1. Introduction

Quantum cascade lasers (QCLs) [1] have attracted much attention as a leading compact and practical solid state source in mid-infrared (IR) and terahertz frequency regions. Potential applications of QCLs include environmental monitoring, spectroscopy, free space optical communication, and imaging for security. Recently new classes of QCLs with small optical volume, such as microdisk cavities [2], Fabry-Perot (FP) cavities with one-dimensional distributed bragg reflectors [3], FP cavities with two-dimensional (2D) photonic crystal (PC) reflectors [4], and 2D PC surface-emitting microcavities [5], have attracted a lot of interest because they enable single-mode operation, allow easy monolithic integration of the lasers with other components, and have very low energy consumption. Photonic crystal [6] defect-mode microcavity is an important class of microcavity with a smaller mode volume, which further reduces the QCL threshold current. In addition, taking advantage of the design flexibility of PC defect cavities, one can tailor the output direction and emission pattern to optimize the coupling efficiency into an outer optical system, such as an optical fiber and an optical waveguide. The first PC defect microcavity interband laser was demonstrated by optical pumping at low temperature in 1999 [7]. After that, huge amounts of researches on PC defect cavities with high-Q and small volume have been performed, and which have leaded many high-performance PC nanocavity lasers, such as PC lasers with ultimate small volume [8] and room temperature operated CW PC lasers [9]. Utilizing a sophisticated and complex fabrication scheme, electrically-driven PC nanocavity lasers have also been reported at a wavelength of 1.5 µm [10].

Most of researches on high-Q PC defect cavities are intended to confine photons only with transverse electric (TE) -like polarization. On the other hand, photons are polarized in the transverse magnetic (TM) mode in QCLs, where intersubband transitions are used for light emission. Therefore we have to design high-Q PC defect cavities suitable for TM-like modes to realize PC defect microcavity QCLs. However, only a few detailed investigations regarding the design of PC defect microcavities for TM-like modes have been reported. Two-dimensional PC structures consisting of air holes in a high-index material do not possess a photonic bandgap (PBG) for TM-like modes in a frequency region of interest. Although PC structures with a lattice of rods in air are an alternative, it is difficult to inject current into a defect cavity region, because the high-index material regions are unconnected. Therefore it is challenging to design PC defect cavity structures supporting strongly-localized TM modes with efficient electrical injection availability. In fact, PC defect microcavity QCLs have not been demonstrated yet.

Recently, triangular lattice PCs with triangular-shaped air holes [11, 12], or honeycomb lattice PCs with circular-shaped air holes [13, 14], have been proposed to open a PBG for TM-like modes in PC slabs. The idea has also been proposed to fabricate a conventional PC pattern with circular-shaped air holes into cross-sections of a QC structure, where PC structures recognize photons generated by intersubband transition as TE-like polarized rather than TM-like [15]. However in both the former structures, the Q-factor is quite low (several hundreds) due to a narrow PBG and large radiation losses in the vertical direction. In addition, air-suspended structures [11, 12, and 13] make electrical injection difficult. The Q-factor of the latter structure is large enough to achieve lasing action, but accurate mass-fabrication is difficult.

The structure we focus on in this paper is a PC structure with in-plane perturbations of effective dielectric constant due to gradual changes in the structural parameters. This structure was originally proposed and demonstrated by O. Painter’s group for TE-like modes [16, 17]. The in-plane dielectric perturbation enables one to weaken the couplings between the cavity mode and the leaky modes (radiation modes and waveguide modes), which results in an increase of Q-factor. For cavities in TE-like modes, reduction of the mode coupling with radiation modes plays dominant role for boosting Q-factor, because there is a relatively wide PBG for lateral direction, except for the structures with low dielectric constant of the host material [18] or square lattice structures. On the other hand, the suppression of the couplings with waveguide modes is also critical for realizing high-Q photonic crystal cavities for TM-like modes, because there are not PBGs for TM-like modes in a frequency region of interest. The possibility of mode-decoupling with both radiation and waveguide modes in a graded lattice PC motivated us to utilize the graded PC structure for designing high-Q defect microcavity for QCLs.

