A surface-emitting 3D metal-nanocavity laser: proposal and theory

doi:10.1364/OE.19.013225

« Show journal navigation

A surface-emitting 3D metal-nanocavity laser: proposal and theory

Chien-Yao Lu and Shun Lien Chuang »View Author Affiliations

Optics Express, Vol. 19, Issue 14, pp. 13225-13244 (2011)
http://dx.doi.org/10.1364/OE.19.013225

View Full Text Article

Acrobat PDF (1788 KB)

Journal Search
Article Lookup

Article Tools

Citations

Alert me when this article is cited
Export Citation/Save

Full Text
Article Info
References (35)
Cited By
Figures (19)
Metrics

Abstract

A novel three-dimensional (3D) metal-nanocavity (or nano-coin) semiconductor laser suitable for electrical injection is proposed and analyzed. Our design uses metals as both the cavity sidewall and the top/bottom reflectors (i. e., a fully metal encapsulated nanolaser) and maintains the surface-emitting nature. As a result of the large permittivity contrast between the dielectric and metal, the optical energy can be well-confined inside the metal nanocavity. With a proper design and the choice of the HE₁₁₁ mode, which has the best top surface radiation pattern, a laser with a physical size smaller than 0.01λ₀³ is achievable at 1.55 μm wavelength with a reasonable semiconductor gain at room temperature. We provide a detailed theoretical model starting from the waveguide analysis to full 3D structure simulations by taking into account both geometry and metal dispersion. We show a systematic procedure for analyzing this class of 3D metal-cavity (or nano-coin) lasers with discussions on the optimization of the performance such as light output power, threshold reduction, and output beam shaping.

1. Introduction and proposal of a metal-nanocavity (nano-coin) semiconductor laser

There are a lot of advantages of making ultrasmall lasers such as thresholdless current with almost zero power consumption. Compact nanocavities such as photonic crystals or distributed Bragg reflectors (DBR) have been used to tightly confine the optical field in a tiny volume [1

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(5689), 1444–1447 (2004). [CrossRef] [PubMed]

K. Nozaki, S. Kita, and T. Baba, “Room temperature continuous wave operation and controlled spontaneous emission in ultrasmall photonic crystal nanolaser,” Opt. Express 15(12), 7506–7514 (2007). [CrossRef] [PubMed]

]. Because at least a few pairs of these feedback structures are required to achieve a high quality factor, the sizes are usually several times larger than the wavelength. In the past few years, plasmonic structures have been extensively studied for photonic applications, especially for manipulating optical fields within a subwavelength scale. Metals exhibit a large negative permittivity at the optical frequency due to the interaction of electromagnetic fields with free electrons. Nanolasers with a metal cavity can reduce the optical volume below the diffraction limit with the help of surface plasmon resonance [3

R. M. Ma, R. F. Oulton, V. J. Sorger, G. Bartal, and X. Zhang, “Room-temperature sub-diffraction-limited plasmon laser by total internal reflection,” Nat. Mater. 10(2), 110–113 (2011). [CrossRef] [PubMed]

]. Also, with the use of the metal as an optical isolator, it is possible to have a crosstalk free environment for dense photonic integration. The challenges for practical applications are how to realize continuous-wave operation by electrical injection at room temperature, light collection, performance, and integration. A few types of metal-cavity micro/nanolasers have been analyzed theoretically or demonstrated experimentally [4

C. Y. Lu, S. W. Chang, S. L. Chuang, T. D. Germann, and D. Bimberg, “Metal-cavity surface emitting microlaser at room temperature,” Appl. Phys. Lett. 96(25), 251101 (2010). [CrossRef]

–14

C. Manolatou and F. Rana, “Subwavelength nanopatch cavities for semiconductor plasmon lasers,” IEEE J. Quantum Electron. 44(5), 435–447 (2008). [CrossRef]

]. Designs with metal as parts of the cavity wall and feedback structures to compress the optical mode have been demonstrated by optical pumping from low to high temperature [8

K. Yu, A. Lakhani, and M. C. Wu, “Subwavelength metal-optic semiconductor nanopatch lasers,” Opt. Express 18(9), 8790–8799 (2010). [CrossRef] [PubMed]

S. H. Kwon, J. H. Kang, C. Seassal, S. K. Kim, P. Regreny, Y. H. Lee, C. M. Lieber, and H. G. Park, “Subwavelength plasmonic lasing from a semiconductor nanodisk with silver nanopan cavity,” Nano Lett. 10(9), 3679–3683 (2010). [CrossRef] [PubMed]

,11

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4(6), 395–399 (2010). [CrossRef]

]. A subwavelength metal-clad disk laser [10

J. Huang, S. H. Kim, and A. Scherer, “Design of a surface-emitting, subwavelength metal-clad disk laser in the visible spectrum,” Opt. Express 18(19), 19581–19591 (2010). [CrossRef] [PubMed]

] with a metal reflector under the cavity, similar to that of the nanopan laser [9

] has been analyzed by considering the TE _mnp modes. Room temperature operation of metal-cavity lasers by optical pumping was demonstrated with an optimized plug-in structure for both mode confinement and metal loss reduction [11

]. In 2007, Hill et. al. demonstrated the first metal-cavity nanolaser under CW current injection at 78 K with the idea of cutoff to improve the mode confinement [12

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1(10), 589–594 (2007). [CrossRef]

]; and in 2009, demonstrated a Fabry-Perot type laser which operated under pulsed current injection at 298 K [13

M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y. S. Oei, R. Nötzel, C. Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17(13), 11107–11112 (2009). [CrossRef] [PubMed]

]. Due to the strong distortion of optical fields in certain directions, those metal-cavity nanolasers all suffer from low output power and poor beam shape. With the full coverage of metal, light can only emit through the substrate, which makes the collection inefficient [11

–13

]. Vertical-cavity surface-emitting lasers (VCSELs) have been studied extensively during the past few decades, and significant improvements have been made such as small active volumes (for example, a 1.5 μm diameter micropillar with a volume of ~8 λ₀ ³), high-power arrays, and high speed applications [15

J. L. Jewell, N. A. Olsson, A. Scherer, S. L. McCall, J. P. Harbison, L. T. Florez, and Y. H. Lee, “Surface-emitting microlasers for photonic switching and interchip connections,” Opt. Eng. 29(3), 210–214 (1990). [CrossRef]

–17

R. Safaisini, J. R. Joseph, and K. L. Lear, “Scalable high-CW-power high-speed 980-nm VCSEL arrays,” IEEE J. Quantum Electron. 46(11), 1590–1596 (2010). [CrossRef]

]. To further reduce the size of VCSELs, the high reflection spectral window of DBR will become extremely sensitive to the modal dispersion, and the scattering loss from the sidewall irregularity will be important. Recently, metal-cavity surface-emitting micro/nanolasers with DBR as part of the feedback structure have been proposed and realized under continuous-wave current injection at room temperature [4

C. Y. Lu, S. W. Chang, S. L. Chuang, T. D. Germann, and D. Bimberg, “Metal-cavity surface emitting microlaser at room temperature,” Appl. Phys. Lett. 96(25), 251101 (2010). [CrossRef]

–7

C. Y. Lu, S. L. Chuang, A. Mutig, and D. Bimberg, “Metal-cavity surface-emitting microlasers with hybrid metal-DBR reflectors,” Opt. Lett. (in press). [PubMed]

]. With the surface-emitting configuration, the performance of metal-cavity lasers has been significantly improved, including microwatt output power, ultra-narrow linewidth, circular beam shape, and extremely low thermal impedance. From an application point of view, a compatible platform for advanced stacking with integrated circuits is also an important issue [18

M. Motoyoshi, “Through silicon via (TSV),” Proc. IEEE 97(1), 43–48 (2009). [CrossRef]

]. The metal-cavity surface-emitting nanolasers are substrate-free with the transferability to other platforms. A further improvement with hybrid metal-DBR mirrors of these metal-cavity lasers is demonstrated experimentally [6

S. W. Chang, C. Y. Lu, S. L. Chuang, T. D. Germann, U. W. Pohl, and D. Bimberg, “Theory of metal-cavity surface-emitting microlasers and comparison with experiment,” IEEE J. Sel. Top. Quantum Electron. (in press).

C. Y. Lu, S. L. Chuang, A. Mutig, and D. Bimberg, “Metal-cavity surface-emitting microlasers with hybrid metal-DBR reflectors,” Opt. Lett. (in press). [PubMed]

]. Even though several metal-cavity lasers were demonstrated, the fundamental limit of the smallest realizable metal-cavity nanolaser is still under investigation.

In this paper, we propose and analyze a new class of electrical-injection metal-nanocavity (nano-coin) lasers with full coverage of metal, which preserves the surface-emitting and substrate-free configurations. Our design uses metal as a cavity wall on all sides, which eases the requirement of precise design of optical feedback structures such as DBR [4

C. Y. Lu, S. W. Chang, S. L. Chuang, T. D. Germann, and D. Bimberg, “Metal-cavity surface emitting microlaser at room temperature,” Appl. Phys. Lett. 96(25), 251101 (2010). [CrossRef]

–7

C. Y. Lu, S. L. Chuang, A. Mutig, and D. Bimberg, “Metal-cavity surface-emitting microlasers with hybrid metal-DBR reflectors,” Opt. Lett. (in press). [PubMed]

] or cutoff waveguides [11

–13

]. Moreover, with the help of a full metal coverage, the radiation to adjacent devices can be greatly reduced, and, thus a crosstalk-free environment can be achieved. The proposed structure is transferable and stackable due to its substrate-free configuration. With a proper design, the metal-cavity nanolaser can achieve room temperature operation with a physical cavity volume of only 0.01λ₀ ³. Our proposed three-dimensional metal-nanocavity laser is shown in Fig. 1 , which consists of an In_0.53Ga_0.47As bulk gain material with InP as carrier injectors. The cavity wall is defined by silver metal with silicon nitride (SiN _x ) as both a current blocker to prevent any short circuit and a part of optical waveguide shell to reduce optical leakage to the metal. The metal serves a few purposes including the electrical contacts, the cavity wall for optical mode confinement, shielding from crosstalk among lasers, and, more importantly, an efficient heat sink. The emission comes from the top with a thin silver layer as the reflector. Due to the full coverage of metal on the cavity wall, the modes resemble those in a circular waveguide resonator. In order to capture the general features of the cavity modes resulting from the superposition of two counter-propagating fields, the waveguide modes are analyzed and discussed in section 2. In section 3, cavity modes resulting from the lowest few orders of the waveguide modes are analyzed for laser applications. In section 4 we discuss the optimization through a proper design of the SiN _x and silver reflector layers. A general discussion of the far-field patterns is given for those designs. In section 5, the rate equations are used to analyze the laser performance. The issues of light output power will also be addressed. We then conclude in section 6.

