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We present a prototype of semiconductor lasers with plasmonic periodic structures that only support transverse-magnetic modes at telecommunication wavelengths. The structure does not sustain transverse-electric guided modes which are irrelevant to surface-wave-enhanced applications, and lasing modes must be surface-wave-like. With thin low-index dielectric buffers near the metal surface, the threshold gain is kept at a decent level around the photonic band edge. Thin windows are then opened on the metal surface to let out significant surface fields. This facilitates usages of surface waves for the spectroscopy and sensing.
© 2014 Optical Society of America
1. Introduction
Surface plasmon polaritons (SPPs) have been employed in various areas [1–3]. The intense fields of SPP modes at interfaces of metals and dielectrics make many applications that require a high intensity accessible. This feature also enhances the resolution and sensitivity which are essential for the spectroscopy and sensing [4–14]. Recently, SPP modes have also been utilized in electrically-injected semiconductor lasers (SLs) and amplifiers [15–26]. These active nano-photonic devices provide the advanced probing capability and functionality. Integrating them with present schemes of the surface-wave-enhanced (SW-enhanced) spectroscopy and sensing may further compact measurement setups and increase the signal-to-noise ratio.
Despite demonstrations and developments of plasmonic SLs and amplifiers with planar or metal-grating structures [20–26], a few obstacles remain to be solved. The most critical one is the loss. Due to lossy metals and high fields near dielectric-metal interfaces, propagation distances of transverse-magnetic-like (TM-like) SPP modes are usually limited. One remedy to this problem is the inclusion of other photonic structures, for example, dielectric layers [27,28], as buffers which reduce the metal loss but still preserve the characteristics of SPP waves. Yet, SPP modes are often not the only modes in photonic structures. For instance, transverse-electric-like (TE-like) guided modes may be present. These modes lack significant surface fields that enhance interactions near surfaces. In addition, unless TE-like modes are (nearly) cut off, their lasing thresholds are often lower than those of SPP modes [16, 19]. If a photonic structure supports both the TE-like and TM-like SPP modes, usually TE-like ones dominate in the presence of gain, which is not useful in SW-enhanced applications. To favor SPP modes in these conditions, specific semiconductor nanostructures such as tensile-strained multiple quantum wells which provide the higher TM than TE gains have to be used [24–26]. Another issue on SPP modes is their sensitivity to corrugated metal surfaces and guiding structures [29, 30]. Precise fabrications are required in these cases.
In this work, we present the design of an electrically-injected SL structure which contains a metal-dielectric grating and merely supports SPP-like modes for SW-enhanced applications. Since no TE-like modes are supported, the active medium could simply be a bulk semiconductor that provides the isotropic rather than TM gain at telecommunication wavelengths. We manage the metal loss with a low-index dielectric grating which confines the field and reduces its penetration into the metal [27,28]. The attainable gain of bulk semiconductors is estimated to ensure that it could surpass the propagation loss of SPP modes. The current paths are reserved between dielectric buffers of the grating so that the electrical injection is possible. A significant surface field for SW-enhanced applications is then let out at windows opened on metal surfaces. Since only SPP-like modes are supported in this structure, TE-like modes that are irrelevant for SW-enhanced applications would not dominate in lasing fields.
The remaining part of the paper is organized as follows. The structure of metal-dielectric grating lasers is described in section 2. The basic formulation and some physical parameters that help design the laser structure are briefly introduced in section 3. In section 4, we present the designs and calculations of the grating laser structure. A conclusion is given in section 5.
2. Semiconductor laser structure with metal-dielectric grating
There are various structures of plasmonic lasers and amplifiers [31]. Here, we adopt a grating structure whose unit cell is shown in Fig. 1(a). For realistic fabrications, the structure could be further modified.