In Section 2, we briefly introduce graded PC structures and mode-decoupling mechanism due to the structural modulations. We introduce our calculation model in Section 3, and give calculated results and discussions in Section 4. We have achieved a high Q-factor (~2200) for a surface emitting PC defect cavity for GaAs-based mid-IR QCLs by gradually modulating the air hole radii over several periods. We discuss the origin for the obtained high Q-factor and dependence of cavity characteristics on structural parameters. The effect of material absorption, which is unavoidable in realistic QCL structures, is also discussed. Finally, we show the possibility of the lasing operation with a very low threshold current in our designed microcavity in Section 5.

2. Graded PC structures and mode decoupling due to dielectric perturbations

Here, we briefly introduce the concept of graded PC structures and explain how they boost Q-factor by suppressing the couplings between the cavity mode and the undesired leaky modes [16].

Figure 1(a) schematically illustrates an example of 2D graded PC structures. Structural modulation can be introduced by changing structural parameters (period, air hole radius, etc.) gradually from the center to the edge. In the example, the air-hole radius is used as a modulated parameter. This structural modulation creates an in-plane dielectric perturbation (Δη(r)), where η=1/ε is the inverse of the dielectric profile of the lattice, and r is the in-plane coordinate. Figure 1(b) and (c) show the distributions of the dielectric perturbation in real space and in momentum space $\left(\widetilde{\Delta \eta }\left({\mathbf{k}}_{\perp }\right)\right)$, respectively, where k _⊥ is the in-plane momentum. The distribution of the dielectric perturbation in momentum space is the key to increase Q-factor. Let us consider the mode coupling of the cavity mode with radiation and waveguide modes. The distributions of the cavity, radiation, and waveguide modes in momentum space are schematically illustrated in Fig. 1(d). The coupling amplitude between the modes having difference momentum components is determined by the Fourier amplitude of the dielectric perturbation [19]. Therefore, it is possible to increase the Q-factor by reducing $\widetilde{\Delta \eta }\left({\mathbf{k}}_{\perp }\right)$ at the corresponding vectors from the dominant cavity modes to the leaky modes (i.e., radiation and waveguide modes) in momentum space.

All previous reports related to graded lattice PC structures have focused on design of cavity modes with TE-like polarization. However this mechanism is also useful for increasing the Q-factor even for TM-like modes as we show below.

 

Fig. 1. Graded lattice PC structure and mechanism of mode coupling of a cavity mode with leaky modes in the structure. (a) Schematic illustration of a 2D graded PC structure. Air hole radii are modulated gradually outwards over two periods. (b), (c) Distributions of the dielectric perturbation in real space and momentum space. (d) Schematic illustration of mode distributions of cavity, radiation, and waveguide modes in momentum space.

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3. Model of PC cavity for QCLs ~layer structure and cavity geometry

3.1 Schematic image of the cavity and its layer structure

Figure 2 shows a schematic image of PC microcavity we studied and its layer structure. Since it is necessary to inject electrical current to drive a QCL, the cladding layers were set to be semiconductors. There is a significant difference here from the design of air-bridge PC structures. The layer structure, doping density of each layer, and thickness of cladding and active layers assumed in our calculation are the same as those in Ref. [20]. The refractive index and extinction coefficient of each layer were calculated by the Drude model. We also assumed that air holes are etched down to the interface between the lower cladding layer and the substrate, as shown in Fig. 2.

 

Fig. 2. Schematics of a PC QC microcavity and its layer structure. The active region is sandwiched by low-doped GaAs layers to reduce absorption loss caused by high-doped cladding layers. The doping density and thickness of active and cladding layers are the same as in Ref. [20].