Fig. 1 Our proposed surface-emitting three-dimensional (3D) metal-nanocavity (nano-coin) laser. The active region is composed of a bulk In_0.53Ga_0.47As with a height h in the active region height and a in radius. InP is used as both electron and hole injectors. An InGaAsP layer serves as a contact layer for n-contact. The whole device is encapsulated in silver with SiN _x as a current blocker. (a) A 3D view and (b) a cross sectional view of the structure.

2. Metal-clad waveguide analysis and design consideration

In a metal-clad inhomogeneous circular waveguide, the normal modes are either hybrid mode (HE _mn or EH _mn ) for azimuthal order m ≥ 1 or transverse mode (TE₀ _n or TM₀ _n ) for m = 0, where n represents the order of radial distribution [19

J. A. Kong, Electromagnetic Wave Theory (EWM, 2008).

,20

L. Novotny and C. Hafner, “Light propagation in a cylindrical waveguide with a complex, metallic, dielectric function,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 50(5), 4094–4106 (1994). [CrossRef] [PubMed]

]. Figure 2 shows such a circular waveguide with an inhomogeneous radial distribution (In_0.53Ga_0.47As semiconductor core with a SiN _x shell and a silver metal wall) and its corresponding mode patterns (in the basis of cos(mϕ) and sin(mϕ)) calculated by the finite element method (FEM). To have a large modal gain, a mode with good overlap between optical field and core gain material should be considered. Figure 2 lists our calculated field distribution (electric field norm: |E|; magnetic field norm: |H|) of the first 5 lowest order modes (HE₁₁, TM₀₁, HE₂₁, TE₀₁, EH₁₁) at 1550 nm with an core radius of 250 nm and a shell thickness of 50 nm. For the transverse mode TM₀₁ (TE₀₁), both the magnetic (electric) field and its resulting time-averaged power along the waveguide direction (P_z) have a node at the center due to the nature of Bessel functions and their lack of longitudinal magnetic (electric) field components. Since the periphery of the waveguide usually degrades during the fabrication, configurations with a node at the center will result in an inefficient interaction between the gain materials and the optical fields when propagating back and forth along the z-direction. Optical modes with m > 1 are not suitable here because of their whispering-gallery behaviors, which usually experience more perturbation from the periphery than the lower order modes and have a power node at the center.

Fig. 2 Mode patterns of a circular metallic waveguide with a core-shell structure inside, calculated by the finite element method (FEM). (Top left:) the cross-sectional view of the waveguide layered structure. Silver is used to surround the circular waveguide with an In_0.53Ga_0.47As core of radius a = 250 nm and a thin SiN _x shell layer of thickness s = 50 nm. Five lowest order mode patterns (|(E)|, |(H)|, and P_z) are plotted correspondingly. Power nodes at the waveguide center are observed in TM₀₁, HE₂₁, and TE₀₁ modes.

A simplified perfect electric conductor (PEC) model can be used to capture the guiding properties of circular metallic waveguide. For a homogeneous metallic waveguide, the propagation constant k_z should satisfy the following equation,

k_{ρ}^{2} + k_{z}^{2} = ω^{2} μ ε_{c o r e} = {(\frac{2 π}{λ_{0}} n_{core})}^{2},

(1)

where k_ρ is equal to χ_mn /b and χ_mn is the root of Bessel functions or their derivatives and b is the outer radius as shown in Fig. 2. ε_core is the permittivity of the homogeneous waveguide core and n _core and λ ₀ are the refractive index of the core and the free space wavelength. The mode cutoff occurs at k_z = 0, or equivalently,

(\frac{2 π}{λ_{c}} n_{core}) = k_{ρ} = \frac{χ_{m n}}{b} .

(2)

The resulting cutoff wavelength (λ_c ) is then equal to 2πbn _core/χ _mn . For a PEC waveguide with a homogeneous cross-section, the value χ_mnλ_c /2πn _core b should be 1.

In the descending order of cutoff wavelengths, the dominant modes are HE₁₁, TM₀₁, HE₂₁, TE₀₁, and EH₁₁. Figure 3(a) shows the cutoff wavelengths (λ_c ) as a function of the outer radius (b) calculated with a SiN _x shell thicknesses of 50 nm and material (metal and semiconductor) permittivities from [21

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

,22

S. Adachi, “Refractive indices of III-V compounds: key properties of InGaAsP relevant to device design,” J. Appl. Phys. 53(8), 5863–5869 (1982). [CrossRef]

]. To ensure a propagating behavior inside the cavity within the window of material emission spectrum, the radius with a cutoff wavelength longer than the emission wavelength should be chosen. For example, at around 1550 nm (corresponding to In_0.53Ga_0.47As emission), the HE₁₁ (EH₁₁) mode can only propagate when the radius (b) is larger than 130 nm (260 nm) as shown in Fig. 3(a). The fundamental mode HE₁₁ has the longest cutoff wavelength for the same size, and, thus can potentially be a good choice in designing an optical cavity with a minimal transverse dimension [23

S. W. Chang, T. R. Lin, and S. L. Chuang, “Theory of plasmonic fabry-perot nanolasers,” Opt. Express 18(14), 15039–15053 (2010). [CrossRef] [PubMed]

Fig. 3 (a) Cutoff wavelengths of different modes as a function of core radius. The fundamental mode HE₁₁ has the longest cutoff frequency among all the other modes and can be used to design a cavity of a minimal radial dimension. (b) χ _mn λ _c/2πn _core(b + Δ) as function of core radius. χ _mn is the root of Bessel functions or their derivatives (HE₁₁: 1.84, TM₀₁: 2.405, HE₂₁: 3.05, TE₀₁: 3.83, and EH₁₁: 3.83). The dashed line represents the prediction by the use of homogeneous waveguide with a receded PEC wall by a skin depth Δ. The actual cutoff wavelength will be close to the prediction when the wavelengths are long enough such that the thin cladding layer becomes negligible.

Figure 3(b) plots the value χ_mnλ_c /[2πn _core(b + Δ)] as a function of the outer radius b, where χ_mn is the root of Bessel functions or their derivatives (HE₁₁: 1.84, TM₀₁: 2.405, HE₂₁: 3.05, TE₀₁: 3.83, and EH₁₁: 3.83), Δ is the penetration skin depth, and n _core is the core (In_0.53Ga_0.47As) refractive index. The values deviate from unity as the radius decreases. It means that the approximation by a homogeneous circular waveguide with a real dispersive metal replaced by a PEC wall and receding by the penetration depth fails in this region. This discrepancy comes from the non-negligible permittivity transition from semiconductor core to the insulator SiN _x shell. Moreover, for hybrid modes which should not occur in a homogeneous waveguide with a PEC wall, the approximation will hold only when the radius is large enough for the hybrid modes to be considered as transverse modes and the cutoff wavelength is long enough to ignore the shell thickness.

In an inhomogeneous waveguide such as a core-shell structure (assuming n _core > n _shell in this paper) with a real metal surrounding in Fig. 2, there are three propagating modes in the waveguide: surface (plasmonic) mode, core-shell mode, and core (dielectric or fiber-optical) mode. The surface mode exists when the propagation constant k _z of the mode is larger than k _core = 2πn _core/λ₀, where λ₀ is the free space wavelength, or equivalently, the effective index n _eff (defined by k _z = (2πn _eff /λ₀)) is larger than the material indices inside the waveguide (n _shell < n _core < n _eff). In surface mode, the optical energy is concentrated near the metal-dielectric interface and decays exponentially into both regions. As a result of large field penetration into metal, this mode usually suffers from a high propagation loss. The core-shell mode exists when n _eff < n _shell, and in this case, a significant field leaks out of the core region and propagates (with a real k_ρ ) inside the shell region. The optical power is actually guided in both the core and the shell regions. The core mode represents n _shell < n _eff < n _core, which is the normal propagation mode (or dielectric mode or fiber-optical mode) with power propagation mostly inside the core with an evanescent decay to the shell region, and is expected to be the most lossless among all three modes. The design with the core mode not only helps reduce the sidewall metal loss but also concentrate the energy more into the center region. Figure 4(a) shows a two-dimensional (2D) plot of the effective index n _eff as a function of radius a with a constant SiN _x thickness (s = 50 nm) and the wavelength. Figure 4(b) shows a 2D plot of the guiding wavelength as a function of the core radius (a) and the effective index.

Fig. 4 (a) The effective index as a function of core radius a and the guiding wavelength of the HE₁₁ mode of a silver-coated circular waveguide with an In_0.53Ga_0.47As core and a SiN _x shell (50 nm) as shown in Fig. 2. (b) The guiding wavelength as a function of core radius a and effective index. The wavelength is plotted only in the window of 1000-2000 nm. The white line represents the cutoff wavelength of the HE₁₁ mode, on which the effective index and the propagation constant Re(k_z ) equal zero. To have a mode guiding inside the core region, the effective index has to be larger than the refractive index of SiN _x (~2.0: dashed line in (a)). This also regulates the choice of cavity radius to be larger than ~70 nm.

