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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 26 — Dec. 22, 2008
  • pp: 21321–21332
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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  »View Author Affiliations


Optics Express, Vol. 16, Issue 26, pp. 21321-21332 (2008)
http://dx.doi.org/10.1364/OE.16.021321


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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

1. Introduction

Quantum cascade lasers (QCLs) [1

1. J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553–556 (1994). [CrossRef] [PubMed]

] 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

2. L. A. Dunbar, R. Houdre, G. Scalari, L. Sirigu, M. Giovannini, and J. Faist, “Small optical volume terahertz emitting microdisk quantum cascade lasers,” Appl. Phys. Lett. 90141114 (2007). [CrossRef]

], Fabry-Perot (FP) cavities with one-dimensional distributed bragg reflectors [3

3. S. Hofling, J. Seufert, J. P. Reithmaier, and A. Forchel, “Room temperature operation of ultra-short quantum cascade lasers with deeply etched Bragg mirrors,” Electron. Lett. 41, 704–705 (2005). [CrossRef]

], FP cavities with two-dimensional (2D) photonic crystal (PC) reflectors [4

4. J. Heinrich, R. Langhans, J. Seufert, S. Hofling, and A. Forchel, “Quantum cascade microlasers with two-dimensional photonic crystal reflectors,” IEEE Photon. Technol. Lett. 19, 1937–1939 (2007). [CrossRef]

], and 2D PC surface-emitting microcavities [5

5. R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, “Quantum cascade surface-emitting photonic crystal laser,” Science 302, 1374–1377 (2003). [CrossRef] [PubMed]

], 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

6. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef] [PubMed]

] 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

7. O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999). [CrossRef] [PubMed]

]. 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

8. K. Nozaki, S. Kita, and T. Baba, “Room temperature continuous wave operation and controlled spontaneous emission in ultrasmall photonic crystal nanolaser,” Opt. Express15, 7506–7514 (2007), www.opticsinfobase.org/abstract.cfm?URI=oe-15-12-7506. [CrossRef] [PubMed]

] and room temperature operated CW PC lasers [9

9. M. Nomura, S. Iwamoto, K. Watanabe, N. Kumagai, Y. Nakata, S. Ishida, and Y. Arakawa, “Room temperature continuous-wave lasing in photonic crystal nanocavity,” Opt. Express14, 6308–6315 (2006), www.opticsinfobase.org/abstract.cfm?URI=oe-14-13-6308. [CrossRef] [PubMed]

]. Utilizing a sophisticated and complex fabrication scheme, electrically-driven PC nanocavity lasers have also been reported at a wavelength of 1.5 µm [10

10. H. G. Park, S. H. Kim, S. H. Kwon, Y. G. Ju, J. K. Yang, J. H. Baek, S. B. Kim, and Y. H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444–1447 (2004). [CrossRef] [PubMed]

].

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

11. S. Takayama, H. Kitagawa, Y. Tanaka, T. Asano, and S. Noda, “Experimental demonstration of complete photonic band gap in two-dimensional photonic crystal slabs,” Appl. Phys. Lett. 87, 061107 (2005). [CrossRef]

, 12

12. L. C. Andreani and D. Gerace, “Photonic-crystal slabs with a triangular lattice of triangular holes investigated using a guided-mode expansion method,” Phys. Rev. B 73, 235114 (2006). [CrossRef]

], or honeycomb lattice PCs with circular-shaped air holes [13

13. P. Ma, F. Robin, and H. Jackel, “Realistic photonic bandgap structures for TM-polarized light for all-optical switching,” Opt. Express14, 12794–12802 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-26-12794. [CrossRef] [PubMed]

, 14

14. M. Bahriz, V. Moreau, R. Colombelli, O. Crisafulli, and O. Painter, “Design of mid-IR and THz quantum cascade laser cavities with complete TM photonic bandgap,” Opt. Express15, 5948–5965 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-10-5948. [CrossRef] [PubMed]

], 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

15. M. Loncar, B. G. Lee, L. Diehl, M. Belkin, F. Capasso, M. Giovannini, J. Faist, and E. Gini, “Design and fabrication of photonic crystal quantum cascade lasers for optofluidics,” Opt. Express15, 4499–4514 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-8-4499. [CrossRef] [PubMed]

]. 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

11. S. Takayama, H. Kitagawa, Y. Tanaka, T. Asano, and S. Noda, “Experimental demonstration of complete photonic band gap in two-dimensional photonic crystal slabs,” Appl. Phys. Lett. 87, 061107 (2005). [CrossRef]

