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Strip-loaded horizontal slot waveguide for routing microdisk laser emission

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Abstract

A strip-loaded horizontal slot waveguide is used for routing the emission of a microdisk laser. The active region of the laser contains InAs/InGaAs/GaAs quantum dots operating in CW mode under optical pumping. Although the quantum dots emit in a broad range, the laser emission is single mode with $\lambda = 1290\;\text{nm}$. Atomic layer deposition is used for the fabrication of the ${\text{TiO}_2}/{\text{SiO}_2}/{\text{TiO}_2}$ slot waveguide covering the 10-µm-diameter disk. The horizontal slot waveguide covers the microdisk and traps the emission in the fundamental TM mode of the three-stack layer. Once light is localized inside this planar waveguide, a loading strip creates an effective index difference, which is sufficient to confine the mode laterally and leads to the guiding of the lasing emission. An analytical model for the five-layer horizontal planar slot waveguide was developed in order to optimize the slot waveguide according to the operation wavelength of the microlaser and the different materials involved. Scanning near-field optical microscopy is used to observe the propagation of light within the waveguide and confirm the mode coupling into it and therefore the guiding along the loading strip.

© 2020 Optical Society of America

1. INTRODUCTION

The lateral emission of semiconductor microdisk (MD) lasers and their compatibility with planar photonic networks make these active components of special interest as coherent light sources for on-chip optical circuits [1,2]. In an MD cavity, the circular symmetry of the resonator modes with the highest quality ($Q$) factor leads to a localized field near the edge of the cavity. Such modes are called whispering gallery modes (WGMs). $Q$ factors as high as $6 \times {10^5}$ [3] and $6 \times {10^6}$ [4] have been achieved in micrometer-size GaAs MD cavities. MD lasers based on the InAs/InGaAs/GaAs quantum dot (QD) active region have already proven their ability to operate in CW mode up to 100°C heatsink free [5]. The minimal diameter at which lasing was achieved for operation at room temperature (under optical pump) was as small as 1 µm [6]. QD MD lasers grown monolithically on silicon substrates and capable of operating at room and elevated temperatures have been recently reported in [7], and a 3 dB bandwidth of 6.5 GHz was achieved. Semiconductor WGM cavities are excellent candidates for low-threshold laser sources. However, the rotational symmetry of the resonator results in an isotropic laser emission output at all azimuthal angles, and therefore typically leads to a low external efficiency as the light power emerges into free space. Optical communications require a direct emission outcoupling of the laser source to a device to perform in an optimal manner. This implies a directional emission of the source. To achieve this, a deformation of the cavity geometry was initially proposed [812]. Although the introduction of smooth deformations of the MDs results in directional outputs with a reduced in-plane divergence angle of the output beam, the out-of-plane (vertical) divergence remains large, since the thickness of the MD is comparable to the emission wavelength. The deformation of a cavity can also cause significant $Q$ spoiling.

Generally, applications of MD lasers require the effective coupling of their emission to other on-chip devices. Thus an approach to coupling an MD laser with a bus optical waveguide, which channels the emission in a predetermined way, e.g., to another on-chip device, needs to be developed. The directional light outcoupling from the MD can be provided by several solutions, for instance, by bonding an InP-based MD on the top of a silicon-on-insulator waveguide layer [13,14], by coupling to the tapered fiber [15], and by aligning with a closely spaced optical waveguide [16]. Therefore, finding a robust method for efficiently collecting emission and delivering it to another device is of great importance for applications of MD lasers in on-chip optical circuits, and represents a crucial step and a challenge in highly integrated photonic circuits. In this paper, we present a configuration in which the MD is embedded in a low-refractive-index medium covered by a slab optical waveguide (here a slab slot waveguide) with a higher effective index. The evanescent tail of the laser resonator mode penetrates into the waveguide, where the penetrated light can be captured, for the waveguide covers the upper surface of the resonator. Once light is trapped in the slab waveguide, one can manipulate it by adding a pattern on top of the slab and use the strip-loading mechanism to achieve directed guiding; in addition, some other functions (filtering, modulation, splitting, etc.) can be provided by this construction. In our opinion, this possibility gives an advantage to the proposed structure over those presented in Refs. [1417].

Recently, strip-loaded horizontal slot waveguides (SLSWs) based on dielectric ${\text{TiO}_2}/{\text{SiO}_2}/{\text{TiO}_2}$ layers have been proposed for the integration of photonic devices [17]. This concept offers extraordinary field confinement in a slot region of a waveguide a few tens of nanometers thick and an overall tiny vertical cross section. The total thickness of the slot waveguide is about 360 nm, which is close to the MD emission region thickness. The SLSWs can be fabricated by means of mass-production-compatible techniques, such as atomic layer deposition (ALD) and nanoimprint replication, and provides low losses (propagation loss ${\lt}{1.2}\;\text{dB/cm}$) [18]. Waveguides of this type preferably support propagation of TM modes.

In this paper, we consider the use of SLSWs for routing MD lasers emission. The paper is organized as follows: First, we present an analytical model of a five-layer horizontal planar structure that can be used for the calculation of the field distribution in the SLSWs to be coupled to the MD laser. After that, we describe the experimental methods used in this study. Then we present the lasing parameters of the MD laser, which allow one to estimate $Q$ factors and threshold powers, and the studies of the MD mode coupling to the SLSWs by means of scanning near-field optical microscopy (SNOM).

Figure 1 illustrates the subject of this study. The substrate with a MD semiconductor laser is planarized with the use of SU-8 photoresist, and on the top of this structure a planar (horizontal) slot optical waveguide composed of a layer of low-index dielectric (silica) placed between two layers of high-index dielectric (titania) is deposited. To provide directional propagation of light in the waveguide a polymer strip is patterned on the surface of the horizontal slot waveguide, which transforms it to the strip-loaded channel optical waveguide. The MD laser is optically pumped with the second harmonic of a YLF:Nd laser radiation ($\lambda = 527\;\text{nm}$), and SNOM is used to characterize the spatial distribution of light generated by the MD laser.

 figure: Fig. 1.

Fig. 1. Concept and initial considerations. (a) Sketch of a microdisk laser under a strip-loaded horizontal slot waveguide structure. The MD is pumped with an external laser source, and the response is observed using scanning near-field optical microscopy. (b) Cross section of the waveguide device in the center of the MD. The waveguide is a five-layer horizontal slot waveguide including the rail layers and the slot. The MD laser is placed in another low-index dielectric material, SU-8. The loading strip is used to provide the lateral light confinement and its propagation in the $z$ direction. (c) Calculated distribution of the normal component of the electric field (${E_x}$) of the fundamental TM mode propagating in the $z$ direction. Calculations were made using the following parameters: ${\text{TiO}_2}$ layer widths are ${h_1} = 135\; \text{nm}$ and ${h_2} = 160\; \text{nm}$, ${\text{SiO}_2}$ slot width is ${h_s} = 70\; \text{nm}$, wavelength is 1290 nm, refractive indices are ${n_h} = 2.30$, ${n_s} = 1.44$, ${n_c} = 1.60$, and ${n_{ph}} = 1.55$.

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2. MATERIALS AND METHODS

A. Analytical Model for a Five-Layer Horizontal Planar Structure

To design the waveguide enabling optical coupling with the laser and supporting light propagation, we performed analytical calculations of the electric field distribution in the proposed five-layer horizontal and asymmetric planar structure. In the calculations, we neglected the width of the loading strip and used the model of a horizontal planar structure, considering that the loading strip influences the vertical optical field distribution only weakly. In other words, we consider a fully planar waveguide with invariance along $y$ and propagation of light along $z$. To our knowledge, the only analytical model for the five-layer structure (two rails of ${\text{SiO}_2}$ in the air environment) was the model proposed by Almeida et al. [19] for a symmetric vertical slot waveguide. We consider the asymmetric case here.

Figure 1(b) depicts the geometry of the model. The slot waveguide consists of two rails with a high refractive index ${n_{ h}}$, the widths of the rails being ${h_1}$ and ${h_2}$. The slot has a width ${h_{ s}}$ with a low refractive index ${n_{s}}$ and is placed between the rails. The substrate is a dielectric material (photoresist, SU-8) with a refractive index ${n_{{ ph}}}$, and the cover material is another dielectric of refractive index ${n_{ c}}$. Both the photoresist and cover dielectric are considered semi-infinite in the $x$ direction. Dotted lines in Fig. 1(b) show the MD position in the photoresist with respect to the SLSW geometry. An SLSW structure can sustain both quasi-TE and quasi-TM propagating modes. However, only the quasi-TM fundamental mode shows a high enhancement in the low-refractive-index slot region of the waveguide because of the discontinuity of the electric field at the interface between the rails and the slot. Considering that the MD cavity supports TM modes, we will carry the calculations only for TM propagation. Because of the low-contrast waveguide, we also consider invariance in the $y$ direction (slab waveguide). For a TM mode (${H_y},{E_x},{E_z}$) propagating in the $z$ direction with a propagation constant $\beta$, the problem can be solved by considering only the magnetic field as ${{\bf H}_{\bf y}} = {{\bf e}_{\bf y}}{H_y}\exp [i(\beta z - \omega t)]$, where ${{\bf e}_{\bf y}}$ is the unit vector along $y$. The amplitudes of the magnetic and electric fields ${H_y}$ and ${E_x}$ can be found from the Helmholtz equation [20].