3. Theory of metal-nanocavity lasers: modal analysis

3.1. Cavity structures and resonant modes

A cavity can be formed by shorting both ends of a waveguide with a conductive wall. For a given wavelength, to form a standing wave in the longitudinal direction, a minimum length satisfying the round-trip phase condition (2π) is required. (An exception is the TM₀ _n mode which lacks the longitudinal magnetic field (H _z = 0) and the azimuthal variation, and, therefore, the boundary conditions can be satisfied at both ends with E _z ≠ 0 at an arbitrary length.) In the PEC model shown in Fig. 5 , the fields of the mode (mnp) in the cavity can be written as a sum of two counter-propagating waves,

Ε_{m n p} (ρ, φ, z) = Ε_{m n}^{+} (ρ, φ) e^{i k_{z} z} + Ε_{m n}^{-} (ρ, φ) e^{- i k_{z} z},

(3a)

H_{m n p} (ρ, φ, z) = H_{m n}^{+} (ρ, φ) e^{i k_{z} z} + H_{m n}^{-} (ρ, φ) e^{- i k_{z} z},

(3b)

where E _mnp is the electric field, H _mnp is the magnetic field of longitudinal order p, k _z = pπ/L (L: the cavity height),

E_{m n}^{+}

and

E_{m n}^{-}

(

H_{m n}^{+}

and

H_{m n}^{-}

) are the electric (magnetic) fields of counterpropagating modes along the waveguide. The translation invariant symmetry of the waveguide along the z axis leads to the following relations [24

S. W. Lee, S. L. Chuang, and C. S. Lee, “Normal modes in an overmoded circular waveguide coated with lossy material,” IEEE Trans. Microw. Theory Tech. 34(7), 773–785 (1986). [CrossRef]

–26

S. L. Chuang, Physics of Optoelectronic Devices, 1st ed. (Wiley, 1995), Appendix H.

E_{m n}^{+} (ρ, φ) = E_{m n}^{t} (ρ, φ) + E_{m n}^{z} (ρ, φ),

(4a)

E_{m n}^{-} (ρ, φ) = E_{m n}^{t} (ρ, φ) - E_{m n}^{z} (ρ, φ),

(4b)

H_{m n}^{+} (ρ, φ) = H_{m n}^{t} (ρ, φ) + H_{m n}^{z} (ρ, φ),

(4c)

H_{m n}^{-} (ρ, φ) = - H_{m n}^{t} (ρ, φ) + H_{m n}^{z} (ρ, φ),

(4d)

where superscripts represent transverse (t) or longitudinal (z) components. The field components inside the cavity can be further explicitly written as,

E -profile : {\begin{cases} E_{z, m n p} ~ E_{m n}^{z} (ρ, φ) \cos (\frac{p π}{L} z) \\ E_{ρ, m n p} ~ E_{m n}^{ρ} (ρ, φ) \sin (\frac{p π}{L} z) \\ E_{φ, m n p} ~ E_{m n}^{φ} (ρ, φ) \sin (\frac{p π}{L} z) \end{cases},

(5a)

H -profile: {\begin{cases} H_{z, m n p} ~ H_{m n}^{z} (ρ, φ) \sin (\frac{p π}{L} z) \\ H_{ρ, m n p} ~ H_{m n}^{ρ} (ρ, φ) \cos (\frac{p π}{L} z) \\ H_{φ, m n p} ~ H_{m n}^{φ} (ρ, φ) \cos (\frac{p π}{L} z) \end{cases} .

(5b)

A field node of the waveguide mode will also result in a node in the corresponding cavity mode formed with the same transverse field distributions.

Fig. 5 (a) The cross-section of a core-shell waveguide with perfect electrical conductor (PEC) surroundings. (b) A cavity formed of shorting both ends of a waveguide in (a) with a cavity length of L.

To investigate which cavity mode is the best one for laser application, the full structure simulation with real experimental material parameters is performed. Figure 6(a) shows the resonance wavelengths of the six lowest cavity modes (TM₀₁₀, HE₁₁₁, TM₀₁₁, HE₂₁₁, TE₀₁₁, EH₁₁₁) calculated by finite-difference time-domain (FDTD) method [27

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech, 2000).

]. The structure is shown in Fig. 1(a) with vertical 160-nm In_0.53Ga_0.47As, 30-nm n- and p-InP, and 20-nm InGaAsP layer thickness, and a conformal insulator layer of 50-nm SiN _x . The material parameters can be found in [21

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

,22

S. Adachi, “Refractive indices of III-V compounds: key properties of InGaAsP relevant to device design,” J. Appl. Phys. 53(8), 5863–5869 (1982). [CrossRef]

]. According to the waveguide analysis in section 2, all modes except HE₁₁₁ have a node in either the electric field or the power (P _z ) distribution. This node will make a poor overlap between the gain region and the optical field. The dotted line indicates that the resonant wavelength is in the cutoff region of the corresponding waveguide mode. The fundamental mode TM₀₁₀ has a wavelength far above the rest due to the zeroth order in z direction.

Fig. 6 Calculated resonance wavelengths of the higher order modes by a full 3D structure FDTD simulation. (a) The resonance wavelengths as a function of the core radius a of the six lowest order modes. (b) Mode chart of the structure curves with the same slope corresponding to the same longitudinal mode number (p/2n_op )².

For a PEC homogeneous cavity (simplified model for understanding), k_z is quantized to pπ/L, and Eq. (1) leads to the modal characteristic equation,

{(\frac{b}{λ})}^{2} = {(\frac{p}{2 n_{core}})}^{2} {(\frac{b}{L})}^{2} + {(\frac{χ_{m n}}{2 π n_{core}})}^{2},

(6)

and the resonant wavelength can be written as,

λ_{m n p} = \frac{2 π n_{core}}{\sqrt{{(\frac{χ_{m n}}{b})}^{2} + {(\frac{p π}{L})}^{2}}} = \frac{2 π n_{core}}{\sqrt{{(\frac{χ_{m n}}{a + s})}^{2} + {(\frac{p π}{L})}^{2}}} .

(7)

The mode chart for the realistic core-shell-metal planar structure is plotted in Fig. 6(b) with the slope representing (p/2n_op )², where n_op is the effective index (instead of n _core) seen by the optical field. In Fig. 6(b), the outer radius (core radius a plus shell thickness s) is labeled as b.Although the resonant wavelength of HE₁₁₁ mode is slightly larger than those of the other modes (except TM₀₁₀) and will result in a slightly larger cavity size, the corresponding waveguide mode (HE₁₁) has the longest cutoff wavelength as shown in Fig. 3(a) and allows for a wider range of detuning. The nonzero slope of TM₀₁₀ mode comes from the inhomogeneous permittivity inside the cavity and the asymmetry in z direction. Moreover, HE₁₁₁ mode is the most suitable mode for achieving a high gain and optical field overlap. In the following analysis, we will focus on the design for the HE₁₁₁ mode.

For an inhomogeneous waveguide with a small permittivity variation, perturbation method can be applied to calculate the resonance wavelength shift from that of a homogeneous waveguide [19

J. A. Kong, Electromagnetic Wave Theory (EWM, 2008).

,23

S. W. Chang, T. R. Lin, and S. L. Chuang, “Theory of plasmonic fabry-perot nanolasers,” Opt. Express 18(14), 15039–15053 (2010). [CrossRef] [PubMed]

]. However, the permittivity contrast between insulator and semiconductor are usually too large to be considered as a perturbation. To properly include the effect of the shell layer, a model with a multilayered structure should be used. As a result of a large negative real part of silver permittivity at near IR region, the optical modes inside the nanolaser surrounded by silver will be similar to those surrounded by a PEC wall. Figure 7 shows the theoretical calculation of the cavity resonance wavelength of the HE₁₁₁ mode in Fig. 6, based on the PEC model with the metal wall receding by a skin depth, as a function of the core radius a for various cavity lengths L = 180 to 320 nm. The resonance wavelength gradually becomes shorter and approaches the waveguide cutoff as the geometry shrinks. A transition from core (dielectric) mode (solid lines) to core-shell mode (dashed lines) is indicated by the dispersion curve with the effective index n _eff = n _shell = 2.0. This transition marks the onset of propagating fields inside the shell region. In the design with real metals, the transition also means more energy penetration into the metal plasma and, thus, results in a higher propagation loss while propagating along the waveguide.

Fig. 7 The cavity resonance wavelength as a function of the core radius a of various cavity heights L from 180 to 320 nm. The SiN _x shell thickness is fixed at 50 nm. The resonance wavelength with a solid curve is the core (dielectric) mode with n _eff > n _shell. Fields concentrate inside the core region. The resonance wavelength with a dashed curve is the shell mode with n _eff < n _shell. Fields leak out of the core region and the wave can propagate inside the shell region. Lines of waveguide cutoff and transition from core mode to shell mode (n _eff = n _shell = 2.0) are plotted. A PEC model (Section 3) with the metal wall receding by a skin-depth is used in these calculations.

3.2. Simulation results of the cavity properties

To analyze the full nanolaser structure in Fig. 1 with specific material properties, we carried out the simulation using the FDTD method. In our design, In_0.53Ga_0.47As is used as the active material with p- and n- doped InP layers as cladding for contact. The thickness is fixed at 30 nm for both n- and p-doped regions. Also, an n-doped 20-nm InGaAsP layer is used to support the contact area. Silver is used to encapsulate the whole cavity with a low-index material SiN _x in between the semiconductor and metal as a current blocker. At the output, a thin silver is coated to increase the reflectivity and serves as a part of the contact. The material parameters used in the simulation are taken from [21

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

,22

S. Adachi, “Refractive indices of III-V compounds: key properties of InGaAsP relevant to device design,” J. Appl. Phys. 53(8), 5863–5869 (1982). [CrossRef]

]. Drude model is adopted to fit the metal permittivity for simplicity.

In Fig. 8 , all HE₁₁₁ field plots of our proposed cavity with a radius a = 190 nm, b = 240 nm, and cavity height L = 240 nm are shown. The top view in (a) with the z-component of both E and H fields at the center of the cavity shows their m = 1 azimuthal distribution (in the form of sin(mϕ) and cos(mϕ)). Also, the lack of nodal point along the ρ-direction represents the radial order n = 1. In Fig. 8(b), the side view of the field components follows Eq. (3) in their z distribution. Note in Fig. 8(b) plots, the azimuthal dependence has been suppressed for simplicity. The symmetry follows the model with p = 1 and the field patterns closely resemble those of the PEC waveguide.

Fig. 8 (a) Top view of z component of electric and magnetic fields of HE₁₁₁ mode. (b) Side view of all field components inside the cavity. Note the azimuthal dependence for (b) has been suppressed for simplicity.

Figure 9 shows the calculated resonance wavelengths and the corresponding quality factors with various heights (220 nm - 300 nm). The thicknesses of SiN _x and the thin silver reflector are both set to 50 nm. The effect of varying these values will be discussed in section 4. In Fig. 9(a) the resonance wavelength of various cavity heights as a function of the core radius can be modeled using the structure analyzed in section 3 by receding the metal wall with a penetration (skin) depth to account for the finite permittivity in real metals. The dashed lines in Fig. 9(a) show the result from the model. Compared to the full structure calculation by the FDTD method, the simplified model agrees very well except at the small radius region where the field leaks into SiN _x and InGaAsP layers. It starts to dominate especially when the effective index becomes smaller than that of the shielding SiN _x layer.