, 12

12. L. C. Andreani and D. Gerace, “Photonic-crystal slabs with a triangular lattice of triangular holes investigated using a guided-mode expansion method,” Phys. Rev. B 73, 235114 (2006). [CrossRef]

, and 13

13. P. Ma, F. Robin, and H. Jackel, “Realistic photonic bandgap structures for TM-polarized light for all-optical switching,” Opt. Express14, 12794–12802 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-26-12794. [CrossRef] [PubMed]

] 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

16. K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express10, 670–684 (2002), http://www.opticsinfobase.org/abstract.cfm?URI=oe-10-15-670. [PubMed]

, 17

17. K. Srinivasan, P. E. Barclay, O. Painter, J. X. Chen, A. Y. Cho, and C. Gmachl, “Experimental demonstration of a high quality factor photonic crystal microcavity,” Appl. Phys. Lett. 83, 1915–1917 (2003). [CrossRef]

]. 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

18. L. Martiardonna, L. Carbone, A. Tandaechanurat, M. Kitamura, S. Iwamoto, L. Manna, M. D. Vittorio, R. Cingolani, and Y. Arakawa, “Two-dimensional photonic crystal resist membrane nanocavity embedding colloidal dot-in-a-rod nanocrystals,” Nano Lett. 8, 260–264 (2008). [CrossRef]

] 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

16. K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express10, 670–684 (2002), http://www.opticsinfobase.org/abstract.cfm?URI=oe-10-15-670. [PubMed]

].

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 (Δη˜(k)), 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

19. O. Painter, K. Srinivasan, and P. E. Barclay, “Wannier-like equation for the resonant cavity modes of locally perturbed photonic crystals,” Phys. Rev. B 68, 035214 (2003). [CrossRef]

]. Therefore, it is possible to increase the Q-factor by reducing Δη˜(k) at the corresponding vectors from the dominant cavity modes to the leaky modes (i.e., radiation and waveguide modes) in momentum space.

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.

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

20. H. Page, C. Becker, A. Robertson, G. Glastre, V. Ortiz, and C. Sirtori, “300 K operation of a GaAs-based quantum-cascade laser at lambda approximate to 9µm,” Appl. Phys. Lett. 78, 3529–3531 (2001). [CrossRef]

]. 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].

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

16. K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express10, 670–684 (2002), http://www.opticsinfobase.org/abstract.cfm?URI=oe-10-15-670. [PubMed]

, 17

17. K. Srinivasan, P. E. Barclay, O. Painter, J. X. Chen, A. Y. Cho, and C. Gmachl, “Experimental demonstration of a high quality factor photonic crystal microcavity,” Appl. Phys. Lett. 83, 1915–1917 (2003). [CrossRef]

], rather than a hexagonal lattice [21

21. K. Srinivasan and O. Painter, “Fourier space design of high-Q cavities in standard and compressed hexagonal lattice photonic crystals,” Opt. Express11, 579–593 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-6-579. [CrossRef] [PubMed]

], 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

16. K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express10, 670–684 (2002), http://www.opticsinfobase.org/abstract.cfm?URI=oe-10-15-670. [PubMed]

]) 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.

4. Results and discussions

4.1 Improvement of Q-factor in a graded lattice PC

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 (EZ˜) in the xy plane (z=0). Solid and broken lines represent a light line of cladding layer and that of substrate, respectively.
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.

Next, we investigate the difference of the dielectric perturbations in momentum space (Δη˜(k)) in both cavities, which are shown in Fig. 6(a), (b). Dominant components of Δη˜(k) 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 Δη˜(k) 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 Δη˜(k) around the vector from X-point to fundamental waveguide mode (yellow shadowed region), respectively. As shown in Fig. 6(d), the amplitudes of Δη˜(k) 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).

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

14. M. Bahriz, V. Moreau, R. Colombelli, O. Crisafulli, and O. Painter, “Design of mid-IR and THz quantum cascade laser cavities with complete TM photonic bandgap,” Opt. Express15, 5948–5965 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-10-5948. [CrossRef] [PubMed]

], 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)

4.2 Effect of material absorption on total Q factor

1Qmat=1Qtotal1Qpass,
(1)

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.

