1. Introduction

Research on cavity/defect modes based on photonic crystal microcavities (PCMCs) has attracted significant attention in optoelectronics and optical communications [1–8] due to promising applications of their associated high quality factor Q and ultra-small modal volume V_m. These two conditions pave a way to achieve dramatic enhancement of light-matter interaction over a compact space that is critical for optical nonlinear effects such as second harmonic generation (SHG). It is therefore expected to significantly enhance the frequency conversion efficiency, for instance, if spectrally tuning the fundamental frequency to a cavity/defect mode in a one-dimensional (1D) photonic crystal (PC) [9]. Moreover, the SHG efficiency can be further enhanced by engineering the PCMC in a way that the fundamental wave (FW) and second harmonic (SH) are localized within the same defect region simultaneously [6]. Although this dual-localization strategy can significantly enhance the SHG efficiency, more efficiently utilizing the input optical energy within a broader frequency regime still remains challenging. The localized modes of defects are ideally discrete and their densities of states are a series of Dirac peaks, but the presence of leakage mechanism transforms the resonant transmission into a series of Lorentzian peaks with a finite width [10]. Such narrow transmission windows would drastically reduce the SHG efficiency since a large portion of the input light may be filtered out by the cavity [11, 12].

To achieve a robust high-efficiency optical nonlinear process, the cavity mode should simultaneously support both strong field enhancement and a relatively broad (or appropriate) transmission bandwidth. However, these two requirements are in general contradictory. Past studies on PCMCs provide a simple method to broaden the transmission peak while maintaining strong field enhancement in cavities: a single cavity mode will generally split into two modes (bonding and antibonding states) when two defects are coupled to form a PC molecule [12–16]. However, most of these studies have focused on the corresponding formation mechanism of flat-top impurity bands for their potential in ultrashort pulse transmission, but paid insufficient attention on their application to achieve a broadband nonlinear optical process. Moreover, their design of complex PC molecule is usually based on direct periodic repetition of single PC atoms, which is invalid for nonlinear optical enhancement due to the dramatic reduction of nonlinear conversion efficiency if the flat-top impurity band requires a small period number or a zero cavity length [12]. In this paper, we engineer the flat-top (or quasiflat-top) resonance transmission peaks using a cascaded PCMC structure. The spectra re-shaping of cavity modes is simply controlled by the thickness of joint layers, instead of the size of each cavity, which offers great freedom to increase the transmission bandwidth and maintain a high-efficiency SHG process simultaneously. The improved spectral stability for the SHG process also has great potential applications in relaxing the critical or rigorous requirement for ultrahigh-Q microcavities as it allows quite large tolerance in fabrication, design scale inaccuracy, thermal variation, and wavelength detuning.

2. Method and design

A single PCMC with the structure of (HL)^PND(LH)^PN (Fig. 1(a) ) originates from a matrix of (HL)^PNH(LH)^PN, provided that the central layer H is changed into the cavity layer D (the geometric length is thus changed from d_H to d_D), where H and L stand for the high- and low-refractive-index layers, respectively. The structure of (HL)^PN or (LH)^PN is a distributed feedback Bragg reflector and has a period number of PN. They are composed of alternately stacked H and L layers, satisfying the condition of a quarter-wavelength optical thickness of n_Hd_H = n_Ld_L = λ₀/4, where n_H(L) represents the refractive index of H(L) layer and λ₀ is the central wavelength of the photonic bandgap of the Bragg reflectors. To generate a cavity mode at the wavelength of λ₀, the cavity layer D can be designed with a half-wavelength optical thickness (i.e. d_D = 2d_H). As shown in Fig. 1(b), a double-cascaded PCMC structure is formed by connecting two identical single PCMCs using a joint layer (layer J). If we cascade N identical S-PCMCs through N-1 joint layers with the same thickness of d_J, a multiple-cascaded PCMC can be constructed with the arrangement of (HL)^PND(LH)^PN[J(HL)^PND(LH)^PN]^N-1 as depicted in Fig. 1(c). When N = 1, the proposed structure degenerates into a typical single PCMC. In this paper, we consider the cascaded PCMC structures formed by high-index dielectric layers LiNbO₃ embedded in a low-index background of the air, i.e. the materials of H/L/D/J layer are LiNbO₃/air/LiNbO₃/air, respectively. This kind of multilayer structure is able to be fabricated using the top-down lithography. The dispersion of LiNbO₃ is temperature dependent and the effective nonlinear coefficient d_eff is 43.9 pm/V as described in Ref. 6. The interested fundamental wavelength λ₀ here is 1064 nm, a typical lasing generation from Nd:YAG and Nd:YVO₄ which are extensively used in many fields.