Fig. 9 (a) Calculated resonance wavelengths of various cavity heights as a function of radius a using the FDTD method (solid curves) for the real structure (including the complex permittivity of the metal) compared with those using the PEC model (dashed curves) presented in Section 3. (b) Calculated cavity quality factors as a function of radius. The maximum quality factor is around 275 for all cavity heights.

By taking the Fourier transform of the time varying field inside the cavity, the complex resonant frequency (ω _res = ω _r – iω _i) of the cavity with metal dispersions can be obtained. ω _r represents the oscillation frequency of the optical field and ω _i is the attenuation due to the loss (both material and radiation). The cold cavity quality factor can be expressed as Q = ω _r/2ω _i [28

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1989).

]. Figure 9(b) shows the corresponding ideal cold cavity quality factor Q. A clear drop of Q below 100-nm core radius is caused by the energy leakage to SiN _x and InGaAsP layers as described above. The quality factor can be separated into two parts: Q_rad from radiation out of the cavity and Q_mat from material loss (metal loss). With the calculated field components from the FDTD method, Q_rad can be evaluated from the definition [19

J. A. Kong, Electromagnetic Wave Theory (EWM, 2008).

,28

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1989).

\begin{array}{l} Q_{r a d} = 2 π \frac{energy
                                     
                                    stored
                                     
                                    in
                                     
                                    cavity}{radiated
                                     
                                    energy
                                     
                                    per
                                     
                                    optical
                                     
                                    period} = \frac{ω W}{P_{r a d}} \\ = \frac{ω \int_{V} d r \frac{ε_{0}}{4} [ε_{g} (r) + ε_{R} (r)] | E (r) |^{2}}{\int_{S} d s \frac{1}{2} Re [E (r) \times H^{*} (r)] \cdot \hat{n}}, \end{array}

(8)

where W is the total electromagnetic energy inside the cavity, P_rad is the radiated power outside of the cavity surface,

\hat{n}

is the unit normal to the surface, ε_R ( r ) and ε_g ( r ) are, respectively, the real part of the relative permittivity and the relative group permittivity, which is defined as [29

S. W. Chang and S. L. Chuang, “Fundamental formulation for plasmonic nanolasers,” IEEE J. Quantum Electron. 45(8), 1014–1023 (2009). [CrossRef]

ε_{g} (r) = \frac{\partial [ω ε_{R} (r)]}{\partial ω} .

(9)

The material quality factor Q_mat due to absorption can then be obtained from the relation [24

S. W. Lee, S. L. Chuang, and C. S. Lee, “Normal modes in an overmoded circular waveguide coated with lossy material,” IEEE Trans. Microw. Theory Tech. 34(7), 773–785 (1986). [CrossRef]

Q^{- 1} = Q_{m a t}^{- 1} + Q_{r a d}^{- 1} .

(10)

The results are shown in Fig. 10 . The radiation loss has a minimum which originates from minimum energy coupling to a resonant circular slot structure near the output facet [30

R. S. Elliott, Antenna Theory and Design (Prentice-Hall, 1981)

]. Due to the coupling at this radius, the radiation loss through the composite waveguide of InGaAsP and SiN _x is significantly reduced. According to the result of Q_mat , the material loss has a smooth change and reduces as the radius decreases. This reduction comes from the effective increase of shell layer portion inside the cavity, which buffers the field penetration into metal [31

A. Mizrahi, V. Lomakin, B. A. Slutsky, M. P. Nezhad, L. Feng, and Y. Fainman, “Low threshold gain metal coated laser nanoresonators,” Opt. Lett. 33(11), 1261–1263 (2008). [CrossRef] [PubMed]

]. One should note the sharp increase of material loss below radius 100 nm. The sharp transition region represents the transition from core (dielectric) mode of the waveguide to its shell mode as discussed before.

Fig. 10 (a) The radiation quality factor Q_rad of cavities as a function of the core radius a for various heights L = 220 to 300 nm. The peaks represent the coupling to a resonance structure. (b) The material quality factor Q_mat as a function of radius. The material quality factor has only a minimum change among different cavity heights and radii. A sharp drop at around R = 100 nm depicts the transition from core modes to shell modes.

In dealing with small lasers, especially with dispersive plasmonic materials like metals, the confinement factor associated with energy should be used to better account for the dispersive and plasmonic effects. The use of the energy confinement factor corrects the improper use of the negative power flow and negative energy resulting from negative permittivity of metals. The energy confinement factor is defined as [29

S. W. Chang and S. L. Chuang, “Fundamental formulation for plasmonic nanolasers,” IEEE J. Quantum Electron. 45(8), 1014–1023 (2009). [CrossRef]

Γ_{E} = \frac{\int_{V_{a}} d r \frac{ε_{0}}{4} [ε_{g} (r) + ε_{R} (r)] | E (r) |^{2}}{\int_{V} d r \frac{ε_{0}}{4} [ε_{g} (r) + ε_{R} (r)] | E (r) |^{2}},

(11)

where V_a represents the volume of the active material. In Fig. 11(a) , the energy confinement factor is shown for various cavity heights. The energy confinement factor associated with the percentage of active material over the whole cavity increases as both the core radius and height increase. As the radius shrinks, especially when n _eff < n _SiN _x , the energy starts to leak out and results in a reduction in the energy confinement. The overall confinement factors are more than 60% for devices with a radius larger than 100 nm. For a given height, the energy confinement factor approaches a constant and corresponds to the case that the SiN _x has a minimal effect. Compared to conventional dielectric lasers, which usually have lower values due to poor dielectric confinement, our proposed laser structure with the full coverage by metal, helps confine more energy inside the cavity. Figure 11(b) shows the extraction efficiency of the cavity, which is a measurement of how much loss contributes to the radiation or light output. It is defined as,

Fig. 11 The energy confinement factor (a) and extraction efficiency (b) of the cavity as a function of the core radius a with different cavity heights (220 nm – 300 nm). In the inner mode region, Γ_E is above 0.6.

η_{e x t} = \frac{Q_{r a d}^{- 1}}{Q^{- 1}} .

(12)

A special care has to be taken to get a decent light output. Observing from Fig. 11(b), the design with a maximum radiation Q_rad can lead to a poor extraction efficiency even though the condition of lasing can be achieved more easily like photonic crystal lasers [1

]. The physical sizes are summarized in Fig. 12(a) . Due to the nature of standing wave formation in HE₁₁₁ mode, the size should be around the minimum standing wave volume or the diffraction limit (λ ₀/2n)³, where n represents the effective index of the optical mode. With the knowledge of cavity Q, confinement factor Γ_E, and the corresponding resonance frequency ω, the threshold material gain g_th can be formulated as [29

S. W. Chang and S. L. Chuang, “Fundamental formulation for plasmonic nanolasers,” IEEE J. Quantum Electron. 45(8), 1014–1023 (2009). [CrossRef]

g_{t h} = \frac{2 π n_{g}}{Q Γ_{E} λ},

(13)

where n _g is the group index of the active region and λ is the wavelength in vacuum. The values of g_th shown in Fig. 12(b) are around 600 cm⁻¹ – 1,200 cm⁻¹ for cavity heights (240 - 300 nm) and a≥ 250 nm and should be achievable at room temperature within the emission spectrum of In_0.53Ga_0.47As materials.

Fig. 12 (a) The physical cavity volume and (b) the threshold material gain for different cavity heights L = 220 to 300 nm. The cavity volume approaches the diffraction limit ~(λ ₀/2n)³. The threshold can be as low as 600 cm⁻¹ for a≥ 250 nm and cavity heights L≥ 240 nm.

4. Design optimization of metal-nanocavity lasers

4.1. Effects of shell insulator layer

The shell insulator layer plays a very important role in the design of electrical injection lasers. It is not only an electrical buffer layer but also an effective optical buffer layer for reducing metal loss. Recently, a proper design of the insulator thickness was proven to be useful for reducing the laser threshold by optimizing the material loss from metal and its optical confinement factor [31

A. Mizrahi, V. Lomakin, B. A. Slutsky, M. P. Nezhad, L. Feng, and Y. Fainman, “Low threshold gain metal coated laser nanoresonators,” Opt. Lett. 33(11), 1261–1263 (2008). [CrossRef] [PubMed]

]. The theoretical results of major physical parameters for a cavity with a cavity height L = 240 nm and various SiN _x thicknesses from 20 nm – 80 nm are shown in Fig. 13 . Since the cavity boundary is defined by metal, it is more illustrative to plot the result as a function of outer radius b (i.e. the radius of the metal wall or the sum of the core radius a and the SiN _x shell thickness s). Since the boundary remains the same for each b, a smaller change in the resonance wavelength is expected in larger radius case, where the SiN _x layer occupies only a small portion of the cavity and the field inside SiN _x layer is reduced. In Fig. 13(a), the resonance wavelength decreases with increasing SiN _x layer thickness from 20 nm to 80 nm. This is caused by the increase of the average material index inside the metal cavity, or more physically, the decrease of effective wavelength. The cavity quality factors Q in Fig. 13(b) are similar in all cases except in larger radius region. The decrease of quality factor in large radius region as b increases comes mainly from their material loss, or field penetration into metal as can be seen from Figs. 13(c) and 13(d). For longer wavelengths, the effect of SiN _x layer as a buffer layer for reducing loss becomes smaller, as a result, cavities with the same outer radii will experience more loss if SiN _x becomes relatively thinner compared with the wavelengths. Besides, a thicker SiN _x can help reduce field into the metal or minimize the field leaking out of the cavity as radiation loss.

Fig. 13 The physical properties for different SiN _x shell thicknesses s = 20 to 80 nm. (a) The resonance wavelength. (b) The quality factor of the cavity Q. (c) The computed radiation quality factor Q_rad . (d) The material quality factor Q_mat . (e) The energy confinement factor Γ_E. (f) The threshold material gain g_th .