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

gΓ=2eE32Z322Npħcε0neffLpγ32ηinτ3(1τ21τ32)J.Γ,
(2)

Q=2πneffλα,
(3)

Ith=2πneff2λQg0Γ×S,
(4)

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

25. S. Chakravarty, P. Bhattacharya, J. Topol’ancik, and Z. Wu, “Electrically injected quantum dot photonic crystal microcavity light emitters and microcavity arrays,” J. Phys. D Appl. Phys. 40, 2683–2690 (2007). [CrossRef]

], 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.

table-icon
View This Table

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.

Acknowledgments

This work was supported by the Specially Appointed Funds for Promoting Science and Technology.

References and links

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2.

L. A. Dunbar, R. Houdre, G. Scalari, L. Sirigu, M. Giovannini, and J. Faist, “Small optical volume terahertz emitting microdisk quantum cascade lasers,” Appl. Phys. Lett. 90141114 (2007). [CrossRef]

3.

S. Hofling, J. Seufert, J. P. Reithmaier, and A. Forchel, “Room temperature operation of ultra-short quantum cascade lasers with deeply etched Bragg mirrors,” Electron. Lett. 41, 704–705 (2005). [CrossRef]

4.

J. Heinrich, R. Langhans, J. Seufert, S. Hofling, and A. Forchel, “Quantum cascade microlasers with two-dimensional photonic crystal reflectors,” IEEE Photon. Technol. Lett. 19, 1937–1939 (2007). [CrossRef]

5.

R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, “Quantum cascade surface-emitting photonic crystal laser,” Science 302, 1374–1377 (2003). [CrossRef] [PubMed]

6.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef] [PubMed]

7.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999). [CrossRef] [PubMed]

8.

K. Nozaki, S. Kita, and T. Baba, “Room temperature continuous wave operation and controlled spontaneous emission in ultrasmall photonic crystal nanolaser,” Opt. Express15, 7506–7514 (2007), www.opticsinfobase.org/abstract.cfm?URI=oe-15-12-7506. [CrossRef] [PubMed]

9.

M. Nomura, S. Iwamoto, K. Watanabe, N. Kumagai, Y. Nakata, S. Ishida, and Y. Arakawa, “Room temperature continuous-wave lasing in photonic crystal nanocavity,” Opt. Express14, 6308–6315 (2006), www.opticsinfobase.org/abstract.cfm?URI=oe-14-13-6308. [CrossRef] [PubMed]

10.

H. G. Park, S. H. Kim, S. H. Kwon, Y. G. Ju, J. K. Yang, J. H. Baek, S. B. Kim, and Y. H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444–1447 (2004). [CrossRef] [PubMed]

11.

S. Takayama, H. Kitagawa, Y. Tanaka, T. Asano, and S. Noda, “Experimental demonstration of complete photonic band gap in two-dimensional photonic crystal slabs,” Appl. Phys. Lett. 87, 061107 (2005). [CrossRef]

12.

L. C. Andreani and D. Gerace, “Photonic-crystal slabs with a triangular lattice of triangular holes investigated using a guided-mode expansion method,” Phys. Rev. B 73, 235114 (2006). [CrossRef]

13.

P. Ma, F. Robin, and H. Jackel, “Realistic photonic bandgap structures for TM-polarized light for all-optical switching,” Opt. Express14, 12794–12802 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-26-12794. [CrossRef] [PubMed]

14.

M. Bahriz, V. Moreau, R. Colombelli, O. Crisafulli, and O. Painter, “Design of mid-IR and THz quantum cascade laser cavities with complete TM photonic bandgap,” Opt. Express15, 5948–5965 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-10-5948. [CrossRef] [PubMed]

15.

M. Loncar, B. G. Lee, L. Diehl, M. Belkin, F. Capasso, M. Giovannini, J. Faist, and E. Gini, “Design and fabrication of photonic crystal quantum cascade lasers for optofluidics,” Opt. Express15, 4499–4514 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-8-4499. [CrossRef] [PubMed]

16.

K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express10, 670–684 (2002), http://www.opticsinfobase.org/abstract.cfm?URI=oe-10-15-670. [PubMed]

17.

K. Srinivasan, P. E. Barclay, O. Painter, J. X. Chen, A. Y. Cho, and C. Gmachl, “Experimental demonstration of a high quality factor photonic crystal microcavity,” Appl. Phys. Lett. 83, 1915–1917 (2003). [CrossRef]

18.

