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Abstract
We address theoretically lateral localization of surface-trapped optical modes in microcavities formed at a surface of a distributed Bragg reflector (DBR). GaAs-GaAlAs materials are considered as a model system. We analyze such modes and demonstrate that thick metal bars mounted on top of the DBR form a lateral cavity strongly localizing in plane the modes which exhibit in the vertical direction an evanescent decay in the air and an oscillatory decay in the DBR. Such TM-polarized modes are strongly confined between the bars, the fraction of the optical energy of the mode located in the air reaching ∼90%. We show that the wavelength of such modes is nearly temperature-insensitive, the thermal shift can be as small as below 0.005 nm/K, which is an order-of-magnitude smaller than the typical value for vertical cavity surface emitting lasers (VCSELs). TE-polarized modes are observed only if the top layer of the DBR sequence has a thickness different from λ/4 and a cavity layer is formed at the surface. The surface-trapped modes enable near-field outcoupling to an external waveguide or to an optical fiber placed closed to the DBR surface. In DBR structures incorporating an active medium these modes can be employed for construction of microlasers as well as for resonant semiconductor optical amplifiers (SOAs) having nearly temperature-insensitive lasing/resonance wavelength. In another approach applying a reverse bias to the active medium one can realize resonant intensity modulators. Surface-trapped modes can be employed in all-dielectric DBRs enabling low loss waveguides for silicon photonics. Further applications include using such modes at interfaces between a semiconductor DBR and a dielectric medium having a lower refractive index.
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1. Introduction
A variety of surface electromagnetic waves propagating along a boundary between two media and decaying in both directions away from the boundary is of a strong fundamental and practical interest. In most cases such waves exist on surfaces/interfaces of very specific media. For example, an interface wave propagates along a boundary between two homogeneous media only if the dielectric permittivity ε of one medium is negative [1,2], which can be provided by the plasma contribution to ε [3], e. g., in metals. Other examples include a surface of a photonic crystal [4], a surface containing a grating or ridges [5], a surface of a chiral material [6], a boundary of a gyrotropic material and a negative phase velocity medium [7], a surface of an optically anisotropic medium with optical axes tilted with respect to the surface [8].
Arising new applications of semiconductor diode lasers, light-emitting diodes, single photon emitters, etc. and continuously growing level of integration of opto-electronic circuits require devices with substantially advanced performance. Recently devices with various new functionality have been proposed [9–32] based on novel design concepts. In this connection, possible electromagnetic waves bounded to a surface of conventional semiconductors, beyond complex systems of [1–8], can be of high importance for potential use in practical devices.
A periodic layer structure of semiconductor materials, e. g., a distributed Bragg reflector (DBR) is likely to be the simplest structure of this type. A possibility of the localization of electromagnetic waves at the surface of such structure was shown in [33]. However, the optical modes calculated and observed in [33] are rather asymmetric modes of a multilayer waveguide than surface modes. Those modes have effective mode indices in the range 2.4–3.4 and hence very short penetration into the air. Moreover, the mode wavelength is far from any reflectivity stopband of the DBR.
In our earlier work [34], the light emission was studied from an epitaxial structure designed for an antiwaveguiding Vertical Cavity Surface Emitting Laser (VCSEL) processed as a rectangular chip with a contact grid on top and emitting light through openings in the top metal contact. The study demonstrated a wavelength-stabilized mode identified as a result of the coupling of a vertical cavity mode and a surface-trapped TM-polarized mode. It was then shown in [35,36] that an alone standing DBR, without a resonant cavity supports a surface-trapped mode thus creating a virtual cavity at a boundary to a homogeneous medium, the mode wavelength lying within a reflectivity stopband of the DBR. The effective mode index of the surface-trapped mode is close to the refractive index of the neighboring homogeneous medium, i. e., close to 1 for a structure bounded by the air. Thus the surface-trapped mode penetrates into the air over a distance of the order of the mode wavelength.