For a five-layer horizontal structure, the solution for the electric field (the details of the derivation are given in Appendix A) is described by

$$\begin{split}&{E_x} = {A_0}\\&\left\{{\begin{array}{*{20}{l}}\!{\frac{1}{{n_{{ph}}^2}}{e^{{\gamma _{{ ph}}}x}},}&{x \lt 0}\\[10pt]{\frac{1}{{n_h^2}}[\cos ({k_h}x) + {B_2}\sin ({k_h}x)],}&{0 \lt x \lt a}\\[9pt]{\frac{1}{{n_s^2}}[{A_1}\cosh ({\gamma _s}(x - a)) + {A_2}\sinh ({\gamma _s}(x - a))],}&{a \lt x \lt b}\\[9pt]{\frac{1}{{n_h^2}}[{C_1}\cos ({k_h}(x - b)) + {C_2}\sin ({k_h}(x - b))],}&{b \lt x \lt c}\\[9pt]{\frac{1}{{n_c^2}}E{e^{- {\gamma _c}(x - c)}},}&{x \gt c}\end{array}} \right.\!\!,\end{split}$$
where $a$, $b$, and $c$ are coordinates corresponding to the interfaces between the layers [see Fig. 1(b)]. We also define $\gamma _{{ph}}^2 = {\beta ^2} - k_0^2n_{{ph}}^2$, $k_{ h}^2 = k_0^2n_{{ph}}^2 - {\beta ^2}$, $\gamma _{s}^2 = {\beta ^2} - k_0^2n_{s}^2$, and $\gamma _{c}^2 = {\beta ^2} - k_0^2n_{c}^2$, where ${k_0} = \frac{{2\pi}}{\lambda}$ is the vacuum wavenumber. ${A_0}$ is a normalizing constant, and ${A_1}$, ${A_2}$, ${B_2}$, ${C_1}$, ${C_2}$, $E$ are the coefficients to be found from the boundary conditions at the interfaces. These parameters are given in Eqs. (A5)–(A10) of Appendix A.

The refractive indices of most of the materials involved in the device have been measured by ellipsometry as ${n_{ h}} = 2.30$, ${n_{ s}} = 1.44$, and ${n_{ c}} = 1.60$ at the MD operating wavelength. The refractive index of SU-8 is ${n_{{ ph}}} = 1.55$ according to the manufacturer [21]. The optimization of the SLSW structure at the emission wavelength of the MD ($\lambda = 1290\;\text{nm}$) was performed using the resolution of the system of Eq. (1). ${\text{TiO}_2}$ layer widths are ${h_1} = 135\;\text{nm}$, ${h_2} = 160\;\text{nm}$, and the ${\text{SiO}_2}$ slot width is ${h_{s}} = 70\;\text{nm}$. The resulting amplitude distribution for ${E_x}$ is shown in Fig. 1(c). These results are in excellent agreement with previous studies, where solutions were obtained with a mode solver [17,22].

B. Experiment Description

The semiconductor structure was grown using molecular beam epitaxy on a semi-insulating GaAs (100) substrate. The active region comprises five layers of $\text{InAs}/{\text{In}_{0.15}}{\text{Ga}_{0.85}}\text{As/GaAs}$ QDs with a 30-nm-thick GaAs spacer between each layer. The spectral position of the QD ground-state transition is located around $\lambda = 1.3\;\unicode{x00B5}\text{m}$ at room temperature. The active region is placed in the middle of the GaAs layer, confined from both sides with 10-nm-thick ${\text{Al}_{0.3}}{\text{Ga}_{0.7}}\text{As}$ barriers. The total thickness of the vertical waveguide is 365 nm. This waveguide layer was grown on top of a 400-nm-thick ${\text{Al}_{0.98}}{\text{Ga}_{0.02}}\text{As}$ layer. The ${\text{Al}_{0.98}}{\text{Ga}_{0.02}}\text{As}$ layer was then transformed into an ${\text{(AlGa)}_x}{\text{O}_y}$ layer by a selective oxidation process to ensure the optical confinement at the substrate side. MDs 10 µm in diameter were fabricated using photolithography and argon ion beam etching.

To ease the integration of the MD with the waveguide, planarization of the MDs was done using the epoxy resist SU-8. The waveguide layers are deposited by an ALD system (Beneq TFS 200). The waveguide is a horizontal slot waveguide made of ${\text{TiO}_2}/{\text{SiO}_2}/{\text{TiO}_2}$, which can be strip loaded. According to the modeling, these parameters ensure the propagation of the MD radiation in the operating wavelength range of the laser, i.e., $\lambda = 1290\;\text{nm}$. The effective index measurement of the fundamental quasi-TM mode of the horizontal slot waveguide before the strip deposition confirmed the validity of the chosen thicknesses of the layers. The lateral confinement of light is done by strip loading. The loading strips were made of the negative electronic resist AZ nLOF 2070 with a thickness of 250 nm and a width of 15 µm using electron-beam lithography (Fig. 2). Testing of the SLSWs using a rotating polarization analyzer demonstrated its predominant guiding of modes with TM polarization, as anticipated [17]. In this waveguide, the measured propagation losses were less than 1 dB/cm (using a Metricon prism coupler). This value is an overestimate, because by using this technique we have not been able to measure the “propagation loss” due to the scattering at the surface, and we think that the uncertainty of the loss measurement is responsible for the high value. It is worth noting that both the materials used for the slot waveguide and the low-index SU-8 resist have a relatively low dispersion around 1.3 µm wavelength, and the mode dispersion of the waveguiding structure around the stable region connected to the geometrical parameters is also low [17,23]. These enable the functionality of the considered structure in the spectral range around 1300 nm covered by this type of lasers.

 figure: Fig. 2.

Fig. 2. Top view of the waveguide-integrated microdisk. Optical microscope image of a 10 µm MD laser under the strip-loaded horizontal slot waveguide. The ring surrounding the MD laser is a result of the difference in etching rate between the materials of the QDs and the bottom ${(\text{AlGa})_x}{\text{O}_y}$ layer insulating the cavity from the substrate.

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Optical measurements were carried out at room temperature. Photoluminescence (PL) and SNOM spectra were observed and measured while illuminating the samples with the second harmonic of a CW-operating YLF:Nd laser ($\lambda = 527\;\text{nm}$). To test the laser operation, an exciting laser beam (pump) was focused on the MD’s surface by a ${100 \times}$ microscope objective (Mitutoyo M Plan Apo NIR HR) with the numerical aperture $\text{NA} = {0.5}$. Emitted light was collected from the disk by the same objective under normal incidence. Collected light was focused on the entrance slit of a monochromator (Sol Instruments MS5204i) and detected by an InGaAs CCD camera (Andor iDUS). A cantilever-based scanning near-field optical microscope (NT-MDT NTEGRA Spectra) measured the spatial distribution of light at the surface of the device. The aperture of the cantilever (SNOM_NC) was ${\sim}100\;\text{nm}$. For SNOM measurements of the mode propagation along the waveguide, the YLF:Nd laser emission was focused on a spot of about 20 µm diameter using a lensed fiber at an incident angle of 45° [Fig. 1(a)]. The sample was mounted on a piezo-driven translator for near-field mapping in the $y - z$ plane and simultaneous measurement of the sample morphology by atomic force microscopy (AFM) ($x$ direction).

3. RESULTS AND DISCUSSION

A. Lasing in the MD

PL spectra of the initial MD laser uncoated by dielectric layers comprise the broadband spontaneous emission of the QD active region modulated by low-$Q$ resonator modes (so-called Fabry–Perot modes) and narrow lines corresponding to high-$Q$ WGMs of the resonator. Even a few-micrometer-sized semiconductor MD cavity supports a great number of WGMs, whereas the QD optical transition is relatively broad. As a result, one can simultaneously observe in Fig. 3(a) (upper graph) more than five narrow lines corresponding to WGMs of different azimuthal and radial orders in the wavelength range from 1260 to 1300 nm. With increasing pumping level, the dominant WGM line intensity grows and single-mode lasing takes place [see lower graph in Fig. 3(a)]. After this observation, we deposited the SU-8 and SLSWs and observed that single-mode generation was preserved under intensive optical pumping. The lasing behavior of the structure with SLSWs is confirmed by a distinct kink at the dependence of a WGM line intensity versus optical pump power and by the narrowing of the linewidth. The light–light curve and lasing mode linewidth against excitation power are presented in Fig. 3(b). The estimated threshold power is ${P_{\text{th}}} \sim 750 \pm 30\;\unicode{x00B5} \text{W}$, and $Q = \lambda /\Delta \lambda = 26000$.

 figure: Fig. 3.