For the confinement factor Γ_E in Fig. 13(e), reducing the portion of SiN _x inside the cavity will result in the increase of Γ_E for small radius. For a large radius, the dependence of confinement factor on the SiN _x thickness becomes less obvious except for 20 nm and 30 nm cases in which the wavelengths are longer than the other cases and the radial electric field starts to dominate over the other components (k _ρ increases), and, thus experiences more field penetration into the metal. Because of the large permittivity difference between Ag and SiN _x , the field inside the SiN _x builds up as the thickness becomes thinner and this results in the reduction of Γ_E. Figure 13(f) shows the corresponding threshold material gain. The minimum threshold material gain for each case has only a small dependence on SiN _x thicknesses.

The effect of insulator layer thickness has only a small change on the resonance wavelength. The optimization should be focused on the confinement factor and material loss issues. In general, the confinement factor will increase with thinner insulator layers for b < 250 nm with a sacrifice of reducing the material quality factor.

4.2. Effects of the output metal reflector layer

The metals on the output (or top) facet provides both optical feedback and mode confinement of the cavity. Designs with a thin metal usually suffer from both radiation loss through the metal and ohmic loss dissipated inside the metal [24

S. W. Lee, S. L. Chuang, and C. S. Lee, “Normal modes in an overmoded circular waveguide coated with lossy material,” IEEE Trans. Microw. Theory Tech. 34(7), 773–785 (1986). [CrossRef]

,32

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B 61(15), 10484–10503 (2000). [CrossRef]

]. However, designs with a thick metal will regulate the output power. Unlike the other metal surrounding of the cavity, which usually has little constraint on the thickness, the metal reflector on the output facet plays an important role as a cavity wall and an output coupler. Figures 14(a) and 14(b) show the resonance wavelengths and their corresponding cavity quality factors of a 240-nm height cavity with different silver reflector thicknesses. The cavity resonance wavelength, which is determined by the boundary conditions, has little dependence on the metal thickness, especially when the thickness is close to or thicker than its skin depth (for example 15~30 nm for silver at this optical frequency). The quality factor in Fig. 14(b) shows a more obvious dependence on the metal thickness. The separation of Q into Q_rad and Q_mat is plotted in Figs. 14(c) and 14(d). The material quality factor shows a strong dependence on the metal thickness. As a result of a large permittivity difference between semiconductor and metal and/or air and metal, the optical field tends to accumulate near the interface. As the metal becomes thinner, fields from both interfaces start to couple and result in a strong field distribution across the metal which in turn contributes to the high material loss for thin metals [24

S. W. Lee, S. L. Chuang, and C. S. Lee, “Normal modes in an overmoded circular waveguide coated with lossy material,” IEEE Trans. Microw. Theory Tech. 34(7), 773–785 (1986). [CrossRef]

,32

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B 61(15), 10484–10503 (2000). [CrossRef]

]. In Fig. 14(c), the radiation Q_rad is shown for various metal thicknesses. Also, as shown in Fig. 14(e), the energy confinement factor has an almost unchanged behavior due to the quick decaying tail inside the metal, which usually contains a small portion of the total stored energy in the cavity.

Fig. 14 The cavity properties with different metal reflector thicknesses from 20 nm to 80 nm. (a) The resonance wavelength. (b) The quality factor of the cavity Q. (c) The computed radiation quality factor Q_rad . (d) The material quality factor Q_mat . (e) The energy confinement factor Γ_E. (f) The threshold material gain g_th .

In general, the radiation loss is not the main path of loss out of the cavity. However, in order to get a reasonable output power, the thickness should not be too large. A thicker metal will hinder the light penetration through, or equivalently worsen the extraction efficiency, and has only a minimal increase in the reflectivity. The threshold material gains for different metal thicknesses are shown in Fig. 14(f). When the thickness is over 40 nm, the required threshold material gain reduces to below 1,000 cm⁻¹ and is achievable for bulk materials at room temperature. According to the calculated result, a design with a metal thickness of around 40 nm to 60 nm should meet the requirement.

4.3 Far field radiation pattern

The radiation pattern depends strongly on the cavity mode. For a laser with dimensions comparable or even smaller than the wavelength, the emission pattern usually is divergent since the fields are strongly localized in the cavity with large wave vectors. In the waveguide case with an open end, the HE₁₁ mode preserves the most useful radiation pattern compared to others since it has the smallest k_ρ (~χ₁₁/b). A cavity mode design associated with HE₁₁ mode will also prevent a null at the center of the circular waveguide. The radiation pattern I(θ,ϕ) can be defined as [19

J. A. Kong, Electromagnetic Wave Theory (EWM, 2008).

,30

R. S. Elliott, Antenna Theory and Design (Prentice-Hall, 1981)

I (θ, φ) = \frac{P (r, θ, φ) \cdot \hat{r}}{Max[P (r, θ, φ) \cdot \hat{r}]},

(14)

where P(r,θ,ϕ) is the Poynting vector in spherical coordinates and Max[·] represents the maximum value over the hemisphere with a constant radius r in the far field. In Fig. 15 , calculated radiation fields of two dominant components (H_z, E_ρ) of the HE₁₁₁ mode with cavity parameter L = 240 nm, R = 190 nm, R _out = 240 nm, and metal thickness = 50 nm are shown. The radiation comes from top metal reflector facet with a high uniform spreading toward the hemispherical space. In the E_ρ plot, a clear guided wave along the radial direction can be observed in the structure composed of Ag/InGaAsP/SiN _x /Ag hybrid waveguides. These fields do not contribute to the emission into the hemispherical region and have a minimum effect on the radiation pattern.

Fig. 15 Computed far field radiation pattern of H_z and E_ρ components. Note the azimuthal dependence (e ^± ^iϕ ) has been suppressed.

The calculated radiation patterns as a function of θ are shown in Fig. 16 for cavities with different radii. Note the azimuthal dependence has been assumed to be in e ^{± iϕ} convention. The radiation pattern narrows as the radius becomes larger. The divergence comes from large radial wave component k_ρ at a small radius. A small tail near the interface (θ ~90°) comes from the surface plasma propagation along the Ag/air interface. Compared with nanolasers of other designs such as nanopatchs [8

K. Yu, A. Lakhani, and M. C. Wu, “Subwavelength metal-optic semiconductor nanopatch lasers,” Opt. Express 18(9), 8790–8799 (2010). [CrossRef] [PubMed]

,14

C. Manolatou and F. Rana, “Subwavelength nanopatch cavities for semiconductor plasmon lasers,” IEEE J. Quantum Electron. 44(5), 435–447 (2008). [CrossRef]

] in which the output comes from the interference of fields from the opening sides, our design of the HE₁₁₁ mode with a direct facet output has the advantage of less sensitive radiation pattern dependence on the geometry.

Fig. 16 Far-field radiation patterns of cavity with different radii, a, from 130 to 370 nm. The azimuthal dependence has been assumed to be e ^± ^iϕ . The radiation pattern narrows as the aperture size (radius) increases.

5. Threshold analysis and the light output power (L-I curve)

To investigate the possibility of achieving an injection laser and the performance of our proposed metal-cavity structure, we use the rate equations to perform the analysis [23

S. W. Chang, T. R. Lin, and S. L. Chuang, “Theory of plasmonic fabry-perot nanolasers,” Opt. Express 18(14), 15039–15053 (2010). [CrossRef] [PubMed]

,29

S. W. Chang and S. L. Chuang, “Fundamental formulation for plasmonic nanolasers,” IEEE J. Quantum Electron. 45(8), 1014–1023 (2009). [CrossRef]

\frac{\partial n}{\partial t} = η \frac{I}{q V_{a}} - R_{n r} (n) - R_{s p, c o n t} (n) - R_{s p} (n) - R_{s t} (n) S,

(15a)

\frac{\partial S}{\partial t} = - \frac{S}{τ_{p}} + Γ_{E} R_{s p} (n) + Γ_{E} R_{s t} (n) S,

(15b)

R_{n r} (n) = A n + C n^{3},

(15c)

R_{s p, c o n t} (n) = \frac{1}{τ_{r a d} V_{a}} \sum_{c, v, k} f_{c, k} (1 - f_{v, k}),

(15d)

R_{s t} = υ_{g} g (n) .

(15e)

Here n (cm⁻³) is the carrier density, S (cm⁻³) is the photon density, η is the injection efficiency, q is the charge of electron, V_a (cm³) is the volume of active region, R_nr (cm⁻³s⁻¹) is the non-radiative recombination rate, R_sp,cont (cm⁻³s⁻¹) and R_sp (cm⁻³s⁻¹) are the spontaneous emission rate into continuum modes and cavity mode, respectively, R_stS (cm⁻³s⁻¹) is the stimulated emission rate, and g(n) (cm⁻¹) is material gain. The non-radiative recombination rate can be modeled as a sum of surface recombination rate (An) plus Auger recombination rate (Cn ³). The coefficient A is given by ν_sA_s, where ν_s is the surface recombination velocity and is set to be 2.0 × 10² ms⁻¹ [33

E. Yablonovitch, R. Bhat, C. E. Zah, T. J. Gmitter, and M. A. Koza, “Nearly ideal InP/In_0.53Ga_0.47As heterojunction regrowth on chemically prepared In_0.53Ga_0.47As surfaces,” Appl. Phys. Lett. 60(3), 371–373 (1992). [CrossRef]

] and A_s is the surface area of the active region. The Auger coefficient is set to be 4.67 × 10²⁷ cm⁶s⁻¹ [34

Y. Zou, J. S. Osinski, P. Grodzinski, P. Dapkus, W. C. Rideout, W. F. Sharfin, J. Schlafer, and F. D. Crawford, “Experimental study of Auger recombination, gain, and temperature sensitivity of 1.5 μm compressively strained semiconductor lasers,” IEEE J. Quantum Electron. 29(6), 1565–1575 (1993). [CrossRef]

]. The spontaneous emission rate in to continuum modes R_sp,cont is calculated by the formula in Eq. (15d), where τ_rad is an effective radiation lifetime (set to be 5 μs) [23

S. W. Chang, T. R. Lin, and S. L. Chuang, “Theory of plasmonic fabry-perot nanolasers,” Opt. Express 18(14), 15039–15053 (2010). [CrossRef] [PubMed]

] and f_c, _k and f_v, _k represent the occupation numbers at wave vector k in conduction and valance bands.

