1. Introduction

Nanolasers are promising coherent light sources for densely integrated photonic integrated circuits due to their extremely low power consumption [1,2], fast modulation [3] and ultra-compact size [4–8]. In particular, surface-plasmon-polariton (SPP) lasers open new fields of research as they can confine light below the diffraction limit [6–9], whereas nanolasers based on dielectric cavities, such as photonic crystals [1,10], nanowires [11,12] and metal-cladding cavities [2,4,5,13], have sizes physically limited by the wavelength dimension. However, losses from metallic absorption and cavity radiation generally prevent SPP lasing at room temperature; their reduction would allow new practical applications. Several SPP lasers have been reported in metallic cavities. For example, CdS nanowires or nanosquares on 5 nm-thick magnesium fluoride/silver substrates showed SPP lasing at low or room temperatures [6,7], however, subwavelength cavity size reduction was achieved only in one or two dimensions. Nanodisk/nanopan structure has been reported to allow fully three-dimensional (3D) sub-diffraction-limited mode confinement and to demonstrate SPP lasing at a cryogenic temperature [8]. However, novel SPP cavity designs are still required to realize practical, ultra-small coherent light sources that can operate at room temperature. Strong cavity feedback and low metallic loss need to be achieved while the cavity mode is confined in three dimensions at sub-diffraction-limited sizes. Room-temperature SPP laser can be operated only if effective cavity feedback and low metallic absorption loss are achieved [7]. Therefore, it is important to minimize both optical loss and metallic absorption loss. Our SPP cavity has high optical Q, >1.0 × 10⁹, and high metallic Q, >300, at room temperature and hence we expect that these optical properties can enable SPP lasing operation in combination with high gain materials such as InAsP or InGaAsP quantum wells [1,8].

This letter proposes a subwavelength-scale SPP cavity consisting of channel waveguides of different widths. A low-index dielectric layer is introduced to the cavity to minimize metallic absorption loss. SPP modes are strongly confined in the cavity by a mode gap originating from the difference of waveguide width. Numerical simulations show a 3D ultra-small mode volume of λ³/1000, which is two orders of magnitude smaller than the mode volumes of conventional dielectric cavities [1] and comparable to the mode volumes of the smallest plasmonic cavities [14–16]. On the other hand, our SPP cavity shows ~3 times higher Q at room temperature and ~10⁴ times higher Q at 4 K, compared to other SPP cavities.

2. Plasmonic cavity modes confined by mode-gap mirrors

The plasmonic cavity is formed by combining two silver-air channel waveguides of different widths (Fig. 1 ). Wide (width d_c) and narrow (width d_m) air slots are constructed. The sides and bottom walls of the silver are coated with a low-index dielectric layer (thickness t_low, cyan in Fig. 1) and a high-index dielectric slab (thickness t, red) is added. The high-index dielectric slab, consisting of such as InGaAsP, can act as an active material [17]. This cavity structure is optically accessible because one side of the cavity is open to air. Hence, it can efficiently emit photoluminescence by optical pumping from the top, and SPPs can be excited at the bottom silver surface, which is strongly confined within the wide waveguide (width, d_c; and cavity length, L_c). The narrow channel waveguides (width d_m) at both sides of the cavity prevent the propagation of SPP modes in the y-direction.

 

Fig. 1 The proposed SPP cavity structure. SPP modes are confined in the cavity region (width, d_c; length, L_c). A high-index slab (red) and low-index layer (cyan) are also introduced in the structure.

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To understand quantitatively how SPPs are confined in this structure, a 3D finite-difference time-domain (FDTD) simulation was used to calculate dispersion curves of the fundamental SPP mode excited in an infinitely long silver-air channel waveguide of width d (Fig. 2 ). Two values of d (100 nm and 250 nm) were used and the refractive index of the high-index slab was set to 3.4. The Drude model described silver: ε(ω) = ε_∞ - ω_p ² / (ω² + iγω). Here, the background dielectric constant, ε_∞; the plasma frequency, ω_p; and the collision frequency, γ, at room temperature were set to 3.1, 1.4 × 10¹⁶ s⁻¹, and 3.1 × 10¹³ s⁻¹, respectively, by fitting the experimentally determined dielectric function of silver [18]. First, t_low was set to 0 to investigate the confinement mechanism of the SPP waveguide modes.

 

Fig. 2 Dispersion curves of the fundamental SPP modes for waveguides of different widths, 250 nm (black) and 100 nm (red). The right inset illustrates a channel waveguide consisting of a high-index dielectric slab and an air slot with a rectangular cross-section. The electric field intensity profile of the SPP waveguide mode is shown in the left inset.

