. We chose the two resonant modes in each cavity: λ = 1.629μm and 1.510μm in Cavity I, λ = 1.504μm and 1.540μm in Cavity II, λ = 2.076μm and 1.540μm in Cavity III.

Figure 5(a) represents the field intensity of two resonant modes in DBR Cavity III, λ = 2.076μm and 1.540μm, where the dielectric index profile is also plotted with a black thin line. For a classical DBR cavity, the field intensity inside the cavity is always larger than that in mirrors, and the field gradually decays into the mirrors. However, it is not the case for Cavity I and II based on Fano or guided mode resonances. In Cavity I at L_c1 = 5.4μm, for the two resonant modes (λ = 1.629μm and 1.510μm) shown in Fig. 5(b), the magnetic field intensity inside cavity is much smaller than that in mirrors. It is because the guide modes are excited into the mirrors. Outside the cavity, the field intensity decays rapidly, which is in consistent with the very small L _e obtained earlier. Figure 5(c) and Fig. 5(d) correspond to the case for Cavity II. It can be seen that the electric field intensity of the two resonant modes (λ = 1.504μm and 1.540μm) inside cavity is much larger than that in reflector slabs and the field outside cavity also decays very fast. Cases shown in Figs. 5(b)–5(d) can exist in both 1D SWG and 2D PCS based FP cavities. However, the ratio of the intensity inside the reflector slabs to the intensity inside the cavity (between two reflector slabs) is still larger than that of Cavity III, mostly due to nature of Fano or guide mode resonance excitation inside reflector slabs. Nevertheless, it is expected a very high modal confinement inside the cavity with high field intensity.

 

Fig. 5 Field intensity distributions of the resonant modes inside the cavities (a) Cavity III with L_c3 = 5.4μm, (b) Cavity I with L_c1 = 5.4μm, and (c) Cavity II with L_c2 = 5.2μm, and (d) Cavity II with L_c2 = 5.4μm.

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5. Conclusion

In conclusion, we have numerically investigated phase and energy penetration depths, and field distributions of 1D SWG and 2D PCS reflectors based on Fano or guided mode resonances. Comparing to the DBR reflectors, these new types of single layer ultra-compact broadband reflectors can have more complicated larger phase delays and smaller energy penetration properties, which can be engineered via dispersion engineering for large spectral dependent phase delays, and ultra-small energy penetration depths. The work reported here is mostly based on one set of optimized design parameters for 1550nm band reflectors. Following similar procedures, other design parameters can be used for reflectors with different reflection requirements, as well as different phase delays, energy penetrations, and field distributions. All the results and conclusions can be very helpful for the design of resonant cavities for a wide range of photonic applications.