Figure 17 shows the carrier-dependent material gain spectra of the bulk In_0.53Ga_0.47As semiconductor calculated by Luttinger-Kohn model at room temperature [35

S. L. Chuang, Physics of Photonic Devices, 2nd ed. (Wiley, 2009), Chap. 4.

]. Compared with the required threshold material gain for various structures discussed in the context (~700 cm⁻¹ - 1000 cm⁻¹), In_0.53Ga_0.47As is suitable for achieving room temperature operation at reasonable carrier densities. In Fig. 18(a) the light output power as a function of the injection current (L-I curve) of a device with a cavity parameter L = 240 nm, a = 190 nm, SiN _x shell thickness of 50 nm, b = 240 nm, and metal thickness = 50 nm is shown. According to the calculation, the output power is predicted to be 22 μW at 1.0 mA current injection with a threshold below 0.1 mA. In Fig. 18(b), the transition rates are shown at low current injection range. When the current passes 0.12 mA, the stimulated emission rate starts to take over and the injected power contributes mostly to the lasing mode. The inset shows a turning point of stimulate emission rate (red curve) from stimulated absorption to stimulate emission at ~30 μA. The turning point also represents the injection level where the transparent carrier density injection condition is reached (i.e., the net material gain is zero).

Fig. 17 Material gain spectra of bulk In_0.53Ga_0.47As at different injected carrier densities at room temperature.

Fig. 18 (a) The L-I curve at room temperature of a device with a = 190 nm, SiN _x shell thickness = 50 nm, b = 240 nm, metal reflector thickness = 50 nm, and cavity height = 240 nm. (b) The transition rates of the corresponding device. The stimulated emission rate starts to take over after 0.12 mA. (Inset) The turning point from stimulated absorption to stimulated emission (red).

To illustrate the effect of metal reflectors, a calculation with various metal reflector thicknesses is performed. As shown in Fig. 19 , with a thinner metal reflector (30 nm), the output can be as high as 100 μW at the expense of a higher threshold current due to its lower quality factor. Further increase of the metal thickness will block the output power but reduce the threshold. After 50 nm, the turning point depicting the stimulate emission rate equal to the non-radiative recombination rate, which remains pinned with increasing current injection. Therefore, further increase of metal thickness will not change the threshold but simply reduce the output power. For an optimized design of power and low threshold, a nominal metal thickness should be around 40 to 60 nm. In real situation, not only the thickness of the metal but also the contact resistance will affect the performance. Optimization by inserting a low bandgap material such as InGaAsP in between the p-InP and silver interface can efficiently reduce the contact resistance; and, therefore, minimize the heating effect.

Fig. 19 The light output power as a function of the injected current (L-I curves) of devices with different metal reflector thicknesses from 30 nm to 80 nm. A transition point when the stimulated emission rate equals the non-radiative recombination rate is plotted as the green symbols. Also, the transition point when the stimulated emission rate equals the spontaneous emission is plotted as the orange symbols.

6. Conclusions

We have proposed and presented a new class of three-dimensional metal-nanocavity (nano-coin) semiconductor lasers, which will potentially operate at room temperature with electrical injection and emit more than micro-Watt light output power. A model with a multilayered core-shell structure with a perfect metal conductor coating is used to predict the resonance wavelengths and mode patterns inside the cavity. A numerical simulation using the FDTD method taking into account the lossy metal plasma is also performed to confirm our model. Our design is based on the fundamental waveguide mode (HE₁₁) with both ends bounded by metals. The conditions for making a laser cavity have been analyzed, including waveguide cutoff wavelength, optical field, optical energy confinement factor, and output power optimizations. The effects of insulator layers and metal reflectors are discussed and optimized. We show how the insulator thickness affects the material loss and the corresponding shift in the resonance wavelength. In addition, an optimized thickness of the top metal reflector has been proposed, which optimizes the output power and threshold current. The theoretical treatments are universal and suitable for all metal-cavity nanolasers. The laser device performance and operation principles are presented through the analysis of rate equations. Our proposed surface-emitting 3D metal-nanocavity semiconducator laser and theoretical model will be useful for future size shrinkage of electrical-driven plasmonic nanolasers.

Appendices

Appendix: theory of the HE₁₁ _p mode in a circular core-shell-metal waveguide

The field expressions of HE₁₁ mode in a PEC-surrounded waveguide shown in Fig. 5 are given by [19

J. A. Kong, Electromagnetic Wave Theory (EWM, 2008).

,24

S. W. Lee, S. L. Chuang, and C. S. Lee, “Normal modes in an overmoded circular waveguide coated with lossy material,” IEEE Trans. Microw. Theory Tech. 34(7), 773–785 (1986). [CrossRef]

]

Core
                      
                     Region: E_{z}^{c} = A F_{1} (ρ) \sin (m φ) e^{i k_{z} z},

(16a)

H_{z}^{c} = B F_{2} (ρ) \cos (m φ) e^{i k_{z} z},

(16b)

E_{ρ}^{c} = \frac{1}{k_{ρ c}^{2}} [A i k_{z} F'_{1} (ρ) - B \frac{i ω μ m}{ρ} F_{2} (ρ)] \sin (m φ) e^{i k_{z} z},

(16c)

E_{φ}^{c} = \frac{1}{k_{ρ c}^{2}} [\frac{A i k_{z} m}{ρ} F_{1} (ρ) - B i ω μ k_{ρ c} F'_{2} (ρ)] \cos (m φ) e^{i k_{z} z},

(16d)

H_{ρ}^{c} = \frac{1}{k_{ρ c}^{2}} [- A \frac{i ω ε_{core} m}{ρ} F_{1} (ρ) + B i k_{z} k_{ρ c} F'_{2} (ρ)] \cos (m φ) e^{i k_{z} z},

(16e)

H_{φ}^{c} = \frac{1}{k_{ρ c}^{2}} [A i ω ε_{c o r e} k_{ρ c} F'_{1} (ρ) - \frac{B i k_{z} m}{ρ} F_{2} (ρ)] \sin (m φ) e^{i k_{z} z},

(16f)

Shell
                      
                     Region: E_{z}^{s} = C F_{3} (ρ) \sin (m φ) e^{i k_{z} z},

(17a)

H_{z}^{s} = D F_{4} (ρ) \cos (m φ) e^{i k_{z} z},

(17b)

E_{ρ}^{s} = \frac{1}{k_{ρ s}^{2}} [C i k_{z} F'_{3} (ρ) - D \frac{i ω μ m}{ρ} F_{4} (ρ)] \sin (m φ) e^{i k_{z} z},

(17c)

E_{φ}^{s} = \frac{1}{k_{ρ s}^{2}} [\frac{C i k_{z} m}{ρ} F_{3} (ρ) - D i ω μ k_{ρ s} F'_{4} (ρ)] \cos (m φ) e^{i k_{z} z},

(17d)

H_{ρ}^{s} = \frac{1}{k_{ρ s}^{2}} [- C \frac{i ω ε_{s} m}{ρ} F_{3} (ρ) + D i k_{z} k_{ρ s} F'_{4} (ρ)] \cos (m φ) e^{i k_{z} z},

(17e)

H_{φ}^{s} = \frac{1}{k_{ρ s}^{2}} [C i ω ε_{s} k_{ρ s} F'_{3} (ρ) - \frac{D i k_{z} m}{ρ} F_{4} (ρ)] \sin (m φ) e^{i k_{z} z},

(17f)

where

{\begin{cases} k_{ρ s}^{2} + k_{z}^{2} = ω^{2} μ ε_{s}, \\ k_{ρ c}^{2} + k_{z}^{2} = ω^{2} μ ε_{c o r e} . \end{cases}

(18)

In the equations above, s represents shell region and c is the core region, k_ρ is the corresponding radial wavevector, k_z is the propagation constant, and ω is the angular frequency. We have to choose proper F_i (ρ) to satisfy proper conditions. First, inside the core (ρ < a), the wave solution should be a regular function (no singularity), we find

F_{1} (ρ) = J_{m} (k_{ρ c} ρ),

(19a)

F_{2} (ρ) = J_{m} (k_{ρ c} ρ) .

(19b)

Second, the wave solutions in the shell layer consists of two outgoing and incoming cylindrical decaying waves, which satisfy the boundary conditions that E_z and E_ϕ vanish at the outer wall (ρ = b), i.e.,

F_{3} (ρ) = K_{m} (k_{ρ s} ρ) I_{m} (k_{ρ s} b) - I_{m} (k_{ρ s} ρ) K_{m} (k_{ρ s} b),

(20a)

F_{4} (ρ) = K_{m} (k_{ρ s} ρ) I'_{m} (k_{ρ s} b) - I_{m} (k_{ρ s} ρ) K'_{m} (k_{ρ s} b),

(20b)

where J_m is the Bessel function of the first kind and I_m and K_m are the modified Bessel function of the first and second kinds, respectively. Matching the boundary conditions of tangential field components at the core/shell interface, ρ = a, leads to,

A F_{1} (a) = C F_{3} (a),

(21a)

B F_{2} (a) = D F_{4} (a),

(21b)

\frac{1}{k_{ρ c}^{2}} [- \frac{B k_{z} m}{a} F_{2} (a) + A ω ε_{c o r e} k_{ρ c} F'_{1} (a)] = \frac{1}{k_{ρ s}^{2}} [- \frac{D k_{z} m}{a} F_{4} (a) + C ω ε_{s} k_{ρ s} F'_{3} (a)],

(21c)

\frac{1}{k_{ρ c}^{2}} [\frac{A k_{z} m}{a} F_{1} (a) - B ω μ k_{ρ c} F'_{2} (a)] = \frac{1}{k_{ρ s}^{2}} [\frac{C k_{z} m}{a} F_{3} (a) - D ω μ k_{ρ s} F'_{4} (a)] .

(21d)

The non-trivial solution for coefficients A, B, C, D exists when the determinant is zero, i.e.,

| \begin{matrix} F_{1} (a) & 0 & - F_{3} (a) & 0 \\ 0 & F_{2} (a) & 0 & - F_{4} (a) \\ \frac{ω ε_{c o r e} F'_{1} (a)}{k_{ρ c}} & - \frac{k_{z} m}{a k_{ρ c}^{2}} F_{2} (a) & - \frac{ω ε_{s} F'_{3} (a)}{k_{ρ s}} & \frac{k_{z} m}{a k_{ρ s}^{2}} F_{4} (a) \\ \frac{k_{z} m}{a k_{ρ c}^{2}} F_{1} (a) & - \frac{ω μ F'_{2} (a)}{k_{ρ c}} & - \frac{k_{z} m}{a k_{ρ s}^{2}} F_{3} (a) & \frac{ω μ F'_{4} (a)}{k_{ρ s}} \end{matrix} | = 0.