Fig. 3. (a) MD laser spectrum at low optical pump and (b) integrated intensity of the dominant mode line (squares) and mode linewidth (circles) versus the optical pump power for the 10-µm-diameter microlaser of the virgin MD without the SLSW (half-filled) and under the SLSW. The arrow denotes the “kink” showing the lasing threshold.

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The observed single-mode lasing is highly desirable for a variety of applications of lasers coupled to a waveguide to exclude possible mode leaking and other unwanted effects. The influence of dielectical layers on microlasers has been previously observed and reported [24].

B. SNOM for Spectral and Spatial Mapping

We measured the spatial distribution of the MD laser emission using SNOM. The overall setup is shown in Fig. 1(a). The objective of this measurement is to confirm the coupling of the high-$Q$-factor resonator mode to the SLSW structure.

The pump power was chosen to be well above the lasing threshold to achieve a stable SNOM signal, and the device was measured over a broad spectral range ($1280\;\text{nm} \lt \lambda \lt 1315\;\text{nm}$). The morphology of the sample surface, obtained simultaneously by AFM [Fig. 4(a)], helped to determine the exact locations of the loading strip and the microlaser on the sample surface. The spontaneous emission from the InAs/InGaAs/GaAs QDs is observed exactly at the microlaser location [Fig. 4(b)]. The slight change in intensity observed across the disk can be attributed to the roughness of the loading strip surface, which modifies the light outcoupling to the SNOM probe. Despite the method used for the material deposition (ALD and spin coating), the numerous steps of the fabrication process may have induced some roughness that the ALD coating follows [25] in a conformal way. One can see in Fig. 4(a) that, for instance, the contour of the MD is higher than the central part. The diameter of the obtained bright spot corresponds to the MD resonator diameter (${\sim}10\;\unicode{x00B5}\text{m}$), as shown in Fig. 4(b) in the cross section of the intensity distribution. The near-field distribution of the lasing WGM at $\lambda = 1290\;\text{nm}$ [Fig. 4(c)] is quite different from that of the entire spectrum. The lasing emission is not only distributed over the MD location but also penetrates above (along the SLSWs), demonstrating the coupling of the lasing emission from the MD laser to the waveguide and not only the predominant out-of-plane radiation at other wavelengths. The decrease in intensity along the $z$ direction in Fig. 4(c) comes from the fact that the waveguide is buried. And the evanescent tail of the slot mode is weak above the loading strip compared to the large emission just above the MD. At the MD region the guided mode is not yet established in the slot waveguide, but the field is confined in the entire dielectric stack [17,22,23]. A clear preferential direction of propagation is observed along the loading strip, proving the confinement of light within the slot (decay in intensity) as well as the guiding, and therefore the validity of the concept.

 figure: Fig. 4.

Fig. 4. Spectral and spatial near-field investigation of the microlaser and its coupling to the strip-loaded slot waveguide. Maps of (a) AFM and (b), (c) SNOM (top images), together with (a) height and (b), (c) intensity profiles of the 10 µm microdisk laser under the SLSW structure. SNOM images are taken (b) for the entire spontaneous emission spectral range ($1280\; \text{nm} \lt \lambda \lt 1285\; \text{nm}$) and (c) for the lasing mode only ($\lambda = 1290\;\text{nm}$). The location of the MD is shown by blue arrow in (a), and the profile measurement lines are marked with solid white lines on the corresponding top maps.

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As a final proof, we tilted the structure from the vertical orientation by ${\sim}{7^ \circ}$ in the clockwise direction [Fig. 5(a)]. The near-field signal intensity distribution was measured above an area including two adjacent microlasers pumped simultaneously and clearly observable in Fig. 5. The QD spontaneous emissions, observed from both microlasers, are separated from each other by a distance of 50 µm. This indicates the positions of the MDs on the sample [see Fig. 5(b) with the corresponding cross section]. The measured spatial distribution of the laser radiation [at $\lambda = 1290\;\text{nm}$; see Fig. 5(c)] is also tilted by the same angle as the loading strip. This confirms that the lasing WGM is coupled to the ${\text{TiO}_2}/{\text{SiO}_2}/{\text{TiO}_2}$ optical waveguide and guided under the loading strip.

 figure: Fig. 5.

Fig. 5. Near-field investigation for a 7°-tilted sample. (a) An optical microscope image of two 10 µm microdisk lasers under the strip-loaded slot waveguide structure (top) and a near-IR camera image taken from the cleaved optical waveguide. (b) Near-field signal map of spontaneous emission from InAs/InGaAs/GaAs quantum dots over the emission spectral range of the QDs ($1280\; \text{nm} \lt \lambda \lt 1285\; \text{nm}$) (top) and photoluminescence signal observed in a cross section along the loading strip (white line). (c) Near-field mapping of the lasing WGM ($\lambda = 1290\; \text{nm}$) line (top) and its corresponding profile. The contour of the loading strip is highlighted by dashed white lines and the cross sections are marked with solid lines on the maps.

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From the intensity distribution measured by SNOM, in both Figs. 4(c) and 5(c), one can see that the SNOM signal rapidly decreases with the distance from the disk resonator. This behavior is expected because the guided mode is buried, and after its establishment in the waveguide one cannot detect it even by SNOM because the exponentially decaying tail of the guided mode is not long enough to be scattered by the SNOM probe. This can be seen from the calculations illustrated by Fig. 1(c). We consider here a buried waveguide with a high field confinement, and the 250-nm-thick loading strip prevents the evanescent field from being sensed, except that more scattering/radiation occurs before the modal distribution is established. Actually, a distance of a few wavelengths is sufficient for the mode to be established, even when light is injected into it far from the center of the waveguide cross section. At longer distances from the waveguide excitation region, SNOM does not register the propagating light, because no electric field penetrates further than the polymer strip. In this situation, the existence of the waveguide mode can be confirmed either by introducing an additional coupler, e.g., a grating, or by observing the light at the waveguide cut. The latter is presented by the image in Fig. 5(a), which has been captured with a near-IR camera and clearly shows the quasi-TM mode at the output of the cleaved sample.

4. CONCLUSION

In this paper, we proposed an innovative concept for the coupling and routing light emitted by an MD laser. The use of an SLSW allows high confinement of the field inside a very thin layer. The main advantage of the SLSW, compared to a simple high-refractive-index film, is the isolation from the substrate and different cover materials that the slot rails offer. The mode is highly buried but cannot leak to the substrate, and thus presents low loss and high coupling. This opens up a totally new path (to our knowledge) to greater integration of QD-based light sources on-chip for low-loss devices. We have demonstrated the concept using the SNOM technique. Some samples cut after the MDs made the output mode observable. However, the clear demonstration of the concept comes from the near field. To design a waveguide enabling optical coupling with the laser and supporting light propagation, an analytical model of the electric field distribution in the five-layer horizontal and asymmetric planar structure was developed. Finally, it should be noted that the loading strip used in this work led to a multimode waveguide. Several geometries of waveguides combined with a circular one-dimensional photonic crystal in the coupling area are already under investigation to provide more efficient coupling and single-mode behavior of the waveguide.

APPENDIX A: ANALYTICAL DESCRIPTION OF MODAL FIELD DISTRIBUTION IN A HORIZONTAL SLOT WAVEGUIDE

We consider the planar waveguiding structure depicted in Fig. 1(b) and lying in the $y - z$ plane.