 

Fig. 1 Schematic diagram of PCMC structures. (a) Single PCMC, (b) double-cascaded PCMC, and (c) multiple-cascaded PCMC.

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3. Results and discussion

We have firstly done numerical investigations on the properties of the cavity modes in linear transmission spectra of a double-cascaded PCMC as shown in Fig. 2 . The period number is taken as PN = 5 and a normal incidence is considered for simplicity. When d_J = 0, the cavity modes appear in the transmission spectrum as separated Lorentzian profiles. As the air gap d_J varies from 0 to d_L, the antibonding mode located at the shorter resonance wavelength exhibits a red-shift from 1058.61 nm to λ₀, while the bonding mode is almost insensitive to the variation of d_J and keeps its resonance peak at the wavelength of λ₀. When d_J keeps increasing and deviates from d_L to 2d_L, the antibonding mode is standing at the wavelength of λ₀, while the bonding mode moves from λ₀ to 1067.72 nm. In the special case of d_J = d_L, two cavity modes are almost degenerate at λ₀ = 1064 nm and difficult to be distinguished. Under this condition, the original Lorentzian transmission profile becomes a flat-top window. The red and black dotted lines in Fig. 2 show the evolution of the two resonance modes with different thickness of joint layer (d_J), where two modes are continuously modulated. It is worth noting that one of the resonance peaks is always centered at λ₀ = 1064 nm for any value of d_J, provided that the refractive index difference between layer J and background is rather small, which is crucial to observe flat-top unit transmission spectra. Otherwise, the antibonding and bonding modes can be clearly distinguished even in the case of d_J = d_L. Further simulations with d_J > 2d_L indicate that the mode splitting have periodicity of d_J with a period of 2d_L, implying the phase correlation of two cavity modes.

 

Fig. 2 The dependence of the resonance peaks on the air gap distance d_J in a double-cascaded PCMC with PN = 5. The transmission spectra are shifted upwards for clarity with offset = 1.

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The flat-top transmission characteristics of cascaded PCMCs with d_J = d_L are further studied for different N as shown in Fig. 3 , where the phase and transmission spectra of coupled cavity modes for three cascaded structures with N = 2, 3, and 4, as well as a single PCMC (N = 1) for reference are calculated. The phase of the modes in Fig. 3(a) is relative to that of the incident field, i.e. a phase of zero is assumed for the incident field. It can be seen that the phase near the antibonding mode drastically shifts from -π to + π, while it always remains zero for the bonding state. As mentioned above, in a double-cascaded PCMC with d_J = d_L, a flat-top window can be obtained in the transmission spectrum (red dotted line in Fig. 3(b)) due to cavity modes degeneration. The flat-top profile can be well analyzed by a differentiator model, which is synthesized as a linear superposition of original Lorentzian profiles and their successive derivatives [11, 17]. For a triple-cascaded PCMC with N = 3, a quasiflat-top resonance window appears around λ₀ and three coupled modes can be distinguished by the two shallow valleys near the band edges (green solid line in Fig. 3(b)). The two side modes are antibonding states with quick phase shifts, and the middle resonance at λ₀ belongs to bonding state with a phase close to zero. For a quadruple-cascaded PCMC with N = 4, there are four modes in the spectrum (blue dashed line in Fig. 3(b)), corresponding to antibonding/bonding/antibonding/bonding states from shorter to longer wavelengths in the frequency range of our study. It is clear that when the value of N increases, the coupled resonance peak gradually approaches to a rectangle shape but with deeper valleys near the impurity band edges, which may introduce non-negligible variation to the optical nonlinear process.

 

Fig. 3 (a) The simulated phase and (b) transmission spectra of the coupled cavity modes in single and cascaded PCMCs with N = 1 to 4. The structures are with PN = 5 and d_J = d_L.