(22)

The resonant wavelength of the cavity associated with certain modes (mnp) can be obtained by solving equations Eqs. (18), (21), and (22) simultaneously. For modes with no azimuthal variation (m= 0), the characteristic equation Eq. (22) can be further simplified to

(\frac{ε_{s}}{k_{ρ s}} F_{1} F'_{3} - \frac{ε_{c o r e}}{k_{ρ c}} F'_{1} F_{3}) = 0 for {TM}_{0 n},

(23a)

and

(\frac{1}{k_{ρ s}} F_{2} F'_{4} - \frac{1}{k_{ρ c}} F'_{2} F_{4}) = 0 for {TE}_{0 n} .

(23b)

Acknowledgments

This work was sponsored by the DARPA NACHOS Program under Grant No. W911NF-07-1-0314. We thank many insightful discussions with Dr. Shu-Wei Chang formerly at UIUC and Professor Dieter Bimberg at Technical University of Berlin.

References and links

1.	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(5689), 1444–1447 (2004). [CrossRef] [PubMed]
2.	K. Nozaki, S. Kita, and T. Baba, “Room temperature continuous wave operation and controlled spontaneous emission in ultrasmall photonic crystal nanolaser,” Opt. Express 15(12), 7506–7514 (2007). [CrossRef] [PubMed]
3.	R. M. Ma, R. F. Oulton, V. J. Sorger, G. Bartal, and X. Zhang, “Room-temperature sub-diffraction-limited plasmon laser by total internal reflection,” Nat. Mater. 10(2), 110–113 (2011). [CrossRef] [PubMed]
4.	C. Y. Lu, S. W. Chang, S. L. Chuang, T. D. Germann, and D. Bimberg, “Metal-cavity surface emitting microlaser at room temperature,” Appl. Phys. Lett. 96(25), 251101 (2010). [CrossRef]
5.	C. Y. Lu, S. W. Chang, S. L. Chuang, T. D. Germann, U. W. Pohl, and D. Bimberg, “CW substrate-free metal-cavity surface microemitters at 300 K,” Semicond. Sci. Technol. 26(1), 014012 (2011). [CrossRef]
6.	S. W. Chang, C. Y. Lu, S. L. Chuang, T. D. Germann, U. W. Pohl, and D. Bimberg, “Theory of metal-cavity surface-emitting microlasers and comparison with experiment,” IEEE J. Sel. Top. Quantum Electron. (in press).
7.	C. Y. Lu, S. L. Chuang, A. Mutig, and D. Bimberg, “Metal-cavity surface-emitting microlasers with hybrid metal-DBR reflectors,” Opt. Lett. (in press). [PubMed]
8.	K. Yu, A. Lakhani, and M. C. Wu, “Subwavelength metal-optic semiconductor nanopatch lasers,” Opt. Express 18(9), 8790–8799 (2010). [CrossRef] [PubMed]
9.	S. H. Kwon, J. H. Kang, C. Seassal, S. K. Kim, P. Regreny, Y. H. Lee, C. M. Lieber, and H. G. Park, “Subwavelength plasmonic lasing from a semiconductor nanodisk with silver nanopan cavity,” Nano Lett. 10(9), 3679–3683 (2010). [CrossRef] [PubMed]
10.	J. Huang, S. H. Kim, and A. Scherer, “Design of a surface-emitting, subwavelength metal-clad disk laser in the visible spectrum,” Opt. Express 18(19), 19581–19591 (2010). [CrossRef] [PubMed]
11.	M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4(6), 395–399 (2010). [CrossRef]
12.	M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1(10), 589–594 (2007). [CrossRef]
13.	M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y. S. Oei, R. Nötzel, C. Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17(13), 11107–11112 (2009). [CrossRef] [PubMed]
14.	C. Manolatou and F. Rana, “Subwavelength nanopatch cavities for semiconductor plasmon lasers,” IEEE J. Quantum Electron. 44(5), 435–447 (2008). [CrossRef]
15.	J. L. Jewell, N. A. Olsson, A. Scherer, S. L. McCall, J. P. Harbison, L. T. Florez, and Y. H. Lee, “Surface-emitting microlasers for photonic switching and interchip connections,” Opt. Eng. 29(3), 210–214 (1990). [CrossRef]
16.	W. H. Hofmann, P. Moser, P. Wolf, A. Mutig, M. Kroh, and D. Bimberg, “44 Gb/s VCSEL for optical interconnects,” Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPC5.
17.	R. Safaisini, J. R. Joseph, and K. L. Lear, “Scalable high-CW-power high-speed 980-nm VCSEL arrays,” IEEE J. Quantum Electron. 46(11), 1590–1596 (2010). [CrossRef]
18.	M. Motoyoshi, “Through silicon via (TSV),” Proc. IEEE 97(1), 43–48 (2009). [CrossRef]
19.	J. A. Kong, Electromagnetic Wave Theory (EWM, 2008).
20.	L. Novotny and C. Hafner, “Light propagation in a cylindrical waveguide with a complex, metallic, dielectric function,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 50(5), 4094–4106 (1994). [CrossRef] [PubMed]
21.	P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]
22.	S. Adachi, “Refractive indices of III-V compounds: key properties of InGaAsP relevant to device design,” J. Appl. Phys. 53(8), 5863–5869 (1982). [CrossRef]
23.	S. W. Chang, T. R. Lin, and S. L. Chuang, “Theory of plasmonic fabry-perot nanolasers,” Opt. Express 18(14), 15039–15053 (2010). [CrossRef] [PubMed]
24.	S. W. Lee, S. L. Chuang, and C. S. Lee, “Normal modes in an overmoded circular waveguide coated with lossy material,” IEEE Trans. Microw. Theory Tech. 34(7), 773–785 (1986). [CrossRef]
25.	S. L. Chuang, “A coupled-mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. 5(1), 5–15 (1987). [CrossRef]
26.	S. L. Chuang, Physics of Optoelectronic Devices, 1st ed. (Wiley, 1995), Appendix H.
27.	A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech, 2000).
28.	J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1989).
29.	S. W. Chang and S. L. Chuang, “Fundamental formulation for plasmonic nanolasers,” IEEE J. Quantum Electron. 45(8), 1014–1023 (2009). [CrossRef]
30.	R. S. Elliott, Antenna Theory and Design (Prentice-Hall, 1981)
31.	A. Mizrahi, V. Lomakin, B. A. Slutsky, M. P. Nezhad, L. Feng, and Y. Fainman, “Low threshold gain metal coated laser nanoresonators,” Opt. Lett. 33(11), 1261–1263 (2008). [CrossRef] [PubMed]
32.	P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B 61(15), 10484–10503 (2000). [CrossRef]
33.	E. Yablonovitch, R. Bhat, C. E. Zah, T. J. Gmitter, and M. A. Koza, “Nearly ideal InP/In_0.53Ga_0.47As heterojunction regrowth on chemically prepared In_0.53Ga_0.47As surfaces,” Appl. Phys. Lett. 60(3), 371–373 (1992). [CrossRef]
34.	Y. Zou, J. S. Osinski, P. Grodzinski, P. Dapkus, W. C. Rideout, W. F. Sharfin, J. Schlafer, and F. D. Crawford, “Experimental study of Auger recombination, gain, and temperature sensitivity of 1.5 μm compressively strained semiconductor lasers,” IEEE J. Quantum Electron. 29(6), 1565–1575 (1993). [CrossRef]
35.	S. L. Chuang, Physics of Photonic Devices, 2nd ed. (Wiley, 2009), Chap. 4.

OCIS Codes
(140.3410) Lasers and laser optics : Laser resonators
(230.7370) Optical devices : Waveguides
(260.3910) Physical optics : Metal optics
(140.3945) Lasers and laser optics : Microcavities
(250.5960) Optoelectronics : Semiconductor lasers

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: April 18, 2011
Revised Manuscript: June 9, 2011
Manuscript Accepted: June 11, 2011
Published: June 23, 2011

Citation
Chien-Yao Lu and Shun Lien Chuang, "A surface-emitting 3D metal-nanocavity laser: proposal and theory," Opt. Express 19, 13225-13244 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-14-13225