To get a deeper insight in the properties of the surface-trapped modes, we consider a distributed Bragg reflector (DBR) formed of 30 pairs of layers alternating in the vertical z direction, namely layers of a high (n1) and a low (n2) refractive indices with thickness of one quarter of the material wavelength, i. e., with d1=λ0/(4n1) and d2=λ0/(4n2), where λ0 is the wavelength of the peak DBR reflectivity. The DBR terminates with a λ/4-thick layer with a higher refractive index. For specific modeling we choose a DBR structure typical to those employed in VCSELs used in Datacom. Namely, we take λ0 = 850 nm and consider the DBR grown on a GaAs substrate, formed of alternating layers of Ga0.10Al0.90As and Ga0.88Al0.12As and terminated by the air from the top at the surface z = z0.
We calculated the optical power transmittance of the TE- and TM-polarized plane electromagnetic waves versus the wavelength and the angle of incidence extending the small range of angles of incidence addressed in [35] over the entire range of angles using the transfer matrix method [37] and the materials refractive indices from [38], for clarity neglecting absorption in the materials, if any. Results for the TE modes (Figs. 1(a), 1(c)) demonstrate a featureless stopband of extra-low transmittance, whereas the transmittance spectrum for the TM modes (Figs. 1(b), 1(d)) reveals within a broad stopband a narrow spectral range of high transmittance close to 810 nm indicating the presence of a localized optical mode.
Fig. 1. Optical power transmittance of a distributed Bragg reflector at different angles of incidence. (a) TE polarization, all angles. (b) TM polarization, all angles. (c) TE polarization, oblique incidence. (d) TM polarization, oblique incidence. Spectral range of a high transmission of the TM mode (pointed by the arrow) within the reflectivity stopband indicates the existence of an optical mode localized at the surface.
Focusing on the TM-polarized modes we consider a monochromatic TM wave propagating along the x-direction and having only one component of the magnetic field in the y-direction,
Fig. 2. (a)–(c) Vertical profiles of the surface-trapped TM-polarized modes at three different wavelengths. Note that the overall extension of the optical mode is significantly larger in the air than in the DBR in case (c). (d) Real and imaginary parts of the effective mode index versus the mode wavelength. (e) Optical confinement factor of the surface-trapped mode versus the mode wavelength.
The red curve in Fig. 2(d) displays the value $n{^{\prime}_{\textrm{eff}}} - 1$ which rapidly vanishes as the wavelength increases. Then the surface-trapped mode becomes more and more extended to the air until it ceases to exist at ∼819 nm. As we do not consider absorption, the only contribution to $n^{\prime}{^{\prime}_{\textrm{eff}}}$ is related to the leakage of the optical mode to the substrate. Upon wavelength increase, the losses decrease as the optical mode becomes more extended to the air and less to the DBR, as shown also in [39].
Figure 2(e) displays the confinement factor versus the mode wavelength. For definiteness, we have chosen the active medium consisting of four quantum wells of 6 nm thickness separated by 6 nm–thick barriers placed in a local maximum of the in-plane component of the electric field close to the surface, similar to Fig. 4 of [39]. The confinement factor drops at longer wavelengths once the mode becomes more extended to the air and less localized in the semiconductor layers, and also decreases at shorter wavelengths once the wavelength approaches the edge of the DBR stopband.
To address the sensitivity of the surface-trapped modes to temperature, the mode characteristics in Figs. 2(d) and 2(e) are plotted for two temperatures, namely, for room temperature ${T_0}$ (solid lines) and for a higher temperature ${T_0} + 100\;\textrm{K}$ (dashed lines). Temperature dependence of the refractive indices of Ga1-xAlxAs was taken from [40,41]. Upon temperature increase by 100 K, the curves in Figs. 2(d) and 2(e) shift approximately by 5 nm, which implies a characteristic temperature shift of the mode wavelengths at a rate ∼0.05 nm/K. This shift is due to the thermal increase of the semiconductor refractive indices. We note that this rate could be reduced if a major part of the optical energy of the mode is located not in the semiconductor structure, but in the air.