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3.2 Cavity geometry in a square lattice PC

Figure 3 shows the in-plane pattern of our graded PC structure. We adopted a PC structure with a square lattice [16, 17], rather than a hexagonal lattice [21], though it is possible to apply the momentum space design rule to both of them. This is because the square lattice has lower symmetry, and hence simplifies the cavity design. There are three symmetry points in a square lattice where a defect can be introduced. Two of them are the C _4v symmetry points (labeled “d” and “f” in Fig. 2, in Ref. [16]) and the other is a C _2v symmetry point (labeled “e”). In our cavity, a defect is centered at the C _4v symmetry point corresponding to 90° rotational symmetry surrounded by 4 innermost holes (point “f”). The cavity is formed without removing any holes. The air hole radii are increased quadratically outwards over six periods from r=0.20 a to 0.34 a, where r is the air hole radius and a is a lattice constant. The radius of the second nearest air holes from the center was increased a little bit. Too quick jump of the radius leads to decrease of the Q-factor as a result of broadening of the cavity mode in momentum space, due to a strong in-plane confinement in real space. Therefore, it is important to design the size of the jump appropriately. Radius of the second nearest air hole is enlarged only 5% from the value of the quadratic distribution. The modulated area is surrounded by nine periods of PC lattice with a fixed air hole radius of 0.4 a.

In order to verify the effects of a graded structure, we also examined a conventional cavity structure, which is formed just by removing the air holes over six periods from the center (see Fig. 4(e)). As same as the former structure, the cavity area is surrounded by nine periods of lattice with r/a of 0.4.

 

Fig. 3. Investigated graded lattice PC pattern, where grey region is high index material and white circles are air holes. Air holes radii are increased quadratically outwards over six periods from r=0.20 a to 0.34 a and the modulated area is surrounded by nine periods of PC lattice with a fixed air hole radius of 0.4 a.

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4. Results and discussions

4.1 Improvement of Q-factor in a graded lattice PC

Firstly, we investigated cavity characteristics by omitting material absorption. This enables us to compare our results with those in previous report. Absorption effects in realistic QCL structure will be discussed later. Cavity characteristics without absorption for both cavity geometries with the core thickness (d) of 5 a are summarized in Fig. 4. All results were calculated by three-dimensional finite-difference time-domain (FDTD) method [22]. Details on the method have been reported in Ref. [23]. In Fig. 4, we show the field distributions of the vertical component of electric field (E _z) of the fundamental cavity modes for the graded lattice PC cavity (A, see Fig. 4(a)) and the conventional defect cavity (B, see Fig. 4(e)), whose normalized frequencies (f _norm) are 0.181 and 0.179 [a/λ], where a is lattice constant and λ is wavelength. Figure 4(b), (f) are the in-plane (xy cut plane) views at the center of the slab, and (c), (g) are the cross-sectional (xz cut plane) views at y=-a/2 in real space. Figure 4(d), (h) show E _z field distributions in momentum space on the xy cut plane. The dominant momentum components of the cavity mode locate around X-point in both cavity structures. Q-factors in these PC cavities are 2100 and 900, respectively, which are divided to be the vertical Q (Q _⊥) (2900, 1300) and the lateral Q (Q _//) (7900, 3100). In addition, Q _⊥ are divided into the Q-factor to the upper direction (3.7×10⁵, 1.2×10⁴) and that to the bottom direction (2950, 1400). Q _⊥ is correlated with the radiation losses of the cavity mode through the coupling with the radiation modes, while Q _‖ is related to the waveguide losses through the coupling with the waveguide modes. Both Q _⊥ and Q _‖ of the graded lattice PC are higher than those of the conventional PC cavity. These results indicate that the graded lattice structure is useful to improve Q-factors even for TM-like modes. Details on the reason why Q-factor of the gradually modulated PC cavity is higher than that of the non-modulated PC cavity are discussed in the followings.

