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Design of high-Q photonic crystal microcavities with a graded square lattice for application to quantum cascade lasers

Y. Wakayama, A. Tandaechanurat, S. Iwamoto, and Y. Arakawa

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Abstract

A high-Q photonic crystal (PC) microcavity for TM-like modes, which can be applied to quantum cascade lasers (QCLs), was successfully designed in an air-hole based PC slab with semiconductor cladding layers. In spite of no photonic badgaps for TM-like modes in air-hole based PC slabs, cavity Q reached up to 2,200 by utilizing a graded square lattice PC structure. This is ~18 times higher than those previously reported for PC defect-mode microcavities for QCLs. This large improvement is attributed to a suppression of the coupling between the cavity mode and the leaky modes thanks to the dielectric perturbation in the graded structure. We also predicted a dramatic reduction of the threshold current in the designed cavity down to one-fifteenth of that of a conventional QCL, due to a decreased optical volume.

©2008 Optical Society of America

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Figures (8)
Fig. 1.
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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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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Fig. 3.
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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Fig. 4.
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 ( ∣ E Z ˜ ∣ ) 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.
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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Fig. 6.
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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Fig. 7.
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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Fig. 8.
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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Tables (1)

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

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Equations (4)

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1 Q mat = 1 Q total − 1 Q pass ,
g Γ = 2 e E 32 ∣ Z 32 ∣ 2 N p ħ c ε 0 n eff L p γ 32 η in τ 3 ( 1 − τ 21 τ 32 ) J . Γ ,
Q = 2 π n eff λ α ,
I th = 2 π n eff 2 ⁄ λ Q g 0 Γ × S ,
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