In the present paper we exploit the behavior of the surface-trapped optical modes in short microcavities seeking a possibility to reduce the thermal shift of the mode wavelength. First we focus on exactly tractable system of a microcavity bounded at the side surfaces by a perfect magnetic conductor, to elucidate the main factors governing the thermal shift of the mode wavelengths. Then we solve a two-dimensional problem for the optical modes of a microcavity formed by thick metal bars on top of the semiconductor structure. We demonstrate that wavelengths of the optical modes in such microcavity can be nearly temperature insensitive, exhibiting a thermal shift as low as ∼0.005 nm/K that is an order of magnitude smaller than typical wavelength shifts in VCSELs or distributed feedback (DFB) lasers. We show a possibility of the outcoupling of a temperature-insensitive surface-trapped optical mode to an external waveguide or to an optical fiber and make an overview of possible applications of such modes.
2. Propagation of a TM-polarized surface-trapped mode along the surface
To build a bridge between a one-dimensional (1D) consideration addressing waveguides infinitely extended in the lateral direction to a two-dimensional (2D) microcavity bounded in the lateral plane by mirrors, it is worth considering the 2D problem of the propagation of the surface-trapped mode along an infinitely extended surface. We seek a monochromatic two-dimensional (2D) field ${H_y}({x,z} )$ at a fixed wavelength satisfying the wave equation:
The finite-difference frequency-domain (FDFD) method [42] was applied to Eq. (3), whereas a 1D profile of a surface-trapped mode ${H_y}(z )$ in a cross-section plane $x = 0$ was assumed as a source field of Eq. (3). Figure 3(a) illustrates the solution of Eq. (3) for the spatial profile of a propagating surface-trapped mode at the mode wavelength 795 nm. Figure 3(b) displays a decrease of the mode energy density along the waveguide due to the mode leakage to the substrate. The leakage increases at shorter wavelengths, in agreement with the mode intensity decay along the propagation direction ${\sim} \exp ({ - 2n^{\prime}{^{\prime}_{\textrm{eff}}}{k_0}x} )$, where $n^{\prime}{^{\prime}_{\textrm{eff}}}$ is displayed in Fig. 2(d). For the mode wavelength 795 nm, the energy density decrease by ∼10% implies ∼5% reduction of the field amplitude which is hardly visible in Fig. 3(a).
Fig. 3. (a) Spatial profile of the surface-trapped TM-polarized optical mode propagating along the surface. (b) Decrease of the energy density of the surface-trapped mode along the distance.
3. TM-polarized surface-trapped modes in a microcavity bounded by a perfect magnetic conductor
The optical modes of a rectangular 2D microcavity, infinitely extended in the y-direction, can be found as eigenmodes of a 2D wave equation, which differs from (3) as the dielectric function $\varepsilon$ now depends on two coordinates $\varepsilon = \varepsilon ({x,z} )$,
To get a deeper insight into the optical modes of a 2D microcavity it is worth to start with an exactly tractable model structure. Such an example is a cavity bounded at two vertical planes $x ={\pm} D/2$ by a perfect magnetic conductor (PMC) [43], where the tangential to the boundary magnetic field component obeys the PMC boundary conditions, $\partial {H_y}({x,z} )/\partial x = 0\;$. Then the magnetic field profile of the eigenmodes can be found in the factorized form
for symmetric modes like that of Fig. 4(a) and for antisymmetric modes like that of Fig. 4(b). By applying boundary conditions and taking into account that the mode effective index is a function of the mode wavelength according to Fig. 2(d), we obtain the equations for the mode wavelengths, namely for symmetric modes and for antisymmetric modes, where m=1,2,3…Fig. 4. (a), (b) Spatial profiles of the TM-polarized surface-trapped optical modes in a half of a microcavity bounded by the walls of a perfect magnetic conductor. (a) Symmetric mode. (b) Antisymmetric mode. (c) Mode wavelengths at different intervals of the cavity length. Red curves refer to symmetric modes and blue curves are related to antisymmetric modes.