For a TM mode (${H_y},{E_x},{E_z}$) propagating in the $z$ direction with a propagation constant $\beta$,

$${{\bf H}_{\bf y}} = {{\bf e}_{\bf y}}{H_y}\exp \,[i(\beta z - \omega t)],$$
and the amplitude of the magnetic field ${H_y}$ can be found from [19]
$$\frac{{{\partial ^2}{H_y}}}{{\partial {z^2}}} - ({\beta ^2} - k_0^2{\varepsilon _i}){H_y} = 0,$$
where ${\varepsilon _i} = n_i^2$ for each layer of the structure and ${k_0} = 2\pi \lambda$ is a wave vector in vacuum. Equation (A1) coincides with the well-known Schrodinger equation for a particle in a quantum well, and has a solution in the form of a sum of oscillating or evanescent functions, depending on whether (${\beta ^2} - k_0^2{\varepsilon _i}$) is negative or positive. Taking into account the fact that for a propagating mode, the propagation constant $\beta$ should satisfy the relation [19]
$$k_0^2n_{ s}^2,k_0^2n_{ c}^2,k_0^2n_{{ ph}}^2 \lt {\beta ^2} \lt k_0^2n_{ h}^2,$$
and the amplitude of the electric field normal to the interface, ${E_x}$, is related to ${H_y}$ as
$${E_x} = \frac{\beta}{{\omega {\varepsilon _0}{\varepsilon _i}}}{H_y},$$
one has the solution to Eq. (A1) as
$${E_x} = {A_0}\left\{{\begin{array}{*{20}{l}}{\frac{1}{{n_{{ph}}^2}}{e^{{\gamma _{{ph}}}x}},}&{x \lt 0}\\[9pt]{\frac{1}{{n_h^2}}[\cos ({k_h}x) + {B_2}\sin ({k_h}x)],}&{0 \lt x \lt a}\\[9pt]{\frac{1}{{n_s^2}}[{A_1}\cosh ({\gamma _s}(x - a)) + {A_2}\sinh ({\gamma _s}(x - a))],}&{a \lt x \lt b}\\[9pt]{\frac{1}{{n_h^2}}[{C_1}\cos ({k_h}(x - b)) + {C_2}\sin ({k_h}(x - b))],}&{b \lt x \lt c}\\[9pt]{\frac{1}{{n_c^2}}E{e^{- {\gamma _c}(x - c)}},}&{x \gt c}\end{array}} \right.,$$
where $a$, $b$, and $c$ are coordinates corresponding to the interfaces between the layers [see Fig. 1(b)] and ${A_0}$ is a normalizing constant. ${A_1}$, ${A_2}$, ${B_2}$, ${C_1}$, ${C_2}$, and $E$ can be found from the boundary conditions at the interfaces and are as follows:
$${B_2} = \frac{{{\gamma _{{ ph}}}n_{ h}^2}}{{n_{{ ph}}^2{k_{ h}}}},$$
$${A_1} = \cos ({k_{ h}}a) + \frac{{{\gamma _{{ph}}}n_{ h}^2}}{{n_{{ ph}}^2{k_{ h}}}}\sin ({k_{ h}}a),$$
$${A_2} = \frac{{{k_{ h}}n_{ s}^2}}{{n_{h}^2{\gamma _{s}}}}\left(- \sin ({k_{h}}a) + \frac{{{\gamma _{{ph}}}n_{h}^2}}{{n_{{ ph}}^2{k_{ h}}}}\cos ({k_{h}}a)\right),$$
$${C_1} = \left(\cos ({k_{h}}a) + \frac{{{\gamma _{{ph}}}n_{h}^2}}{{n_{{ph}}^2{k_{h}}}}\sin ({k_{h}}a)\right)\cosh ({\gamma _{s}}(b - a)) \\ + \left(\frac{{{k_{h}}n_{s}^2}}{{n_{h}^2{\gamma _{s}}}}\left(- \sin ({k_{h}}a) + \frac{{{\gamma _{{ph}}}n_{h}^2}}{{n_{{ph}}^2{k_{h}}}}\cos ({k_{h}}a)\right)\right)\sinh ({\gamma _{s}}(b - a)),$$
$${C_2} = \frac{{{\gamma _{s}}n_{h}^2}}{{n_{s}^2{k_{h}}}}\left[\left(\!\cos ({k_{h}}a) + \frac{{{\gamma _{{ph}}}n_{h}^2}}{{n_{{ph}}^2{k_{h}}}}\sin ({k_{h}}a)\!\right)\sinh ({\gamma _{s}}(b - a))+\left(\frac{{{k_{h}}n_{s}^2}}{{n_{h}^2{\gamma _{s}}}}(- \sin ({k_{h}}a) + \frac{{{\gamma _{{ph}}}n_{h}^2}}{{n_{{ph}}^2{k_{h}}}}\cos ({k_{h}}a)\!\right)\cosh ({\gamma _{s}}(b - a))\!\right],$$
$$\begin{split}E &= \left[\left(\!\cos ({k_{h}}a) + \frac{{{\gamma _{{ph}}}n_{h}^2}}{{n_{{ph}}^2{k_{h}}}}\sin ({k_{h}}a)\!\right)\cosh ({\gamma _{s}}(b - a)) + \left(\frac{{{k_{h}}n_{s}^2}}{{n_{h}^2{\gamma _{ s}}}}\left(\!- \sin ({k_{h}}a) + \frac{{{\gamma _{{ph}}}n_{h}^2}}{{n_{{ph}}^2{k_{h}}}}\cos ({k_{h}}a)\!\right)\sinh ({\gamma _{s}}(b - a))]\cos ({k_{h}}(c - b)\!\right) \right.\\&\quad + \left[\frac{{{\gamma _{s}}n_{h}^2}}{{n_{s}^2{k_{h}}}}\left(\cos ({k_{h}}a) + \frac{{{\gamma _{{ph}}}n_{h}^2}}{{n_{{ph}}^2{k_{h}}}}\sin ({k_{h}}a)\right)\sinh ({\gamma _{s}}(b - a)) + \left(\frac{{{k_{h}}n_{s}^2}}{{n_{h}^2{\gamma _{s}}}}\left(- \sin ({k_{h}}a) + \frac{{{\gamma _{{ph}}}n_{h}^2}}{{n_{{ph}}^2{k_{h}}}}\cos ({k_{h}}a)\right)\cosh ({\gamma _{s}}(b - a)\right) \right]\\&\quad\times\sin ({k_{h}}(c - b)).\end{split}$$

Finally, the propagation constant $\beta$ is determined with the transcendental equation

$${-}\frac{{{k_{h}}n_{c}^2}}{{n_{h}^2{\gamma _{c}}}} = \tan \left[{k_{h}}(c - a) + \arctan \left(\frac{{n_{s}^2{k_{h}}}}{{{\gamma _{s}}n_{h}^2}}\frac{{\frac{{n_{h}^2{\gamma _{s}}}}{{{k_{h}}n_{s}^2}}\tan \left({k_{h}}a + \arctan \left(\frac{{n_{{ph}}^2{k_{h}}}}{{{\gamma _{{ph}}}n_{h}^2}}\right)\right) + \tanh ({\gamma _{s}}(b - a))}}{{1 + \frac{{n_{{ph}}^2{k_{h}}}}{{{\gamma _{{ph}}}n_{h}^2}}\tan \left({k_{h}}a + \arctan \left(\frac{{n_{{ph}}^2{k_{h}}}}{{{\gamma _{{ph}}}n_{h}^2}}\right)\right)\tanh ({\gamma _{s}}(b - a))}}\right)\right].$$

Funding

Academy of Finland Flagship PREIN (320166); Academy of Finland project (323052); Russian Foundation for Basic Research (18-29-20063, 20-02-00334A); Ministry of Science and Higher Education of the Russian Federation; Fundamental Research Programme of the Presidium of the Russian Academy of Sciences; Horizon 2020 Framework Programme (813159); Support from the Basic Research Program of the National Research University Higher School of Economics.

Acknowledgment

The authors are grateful to Janne Laukkanen for help with ALD and e-beam lithography and Khairul Alam for e-beam alignment.

Disclosures

The authors declare no conflicts of interest.

REFERENCES

1. Q. Song, “Emerging opportunities for ultra-high Q whispering gallery mode microcavities,” Sci. China Phys. Mech. Astron. 62, 74231 (2019). [CrossRef]  

2. L. He, S. K. Ozdemir, and L. Yang, “Whispering gallery microcavity lasers,” Laser Photon. Rev. 7, 60–82 (2012). [CrossRef]  

3. C. P. Michael, K. Srinivasan, T. J. Johnson, O. Painter, K. H. Lee, K. Hennessy, H. Kim, and E. Hu, “Wavelength- and material-dependent absorption in GaAs and AlGaAs microcavities,” Appl. Phys. Lett. 90, 051108 (2007). [CrossRef]  

4. B. Guha, F. Marsault, F. Cadiz, L. Morgenroth, V. Ulin, V. Berkovitz, A. Lemaitre, C. Gomez, A. Amo, S. Combrie, B. Gerard, G. Leo, and I. Favero, “Surface-enhanced gallium arsenide photonic resonator with quality factor of 6×106,” Optica 4, 218–221 (2017). [CrossRef]  

5. N. Kryzhanovskaya, E. Moiseev, Y. V. Kudashova, F. Zubov, A. Lipovskii, M. Kulagina, S. Troshkov, Y. M. Zadiranov, D. Livshits, M. Maximov, and A. Zhukov, “Continuous-wave lasing at 100°C in 1.3 µm quantum dot microdisk diode laser,” Electron. Lett. 51, 1354–1355 (2015). [CrossRef]  

6. N. V. Kryzhanovskaya, A. E. Zhukov, M. V. Maximov, E. I. Moiseev, I. I. Shostak, A. M. Nadtochiy, Y. V. Kudashova, A. A. Lipovskii, M. M. Kulagina, and S. I. Troshkov, “Room temperature lasing in 1-µm microdisk quantum dot lasers,” J. Sel. Top. Quantum Electron. 21, 709–713 (2015). [CrossRef]  