As mentioned in Section 2, it is necessary to know the distributions of the dominant cavity mode, the leaky modes, and the dielectric perturbations in momentum space, for arguing about the origin of the difference in Q-factors of two types of cavities. We calculated the dispersion relation of electromagnetic waves in the defectless PCs by 3D-FDTD at first, in order to know the distributions of the waveguide modes in momentum space. Figure 5 is the photonic band structure for TM-like modes in a square lattice PC structure with d/a=5 and r/a=0.4, which are the same parameters for outermost lattice in the studied cavities. This figure indicates that there is no PBG for TM-like modes in the defectless square lattice PC, and many higher order waveguide modes fall down due to the large d/a. Therefore, we can not achieve high-Q simply by removing air holes from a conventional square lattice PC. The normalized frequencies of the cavity modes summarized in Fig. 4 are shown in Fig. 5 (Each normalized frequency of the cavity “A” and the cavity “B” is almost the same). In order to improve Q _//, it is necessary to suppress the coupling between the cavity mode and the waveguide mode which occurs at green closed circles in Fig. 5. Since the normalized frequency of the cavity mode is below that of the fundamental waveguide mode at X-symmetry point, the distribution of the fundamental waveguide mode which couples with the cavity mode are annular shape whose center is Γ-point.

 

Fig. 4. Cavity characteristics for a graded lattice PC microcavity (A) and a conventional PC microcavity (B). (a), (e) Photonic crystal patterns with gradually modulated r/a and fixed r/a, respectively. (b), (c), (f), (g) Calculated mode distributions of vertical directional electric field component (E _z) at d/a=5 in the xy plane (z=0) ((b), (f)) and in the xz plane (y=-a/2) ((c), (g)). 1D mode plot along z-direction (x=0, y=-a/2) is inserted. (d), (h) Fourier transformed vertical directional electric field component profile $\left(\mid \widetilde{E_Z}\mid \right)$ in the xy plane (z=0). Solid and broken lines represent a light line of cladding layer and that of substrate, respectively.

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Fig. 5. Photonic band structure for TM-like modes, calculated by 3D FDTD method, in which r/a=0.4 (radius rate of the outermost air holes), and d/a=5. The broken red line is the frequency of the fundamental cavity mode in a graded PC lattice. The green and yarrow circles indicate the coupling of the cavity mode with the waveguide modes, and dominant component of the cavity mode, respectively.

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Next, we investigate the difference of the dielectric perturbations in momentum space $\left(\widetilde{\Delta \eta }\left({\mathbf{k}}_{\perp }\right)\right)$ in both cavities, which are shown in Fig. 6(a), (b). Dominant components of $\widetilde{\Delta \eta }\left({\mathbf{k}}_{\perp }\right)$ are on x (y)-axis. Consequently, it is sufficient to consider only mode couplings between the dominant cavity mode and the leaky modes on Γ-X directions as illustrated in Fig. 6(c). Therefore, in order to weaken the coupling with radiation modes and with waveguide modes, it is necessary to reduce $\widetilde{\Delta \eta }\left({\mathbf{k}}_{\perp }\right)$ in the region where |k _⊥| is larger than the distance from X-point to light line (blue shadowed region) on 1D view scanned along y-axis (Fig. 6 (d)) and to reduce $\widetilde{\Delta \eta }\left({\mathbf{k}}_{\perp }\right)$ around the vector from X-point to fundamental waveguide mode (yellow shadowed region), respectively. As shown in Fig. 6(d), the amplitudes of $\widetilde{\Delta \eta }\left({\mathbf{k}}_{\perp }\right)$ for the graded lattice PC microcavity (A) are lower than those for the conventional defect cavity (B) in both regions. Therefore the coupling strength between cavity mode and leaky modes is weaker in the graded lattice PC microcavity than in the non-graded lattice PC microcavity. These are the reasons why Q-factors in the PC cavity with a graded lattice structure are higher than those in the PC cavity with a conventional defect region, for both vertical and lateral directions.

 

Fig. 6. Dielectric perturbation profiles in the momentum space of PC cavities (a) with the gradually modulated r/a and (b) with the fixed r/a. (c) Illustration showing the mode coupling between a cavity mode and leaky modes in momentum space. (d) Comparison between dielectric perturbations scanned from point “a” to “b” in (a) and (b).