Figure 4(c) displays the wavelengths of the symmetric (red curves) and antisymmetric (blue curves) modes versus cavity length for three different intervals of the cavity lengths. For small cavity lengths, Fig. 4(c), left panel, there exists either no mode for a given cavity length, or only one single mode, and the mode wavelength versus cavity length is described by a line, close to a straight line slightly tilted from the vertical one. Upon an increase in cavity length, the spectrum of the cavity modes becomes denser (central and right panels). For a dense spectrum, one estimates the spectral separation between the neighboring modes, as follows $|{d[{{\lambda^{ - 1}}{n_{\textrm{eff}}}(\lambda )} ]/d\lambda } |D\cdot(\Delta\lambda) = 1/2$. Hence,
Equation (8) yields a smaller spectral separation between the neighboring modes for shorter wavelengths, and a longer one for longer wavelengths, where the optical mode is significantly extended to the air, and the waveguide dispersion nearly vanishes, $d{n_{\textrm{eff}}}(\lambda )/d\lambda \approx 0$. Since the effective mode index is close to 1, the spectral separation is about 3–3.5 times smaller than that in typical edge-emitting semiconductor diode lasers.
Figures 5(a) through 5(c) display the dependence of the mode wavelengths versus overheating over room temperature for three values of the cavity length, 20 µm, 40 µm, and 100 µm, respectively. The wavelength thermal coefficient $d\lambda /dT$ is plotted as a function of the mode number in Figs. 5(d) through 5(f). We note that the maximum value of $d\lambda /dT$ is about 0.035 nm/K, which is about twice smaller than typical values (∼0.07 nm/K) for semiconductor VCSELs for the same spectral region. This reduction occurs due to the extension of the optical mode in the air. Most remarkable is a drastic reduction of the wavelength thermal coefficient to values below 0.005 nm/K for long wavelengths $\ge$ 815 nm whereas optical modes are strongly extended to the air, as marked by a cross in Figs. 5(a) and 5(d).
Fig. 5. (a), (b), (c). Temperature dependence of the TM-polarized mode wavelengths for three different lengths of a microcavity, D=20 µm, D=40 µm, and D=100 µm. (d), (e), (f) Mode wavelength thermal coefficient dλ/dT for different modes at three different lengths of the microcavity.
4. TM-polarized surface-trapped modes in a microcavity formed by thick metal bars
To address a real semiconductor structure we assume an infinite-in-plane semiconductor DBR with a 1 µm–thick metal bar on top. We consider a bar formed of gold having the dielectric constant $\varepsilon ={-} 25.8 + i1.6$ [44] for the wavelengths of interest. A large negative real part of the dielectric function causes a strong decay and very short penetration of the electromagnetic field into the metal. The metal bar thus acts as a mirror, and two metal bars at the two sides of the structure form a cavity in the air.
The modes of the microcavity were found as eigenmodes of the 2D wave equation (4), whereas we imposed at the symmetry plane x=0 the boundary conditions $\partial {H_y}({x,z} )/\partial x = 0\;$ for symmetric modes or ${H_y}({x,z} )= 0\;$ for antisymmetric modes and the PML boundary conditions at the other boundaries of the computation domain.
Figures 6(a) through 6(d) illustrate symmetric solutions for the eigenfunctions of Eq. (4) representing the surface-trapped optical modes confined between the two metal bars. One half of a symmetric microcavity is displayed. We note that the semiconductor DBR structure as well as the metal bars are extended in the lateral plane till the PML terminating the computational domain. As it follows from Figs. 6(a) through 6(d), the surface-trapped modes exist at different cavity lengths, and the significant part of the optical energy is located in the air between the bars. We note that, whereas the optical field in the air which is purely evanescent in the case of an infinitely extended surface, now has some radiative component due to the diffraction at the boundaries of the metal bars. This diffraction, along with the leakage to the substrate, limits the mode lifetime.
Fig. 6. Spatial profiles of the TM-polarized surface-trapped optical modes in a half of a microcavity bounded in the lateral plane by a thick metal bar. White contour depicts the bar. The dashed line shows the boundary between the DBR structure and the substrate. The microcavity length 8 µm (a), 12 µm (b), 16 µm (c), 20 µm (d).