7. N. V. Kryzhanovskaya, E. I. Moiseev, Y. S. Polubavkina, M. V. Maximov, D. V. Mokhov, I. A. Morozov, M. M. Kulagina, Y. M. Zadiranov, A. A. Lipovskii, M. Tang, M. Liao, J. Wu, S. Chen, H. Liu, and A. E. Zhukov, “Elevated temperature lasing from injection microdisk lasers on silicon,” Laser Phys. Lett. 15, 015802 (2017). [CrossRef]  

8. T. Harayama and S. Shinohara, “Two-dimensional microcavity lasers,” Laser Photon. Rev. 5, 247–271 (2011). [CrossRef]  

9. C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998). [CrossRef]  

10. M. Lebental, J. S. Lauret, R. Hierle, and J. Zyss, “Highly directional stadium-shaped polymer microlasers,” Appl. Phys. Lett. 88, 031108 (2006). [CrossRef]  

11. M. Hentschel and T.-Y. Kwon, “Designing and understanding directional emission from spiral microlasers,” Opt. Lett. 34, 163–165 (2009). [CrossRef]  

12. Q. Song, W. Fang, B. Liu, S.-T. Ho, G. S. Solomon, and H. Cao, “Chaotic microcavity laser with high quality factor and unidirectional output,” Phys. Rev. A 80, 041807 (2009). [CrossRef]  

13. J. V. Campenhout, P. Rojo-Romeo, P. Regreny, C. Seassal, D. V. Thourhout, S. Verstuyft, L. D. Cioccio, J.-M. Fedeli, C. Lagahe, and R. Baets, “Electrically pumped InP-based microdisk lasers integrated with a nanophotonic silicon-on-insulator waveguide circuit,” Opt. Express 15, 6744–6749 (2007). [CrossRef]  

14. P. Mechet, F. Raineri, A. Bazin, Y. Halioua, T. Spuesens, T. J. Karle, P. Regreny, P. Monnier, D. V. Thourhout, I. Sagnes, R. Raj, G. Roelkens, and G. Morthier, “Uniformity of the lasing wavelength of heterogeneously integrated InP microdisk lasers on SOI,” Opt. Express 21, 10622–10631 (2013). [CrossRef]  

15. K. Srinivasan, M. Borselli, O. Painter, A. Stintz, and S. Krishna, “Cavity Q, mode volume, and lasing threshold in small diameter AlGaAs microdisks with embedded quantum dots,” Opt. Express 14, 1094–1105 (2006). [CrossRef]  

16. S. Koseki, B. Zhang, K. D. Greve, and Y. Yamamoto, “Monolithic integration of quantum dot containing microdisk microcavities coupled to air-suspended waveguides,” Appl. Phys. Lett. 94, 051110 (2009). [CrossRef]  

17. M. Roussey, L. Ahmadi, S. Pélisset, M. Häyrinen, A. Bera, V. Kontturi, J. Laukkanen, I. Vartiainen, S. Honkanen, and M. Kuittinen, “Strip-loaded horizontal slot waveguide,” Opt. Lett. 42, 211–214 (2017). [CrossRef]  

18. R. Ding, T. Baehr-Jones, W.-J. Kim, B. Boyko, R. Bojko, A. Spott, A. Pomerene, C. Hill, W. Reinhardt, and M. Hochberg, “Low-loss asymmetric strip-loaded slot waveguides in silicon-on-insulator,” Appl. Phys. Lett. 98, 233303 (2011). [CrossRef]  

19. V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29, 1209–1211 (2004). [CrossRef]  

20. E. M. Conwell, “Integrated optics,” Phys. Today 29(5), 48–59 (1976). [CrossRef]  

21. http://microchem.com/pdf/SU-82000DataSheet2025thru2075Ver4.pdf.

22. S. Pélisset and M. Roussey, “Strip-loaded nano-photonics on horizontal slot waveguide,” J. Eur. Opt. Soc. Rapid Publ. 15, 6 (2019). [CrossRef]  

23. S. Pélisset, J. Laukkanen, M. Kuittinen, S. Honkanen, and M. Roussey, “Modal properties of a strip-loaded horizontal slot waveguide,” J. Eur. Opt. Soc. Rapid Publ. 13, 37 (2017). [CrossRef]  

24. I. Reduto, M. Fetisova, N. Kryzhanovskaya, K. Kotlyar, A. Raskhodchikov, S. Scherbak, Y. Guseva, A. Lipovskii, M. Maximov, and A. Zhukov, “Lasing in III–V microdisk core–TiO2 shell lasers,” J. Opt. Soc. Am. B 36, 2285–2291 (2019). [CrossRef]  

25. M. Häyrinen, M. Roussey, A. Bera, M. Kuittinen, and S. Honkanen, “Atomic Layer Re-Deposition for Nanoscale Devices,” in Encyclopedia of Plasma Technology, J. L. Shohet, ed. (Taylor & Francis, 2016), Vol. 1, pp. 143–151.

References

  • View by:

  1. Q. Song, “Emerging opportunities for ultra-high Q whispering gallery mode microcavities,” Sci. China Phys. Mech. Astron. 62, 74231 (2019).
    [Crossref]
  2. L. He, S. K. Ozdemir, and L. Yang, “Whispering gallery microcavity lasers,” Laser Photon. Rev. 7, 60–82 (2012).
    [Crossref]
  3. C. P. Michael, K. Srinivasan, T. J. Johnson, O. Painter, K. H. Lee, K. Hennessy, H. Kim, and E. Hu, “Wavelength- and material-dependent absorption in GaAs and AlGaAs microcavities,” Appl. Phys. Lett. 90, 051108 (2007).
    [Crossref]
  4. B. Guha, F. Marsault, F. Cadiz, L. Morgenroth, V. Ulin, V. Berkovitz, A. Lemaitre, C. Gomez, A. Amo, S. Combrie, B. Gerard, G. Leo, and I. Favero, “Surface-enhanced gallium arsenide photonic resonator with quality factor of 6×106,” Optica 4, 218–221 (2017).
    [Crossref]
  5. N. Kryzhanovskaya, E. Moiseev, Y. V. Kudashova, F. Zubov, A. Lipovskii, M. Kulagina, S. Troshkov, Y. M. Zadiranov, D. Livshits, M. Maximov, and A. Zhukov, “Continuous-wave lasing at 100°C in 1.3 µm quantum dot microdisk diode laser,” Electron. Lett. 51, 1354–1355 (2015).
    [Crossref]
  6. N. V. Kryzhanovskaya, A. E. Zhukov, M. V. Maximov, E. I. Moiseev, I. I. Shostak, A. M. Nadtochiy, Y. V. Kudashova, A. A. Lipovskii, M. M. Kulagina, and S. I. Troshkov, “Room temperature lasing in 1-µm microdisk quantum dot lasers,” J. Sel. Top. Quantum Electron. 21, 709–713 (2015).
    [Crossref]
  7. N. V. Kryzhanovskaya, E. I. Moiseev, Y. S. Polubavkina, M. V. Maximov, D. V. Mokhov, I. A. Morozov, M. M. Kulagina, Y. M. Zadiranov, A. A. Lipovskii, M. Tang, M. Liao, J. Wu, S. Chen, H. Liu, and A. E. Zhukov, “Elevated temperature lasing from injection microdisk lasers on silicon,” Laser Phys. Lett. 15, 015802 (2017).
    [Crossref]
  8. T. Harayama and S. Shinohara, “Two-dimensional microcavity lasers,” Laser Photon. Rev. 5, 247–271 (2011).
    [Crossref]
  9. C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
    [Crossref]
  10. M. Lebental, J. S. Lauret, R. Hierle, and J. Zyss, “Highly directional stadium-shaped polymer microlasers,” Appl. Phys. Lett. 88, 031108 (2006).
    [Crossref]
  11. M. Hentschel and T.-Y. Kwon, “Designing and understanding directional emission from spiral microlasers,” Opt. Lett. 34, 163–165 (2009).
    [Crossref]
  12. Q. Song, W. Fang, B. Liu, S.-T. Ho, G. S. Solomon, and H. Cao, “Chaotic microcavity laser with high quality factor and unidirectional output,” Phys. Rev. A 80, 041807 (2009).
    [Crossref]
  13. J. V. Campenhout, P. Rojo-Romeo, P. Regreny, C. Seassal, D. V. Thourhout, S. Verstuyft, L. D. Cioccio, J.-M. Fedeli, C. Lagahe, and R. Baets, “Electrically pumped InP-based microdisk lasers integrated with a nanophotonic silicon-on-insulator waveguide circuit,” Opt. Express 15, 6744–6749 (2007).
    [Crossref]
  14. P. Mechet, F. Raineri, A. Bazin, Y. Halioua, T. Spuesens, T. J. Karle, P. Regreny, P. Monnier, D. V. Thourhout, I. Sagnes, R. Raj, G. Roelkens, and G. Morthier, “Uniformity of the lasing wavelength of heterogeneously integrated InP microdisk lasers on SOI,” Opt. Express 21, 10622–10631 (2013).
    [Crossref]
  15. K. Srinivasan, M. Borselli, O. Painter, A. Stintz, and S. Krishna, “Cavity Q, mode volume, and lasing threshold in small diameter AlGaAs microdisks with embedded quantum dots,” Opt. Express 14, 1094–1105 (2006).
    [Crossref]
  16. S. Koseki, B. Zhang, K. D. Greve, and Y. Yamamoto, “Monolithic integration of quantum dot containing microdisk microcavities coupled to air-suspended waveguides,” Appl. Phys. Lett. 94, 051110 (2009).
    [Crossref]
  17. M. Roussey, L. Ahmadi, S. Pélisset, M. Häyrinen, A. Bera, V. Kontturi, J. Laukkanen, I. Vartiainen, S. Honkanen, and M. Kuittinen, “Strip-loaded horizontal slot waveguide,” Opt. Lett. 42, 211–214 (2017).
    [Crossref]
  18. R. Ding, T. Baehr-Jones, W.-J. Kim, B. Boyko, R. Bojko, A. Spott, A. Pomerene, C. Hill, W. Reinhardt, and M. Hochberg, “Low-loss asymmetric strip-loaded slot waveguides in silicon-on-insulator,” Appl. Phys. Lett. 98, 233303 (2011).
    [Crossref]
  19. V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29, 1209–1211 (2004).
    [Crossref]
  20. E. M. Conwell, “Integrated optics,” Phys. Today 29(5), 48–59 (1976).
    [Crossref]
  21. http://microchem.com/pdf/SU-82000DataSheet2025thru2075Ver4.pdf .
  22. S. Pélisset and M. Roussey, “Strip-loaded nano-photonics on horizontal slot waveguide,” J. Eur. Opt. Soc. Rapid Publ. 15, 6 (2019).
    [Crossref]
  23. S. Pélisset, J. Laukkanen, M. Kuittinen, S. Honkanen, and M. Roussey, “Modal properties of a strip-loaded horizontal slot waveguide,” J. Eur. Opt. Soc. Rapid Publ. 13, 37 (2017).
    [Crossref]
  24. I. Reduto, M. Fetisova, N. Kryzhanovskaya, K. Kotlyar, A. Raskhodchikov, S. Scherbak, Y. Guseva, A. Lipovskii, M. Maximov, and A. Zhukov, “Lasing in III–V microdisk core–TiO2 shell lasers,” J. Opt. Soc. Am. B 36, 2285–2291 (2019).
    [Crossref]
  25. M. Häyrinen, M. Roussey, A. Bera, M. Kuittinen, and S. Honkanen, “Atomic Layer Re-Deposition for Nanoscale Devices,” in Encyclopedia of Plasma Technology, J. L. Shohet, ed. (Taylor & Francis, 2016), Vol. 1, pp. 143–151.