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The simulation shows that the SPP waveguide mode is strongly confined at the bottom silver/high-index slab interface (left inset, Fig. 2) and that the dispersion curve significantly depends on the waveguide width, d. The cutoff frequency at zero wavevector, which appears due to the finite waveguide width, decreases with increasing d [14,19]. The cutoff frequency when d = 100 nm is 1606 THz, larger than that when d = 250 nm, 965 THz. Since SPP modes with frequencies higher than the cutoff are allowed to be transmitted through this waveguide, transmission can be controlled through varying the waveguide width. For example, the SPP mode excited in the waveguide with d = 250 nm cannot propagate into the waveguide with d = 100 nm if the frequency of the mode is between 965 and 1606 THz: the narrower waveguide can be considered to act as a mode-gap mirror. Such a mode-gap mirror formed by differences of waveguide width has both facile fabrication and transmission control of SPP waveguide modes, when compared with mirrors formed by modulating the refractive index of the waveguide [14].

This mode-gap mirror can be used to form a channel-waveguide-type SPP cavity structure comprising a wider waveguide between two narrower waveguides. 3D FDTD simulations were performed to examine how SPP modes are confined in such a cavity (Fig. 3 ). The following structural parameters were used in the simulation: d_c = 250 nm, L_c = 250 nm, d_m = 100 nm and L_m = 500 nm. The high-index slab of thickness t = 200 nm is located inside the air slot at a depth of h = 500 nm. L_c and h were set large enough to achieve strong confinement of the SPP modes. The Q factor of the cavity is saturated if L_c > 250 nm and h > 200 nm. Figures 3(a) and (b) show the top and side views of the electric field intensity profile of the SPP mode strongly confined at the cavity’s bottom dielectric-silver interface. Along the y-axis of the waveguide, the SPP cavity mode with a 1550 nm resonant wavelength (1216 THz) is confined within the subwavelength-scale d_c × L_c waveguide region (250 nm × 250 nm) by the mode gap (965 - 1606 THz). The mode is confined along the x-axis by metallic mirrors. The high-index slab, which can act as an active material and excite SPP cavity modes, is isolated by the air slot of height h = 500 nm between the two metal walls separated by d_c = 250 nm. Since this air slot allows the propagation of light with a wavelength shorter than 2d (i.e. 2 × 250 nm = 500 nm), radiation of the SPP cavity mode into the free space is prevented and it is confined vertically. As a result, the subwavelength SPP mode can be strongly confined in the cavity in three dimensions due to the confinement mechanisms of mode-gap, metal mirrors and air-slot gap. In particular, SPP mode-gap mechanism can enable the ultracompact integration of this cavity with other photonic devices such as SPP waveguides [14] and also provide strong confinement as much as metal mirrors.

 

Fig. 3 (a) Top and (b) cross-sectional views of the electric field intensity profile of the SPP cavity mode. The top view is obtained at the white dotted surface marked in the cross-sectional view. (c) Room-temperature Q factor and resonant wavelength as functions of cavity length, L_c.

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Losses in the metallic cavity can be divided into optical loss and metallic absorption loss. In the proposed cavity, radiation into the free space is strongly forbidden and thus optical loss becomes negligible, providing strong cavity feedback. An ultra-high optical Q factor of 1.2 × 10⁹ was calculated by neglecting metallic absorption in the subwavelength metallic cavity – i.e. by setting the damping constant, γ, to zero [8,14]. Such a situation corresponds to the lowest temperature limit of metallic absorption. The mode volume was calculated to be extremely small, λ³/10000 or 0.0040 (λ/n)³– where λ and n are the wavelength in free space and the refractive index of the high-index slab, respectively. The mode volume was defined as the ratio of the total electric field energy of the mode to the peak energy density, where the effective refractive index of metal was used [2,8,14]. With increasing temperature, metallic absorption loss increases, dominating the total loss of the cavity [8]. The Q factor of the SPP cavity decreases from 1.2 × 10⁹ at 0 K to 125 at room temperature due to metallic absorption. The resonant wavelength and mode volume were calculated to be almost constant regardless of temperature. This significant degradation of Q factor with increasing temperature will be a major obstacle preventing the room-temperature operation of SPP lasers [7,8].