5. Outcoupling of a surface-trapped mode to a waveguide
For practical applications of surface-trapped modes one needs to ensure mode outcoupling to an optical fiber or to a waveguide. Figure 7(a) illustrates a semiconductor DBR structure with a thick metal bar on top of the DBR and a dielectric waveguide with a thickness 0.2 µm and the refractive index n=1.6 placed 0.1 µm above the top boundary of the metal bar. Unlike the structures of Figs. 6(a) through 6(d), the structure of Fig. 7(a) represents a semiconductor mesa bounded in the lateral plane by air at x=8 µm, while the dielectric waveguide extends infinitely in the lateral direction. Figure 7(b) shows the optical field profile in the vicinity of the metal bar and demonstrates that the field penetrates into the dielectric waveguide extended in the plane away from the semiconductor device. Initially a stronger field in the waveguide decreases and then stabilizes as the optical mode of the cavity excites multiple modes of the waveguide, and then only the fundamental mode of the waveguide survives and propagates. Although Figs. 7(a) and 7(b) focus on a simplified arrangement of a semiconductor device and an external waveguide, similar effects of the outcoupling persist also for realistic structures where both the semiconductor chip and the external waveguide are combined in an integrated optical circuit.
Fig. 7. (a) Spatial profile of the surface-trapped TM-polarized optical mode in a half of a microcavity bounded by a thick metal bar and having an external waveguide on top. The vertical white solid line depicts the boundary of the semiconductor mesa, the two parallel white horizontal lines denote the external waveguide, whereas the short dash line indicates the part of the figure shown at a larger magnification in Fig. 7(b). (b) Structure in the vicinity of the bar at a larger magnification.
6. TE-polarized surface-trapped modes in a microcavity
A perfect DBR in question terminated by a λ/4–thick layer with a high refractive index does not localize a TE-polarized surface mode. However, as shown in [36,39], TE surface modes exist for some thicknesses of the top layer, different from conventional λ/4. Figures 8(a) through 8(e) address TE-polarized surface modes in a DBR structure, where the top layer having a higher refractive index is of a thickness λ/2. Figures 8(a) through 8(c) display the 1D profiles of the monochromatic TE mode propagating in the x-direction
Fig. 8. (a)–(c) Vertical profiles of the surface-trapped TE-polarized modes at three different wavelengths. (d) Spatial profile of a TE-polarized surface-trapped mode in a half microcavity, strongly extended to the air. (e) Spatial profile of a TE-polarized surface-trapped mode in a half microcavity strongly localized in the λ/2–thick topmost layer which forms a surface cavity.
Figures 8(d) and 8(e) display solutions for a 2D electric field ${E_y}({x,z} )$ in a microcavity bounded by metal bars, whereas the 2D mode profile is an eigenfunction of a 2D wave equation
Figure 8(d) represents a surface-trapped optical mode strongly extended to the air, similar to the TM-modes of Figs. 6(a) through 6(d). In addition, a different kind of a surface-trapped mode is revealed for the TE polarization. The mode of Fig. 8(e) is less extended to the air and mainly localized in the λ/2–thick topmost layer of the structure which forms a surface cavity. It has a longer lifetime due to a weaker diffraction at the boundaries of the metal bars. Further, such mode has a larger thermal shift of the wavelength as it is more located in the semiconductor structure. We note that the existence of a second mode localized close to the surface impedes possible device applications that require a single mode having a temperature-insensitive wavelength. Modes of such type were not found for the TM polarization, as no layer thicker than λ/4 was present in those structures. Thus, TM-polarized modes are more favorable once modes with temperature-insensitive wavelength are targeted.