2019 (3)

Q. Song, “Emerging opportunities for ultra-high Q whispering gallery mode microcavities,” Sci. China Phys. Mech. Astron. 62, 74231 (2019).
[Crossref]

S. Pélisset and M. Roussey, “Strip-loaded nano-photonics on horizontal slot waveguide,” J. Eur. Opt. Soc. Rapid Publ. 15, 6 (2019).
[Crossref]

I. Reduto, M. Fetisova, N. Kryzhanovskaya, K. Kotlyar, A. Raskhodchikov, S. Scherbak, Y. Guseva, A. Lipovskii, M. Maximov, and A. Zhukov, “Lasing in III–V microdisk core–TiO2 shell lasers,” J. Opt. Soc. Am. B 36, 2285–2291 (2019).
[Crossref]

2017 (4)

S. Pélisset, J. Laukkanen, M. Kuittinen, S. Honkanen, and M. Roussey, “Modal properties of a strip-loaded horizontal slot waveguide,” J. Eur. Opt. Soc. Rapid Publ. 13, 37 (2017).
[Crossref]

B. Guha, F. Marsault, F. Cadiz, L. Morgenroth, V. Ulin, V. Berkovitz, A. Lemaitre, C. Gomez, A. Amo, S. Combrie, B. Gerard, G. Leo, and I. Favero, “Surface-enhanced gallium arsenide photonic resonator with quality factor of 6×106,” Optica 4, 218–221 (2017).
[Crossref]

N. V. Kryzhanovskaya, E. I. Moiseev, Y. S. Polubavkina, M. V. Maximov, D. V. Mokhov, I. A. Morozov, M. M. Kulagina, Y. M. Zadiranov, A. A. Lipovskii, M. Tang, M. Liao, J. Wu, S. Chen, H. Liu, and A. E. Zhukov, “Elevated temperature lasing from injection microdisk lasers on silicon,” Laser Phys. Lett. 15, 015802 (2017).
[Crossref]

M. Roussey, L. Ahmadi, S. Pélisset, M. Häyrinen, A. Bera, V. Kontturi, J. Laukkanen, I. Vartiainen, S. Honkanen, and M. Kuittinen, “Strip-loaded horizontal slot waveguide,” Opt. Lett. 42, 211–214 (2017).
[Crossref]

2015 (2)

N. Kryzhanovskaya, E. Moiseev, Y. V. Kudashova, F. Zubov, A. Lipovskii, M. Kulagina, S. Troshkov, Y. M. Zadiranov, D. Livshits, M. Maximov, and A. Zhukov, “Continuous-wave lasing at 100°C in 1.3 µm quantum dot microdisk diode laser,” Electron. Lett. 51, 1354–1355 (2015).
[Crossref]

N. V. Kryzhanovskaya, A. E. Zhukov, M. V. Maximov, E. I. Moiseev, I. I. Shostak, A. M. Nadtochiy, Y. V. Kudashova, A. A. Lipovskii, M. M. Kulagina, and S. I. Troshkov, “Room temperature lasing in 1-µm microdisk quantum dot lasers,” J. Sel. Top. Quantum Electron. 21, 709–713 (2015).
[Crossref]

2013 (1)

2012 (1)

L. He, S. K. Ozdemir, and L. Yang, “Whispering gallery microcavity lasers,” Laser Photon. Rev. 7, 60–82 (2012).
[Crossref]

2011 (2)

T. Harayama and S. Shinohara, “Two-dimensional microcavity lasers,” Laser Photon. Rev. 5, 247–271 (2011).
[Crossref]

R. Ding, T. Baehr-Jones, W.-J. Kim, B. Boyko, R. Bojko, A. Spott, A. Pomerene, C. Hill, W. Reinhardt, and M. Hochberg, “Low-loss asymmetric strip-loaded slot waveguides in silicon-on-insulator,” Appl. Phys. Lett. 98, 233303 (2011).
[Crossref]

2009 (3)

S. Koseki, B. Zhang, K. D. Greve, and Y. Yamamoto, “Monolithic integration of quantum dot containing microdisk microcavities coupled to air-suspended waveguides,” Appl. Phys. Lett. 94, 051110 (2009).
[Crossref]

M. Hentschel and T.-Y. Kwon, “Designing and understanding directional emission from spiral microlasers,” Opt. Lett. 34, 163–165 (2009).
[Crossref]

Q. Song, W. Fang, B. Liu, S.-T. Ho, G. S. Solomon, and H. Cao, “Chaotic microcavity laser with high quality factor and unidirectional output,” Phys. Rev. A 80, 041807 (2009).
[Crossref]

2007 (2)

J. V. Campenhout, P. Rojo-Romeo, P. Regreny, C. Seassal, D. V. Thourhout, S. Verstuyft, L. D. Cioccio, J.-M. Fedeli, C. Lagahe, and R. Baets, “Electrically pumped InP-based microdisk lasers integrated with a nanophotonic silicon-on-insulator waveguide circuit,” Opt. Express 15, 6744–6749 (2007).
[Crossref]

C. P. Michael, K. Srinivasan, T. J. Johnson, O. Painter, K. H. Lee, K. Hennessy, H. Kim, and E. Hu, “Wavelength- and material-dependent absorption in GaAs and AlGaAs microcavities,” Appl. Phys. Lett. 90, 051108 (2007).
[Crossref]

2006 (2)

2004 (1)

1998 (1)

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[Crossref]

1976 (1)

E. M. Conwell, “Integrated optics,” Phys. Today 29(5), 48–59 (1976).
[Crossref]

Ahmadi, L.

Almeida, V. R.

Amo, A.

Baehr-Jones, T.