Fig. 1 (a) The schematic diagram of a unit cell in the laser cavity. The active region, dielectric buffer, WG cladding, and two metal regions are denoted as Ωa, Ωb, Ωc, and Ωm,1(2), respectively. A window (Ωw, dashed-line region) at the center of Ag covering will be opened to let out the surface field. (b) The limiting case at d3 = 0. The metal is approximated as a PEC. The electric fields of TE modes vanish on the nodal plane. (c) The corresponding TE modes in a symmetric WG twice wider in the core. They are odd TE modes of the symmetric WG and all have finite cutoff frequencies.
The grating has a period Λ. It is based on semiconductors indium phosphide (InP) and indium gallium arsenide (In0.53Ga0.47As) lattice-matched to InP, both of which are III–V materials commonly used at telecommunication wavelengths. Semiconductor InP functions as the upper and lower waveguide (WG) claddings (region Ωc), and In0.53Ga0.47As acts as the WG core and gain material (Ωa). A periodic pattern of dielectric buffers (Ωb) composed of titanium dioxide (TiO2) is introduced into the upper InP layer. The intervening regions of InP between dielectric buffers function as current paths and are covered with gold (Au), which is utilized as ohmic contacts to avoid the Schottky barrier for injected carriers. The whole structure is then deposited with silver (Ag). The regions of Ag and Au are denoted as Ωm,1 and Ωm,2, respectively. The two metals act as plasma materials, and together with TiO2, they form a metal-dielectric grating. The thickness and width of each dielectric buffer in the grating affect field strengths of TM-like modes penetrating into metals [27,28]. Therefore, the sizes of dielectric chunks could be utilized to control the loss. On the other hand, most of the lasing field is blocked by the Ag covering and cannot reach the surface for spectroscopic and sensing applications. Therefore, after the loss is reduced to an acceptable level, a narrow window (Ωw) will be opened at the center of Ag covering to let out the surface field.
The triple layers of InP/In0.53Ga0.47As/InP form a natural slab WG due to the high refractive index of In0.53Ga0.47As. Therefore, the SPP-like modes in this grating structure exhibit both features of guided TM modes from the slab WG and surface waves at interfaces of the metal and dielectric, namely, a hybrid mode. Once carriers are injected into the active region of In0.53Ga0.47As, the hybrid mode experiences the gain there and compensates the ohmic loss near the metal-dielectric interface. The thickness d3 of the upper InP layer could be designed (several tens of nanometers) so that fields of SPP-like hybrid modes are mostly confined near the metal-dielectric interface rather than in the WG core [32,33]. In this way, the spatially sharp surface field is preserved in this grating structure.
The TE-like guided modes are absent if the In0.53Ga0.47As and upper InP layers are sufficiently thin and hence cut off these modes. The cutoff of TE-like guided modes could be expected if one considers the limiting case in which the upper InP layer is absent (d3 = 0), and metal is approximated as a perfect electric conductor (PEC). Under such circumstances, the metal-semiconductor layer structure turns into an asymmetric slab WG shown in Fig. 1(b). The electric fields Ez(y) of TE modes in this asymmetric WG have to vanish at the PEC interface. From the image theory of electromagnetism, these modes are linked to the odd TEm modes (m = 1, 3...) of a symmetric slab WG twice wider in the core (2d4), as indicated in Fig. 1(c). It is known that odd TE modes in a symmetric slab WG are all cut off if the core is thin enough. Equivalently, if the core thickness d4 of the asymmetric WG is sufficiently thin, each guided TE mode in this limiting case disappears. Note that the fundamental TM mode of the asymmetric WG is not constrained by the PEC boundary and hence is not cut off. While the aforementioned scenario is true in the limiting case, it should remain valid if (1) thicknesses d3 and d4 are sufficiently small, and (2) the refractive index of the dielectric grating is decently but not too much lower than that of InP. Condition (2) could further shorten the effective optical length of the upper cladding normal to the planar structure before the index guiding in semiconductor layers due to the low-index dielectric comes into play. Numerical calculations indicate that TE guided modes can be indeed eliminated in this way.