7. On applications of temperature-insensitive surface-trapped modes
In the present Section we give a brief overview on possible applications of the surface-trapped modes enabling advanced opto-electronic devices. It follows from Figs. 5(a) through 5(f) that the thermal shift of the mode wavelength can be <0.005 nm/K, that is an order of magnitude smaller than that in VCSELs or DFB lasers. Such ultra-small thermal shift can be realized at a long wavelength side of the spectrum of the surface-trapped modes, where the modes are significantly extended to the air. In particular, for the structure in question, those are modes at λ>815 nm. Figure 2(e) reveals the optical confinement factor of such modes <0.25%. The lasing is known to have been realized in edge-emitting geometry in structures with the optical confinement factor as low as 1%, e. g., in longitudinal photonic band crystal lasers [45], in super large optical cavity lasers [46], or in tilted wave lasers [47].
We note that the maximum refractive index step $\Delta n = {n_1} - {n_2}$ in the DBR is a prerequisite for the minimum temperature shift of the mode wavelength. A smaller $\Delta n$ implies a weaker decay of the field in the DBR and a larger fraction of the optical energy of the mode located in the semiconductor DBR. E. g., reduction of $\Delta n$ by a factor of 2 would cause an increase of the wavelength thermal shift from 0.005 nm/K to 0.008 nm/K.
As regards the microcavities considered in the present paper, they can be employed as microlasers whereas metal bars may serve as top contacts mounted on top of a doped semiconductor DBR. In these structures the lasing conditions for the modes with a small confinement factor are more favorable than in the edge-emitting lasers of [45,46,47]. Once the mode tends to be more localized in the air and less in the semiconductor structure, the absorption losses in the doped semiconductor layers and the leakage losses to the substrate reduce proportionally to the reduction of the confinement factor. Absorption in the metal contacts is very low due to an extremely small penetration of the electromagnetic field into the metal. The output losses through the side walls of the mesa are very low as the mode is not extended beneath the metal contacts. The role of the diffraction losses at the metal contacts decreases for longer microcavities. The output losses to an external waveguide or to a fiber as in Figs. 7(a) and 7(b) can be controlled by varying the separation between the semiconductor chip and the waveguide or fiber to reach the optimum conditions for the lasing.
While Figs. 5(a) through 5(f) demonstrate the presence of a plurality of modes in a microcavity, the gain/loss conditions for those modes are very different. Modes more extended to the air are reflected mostly by the metal contacts and may have extremely low losses. Modes more confined in a semiconductor structure are more extended beneath the metal contacts and can reach the mesa side walls. The side walls formed typically by etching are rough in contrast to cleaved facets of edge-emitting lasers thus causing high losses for the modes reaching the side walls. Moreover, if the device is integrated in an optical integrated circuit, no mesa walls are present, and special absorbing sections can be used to promote the mode selection in favor of modes more extended to the air.
It should be noted that, in an optical integrated circuit, semiconductor devices are typically capped by a dielectric, e. g., SiO2 with a refractive index ndiel=1.45. Then surface-trapped optical modes have the mode effective index neff only slightly above ndiel. The field amplitude of such mode typically decays over each DBR period by a factor $\sqrt {({n_1^2 - n_{\textrm{eff}}^2} )/({n_2^2 - n_{\textrm{eff}}^2} )}$, this factor showing only a minor difference between the structures bounded by air and those capped by SiO2. Thus the mode characteristics including an ultra-small thermal shift of the wavelength ∼0.005 nm/K will persist also for practical devices in optical integrated circuits.
Comparing several device concepts providing temperature stabilization of the emission wavelengths of edge-emitting lasers with the approach in question the following notes should be given. Tilted cavity laser [48] supports lasing at a certain cavity mode lying within a narrow stopband of specially designed multilayer interference reflector for the tilted optical modes, thus leading to a thermal stabilization of the lasing wavelength. Devices operating at λ∼1 µm demonstrated the wavelength thermal shift ∼0.13 nm/K [49] well below a standard value of ∼0.4 nm/K determined by a thermal shift of the material gain spectrum. Tilted wave laser governed by the interference of the optical wave propagating along the active waveguide and the one leaked to the passive waveguide and returned back reveals even smaller thermal shift of the lasing wavelength ∼0.1 nm/K [50]. The wavelength-stabilized operation of edge-emitting lasers based on a surface-trapped optical mode [36,39] depends on the mode competition governed by an interplay between the optical confinement factor, leakage loss to the substrate and output loss through the facets. This yields a thermal shift of the lasing wavelength at a rate ∼0.07 nm/K, the maximum fraction of the energy of the optical mode located in the air reaching ∼60%. On the contrary, the surface-trapped optical modes in microcavities, bounded by metal contacts, can be realized such that the thermal shift of the lasing wavelength is an order of magnitude smaller, below ∼0.005 nm/K, while about 90% of the optical energy being located in the air. Similar effects occur if the semiconductor structure is not bounded by the air from the top, but is covered by a dielectric having a refractive index significantly lower than those of semiconductor materials, whereas typical indices of the dielectrics in the range 1.4–2.5 meet this requirement.