R. Ding, T. Baehr-Jones, W.-J. Kim, B. Boyko, R. Bojko, A. Spott, A. Pomerene, C. Hill, W. Reinhardt, and M. Hochberg, “Low-loss asymmetric strip-loaded slot waveguides in silicon-on-insulator,” Appl. Phys. Lett. 98, 233303 (2011).
[Crossref]

Baets, R.

Barrios, C. A.

Bazin, A.

Bera, A.

M. Roussey, L. Ahmadi, S. Pélisset, M. Häyrinen, A. Bera, V. Kontturi, J. Laukkanen, I. Vartiainen, S. Honkanen, and M. Kuittinen, “Strip-loaded horizontal slot waveguide,” Opt. Lett. 42, 211–214 (2017).
[Crossref]

M. Häyrinen, M. Roussey, A. Bera, M. Kuittinen, and S. Honkanen, “Atomic Layer Re-Deposition for Nanoscale Devices,” in Encyclopedia of Plasma Technology, J. L. Shohet, ed. (Taylor & Francis, 2016), Vol. 1, pp. 143–151.

Berkovitz, V.

Bojko, R.

R. Ding, T. Baehr-Jones, W.-J. Kim, B. Boyko, R. Bojko, A. Spott, A. Pomerene, C. Hill, W. Reinhardt, and M. Hochberg, “Low-loss asymmetric strip-loaded slot waveguides in silicon-on-insulator,” Appl. Phys. Lett. 98, 233303 (2011).
[Crossref]

Borselli, M.

Boyko, B.

R. Ding, T. Baehr-Jones, W.-J. Kim, B. Boyko, R. Bojko, A. Spott, A. Pomerene, C. Hill, W. Reinhardt, and M. Hochberg, “Low-loss asymmetric strip-loaded slot waveguides in silicon-on-insulator,” Appl. Phys. Lett. 98, 233303 (2011).
[Crossref]

Cadiz, F.

Campenhout, J. V.

Cao, H.

Q. Song, W. Fang, B. Liu, S.-T. Ho, G. S. Solomon, and H. Cao, “Chaotic microcavity laser with high quality factor and unidirectional output,” Phys. Rev. A 80, 041807 (2009).
[Crossref]

Capasso, F.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[Crossref]

Chen, S.

N. V. Kryzhanovskaya, E. I. Moiseev, Y. S. Polubavkina, M. V. Maximov, D. V. Mokhov, I. A. Morozov, M. M. Kulagina, Y. M. Zadiranov, A. A. Lipovskii, M. Tang, M. Liao, J. Wu, S. Chen, H. Liu, and A. E. Zhukov, “Elevated temperature lasing from injection microdisk lasers on silicon,” Laser Phys. Lett. 15, 015802 (2017).
[Crossref]

Cho, A. Y.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[Crossref]

Cioccio, L. D.

Combrie, S.

Conwell, E. M.

E. M. Conwell, “Integrated optics,” Phys. Today 29(5), 48–59 (1976).
[Crossref]

Ding, R.

R. Ding, T. Baehr-Jones, W.-J. Kim, B. Boyko, R. Bojko, A. Spott, A. Pomerene, C. Hill, W. Reinhardt, and M. Hochberg, “Low-loss asymmetric strip-loaded slot waveguides in silicon-on-insulator,” Appl. Phys. Lett. 98, 233303 (2011).
[Crossref]

Faist, J.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[Crossref]

Fang, W.

Q. Song, W. Fang, B. Liu, S.-T. Ho, G. S. Solomon, and H. Cao, “Chaotic microcavity laser with high quality factor and unidirectional output,” Phys. Rev. A 80, 041807 (2009).
[Crossref]

Favero, I.

Fedeli, J.-M.

Fetisova, M.

Gerard, B.

Gmachl, C.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[Crossref]

Gomez, C.

Greve, K. D.

S. Koseki, B. Zhang, K. D. Greve, and Y. Yamamoto, “Monolithic integration of quantum dot containing microdisk microcavities coupled to air-suspended waveguides,” Appl. Phys. Lett. 94, 051110 (2009).
[Crossref]

Guha, B.

Guseva, Y.

Halioua, Y.

Harayama, T.

T. Harayama and S. Shinohara, “Two-dimensional microcavity lasers,” Laser Photon. Rev. 5, 247–271 (2011).
[Crossref]

Häyrinen, M.

M. Roussey, L. Ahmadi, S. Pélisset, M. Häyrinen, A. Bera, V. Kontturi, J. Laukkanen, I. Vartiainen, S. Honkanen, and M. Kuittinen, “Strip-loaded horizontal slot waveguide,” Opt. Lett. 42, 211–214 (2017).
[Crossref]

M. Häyrinen, M. Roussey, A. Bera, M. Kuittinen, and S. Honkanen, “Atomic Layer Re-Deposition for Nanoscale Devices,” in Encyclopedia of Plasma Technology, J. L. Shohet, ed. (Taylor & Francis, 2016), Vol. 1, pp. 143–151.

He, L.

L. He, S. K. Ozdemir, and L. Yang, “Whispering gallery microcavity lasers,” Laser Photon. Rev. 7, 60–82 (2012).
[Crossref]

Hennessy, K.

C. P. Michael, K. Srinivasan, T. J. Johnson, O. Painter, K. H. Lee, K. Hennessy, H. Kim, and E. Hu, “Wavelength- and material-dependent absorption in GaAs and AlGaAs microcavities,” Appl. Phys. Lett. 90, 051108 (2007).
[Crossref]

Hentschel, M.

Hierle, R.

M. Lebental, J. S. Lauret, R. Hierle, and J. Zyss, “Highly directional stadium-shaped polymer microlasers,” Appl. Phys. Lett. 88, 031108 (2006).
[Crossref]

Hill, C.

R. Ding, T. Baehr-Jones, W.-J. Kim, B. Boyko, R. Bojko, A. Spott, A. Pomerene, C. Hill, W. Reinhardt, and M. Hochberg, “Low-loss asymmetric strip-loaded slot waveguides in silicon-on-insulator,” Appl. Phys. Lett. 98, 233303 (2011).
[Crossref]

Ho, S.-T.

Q. Song, W. Fang, B. Liu, S.-T. Ho, G. S. Solomon, and H. Cao, “Chaotic microcavity laser with high quality factor and unidirectional output,” Phys. Rev. A 80, 041807 (2009).
[Crossref]

Hochberg, M.

R. Ding, T. Baehr-Jones, W.-J. Kim, B. Boyko, R. Bojko, A. Spott, A. Pomerene, C. Hill, W. Reinhardt, and M. Hochberg, “Low-loss asymmetric strip-loaded slot waveguides in silicon-on-insulator,” Appl. Phys. Lett. 98, 233303 (2011).
[Crossref]

Honkanen, S.

M. Roussey, L. Ahmadi, S. Pélisset, M. Häyrinen, A. Bera, V. Kontturi, J. Laukkanen, I. Vartiainen, S. Honkanen, and M. Kuittinen, “Strip-loaded horizontal slot waveguide,” Opt. Lett. 42, 211–214 (2017).
[Crossref]

S. Pélisset, J. Laukkanen, M. Kuittinen, S. Honkanen, and M. Roussey, “Modal properties of a strip-loaded horizontal slot waveguide,” J. Eur. Opt. Soc. Rapid Publ. 13, 37 (2017).
[Crossref]

M. Häyrinen, M. Roussey, A. Bera, M. Kuittinen, and S. Honkanen, “Atomic Layer Re-Deposition for Nanoscale Devices,” in Encyclopedia of Plasma Technology, J. L. Shohet, ed. (Taylor & Francis, 2016), Vol. 1, pp. 143–151.

Hu, E.

C. P. Michael, K. Srinivasan, T. J. Johnson, O. Painter, K. H. Lee, K. Hennessy, H. Kim, and E. Hu, “Wavelength- and material-dependent absorption in GaAs and AlGaAs microcavities,” Appl. Phys. Lett. 90, 051108 (2007).
[Crossref]

Johnson, T. J.

C. P. Michael, K. Srinivasan, T. J. Johnson, O. Painter, K. H. Lee, K. Hennessy, H. Kim, and E. Hu, “Wavelength- and material-dependent absorption in GaAs and AlGaAs microcavities,” Appl. Phys. Lett. 90, 051108 (2007).
[Crossref]

Karle, T. J.

Kim, H.

C. P. Michael, K. Srinivasan, T. J. Johnson, O. Painter, K. H. Lee, K. Hennessy, H. Kim, and E. Hu, “Wavelength- and material-dependent absorption in GaAs and AlGaAs microcavities,” Appl. Phys. Lett. 90, 051108 (2007).
[Crossref]

Kim, W.-J.

R. Ding, T. Baehr-Jones, W.-J. Kim, B. Boyko, R. Bojko, A. Spott, A. Pomerene, C. Hill, W. Reinhardt, and M. Hochberg, “Low-loss asymmetric strip-loaded slot waveguides in silicon-on-insulator,” Appl. Phys. Lett. 98, 233303 (2011).
[Crossref]

Kontturi, V.