3. Basic formulation
We employe a frequency-domain method which incorporates sources (gain) into optical modes of periodic photonic structures [34, 35]. In this approach, Maxwell’s equations in the presence of sources (gain) is converted to a generalized eigenvalue (GE) problem for Bloch modes. A multidomain pseudospectral frequency-domain method is also numerically realized in the approach to enhance the convergence [36,37]. Since sources (gain) are built into the formulation, we can directly look into the issues of loss, threshold gain, and their structural dependencies, which are often inferred indirectly in other approaches [38–41].
For the grating in Fig. 1(a), let us consider a source-driven Bloch mode fn,kx (ρ, ω) (ρ = xx̂ + yŷ) labeled with a band index n, wave number kx along the direction of periodicity, and real frequency ω. The wave equation and boundary condition of fn,kx (ρ, ω) are [34]
where ε0, μ0, and c are permittivity, permeability, and speed of light in the vacuum; εr(ρ, ω) is the periodic relative permittivity distribution of the grating structure; js,n,kx (ρ, ω) is the associated source which generates the mode; U(ρ) is an indicator function which is unity in the active region Ωa but zero elsewhere; and Δεr,n,kx (ω) is a proportional factor which derives its dependence on ω through numerical calculations. The relative permittivity εr(ρ, ω) is locally homogeneous and equal to εu(ω) in region Ωu (u = a, b, c, m1, m2, w). From Eq. (1b), js,n,kx (ρ, ω) is only present in the active region Ωa and proportional to mode profile fn,kx (ρ, ω). If we substitute this expression into Eq. (1a), the wave equation becomes a GE problem of fn,kx (ρ, ω) with Δεr,n,kx (ω) acting as the eigenvalue. Once Δεr,n,kx (ω) is obtained, it could be combined with the permittivity εa(ω) of the gain material and interpreted as the amount of permittivity variation which makes the mode self oscillate at frequency ω [34, 35].In region u, we simplify the GE problem with the operator identity ∇ × ∇× = ∇(∇·) − ∇2 in the cartesian coordinate and null-divergence condition. This simplification is allowed because the field is present in a seemingly charge-free domain with an effective homogeneous permittivity εu(ω) + δuaΔεr,n,kx (ω), where δua is the Kronecker delta. Boundary conditions of various field components are utilized to connect fields in different regions together. In this way, the GE problem in region u is rewritten in terms of Hemholtz equation as
Perfectly matched layers [37] are then imposed on the top and bottom of computation domains to absorb outgoing waves along ±ŷ directions in cases of leaky modes.Some useful parameters utilized in designs of the grating structure were derived in previous works [34, 35]. Dispersion relations of Bloch modes (ωn,kx) are obtained by finding the frequency that minimizes the absolute value |ωΔεr,n,kx (ω)| at a given wave number kx. With the frequency ωn,kx and cold-cavity condition (no gain or absorption from interband transitions is included in Im[εa(ω)] of the gain material), we estimate the threshold gain g′th,n,kx of mode (n, kx) without the mirror loss at output facets as follows [35]:
If the contents of two square roots in Eq. (3) are dominated by the positive real part Re[εa(ω)], this threshold gain could be further approximated with binomial expansions asFor simplicity of designs and to obtain a rough picture of modal dispersions first, we would begin with the slab WG in absence of dielectric buffers and Au contacts, namely, TiO2 and Au are replaced with InP and Ag, respectively. The complex propagation constant βn(ω) of mode n in this slab WG and its modal loss αM,n(ω) = 2Im[βn(ω)] is calculated as a function of frequency ω using the cold-cavity condition of In0.53Ga0.47As. The transparency gain gtr,n(ω) which exactly compensates the propagation loss can be estimated from αM,n(ω) as [42–44]
where Γwg,n(ω) is the WG confinement factor; na(ω) is the real part of the refractive index in the active region; η0 is the intrinsic impedance; and En(y, ω) and Hn(y, ω) are the electric and magnetic fields of WG mode n. Equation (5) will be utilized to investigate dissipations and feasibilities of the slab WG and associated variations. The designs related to periodic dielectric buffers, Au contacts, and windows of surface fields are then considered afterward.4. Calculations and discussions
In the calculations hereafter, we utilize the following data of material permittivities. The permittivities εa(ω) (In0.53Ga0.47As) and εc(ω) (InP) are adopted from Refs. [45] and [46], respectively. The two metal permittivities εm,1(ω) (Ag) and εm,2(ω) (Au) are obtained from Ref. [47]. The dielectric permittivity εb(ω) (TiO2) is set to a nondispersive number 6.026.