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To improve Q-factor, it is meaningful not only to design an in-plane PC structure but also to optimize thickness of the core region. Figure 7 shows the Q-factor dependence of the fundamental cavity mode in the graded lattice PC cavity as a function of d/a. As shown in Fig. 7, Q-factor increases with d/a. The origin for this is the increase of the vertical Q with d/a. As d/a increases, the normalized frequency of the cavity mode decreases, which means the radius of the light cone is reduced. Consequently, the momentum components located within the light cone decreases and the vertical Q increases. Contrary to the increase of the vertical Q, the lateral Q slightly decreases with d/a. This is because the higher order guided bands fall and couple with the cavity mode as d/a increases.

The maximum Q-factor of 2200 at d/a=6 is the highest Q-factor reported thus far for TM-like modes in slab-type PC cavities. When compared with a Q-factor of ~120 (where Q _‖ ~240 and Q_⊥~240) for a defect cavity in honeycomb lattice PC [14], we achieved a large improvement of 18 times.

As thickness of the core region (d) is expressed as λ×f _norm×d/a, d should be set to 3.7 µm (10.3 µm) when d/a is 2 (6) in order to match the cavity mode to the wavelength of 9.5 µm.

 

Fig. 7. Q-factor dependence on d/a. Q-factor is divided into vertical directional Q and lateral directional Q components. The red line is the normalized frequency (a/λ)

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4.2 Effect of material absorption on total Q factor

In the previous subsection, we demonstrated the high-Q cavity in a graded PC structure and explained its physical origin, where we did not include the absorption losses in materials. However it is very important to take an effect of material absorption into account because an absorption coefficient in cladding layers is large in actual QCLs. Figure 8, shows the dependence of total Q (Q _total), passive Q (Q _pass), and material Q (Q _mat) on d/a. Total Q, which includes radiation, propagation, and material losses, is calculated by taking both the real and imaginary parts of the refractive index of each layer into account. The Q-factor originating from material absorption (Q _mat) is calculated by:

where Q _pass is the Q-factor without material absorption, discussed in the previous subsection. Q _mat increases with d/a because of the reduction of overlap between the cavity mode and absorptive cladding regions as d/a increases.

 

Fig. 8. Q-factor dependence on d/a. Total Q-factor (Q _total) is divided into passive Q (Q _pass) and material Q (Q _mat) components.

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When d/a is 6, Q _total reaches up to 970. The mode volume of the cavity was also calculated to be ~6.47(λ/n _eff)³, where n _eff is effective refractive index. The reason for the larger mode volume, compared to the conventional nanocavities, is mainly due to the difference in the thickness of core regions. In an intersubband laser, contrary to the conventional interband lasers, the effects of the enhanced radiative transition rate and the increased spontaneous emission coupling factor due to a small mode volume on the threshold current density are negligible because of the fast nonradiative transition rate between intraband sublevels. Details will be explained in the end of the following subsection. Therefore mode volume is not an important parameter, at least in the view point of the threshold current density. Of course, a cavity with smaller mode volume is attractive even for intersubband lasers to reduce the threshold current.

4.3 Possibility of lasing operation with a low threshold current in the designed PC cavity

An important issue to be addressed is whether the obtained Q-factor is sufficient for lasing operation. For this purpose we need to know the modal gain of the quantum cascade structure. We fabricated a Fabry-Perot QCL with the same active region as that assumed in our simulation. The minimum cavity length for lasing operation was 620 µm in the experiment. From this cavity length and the reflectivity of cleaved facet (0.27), we calculated the mirror loss to be 21.1 cm^-1. The internal loss of the fabricated QCL was calculated to be 21 cm^-1, which matched the value obtained from the dependence of the threshold current density on the cavity length. Thus we can conclude that the maximum modal gain in the FP laser was 42.1 cm^-1.