L. Martiardonna, L. Carbone, A. Tandaechanurat, M. Kitamura, S. Iwamoto, L. Manna, M. D. Vittorio, R. Cingolani, and Y. Arakawa, “Two-dimensional photonic crystal resist membrane nanocavity embedding colloidal dot-in-a-rod nanocrystals,” Nano Lett. 8, 260–264 (2008). [CrossRef]

19.

O. Painter, K. Srinivasan, and P. E. Barclay, “Wannier-like equation for the resonant cavity modes of locally perturbed photonic crystals,” Phys. Rev. B 68, 035214 (2003). [CrossRef]

20.

H. Page, C. Becker, A. Robertson, G. Glastre, V. Ortiz, and C. Sirtori, “300 K operation of a GaAs-based quantum-cascade laser at lambda approximate to 9µm,” Appl. Phys. Lett. 78, 3529–3531 (2001). [CrossRef]

21.

K. Srinivasan and O. Painter, “Fourier space design of high-Q cavities in standard and compressed hexagonal lattice photonic crystals,” Opt. Express11, 579–593 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-6-579. [CrossRef] [PubMed]

22.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966). [CrossRef]

23.

A. Tandaechanurat, S. Iwamoto, M. Nomura, N. Kumagai, and Y. Arakawa, “Increase of Q-factor in photonic crystal H1-defect nanocavities after closing of photonic bandgap with optimal slab thickness,” Opt. Express16, 448–455, (2008), http://oe.osa.org/abstract.cfm?URI=oe-16-1-448. [CrossRef] [PubMed]

24.

T. Herrle, S. Haneder, and W. Wegscheider, “Role of excited states for the material gain and threshold current density in quantum wire intersubband laser structures,” Phys. Rev. B 73, 205328 (2006). [CrossRef]

25.

S. Chakravarty, P. Bhattacharya, J. Topol’ancik, and Z. Wu, “Electrically injected quantum dot photonic crystal microcavity light emitters and microcavity arrays,” J. Phys. D Appl. Phys. 40, 2683–2690 (2007). [CrossRef]

26.

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681–681 (1946).

OCIS Codes
(140.4780) Lasers and laser optics : Optical resonators
(140.3948) Lasers and laser optics : Microcavity devices
(230.5298) Optical devices : Photonic crystals
(140.5965) Lasers and laser optics : Semiconductor lasers, quantum cascade

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: September 29, 2008
Revised Manuscript: December 1, 2008
Manuscript Accepted: December 5, 2008
Published: December 10, 2008

Citation
Y. Wakayama, A. Tandaechanurat, S. Iwamoto, and Y. Arakawa, "Design of high-Q photonic crystal microcavities with a graded square lattice for application to quantum cascade lasers," Opt. Express 16, 21321-21332 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-26-21321