4.1. Metal-semiconductor slab waveguide
The metal-semiconductor slab WG is the precursor of the more complicated grating structure. We preliminarily set the thicknesses d3 of upper InP layer, d4 of In0.53Ga0.47As layer, and d5 of Ag covering to 0.19, 0.12, and 0.4 μm, respectively. The heights d1 of Au contacts and d2 of TiO2 buffers are set to zero, and therefore the grating period virtually vanishes (Λ = 0).
The fundamental TE and TM modes are the only two guided modes in this WG around the telecommunication wavelength 1.55 μm. The fundamental TM mode is a SPP-like one. As shown in Fig. 2, its magnitude of the normalized field component |Ey(y)| at a photon energy h̄ω = 0.8 eV (1.55 μm in wavelength, and h̄ is Planck constant) has a sharp and maximal peak at InP side of the semiconductor-metal interface (y = 5.32 μm). In addition, the field component Ey(y) is discontinuous at various material interfaces due to continuity of the displacement field Dy(y) normal to the planar structure. On the other hand, the normalized magnitude |Ez(y)| of the fundamental TE mode in the same figure indicates that it is guided by the In0.53Ga0.47As WG core (y = 5.13 − 5.25 μm). The electric field of this TE mode is continuous. However, it penetrates into the InP substrate significantly and is not well confined in the active region.
Fig. 2 Normalized magnitude profiles of the fundamental TM (|Ey(y)|) and TE (|Ez(y)|) modes at h̄ω = 0.8 eV. The TM mode peaks at the metal-semiconductor interface.
Figure 3(a) shows the dispersion relations Re[βn(ω)] versus photon energy h̄ω of the slab WG. Since the translational invariance of the WG structure can be regarded as a special case of vanishing periods, we may obtain analogous dispersion relations kx versus h̄ωn,kx using the proposed GE approach with Λ → 0. The results are also shown in Fig. 3(a) for comparisons. The two sets of dispersion curves agree well, which is also a validation of the GE method. The dispersion curve of the TE mode is close to the light line (dispersion curve of photons in three-dimensional plane-wave states) of InP. This explains why the TE mode profile in Fig. 2 has a considerable distribution in the substrate.
Fig. 3 (a) Dispersion relations of the fundamental TM and TE modes in the metal-semiconductor slab WG. The results from typical WG calculations and proposed GE approach agree well. The TE dispersion is close to the light line of InP (blue dashed line). (b) The transparency gains gtr,n(ω) as a function of h̄ω for the same two modes. They match well with the curves h̄ωn,kx versus g′th,n,kx from the GE approach. The material gain at N = 4 × 1018 cm−3 is shown in black squares.
Similarly, we may compare the transparency gain gtr,n(ω) as a function of h̄ω with the threshold gain gth,n,kx parameterized by h̄ωn,kx for the two modes. The two sets of curves are illustrated in Fig. 3(b), and they also match well with each other. As pointed out in section 1, TE-like modes often have much lower thresholds than those of TM-like ones in metallic photonic structures, and this metal-semiconductor WG is no exception. In the same figure, we also show the theoretical material gain spectrum of In0.53Ga0.47As at a carrier density N = 4 × 1018 cm−3, which is calculated using a Gaussian lineshape with a half-width-at-half-maximum linewidth of 22.5 meV. This carrier density is about 1.5 to 3 times higher than that in a conventional double-heterojunction SL and is chosen as a benchmark, presuming that the large gain necessary for loss compensations is achievable at a high injection level due to good thermal properties of metals [48]. From Fig. 3(b), even at this level of carrier density, the material gain is still lower than gtr,n(ω) of the fundamental TM mode but much higher than that of the poorly-guided TE mode on most of the gain spectrum. In fact, the material gain may support the propagation of the fundamental TE mode up to h̄ω ≈ 0.95 eV. Under such circumstances, the TM SPP-like mode could hardly compete with the TE one.