To further investigate the optical properties of this SPP cavity, Q factors and resonant wavelengths were calculated at room temperature as functions of cavity length, L_c (Fig. 3(c)). The FDTD simulation shows that the resonant wavelength increases from 1246 to 1879 nm as L_c increases from 50 to 800 nm, while the maximum Q factor is 134 at L_c = 400 nm. The SPP cavity mode can be considered a fundamental Fabry-Perot cavity mode and the resonant wavelength increases with cavity length L_c. A mode gap exists between the cutoff wavelengths of the waveguides with d = 250 nm and d = 100 nm (Fig. 2). As the resonant wavelength approaches 1174 nm (965 THz), the cutoff wavelength of the mirror region, the mode-gap effect becomes weaker and Q factor decreases. As the resonant wavelength increases to the long wavelength of 1879 nm, the height of the air slot between the metal walls effectively becomes shorter. Since this air slot can be considered a vertical mirror, the vertical radiation loss increases at the longer resonance. Consequently, the maximum Q of 134 was achieved at L_c = 400 nm, the optimal compromise of the effects of the mode-gap mirror and the air slot mirror.

3. Plasmonic cavity with lower metallic absorption

Q factors of SPP cavities are generally smaller than those of conventional optical cavities. Large metal absorption is unavoidable due to the field of the SPP mode confined at the dielectric-metal interface. Therefore, to achieve high Q at room temperature, it is necessary to minimize metallic loss of SPP cavities. An efficient way to reduce metallic loss is the introduction a low-index layer at the dielectric-metal interface [20]. Accordingly, a low-index layer with a refractive index of 1.5 (e.g. SiO₂) was introduced at the silver interface (Fig. 4(a) ). This layer can be deposited after etching the high-index dielectric ridge waveguide.

 

Fig. 4 (a) Cross-sectional electric field intensity profile of the SPP cavity with a low index layer of thickness t_low. (b) Electric field intensity distribution along the z-axis. (c) Q factor (black) and confinement factor (red) at room temperature calculated as functions of t_low. Dotted lines indicate the Q and confinement factors of the cavity without low-index material.

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The optical properties of SPP modes excited in this cavity were investigated. The thickness of the low-index layer, t_low, was set to 40 nm. Since the effective index of the cavity mode is reduced due to the low-index layer, the cavity region is enlarged by (d_c, L_c) = (350 nm, 350 nm) to adjust the resonances at telecommunication wavelength, 1550 nm. The other structural parameters were set to be identical to those of Fig. 3. The SPP mode is then strongly confined in the cavity region between the mirror waveguides by the mode-gap confinement mechanism similar to Fig. 3(b). Figures 4(a) and (b) show that most of the electric field intensity is confined in the low-index layer which would lower metallic absorption, unlike in Fig. 3(b). The low-index layer allows the electric fields of the cavity mode inside metal to decrease, while maintaining the 3D subwavelength SPP confinement. In this cavity with a low index layer of 40 nm, the Q factor becomes 300, 2.5 times larger than that of the cavity without the low index layer, while SPP remains three-dimensionally confined within an extremely small mode volume of λ³/1000 or 0.040 (λ/n)³, where n is the refractive index of the high-index slab.

Quantitative examination of the effect of the low-index layer on metallic absorption involved calculating the Q (black) and confinement factor (red) at room temperature as functions of the thickness of the low-index layer, t_low (Fig. 4(c)). The confinement factor is defined by the ratio between the energy in the high-index slab and total energy of the cavity mode. The Q and confinement factors calculated for the cavity in Fig. 3 without a low-index layer are shown for comparison as dotted lines in this figure. The FDTD simulation shows that the Q factor of the cavity increases significantly with increasing t_low because the mode energy in the silver becomes lower (Fig. 4(b)). The Q factor increases to 350, compared with the Q factor of 125 for the cavity without the low-index material. The introduction of the low-index layer decreases the confinement factor with increasing t_low, which is defined by the ratio between the energy in the high-index slab (active material) and the total energy of the cavity mode. Since both high Q and high confinement factors are desirable for lasing, the low-index layer’s thickness should be optimized for the room-temperature operation of SPP lasers.

4. Conclusion

A plasmonic cavity with 3D subwavelength-scale optical confinement is proposed based on a mode-gap mechanism in the channel-waveguide. The mode volume (λ³/10000) is well below the wavelength-scale of light in free space, and the optical Q factor of the cavity mod is very high (1.2 × 10⁹). The proposed width-modulated mode-gap mechanism that permits strong 3D subwavelength optical confinement and good cavity feedback can be applied to the design of novel plamonic devices of very small physical size.

The introduction of a low-index dielectric at the interface of the high-index dielectric and metal decreased metallic loss in the cavity, thereby enhancing the room temperature Q factor from 125 to 350. Optical radiation and metallic absorption losses of the plasmonic cavity could be suppressed simultaneously by the mode-gap and a low-index material, where the mode volume is much smaller than the wavelength-scale. This cavity structure could aid the development of room temperature SPP lasers by using InAsP or GaAs as the active medium. Owing to its simple design that can be fabricated by conventional techniques, the proposed plasmonic cavity is a promising candidate for new coherent sources with ultra-compact and low power consumption.