Sort: Author | Year | Journal | Reset

References

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(5689), 1444–1447 (2004). [CrossRef] [PubMed]
K. Nozaki, S. Kita, and T. Baba, “Room temperature continuous wave operation and controlled spontaneous emission in ultrasmall photonic crystal nanolaser,” Opt. Express 15(12), 7506–7514 (2007). [CrossRef] [PubMed]
R. M. Ma, R. F. Oulton, V. J. Sorger, G. Bartal, and X. Zhang, “Room-temperature sub-diffraction-limited plasmon laser by total internal reflection,” Nat. Mater. 10(2), 110–113 (2011). [CrossRef] [PubMed]
C. Y. Lu, S. W. Chang, S. L. Chuang, T. D. Germann, and D. Bimberg, “Metal-cavity surface emitting microlaser at room temperature,” Appl. Phys. Lett. 96(25), 251101 (2010). [CrossRef]
C. Y. Lu, S. W. Chang, S. L. Chuang, T. D. Germann, U. W. Pohl, and D. Bimberg, “CW substrate-free metal-cavity surface microemitters at 300 K,” Semicond. Sci. Technol. 26(1), 014012 (2011). [CrossRef]
S. W. Chang, C. Y. Lu, S. L. Chuang, T. D. Germann, U. W. Pohl, and D. Bimberg, “Theory of metal-cavity surface-emitting microlasers and comparison with experiment,” IEEE J. Sel. Top. Quantum Electron. (in press).
C. Y. Lu, S. L. Chuang, A. Mutig, and D. Bimberg, “Metal-cavity surface-emitting microlasers with hybrid metal-DBR reflectors,” Opt. Lett. (in press). [PubMed]
K. Yu, A. Lakhani, and M. C. Wu, “Subwavelength metal-optic semiconductor nanopatch lasers,” Opt. Express 18(9), 8790–8799 (2010). [CrossRef] [PubMed]
S. H. Kwon, J. H. Kang, C. Seassal, S. K. Kim, P. Regreny, Y. H. Lee, C. M. Lieber, and H. G. Park, “Subwavelength plasmonic lasing from a semiconductor nanodisk with silver nanopan cavity,” Nano Lett. 10(9), 3679–3683 (2010). [CrossRef] [PubMed]
J. Huang, S. H. Kim, and A. Scherer, “Design of a surface-emitting, subwavelength metal-clad disk laser in the visible spectrum,” Opt. Express 18(19), 19581–19591 (2010). [CrossRef] [PubMed]
M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4(6), 395–399 (2010). [CrossRef]
M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1(10), 589–594 (2007). [CrossRef]
M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y. S. Oei, R. Nötzel, C. Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17(13), 11107–11112 (2009). [CrossRef] [PubMed]
C. Manolatou and F. Rana, “Subwavelength nanopatch cavities for semiconductor plasmon lasers,” IEEE J. Quantum Electron. 44(5), 435–447 (2008). [CrossRef]
J. L. Jewell, N. A. Olsson, A. Scherer, S. L. McCall, J. P. Harbison, L. T. Florez, and Y. H. Lee, “Surface-emitting microlasers for photonic switching and interchip connections,” Opt. Eng. 29(3), 210–214 (1990). [CrossRef]
W. H. Hofmann, P. Moser, P. Wolf, A. Mutig, M. Kroh, and D. Bimberg, “44 Gb/s VCSEL for optical interconnects,” Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPC5.
R. Safaisini, J. R. Joseph, and K. L. Lear, “Scalable high-CW-power high-speed 980-nm VCSEL arrays,” IEEE J. Quantum Electron. 46(11), 1590–1596 (2010). [CrossRef]
M. Motoyoshi, “Through silicon via (TSV),” Proc. IEEE 97(1), 43–48 (2009). [CrossRef]
J. A. Kong, Electromagnetic Wave Theory (EWM, 2008).
L. Novotny and C. Hafner, “Light propagation in a cylindrical waveguide with a complex, metallic, dielectric function,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 50(5), 4094–4106 (1994). [CrossRef] [PubMed]
P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]
S. Adachi, “Refractive indices of III-V compounds: key properties of InGaAsP relevant to device design,” J. Appl. Phys. 53(8), 5863–5869 (1982). [CrossRef]
S. W. Chang, T. R. Lin, and S. L. Chuang, “Theory of plasmonic fabry-perot nanolasers,” Opt. Express 18(14), 15039–15053 (2010). [CrossRef] [PubMed]
S. W. Lee, S. L. Chuang, and C. S. Lee, “Normal modes in an overmoded circular waveguide coated with lossy material,” IEEE Trans. Microw. Theory Tech. 34(7), 773–785 (1986). [CrossRef]
S. L. Chuang, “A coupled-mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. 5(1), 5–15 (1987). [CrossRef]
S. L. Chuang, Physics of Optoelectronic Devices, 1st ed. (Wiley, 1995), Appendix H.
A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech, 2000).
J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1989).
S. W. Chang and S. L. Chuang, “Fundamental formulation for plasmonic nanolasers,” IEEE J. Quantum Electron. 45(8), 1014–1023 (2009). [CrossRef]
R. S. Elliott, Antenna Theory and Design (Prentice-Hall, 1981)
A. Mizrahi, V. Lomakin, B. A. Slutsky, M. P. Nezhad, L. Feng, and Y. Fainman, “Low threshold gain metal coated laser nanoresonators,” Opt. Lett. 33(11), 1261–1263 (2008). [CrossRef] [PubMed]
P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B 61(15), 10484–10503 (2000). [CrossRef]
E. Yablonovitch, R. Bhat, C. E. Zah, T. J. Gmitter, and M. A. Koza, “Nearly ideal InP/In0.53Ga0.47As heterojunction regrowth on chemically prepared In0.53Ga0.47As surfaces,” Appl. Phys. Lett. 60(3), 371–373 (1992). [CrossRef]
Y. Zou, J. S. Osinski, P. Grodzinski, P. Dapkus, W. C. Rideout, W. F. Sharfin, J. Schlafer, and F. D. Crawford, “Experimental study of Auger recombination, gain, and temperature sensitivity of 1.5 μm compressively strained semiconductor lasers,” IEEE J. Quantum Electron. 29(6), 1565–1575 (1993). [CrossRef]
S. L. Chuang, Physics of Photonic Devices, 2nd ed. (Wiley, 2009), Chap. 4.

Appl. Phys. Lett.

C. Y. Lu, S. W. Chang, S. L. Chuang, T. D. Germann, and D. Bimberg, “Metal-cavity surface emitting microlaser at room temperature,” Appl. Phys. Lett. 96(25), 251101 (2010). [CrossRef]
E. Yablonovitch, R. Bhat, C. E. Zah, T. J. Gmitter, and M. A. Koza, “Nearly ideal InP/In0.53Ga0.47As heterojunction regrowth on chemically prepared In0.53Ga0.47As surfaces,” Appl. Phys. Lett. 60(3), 371–373 (1992). [CrossRef]

IEEE J. Quantum Electron.

Y. Zou, J. S. Osinski, P. Grodzinski, P. Dapkus, W. C. Rideout, W. F. Sharfin, J. Schlafer, and F. D. Crawford, “Experimental study of Auger recombination, gain, and temperature sensitivity of 1.5 μm compressively strained semiconductor lasers,” IEEE J. Quantum Electron. 29(6), 1565–1575 (1993). [CrossRef]
S. W. Chang and S. L. Chuang, “Fundamental formulation for plasmonic nanolasers,” IEEE J. Quantum Electron. 45(8), 1014–1023 (2009). [CrossRef]
C. Manolatou and F. Rana, “Subwavelength nanopatch cavities for semiconductor plasmon lasers,” IEEE J. Quantum Electron. 44(5), 435–447 (2008). [CrossRef]
R. Safaisini, J. R. Joseph, and K. L. Lear, “Scalable high-CW-power high-speed 980-nm VCSEL arrays,” IEEE J. Quantum Electron. 46(11), 1590–1596 (2010). [CrossRef]

IEEE J. Sel. Top. Quantum Electron.

S. W. Chang, C. Y. Lu, S. L. Chuang, T. D. Germann, U. W. Pohl, and D. Bimberg, “Theory of metal-cavity surface-emitting microlasers and comparison with experiment,” IEEE J. Sel. Top. Quantum Electron. (in press).

IEEE Trans. Microw. Theory Tech.

S. W. Lee, S. L. Chuang, and C. S. Lee, “Normal modes in an overmoded circular waveguide coated with lossy material,” IEEE Trans. Microw. Theory Tech. 34(7), 773–785 (1986). [CrossRef]

J. Appl. Phys.

S. Adachi, “Refractive indices of III-V compounds: key properties of InGaAsP relevant to device design,” J. Appl. Phys. 53(8), 5863–5869 (1982). [CrossRef]

J. Lightwave Technol.

S. L. Chuang, “A coupled-mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. 5(1), 5–15 (1987). [CrossRef]

Nano Lett.

S. H. Kwon, J. H. Kang, C. Seassal, S. K. Kim, P. Regreny, Y. H. Lee, C. M. Lieber, and H. G. Park, “Subwavelength plasmonic lasing from a semiconductor nanodisk with silver nanopan cavity,” Nano Lett. 10(9), 3679–3683 (2010). [CrossRef] [PubMed]

Nat. Mater.

R. M. Ma, R. F. Oulton, V. J. Sorger, G. Bartal, and X. Zhang, “Room-temperature sub-diffraction-limited plasmon laser by total internal reflection,” Nat. Mater. 10(2), 110–113 (2011). [CrossRef] [PubMed]

Nat. Photonics

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4(6), 395–399 (2010). [CrossRef]
M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1(10), 589–594 (2007). [CrossRef]

Opt. Eng.

J. L. Jewell, N. A. Olsson, A. Scherer, S. L. McCall, J. P. Harbison, L. T. Florez, and Y. H. Lee, “Surface-emitting microlasers for photonic switching and interchip connections,” Opt. Eng. 29(3), 210–214 (1990). [CrossRef]

Opt. Express

Opt. Lett.

A. Mizrahi, V. Lomakin, B. A. Slutsky, M. P. Nezhad, L. Feng, and Y. Fainman, “Low threshold gain metal coated laser nanoresonators,” Opt. Lett. 33(11), 1261–1263 (2008). [CrossRef] [PubMed]
C. Y. Lu, S. L. Chuang, A. Mutig, and D. Bimberg, “Metal-cavity surface-emitting microlasers with hybrid metal-DBR reflectors,” Opt. Lett. (in press). [PubMed]

Phys. Rev. B

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B 61(15), 10484–10503 (2000). [CrossRef]
P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics

L. Novotny and C. Hafner, “Light propagation in a cylindrical waveguide with a complex, metallic, dielectric function,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 50(5), 4094–4106 (1994). [CrossRef] [PubMed]

Proc. IEEE

M. Motoyoshi, “Through silicon via (TSV),” Proc. IEEE 97(1), 43–48 (2009). [CrossRef]

Science

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(5689), 1444–1447 (2004). [CrossRef] [PubMed]

Semicond. Sci. Technol.

C. Y. Lu, S. W. Chang, S. L. Chuang, T. D. Germann, U. W. Pohl, and D. Bimberg, “CW substrate-free metal-cavity surface microemitters at 300 K,” Semicond. Sci. Technol. 26(1), 014012 (2011). [CrossRef]

Other

J. A. Kong, Electromagnetic Wave Theory (EWM, 2008).
W. H. Hofmann, P. Moser, P. Wolf, A. Mutig, M. Kroh, and D. Bimberg, “44 Gb/s VCSEL for optical interconnects,” Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPC5.
S. L. Chuang, Physics of Optoelectronic Devices, 1st ed. (Wiley, 1995), Appendix H.
A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech, 2000).
J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1989).
R. S. Elliott, Antenna Theory and Design (Prentice-Hall, 1981)
S. L. Chuang, Physics of Photonic Devices, 2nd ed. (Wiley, 2009), Chap. 4.

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Click here to see a list of articles that cite this paper

Figures


Fig. 1	Fig. 2	Fig. 3


Fig. 4	Fig. 5	Fig. 6


Fig. 7	Fig. 8	Fig. 9


Fig. 10	Fig. 11	Fig. 12


Fig. 13	Fig. 14	Fig. 15


Fig. 16	Fig. 17	Fig. 18


Fig. 19