To avoid the dominance of the fundamental TE mode in the presence of gain, we attempt to cut it off at h̄ω = 0.95 eV by varying the thickness d3 of the upper InP cladding. Other length parameters remain the same. Once the TE guided mode at this photon energy is cut off, it would not exist in the gain window h̄ω ∈ [0.7, 0.95] eV at the benchmark carrier density. In Fig. 4(a), we show effective indices neff,n(ω) ≡ Re[βn(ω)]/k0 of the fundamental TM and TE modes as a function of d3 at h̄ω = 0.95 eV, where k0 = ω/c is the propagation constant in the vacuum. The cutoff takes place if the effective index is identical to the refractive index nc(ω) = 3.202 of InP at h̄ω = 0.95 eV. In that condition, the mode ceases being confined by the WG core and begins to leak into the InP substrate. From Fig. 4(a), the critical thickness d3 at which the cuttoff occurs is around 0.1 μm. On the other hand, the effective index of the SPP-like TM mode increases as the thickness d3 decreases, which is opposite to the behavior of the TE mode. In fact, the smaller thickness d3 of the upper InP cladding makes the fundamental TM mode more surface-wave-like (SW-like). Meanwhile, the WG core is closer to the peak of the surface wave. This improves the mode confinement in the active region, and the transparency gain of the SPP-like mode is reduced, as shown in Fig. 4(b) for gtr,n(ω) at h̄ω = 0.8, 0.87, and 0.95 eV in the gain window. The improvement of the lower-frequency mode is less significant even though the transparency gain at h̄ω = 0.8 eV could be slightly lower than the material gain at d3 ≈ 0.115 μm, as illustrated in the same figure.
Fig. 4 (a) The effective indices neff,n(ω) of the fundamental TM and TE modes versus d3 at h̄ω = 0.95 eV. The blue dashed line is the refractive index nc(ω) = 3.202 of InP at the same photon energy. (b) The transparency gains gtr,n(ω) of the SPP-like mode versus d3 at h̄ω = 0.8, 0.87, and 0.95 eV. The material gain at h̄ω = 0.8 eV and N = 4 × 1018 cm−3 is shown as the blue dashed line.
As the thickness d3 approaches zero, some portion of the transparency gain gtr,n(ω) in the gain window at h̄ω ∈ [0.7, 0.95] eV would be lower than the material gain. However, a finite thickness d3 of the n-type InP cladding is necessary for proper current injections. A decent choice of d3 is its critical value which just cuts off the fundamental TE mode at h̄ω = 0.95 eV, namely, d3 ≈ 0.1 μm. As can be inferred from Fig. 4(b), with this thickness, the material gain at N = 4 × 1018 cm−3 can marginally support the propagation of SPP-like modes around h̄ω = 0.8 eV. However, the introduction of Au for ohmic contacts is expected to significantly increase the dissipation of SPP-like modes since Au is much more lossy than Ag. The gain margin is not sufficient in the presence of Au, and the dielectric is utilized to further reduce the loss.