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The characteristics of SHG in cascaded PCMCs are then theoretically investigated in comparison with a single PCMC (N = 1). In the inhomogeneous nonlinear dielectric structures as proposed in this work, the quadratic nonlinear interaction between the FW and SH fields can be described by the following nonlinear coupled-wave equations [6]:

We firstly study a double-cascaded PCMC structure by changing the cavity separation through the thickness of joint layer (d_J). The examples under the incident FW intensities of I_0FW = 10 and 100 MW/cm² are shown in Figs. 4(a) and (b) , respectively. The SHG efficiency exhibits a hump-like profile with the wavelength tuning. The relationship between the peak position of the efficiency curve and the thickness of layer J is extracted in Fig. 4(c) to further clarify the dependence of SHG on the thickness of the layer J. By varying d_J from 0 to 2d_L, the efficiency peak related to the antibonding state red-shifts from 529.30 nm to λ₀/2, and another peak associated with the bonding state also red-shifts from λ₀/2 to 533.86 nm. Overall, two main features should be pointed out. (i) The peak positions are almost insensitive to the FW intensity but approximately coincide with one half of the resonance wavelengths in Fig. 2, implying that the splitting of the SHG efficiency peaks originates from the mode splitting of FW; (ii) When d_J = d_L, two efficiency peaks are balanced on both sides of λ₀/2 with approximately an equal distance, indicating that the corresponding cascaded structure can support a broadband SH output centered at λ₀/2.

 

Fig. 4 The SHG efficiency as a function of wavelength tuning in a double-cascaded PCMC with various air gap distance d_J. The incident FW intensities are given as (a) 10 and (b) 100 MW/cm², respectively. The efficiency spectra are shifted upwards for clarity with offset = 0.1. (c) The dependence of peak position of efficiency curve on thickness of layer J for (a) and (b).

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Another important aspect is how the number of coupled cavities (N) in the cascaded structure affects the conversion efficiency at the SH wavelength of 532 nm (i.e. λ₀/2). Figure 5(a) shows the dependence of the conversion efficiency on the incident FW intensity I_0FW for N = 1, 2, 3, and 4 with d_J = d_L. As concluded in Refs. 6 and 19-21, with an increase of incident FW power, the total conversion efficiency including both forward and backward SH is rapidly enhanced till a saturation point and then gradually drops down when the backward FW intensity goes over the forward FW intensity. It is due to that the FW energy mainly flows to backward FW rather than forward and backward SH. In a symmetric single PCMC structure that satisfies the conditions of pump depletion and global phase matching, the SH outputs are balanced in the forward and backward directions, and the maximum of total conversion efficiency can reach 50%. By breaking the symmetry of the structure, the saturation efficiency may increase or decrease depending on the reflectivity or transmittance of the left and right PC mirrors at the both sides of an individual cavity layer [19]. In Fig. 5(a), the saturation efficiency of the symmetric single PCMC is nearly 40%, which is lower than 50% because the field confinement in the case of PN = 5 is not strong enough to ensure a quite small coherent length; thus the dispersive propagation through the multilayer has obvious influence on the efficiency of SHG. Compared to the single PCMC, the efficiency of the cascaded PCMC structure is greatly enhanced mainly due to the added cavities for nonlinear interaction. For instance, under the same incident FW intensity of 460 MW/cm², the quadruple-cascaded PCMC can reach its maximum efficiency up to 59%, while the efficiency of structures with N = 1, 2, and 3 are 37%, 40%, and 57%, respectively. However, the enhancement factor is not simply proportional to the number of cavities. As shown in Fig. 5(a), the efficiency of the triple-cascaded PCMC exhibits more evident enhancement than the structures with N = 2 and 4. It originates from the fact that in the odd-cascaded structures, one of cavity modes exactly takes place at the wavelength of λ₀ and the FW is thus more strongly confined in the cavity layer, resulting in the prominent enhancement of nonlinear process.

 

Fig. 5 (a) The dependence of SHG efficiency on the incident FW intensity for the examples of N = 1, 2, 3, and 4 with d_J = d_L. (b) The comparison of high-efficiency spectral linewidth in PCMC structures with N = 1 to 4 under I_0FM = 200 MW/cm².