We further note here that a possibility of the absolute temperature insensitivity was demonstrated theoretically for tilted cavity lasers in [51]. However, that effect relies to a high extent upon a strong dependence of the refractive index thermal coefficient $dn/dT$ of a semiconductor alloy on the alloy composition once the wavelength is close to the absorption edge for some material used in the structure where the material also exhibits a strong frequency dispersion of the refractive index. Such an effect can hardly be expected for a spectral range far from material resonances. On the contrary, the practical temperature stabilization of the lasing wavelength of the microlasers revealed in the present paper is independent of the particular material properties and thus provides the stabilization mechanism superior over the others.
Although the modeling presented has been performed for a 2D geometry implying a microcavity having a small size in one in-plane direction (x) and a much larger size in another in-plane direction (y), similar surface-trapped modes extended to the air and bounded by a metal contact should exist also in mesas of realistic 3D shape, like circular or rectangular mesas, etc. A 2D modeling performed for cylindrically-symmetric structures confirms the existence of such modes. Thus, microlasers with nearly temperature-insensitive wavelength (e. g., having a thermal shift of the wavelength below 0.005 nm/K) can well be realized.
Further, similar surface-trapped modes should also exist in ring resonators and even without metal bars, whereas the lateral quantization of the mode can occur along the circumference of a circle. The modes mostly located in the air or in the dielectric cap and having a low overlap with the transition sections between the ring and the rest of an optical integrated circuit and, thus, low losses will also exhibit an extremely small wavelength temperature shift.
8. Summary
To conclude, we have presented modeling results demonstrating that the surface-trapped modes existing at a surface of a semiconductor DBR or at an interface between a DBR and a dielectric can be localized in the lateral plane by thick metal bars mounted on top of the DBR. In case of doped semiconductor DBRs the metal bars may serve as contacts. Once a semiconductor structure is not bounded in the lateral plane by cleaved or etched perpendicular facets, only modes significantly localized in the air between the metal contacts can have low losses and can contribute to lasing or amplification. The fraction of the optical energy of such TM-polarized modes located in the air can be about 90%, and the thermal shift of the mode wavelength can be as weak as below 0.005 nm/K, which is an order-of-magnitude smaller than in VCSELs or in edge-emitting lasers also employing surface-trapped modes. This enables microlasers with nearly temperature-insensitive lasing wavelength.
Another application includes a resonant semiconductor optical amplifier (SOA), having a structure similar to that of Fig. 7(a), whereas the optical wave propagating along an external waveguide or an optical fiber is amplified for a narrow spectral interval, in which the surface-trapped optical mode of the structure reaches the waveguide or fiber. As in the case of microlasers, the resonant wavelength of the SOA is also nearly temperature-insensitive. Moreover, no question of mode competition arises for SOAs as only modes sufficiently extended to the air overlap with the waveguide or fiber contributing to the amplification of the optical wave.
Applying a reverse bias to the active medium of a similar structure results in an enhanced absorption in the device thus providing a resonant absorption modulator which can be employed in an integrated optical circuit of the silicon photonics.
Furthermore, using waveguides, in which a substantial part of the optical power propagates not in silicon, but in the air or in a practically lossless dielectric material provides a significant advantage over conventional waveguides in silicon photonics exhibiting significant losses.
Disclosures
The authors claim no conflicts of interests.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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