4.2. Insertion of dielectric layer
To further lower the transparency gain of the SPP-like mode, we insert a TiO2 dielectric layer beneath the Ag covering [27, 28], namely, d2 ≠ 0 in Fig. 1. The resulted structure is still a slab WG. From section 4.1, if the upper WG cladding is composed of InP, and its thickness d3 is smaller than 0.1 μm, the fundamental TE mode is efficiently cut off at h̄ω = 0.8 eV. In this section, the WG cladding would be composed of InP and TiO2. The TE mode is still cut off under such circumstances, but modal features of the SPP-like mode are changed considerably. For related calculations, most of the length parameters are identical to those in the previous section. They are listed as follows: d3 =0.1 μm, d4 =0.12 μm, d5 =0.4 μm, and d1 =W1 =Λ=0.
Figure 5(a) and (b) shows magnitude distributions of the electric field |Ey(y)| and associated displacement field |Dy(y)| of the SPP-like mode around the dielectric buffer layer with a thickness d2 = 0.04 μm, respectively. The counterparts at d2 = 0.07 μm are shown in Fig. 5(c) and (d). The photon energy h̄ω is 0.8 eV. Both of the two profiles |Ey(y)| in Fig. 5(a) and (c) exhibit high field strengths which are close to the Ag covering and are concentrated in respective dielectric buffer layers as a result of the continuity of Dy(y). In a broad sense, these modes are still SW-like. Nevertheless, the localized fields are not only consequences of metal-dielectric interfaces but also results of small dielectric buffers. The larger dielectric thickness d2 usually lowers the portion of the optical field inside metals. However, the field is not necessarily drawn into the dielectric buffer. It may be redistributed into the active region and lower WG cladding if d2 is sufficiently thick, as shown in Fig. 5(c).
Fig. 5 The normalized magnitude distributions of |Ey(y)| and |Dy(y)| for the SPP-like mode in the presence of a dielectric buffer layer whose thickness d2 is 0.04 μm [(a) and (b)] and 0.07 μm [(c) and (d)], respectively. The photon energy h̄ω is 0.8 eV. The thick dielectric layer redistribute fields into the active region and lower WG cladding.
Figure 6(a) shows the transparency gain gtr,n(ω) at h̄ω = 0.8 eV as a function of thickness d3 of the upper cladding layer for different dielectric heights d2 = 0.04, 0.05, and 0.06 μm. These transparency gains are larger at the smaller d3 since the active WG core is closer to the Ag covering. On the other hand, the larger d2 results in the smaller transparency gains because the field strength is reduced in the metal and redistributed to the active region and lower WG cladding, as can be inferred from the field profiles in Fig. 5(c). The comparisons of transparency gain spectra gtr,n(ω) with the material gain of In0.53Ga0.47As (N = 4 × 1018 cm−3) at d3 = 0.1 μm and d2 = 0.04, 0.05, and 0.06 μm are shown in Fig. 6(b). The dielectric layer creates a significant gain margin for the SPP-like mode around the wavelength of 1.55 μm. This margin can be utilized to compensate the loss from Au contacts. On the other hand, we would not make the dielectric layer even thicker. As the openings below Au contacts act as current paths into the grating structure, the semiconductor InP below dielectric buffers plays the role of spreading the injection current into the active region. In views of an already thin cladding (d3 = 0.1 μm), the excessive thickness d2 of the dielectric layer could make the current distribution uneven. This factor should be kept in mind for designs of this grating laser.
Fig. 6 (a) The transparency gains gtr,n(ω) of the SSP-like modes at h̄ω = 0.8 eV with buffer thicknesses d2 = 0.04, 0.05, and 0.06 μm. The larger d2 makes the transparency gain lower. (b) The transparency gain spectra at d2 = 0.04, 0.05, and 0.06 μm. The cladding thickness d3 is 0.1 μm. The gain spectrum at N = 4×1018 cm−3 is shown in black squares. Considerable gain margins are present between the transparency gains and material one.