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Figure 5(b) shows the SHG efficiency with detuned wavelengths, i.e. the dependence of the SH outputs on the SH wavelength with constant input FW intensity I_0FW of 200 MW/cm², where PCMC structures with N = 1 to 4 are simulated. It is evident that the generated SH energy varies with the wavelength tuning. For the single PCMC, the SHG curve exhibits a Lorentzian profile with a maximum of 30% around λ₀/2. With N increasing from 1 to 4, the SHG efficiency at λ₀/2 monotonically increases as described above in Fig. 5(a). In the double-cascaded PCMC structure (N = 2), the SHG efficiency curve exhibits a typical lump-like profile with two peaks occurring at 532 ± 0.03 nm. The corresponding spectral linewidth at 30% (the maximum SHG efficiency from the single PCMC) is 0.095 nm. For the case of triple-cascaded structure (N = 3), it is found that the points corresponding to the maximum efficiencies are located at 532 and 532 ± 0.04 nm in the spectra. The spectral linewidth at 30% is broadened to 0.105 nm. Such a wider SH linewidth is consistent with the wider high-transmission window at the fundamental wavelength in the triple-cascaded PCMC. However, if one wants to get a linewidth of 0.105 nm in the single PCMC structure, only 7% of SHG efficiency could be obtained. These phenomena confirm that the introduction of cascaded structures not only broadens the spectral range of high-efficient frequency conversion but also significantly enhances the efficiency of the nonlinear optical process. For the case of N = 4, the shape of SHG efficiency curve becomes not regular but still exhibits four peaks at the wavelengths of 531.962, 531.968, 532.006, and 532.045 nm, respectively. The spectral linewidth of SHG efficiency at 30% is still wide but drops down to 0.085 nm due to the deeper valleys close to the impurity band edges. These valleys correspond to the peaks of group velocities that obviously are undesirable for the nonlinear interaction [12]. Therefore, in the cascaded structures with even lager N, the broadening effectiveness is unusually reduced due to the stronger fluctuation in the spectra. Moreover, we notice that the third-order optical nonlinearity may affect the overall conversion efficiency under strong pumping conditions. To clarify this issue, we have taken the third harmonic (TH) generation into account in a single PCMC structure with PN = 5 as an example. The output power intensity of TH is six orders of magnitude less than that of SH with the incident fundamental power intensity up to 1000 MW/cm². Therefore, the influence can be ignored when we numerically simulate the SHG process in the single and cascaded PCMC structures.

Finally, we investigate the detailed forward (transmitted) and backward (reflected) output power both at the FW and SH to specialize the energy exchange properties in the cascaded PCMC structures. Two salient features are distinguished in Fig. 6 . (i) In all the structures with N = 1 to 4, the forward FW energy is not only converting into SH, but also strongly reflected or coupling to the backward FW due to the high index contrast between layers. It leads to a consequence that the SH conversion efficiency tends to saturate in both directions. (ii) In the single PCMC structure, the SH outputs in both directions are approximately the same at all times, while in the cascaded PCMC structures the SH output is backward-dominant due to the symmetry breaking. As discussed in our previous work [19], the forward factor (i.e. the SHG efficiency ratio of the forward to the total) can be determined by $T_R^{2\omega }/(T_L^{2\omega }+T_R^{2\omega })$. Here $T_{L(R)}^{2\omega }$ is the transmittance of the left (right) PC mirror at SH. In the multiple-cavity PC structures as proposed in this work, the SH output is believed to be mainly generated from the left-side cavities other than the right-side ones, because the nonlinear interaction will be gradually weakened along the forward propagation direction due to pump depletion. For the cavity on the left side, it has$T_R^{2\omega }<T_L^{2\omega }$, and thus the forward factor is pretty small in the cascaded PCMC structures, which offers an alternative way for designing practical photonic devices that require nonlinear one-way output.

 

Fig. 6 The dependence of (a)-(d) forward and (e)-(h) backward FW and SH output on the incident FW power intensity for the PCMC structures with N = 1 to 4 and d_J = d_L.

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4. Summary

In summary, we proposed an effective approach to broaden the resonance peak of cavity modes by introducing the cascaded photonic crystal microcavities based on the concept of spectra re-shaping. The numerical simulations clearly confirm its potential application in achieving a wider spectral range with high SHG conversion efficiency and other nonlinear optical processes. Such improvement of spectral stability is practically important for high-performance nonlinear optical devices.