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References

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  2. L. A. Dunbar, R. Houdre, G. Scalari, L. Sirigu, M. Giovannini, and J. Faist, "Small optical volume terahertz emitting microdisk quantum cascade lasers," Appl. Phys. Lett. 90141114 (2007). [CrossRef]
  3. S. Hofling, J. Seufert, J. P. Reithmaier, and A. Forchel, "Room temperature operation of ultra-short quantum cascade lasers with deeply etched Bragg mirrors," Electron. Lett. 41, 704-705 (2005). [CrossRef]
  4. J. Heinrich, R. Langhans, J. Seufert, S. Hofling, and A. Forchel, "Quantum cascade microlasers with two-dimensional photonic crystal reflectors," IEEE Photon. Technol. Lett. 19, 1937-1939 (2007). [CrossRef]
  5. R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, "Quantum cascade surface-emitting photonic crystal laser," Science 302, 1374-1377 (2003). [CrossRef] [PubMed]
  6. E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987). [CrossRef] [PubMed]
  7. O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, "Two-dimensional photonic band-gap defect mode laser," Science 284, 1819-1821 (1999). [CrossRef] [PubMed]
  8. K. Nozaki, S. Kita, and T. Baba, "Room temperature continuous wave operation and controlled spontaneous emission in ultrasmall photonic crystal nanolaser," Opt. Express 15, 7506-7514 (2007). [CrossRef] [PubMed]
  9. M. Nomura, S. Iwamoto, K. Watanabe, N. Kumagai, Y. Nakata, S. Ishida, and Y. Arakawa, "Room temperature continuous-wave lasing in photonic crystal nanocavity," Opt. Express 14, 6308-6315 (2006). [CrossRef] [PubMed]
  10. H. G. Park, S. H. Kim, S. H. Kwon, Y. G. Ju, J. K. Yang, J. H. Baek, S. B. Kim, and Y. H. Lee, "Electrically driven single-cell photonic crystal laser," Science 305, 1444-1447 (2004). [CrossRef] [PubMed]
  11. S. Takayama, H. Kitagawa, Y. Tanaka, T. Asano, and S. Noda, "Experimental demonstration of complete photonic band gap in two-dimensional photonic crystal slabs," Appl. Phys. Lett. 87, 061107 (2005). [CrossRef]
  12. L. C. Andreani and D. Gerace, "Photonic-crystal slabs with a triangular lattice of triangular holes investigated using a guided-mode expansion method," Phys. Rev. B 73, 235114 (2006). [CrossRef]
  13. P. Ma, F. Robin, H. Jackel, "Realistic photonic bandgap structures for TM-polarized light for all-optical switching," Opt. Express 14, 12794-12802 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-26-12794. [CrossRef] [PubMed]
  14. M. Bahriz, V. Moreau, R. Colombelli, O. Crisafulli, and O. Painter, "Design of mid-IR and THz quantum cascade laser cavities with complete TM photonic bandgap," Opt. Express 15, 5948-5965 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-10-5948. [CrossRef] [PubMed]
  15. M. Loncar, B. G. Lee, L. Diehl, M. Belkin, F. Capasso, M. Giovannini, J. Faist, and E. Gini, "Design and fabrication of photonic crystal quantum cascade lasers for optofluidics," Opt. Express 15, 4499-4514 (2007), [CrossRef] [PubMed]
  16. http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-8-4499. [PubMed]
  17. K. Srinivasan and O. Painter, "Momentum space design of high-Q photonic crystal optical cavities," Opt. Express 10, 670-684 (2002), http://www.opticsinfobase.org/abstract.cfm?URI=oe-10-15-670. [CrossRef]
  18. K. Srinivasan, P. E. Barclay, O. Painter, J. X. Chen, A. Y. Cho, and C. Gmachl, "Experimental demonstration of a high quality factor photonic crystal microcavity," Appl. Phys. Lett. 83, 1915-1917 (2003). [CrossRef]
  19. L. Martiardonna, L. Carbone, A. Tandaechanurat, M. Kitamura, S. Iwamoto, L. Manna, M. D. Vittorio, R. Cingolani, and Y. Arakawa, "Two-dimensional photonic crystal resist membrane nanocavity embedding colloidal dot-in-a-rod nanocrystals," Nano Lett. 8, 260-264 (2008). [CrossRef]
  20. O. Painter, K. Srinivasan, and P. E. Barclay, "Wannier-like equation for the resonant cavity modes of locally perturbed photonic crystals," Phys. Rev. B 68,035214 (2003). [CrossRef]
  21. H. Page, C. Becker, A. Robertson, G. Glastre, V. Ortiz, and C. Sirtori, "300 K operation of a GaAs-based quantum-cascade laser at lambda approximate to 9μm," Appl. Phys. Lett. 78, 3529-3531 (2001). [CrossRef] [PubMed]
  22. K. Srinivasan and O. Painter, "Fourier space design of high-Q cavities in standard and compressed hexagonal lattice photonic crystals," Opt. Express 11, 579-593 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-6-579. [CrossRef]
  23. K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antennas Propag. AP-14, 302-307 (1966). [CrossRef] [PubMed]
  24. A. Tandaechanurat, S. Iwamoto, M. Nomura, N. Kumagai, and Y. Arakawa, "Increase of Q-factor in photonic crystal H1-defect nanocavities after closing of photonic bandgap with optimal slab thickness," Opt. Express 16,448-455, (2008), http://oe.osa.org/abstract.cfm?URI=oe-16-1-448. [CrossRef]
  25. T. Herrle, S. Haneder, and W. Wegscheider, "Role of excited states for the material gain and threshold current density in quantum wire intersubband laser structures," Phys. Rev. B 73,205328 (2006). [CrossRef]
  26. S. Chakravarty, P. Bhattacharya, J. Topol'ancik, and Z. Wu, "Electrically injected quantum dot photonic crystal microcavity light emitters and microcavity arrays," J. Phys. D Appl. Phys. 40, 2683-2690 (2007).
  27. E. M. Purcell, "Spontaneous Emission probabilities at radio frequencies," Phys. Rev. 69, 681-681 (1946).

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