4.3. Semiconductor laser with metal-dielectric grating
For laser structures of the metal-dielectric grating, the period Λ is set to 0.234 μm, which opens a photonic bandgap at the boundary kx = ±π/Λ of the Brillouin zone (BZ). This period is chosen such that the bandedge of the SPP-like branch below the bandgap (lower SPP branch) is around h̄ω =0.8 eV. We will examine the dispersion relation h̄ωn,kx and threshold gain g′th,n,kx of the lower SPP branch by varying the thickness d2 and width W1 of the dielectric buffer. The SPP-like branch above the bandgap (upper SPP branch) only behaves like guided modes in a small kx space near the BZ boundary. As its frequency increases, the upper SPP branch crosses the light line of InP and is immersed into the continuum modes of the substrate. The corresponding radiation loss toward the InP substrate has a sudden and tremendous increment beyond this turning point, which can be told from the threshold gain g′th,n,kx of the upper SPP branch. Since the upper branch has a relatively small range of working frequencies, we should only focus on the lower branch.
The length parameters which are kept constant are listed as follows: d1 = 60 nm (thickness of Au contacts), d3 = 0.1 μm, d4 = 0.12 μm, and d5 = 0.4 μm.
Since the width W1 of dielectric buffers is not identical to the period Λ, the current injection into the active region is made possible through InP regions below Au contacts. We first set W1 = 0.7Λ. The dispersion relations h̄ωn,kx at d2 = 0.04, 0.05, and 0.06 μm are shown in Fig. 7(a). Variations of the dielectric thickness change the photonic bandgap and slightly shift bandedges of the lower and upper SPP branches. The thicker dielectric buffer blue shifts the bandedge more. As a result, at d2 = 0.06 μm, the upper branch is only guided (below the light line of InP) in quite limited ranges for both the wave number kx and frequency. For the lower SPP branches corresponding to three thicknesses d2, their threshold gains g′th,n,kx as a function of h̄ωn,kx are shown in Fig. 7(b) up to the respective bandedges. The lower SPP branch with the largest thickness d2 still exhibits the lowest threshold. However, the presence of Au contacts make all these threshold gains g′th,n,kx significantly higher than the transparency gains gtr,n(ω) of counterpart modes similar slab WGs [Fig. 6(b)]. For the lower SPP branch, its upper-bound photon energy of the gain window is the corresponding bandedge energy. On the other hand, the lower-bound photon energy marks the crossing between the threshold gain gth,n,kx and material gain at N = 4 × 1018 cm−3. The additional dissipation due to the proximity to Au contacts increases the threshold gain g′th,n,kx and hence narrows the gain window. Meanwhile, the gain margin reserved for the compensation of the mirror loss is also reduced in the window. In Fig. 7(a), the gain window of the lower SPP branches at d2 = 0.06 μm is shown in red. It covers an energy range of about 50 meV.
Fig. 7 (a) The dispersion relations h̄ωn,kx of the three SPP-like modes at W1 = 0.7Λ and d2 = 0.04, 0.05, and 0.06 μm. The three upper SPP branches become leaky modes once they cross the light line of InP (cyan dotted line). The red region marks the gain window of the lower SPP mode at d2 = 0.06 μm. (b) Comparisons between the material gain at N = 4 × 1018 cm−3 (black squares) and threshold gains g′th,n,kx of the three lower SPP branches in (a). The gain margins are smaller than their counterparts in Fig. 6(b)
We further decrease the width W1 of dielectric buffers to investigate its effect on the dispersion relation h̄ωn,kx and threshold gain g′th,n,kx. The reduction of width W1 has a similar effect to that due to the decrease of thickness d2 on dispersion relations. Both of them make dielectric buffers smaller and therefore red shift the dispersion relation. In Fig. 8(a), the dispersion relations at W1 = 0.5, 0.55, 0.6, and 0.7Λ indeed show this trend. Since the area of metal contacts become larger as the width W1 of dielectric buffers turn smaller, the threshold gain g′th,n,kx is enhanced accordingly by the extra loss from Au, as indicated in Fig. 8(b). Both the gain window and margin are reduced as W1 decreases. From Fig. 8(b), as the width W1 of dielectric buffers is reduced to 0.55Λ, the gain window and margin both vanish. In this case, the material gain at N = 4×1018 cm−3 can only marginally support the propagation of SPP-like modes in a narrow frequency range, and there is no room for the compensation of the mirror loss at output facets. The reservation of this gain margin for the mirror loss is important for Bloch modes away from the bandedge. In those cases, the group velocity is on the order of the light speed in InP, and the round-trip time could be short in small laser cavities. This would enhance the loss rate at output facets. The effect has the less impact on bandedge modes due to their low group velocities. Since the gain margin is also large near the bandedge, bandedge modes would possibly lase first, which is also the case in typical SLs with distributed feedback structures.