The modal gain of QCLs at low temperature can be written as [24],

where E ₃₂ is the energy difference between the excited state (subband 3) and the ground state (subband 2), Z ₃₂ is the radiative transition matrix element, N _p is the number of emission layers, n _eff is the effective index, L _p is the length of a period, γ ₃₂ is the full width at half maximum of the laser transition, η _in is the carrier injection efficiency to excited state, τ ₃ is the nonradiative lifetime from subband3, and τ _{32 (21)} is the nonradiative lifetime from subband 3 (2) to subband 2 (1). The differences between a FP and a PC cavity in the above equation are n _eff and Γ. We calculated these values for the designed PC cavity (the fabricated FP cavity) to be 2.91 (3.2), 0.23 (0.31), respectively. The values for the PC cavity were calculated by 3D FDTD, while those for the FP cavity were calculated analytically, using the parallel plate waveguide model. From these results, we estimated the maximum modal gain in the PC microcavity to be about 34.3 cm^-1. This gives us a maximum optical loss inside the cavity. We estimated the minimum total Q-factor of a PC cavity needed for lasing to be about 560 from the relationship between the Q-factor and optical loss per unit length (α), which is expressed by the following equation:

Since this result is lower than the numerically obtained total Q-factor (~970), we expect that the designed cavity enables lasing operation. We would like to emphasize that Q-factor of ~1000 is sufficient for lasing operation, because the modal gain in a QCL is very large due to the large radiative transition matrix element (Z ₃₂) caused by intersubband transition and the large number of emission layers (N _p).

By utilizing the PC cavity, we expect a large reduction in threshold current. Since the modal gain is proportional to the current density, as shown in Eq. (2), the threshold current can be written as:

where I _th is threshold current, S is current injection area, and g ₀ is gn _eff/J. We compare each value for the FP cavity with that for the PC cavity in Table 1. It was found that the threshold current in a PC microcavity laser was reduced to be one-fifteenth, due to a large reduction in optical volume with keeping the Q-factor needed for lasing operation. In this estimation, we assumed that the current was laterally injected to the entire PC region from the electrode surrounding the PC cavity, as shown in Fig. 2, because it is difficult to fabricate structures where current is injected only in a small region in which light is confined. However, if we adopt a sophisticated technique for fabricating micro metal wire connected to the center of the cavity [25], the threshold current will decrease further. Since the area where the optical intensity is larger than 1/e times the maximum intensity is less than one fiftieth of the entire PC region, I _th will be reduced up to 1/750 compared to the conventional lasers.

Table 1. Comparison of characteristics of a Fabry-Perot cavity and a PC cavity.

View Table

In this estimation, the effect of the increased spontaneous emission rate, which is called the Purcell effect [26], is ignored. Strictly speaking, 1/τ ₃ and 1/τ ₃₂, in Eq.(2), should be expressed as 1/τ ₃+F _p/τ ₃₂__rad and 1/τ ₃₂+F _p/τ ₃₂__rad, respectively, where F _p and τ ₃₂__rad are the Purcell factor and radiation lifetime from subband 3 to subband 2. The non-radiative lifetime is generally of the order of ps, while the radiative lifetime is a few hundred ns in a vacuum field. Even if the Purcell factor is larger than a hundred, the non-radiative process in intersubband lasers is still much faster than the radiative process. As a result, the Purcell effect does not change the expression of modal gain.

In the end of this paper, let us discuss the designed structure in the fabrication aspects. Imperfect circular shapes and location disorders of the fabricated air holes will not lead to a major degradation of the designed Q-factor because wavelength used in this work is so long. The imperfections mentioned above are generally in an order of nm using the current nanofabrication technologies. The most critical parameter which affects the Q-factor of our cavities is the depth of the air holes. In addition, verticality of the air hole may also degrade the Q-factor. It is important to develop etching technology for deep and vertical air holes.

5. Conclusion

We have presented a design for high-Q photonic crystal defect microcavities using a graded square lattice for application to quantum cascade lasers. The maximum Q-factor without material absorption exceeded 2200, which is ~18 times higher than the values previously reported for the PC defect microcavity for QCLs. The high Q-factor, despite the lack of PBGs, originates from weak coupling between the cavity mode and the leaky modes (radiation modes and waveguide modes).

The maximum gain obtained in the experiment, the calculated confinement factor, and the calculated effective index predicted the possibility of lasing operation with very low threshold current in QCL. The threshold current of the designed structure is reduced to at least one fifteenth of that of a conventional QCL.