Fig. 8 (a) The dispersion relations h̄ωn,kx at d2 = 0.06 μm for W1 = 0.5, 0.55, 0.6, and 0.7Λ. The smaller width W1 makes dielectric buffers narrower and red shifts the dispersion relations. The light line of InP is shown as the cyan dotted line. (b) The threshold gains g′th,n,kx of the lower SPP branches corresponding to the four widths W1 in (a). The wider Au contacts at the smaller W1 increase the threshold gain. Both the gain window and margin decrease accordingly. The material gain at N = 4 × 1018 cm−3 is shown in black squares.
The normalized field profile |Ey(ρ)| of the bandedge mode (h̄ω ≈ 0.8 eV) at d2 = 0.06 μm and W1 = 0.7Λ is shown in Fig. 9(a). The design ensures that there are sufficient gain margin for the lasing action in this semiconductor grating structure at least at the benchmark carrier density N = 4 × 1018 cm−3. On the other hand, the field inside the dielectric buffer would be blocked by the Ag covering with a thickness d5 in tens of nanometers. It does not reach the ambient environment for SW-enhanced applications. Thinning the Ag covering below the optical penetration depth of about 12.5 nm could bring the field into the free space. However, the associated threshold gain gth,n,kx significantly increases and is too large to achieve. To make SW-enhanced applications plausible, a small window with a width W2 = 0.1Λ ≈ 23.4 nm is opened at the center of the Ag covering (d5 = 85 nm) to let out the surface field. The TE-like modes are still cut off in this condition. The resulted normalized field profile |Ey(ρ)| at the bandedge is shown in Fig. 9(b). This window actually lowers the threshold gain of SPP-like modes slightly since the contact area with metal is reduced (g′th,n,kx = 6.62 × 102 cm−1). In this way, SW-enhanced measurements could be carried out on molecules or nanostructures captured into the window (target sizes smaller than 0.1Λ). With the wider window, detections of the larger objects could also be possible.
Fig. 9 Normalized field distributions |Ey(ρ)| of (a) the lower SPP branch with the full Ag covering at the bandedge, and (b) the counterpart with a window which has a small width W2 = 0.1Λ ≈ 23.4 nm on the Ag covering. The thickness d2 and width W1 are 0.06 μm and 0.7Λ, respectively. With the window, the surface field can reach the ambient environment for SW-enhanced applications of captured molecules and nanostructures.
5. Conclusion
We have presented the design of a SL based on the metal-dielectric grating which only supports SPP-like modes. The TE-like modes that are irrelevant for SW-enhanced applications are cut off through thickness variations of semiconductor-metal layer structures and insertions of dielectric buffers. In addition, the bulk semiconductor rather than specific nanostructures that favor TM-like modes is sufficient for the device scheme. We investigated the dispersion relation and threshold gain of the metal-dielectric grating. The structure was adjusted so that a sufficient gain margin remain for the lasing action of Bloch modes. With surface fields let out through thin windows opened on the metal surface, the laser structure could be utilized for SW-enhanced applications.
Acknowledgments
This work is sponsored by Research Center for Applied Sciences, Academia Sinica, Taiwan, and Ministry of Science and Technology, Taiwan, under grant numbers NSC102-2221-E-001-027 and MOST103-2221-E-001-016.
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