Koseki, S.

S. Koseki, B. Zhang, K. D. Greve, and Y. Yamamoto, “Monolithic integration of quantum dot containing microdisk microcavities coupled to air-suspended waveguides,” Appl. Phys. Lett. 94, 051110 (2009).
[Crossref]

Kotlyar, K.

Krishna, S.

Kryzhanovskaya, N.

I. Reduto, M. Fetisova, N. Kryzhanovskaya, K. Kotlyar, A. Raskhodchikov, S. Scherbak, Y. Guseva, A. Lipovskii, M. Maximov, and A. Zhukov, “Lasing in III–V microdisk core–TiO2 shell lasers,” J. Opt. Soc. Am. B 36, 2285–2291 (2019).
[Crossref]

N. Kryzhanovskaya, E. Moiseev, Y. V. Kudashova, F. Zubov, A. Lipovskii, M. Kulagina, S. Troshkov, Y. M. Zadiranov, D. Livshits, M. Maximov, and A. Zhukov, “Continuous-wave lasing at 100°C in 1.3 µm quantum dot microdisk diode laser,” Electron. Lett. 51, 1354–1355 (2015).
[Crossref]

Kryzhanovskaya, N. V.

N. V. Kryzhanovskaya, E. I. Moiseev, Y. S. Polubavkina, M. V. Maximov, D. V. Mokhov, I. A. Morozov, M. M. Kulagina, Y. M. Zadiranov, A. A. Lipovskii, M. Tang, M. Liao, J. Wu, S. Chen, H. Liu, and A. E. Zhukov, “Elevated temperature lasing from injection microdisk lasers on silicon,” Laser Phys. Lett. 15, 015802 (2017).
[Crossref]

N. V. Kryzhanovskaya, A. E. Zhukov, M. V. Maximov, E. I. Moiseev, I. I. Shostak, A. M. Nadtochiy, Y. V. Kudashova, A. A. Lipovskii, M. M. Kulagina, and S. I. Troshkov, “Room temperature lasing in 1-µm microdisk quantum dot lasers,” J. Sel. Top. Quantum Electron. 21, 709–713 (2015).
[Crossref]

Kudashova, Y. V.

N. Kryzhanovskaya, E. Moiseev, Y. V. Kudashova, F. Zubov, A. Lipovskii, M. Kulagina, S. Troshkov, Y. M. Zadiranov, D. Livshits, M. Maximov, and A. Zhukov, “Continuous-wave lasing at 100°C in 1.3 µm quantum dot microdisk diode laser,” Electron. Lett. 51, 1354–1355 (2015).
[Crossref]

N. V. Kryzhanovskaya, A. E. Zhukov, M. V. Maximov, E. I. Moiseev, I. I. Shostak, A. M. Nadtochiy, Y. V. Kudashova, A. A. Lipovskii, M. M. Kulagina, and S. I. Troshkov, “Room temperature lasing in 1-µm microdisk quantum dot lasers,” J. Sel. Top. Quantum Electron. 21, 709–713 (2015).
[Crossref]

Kuittinen, M.

M. Roussey, L. Ahmadi, S. Pélisset, M. Häyrinen, A. Bera, V. Kontturi, J. Laukkanen, I. Vartiainen, S. Honkanen, and M. Kuittinen, “Strip-loaded horizontal slot waveguide,” Opt. Lett. 42, 211–214 (2017).
[Crossref]

S. Pélisset, J. Laukkanen, M. Kuittinen, S. Honkanen, and M. Roussey, “Modal properties of a strip-loaded horizontal slot waveguide,” J. Eur. Opt. Soc. Rapid Publ. 13, 37 (2017).
[Crossref]

M. Häyrinen, M. Roussey, A. Bera, M. Kuittinen, and S. Honkanen, “Atomic Layer Re-Deposition for Nanoscale Devices,” in Encyclopedia of Plasma Technology, J. L. Shohet, ed. (Taylor & Francis, 2016), Vol. 1, pp. 143–151.

Kulagina, M.

N. Kryzhanovskaya, E. Moiseev, Y. V. Kudashova, F. Zubov, A. Lipovskii, M. Kulagina, S. Troshkov, Y. M. Zadiranov, D. Livshits, M. Maximov, and A. Zhukov, “Continuous-wave lasing at 100°C in 1.3 µm quantum dot microdisk diode laser,” Electron. Lett. 51, 1354–1355 (2015).
[Crossref]

Kulagina, M. M.

N. V. Kryzhanovskaya, E. I. Moiseev, Y. S. Polubavkina, M. V. Maximov, D. V. Mokhov, I. A. Morozov, M. M. Kulagina, Y. M. Zadiranov, A. A. Lipovskii, M. Tang, M. Liao, J. Wu, S. Chen, H. Liu, and A. E. Zhukov, “Elevated temperature lasing from injection microdisk lasers on silicon,” Laser Phys. Lett. 15, 015802 (2017).
[Crossref]

N. V. Kryzhanovskaya, A. E. Zhukov, M. V. Maximov, E. I. Moiseev, I. I. Shostak, A. M. Nadtochiy, Y. V. Kudashova, A. A. Lipovskii, M. M. Kulagina, and S. I. Troshkov, “Room temperature lasing in 1-µm microdisk quantum dot lasers,” J. Sel. Top. Quantum Electron. 21, 709–713 (2015).
[Crossref]

Kwon, T.-Y.

Lagahe, C.

Laukkanen, J.

M. Roussey, L. Ahmadi, S. Pélisset, M. Häyrinen, A. Bera, V. Kontturi, J. Laukkanen, I. Vartiainen, S. Honkanen, and M. Kuittinen, “Strip-loaded horizontal slot waveguide,” Opt. Lett. 42, 211–214 (2017).
[Crossref]

S. Pélisset, J. Laukkanen, M. Kuittinen, S. Honkanen, and M. Roussey, “Modal properties of a strip-loaded horizontal slot waveguide,” J. Eur. Opt. Soc. Rapid Publ. 13, 37 (2017).
[Crossref]

Lauret, J. S.

M. Lebental, J. S. Lauret, R. Hierle, and J. Zyss, “Highly directional stadium-shaped polymer microlasers,” Appl. Phys. Lett. 88, 031108 (2006).
[Crossref]

Lebental, M.

M. Lebental, J. S. Lauret, R. Hierle, and J. Zyss, “Highly directional stadium-shaped polymer microlasers,” Appl. Phys. Lett. 88, 031108 (2006).
[Crossref]

Lee, K. H.

C. P. Michael, K. Srinivasan, T. J. Johnson, O. Painter, K. H. Lee, K. Hennessy, H. Kim, and E. Hu, “Wavelength- and material-dependent absorption in GaAs and AlGaAs microcavities,” Appl. Phys. Lett. 90, 051108 (2007).
[Crossref]

Lemaitre, A.

Leo, G.

Liao, M.

N. V. Kryzhanovskaya, E. I. Moiseev, Y. S. Polubavkina, M. V. Maximov, D. V. Mokhov, I. A. Morozov, M. M. Kulagina, Y. M. Zadiranov, A. A. Lipovskii, M. Tang, M. Liao, J. Wu, S. Chen, H. Liu, and A. E. Zhukov, “Elevated temperature lasing from injection microdisk lasers on silicon,” Laser Phys. Lett. 15, 015802 (2017).
[Crossref]

Lipovskii, A.

I. Reduto, M. Fetisova, N. Kryzhanovskaya, K. Kotlyar, A. Raskhodchikov, S. Scherbak, Y. Guseva, A. Lipovskii, M. Maximov, and A. Zhukov, “Lasing in III–V microdisk core–TiO2 shell lasers,” J. Opt. Soc. Am. B 36, 2285–2291 (2019).
[Crossref]

N. Kryzhanovskaya, E. Moiseev, Y. V. Kudashova, F. Zubov, A. Lipovskii, M. Kulagina, S. Troshkov, Y. M. Zadiranov, D. Livshits, M. Maximov, and A. Zhukov, “Continuous-wave lasing at 100°C in 1.3 µm quantum dot microdisk diode laser,” Electron. Lett. 51, 1354–1355 (2015).
[Crossref]

Lipovskii, A. A.

N. V. Kryzhanovskaya, E. I. Moiseev, Y. S. Polubavkina, M. V. Maximov, D. V. Mokhov, I. A. Morozov, M. M. Kulagina, Y. M. Zadiranov, A. A. Lipovskii, M. Tang, M. Liao, J. Wu, S. Chen, H. Liu, and A. E. Zhukov, “Elevated temperature lasing from injection microdisk lasers on silicon,” Laser Phys. Lett. 15, 015802 (2017).
[Crossref]

N. V. Kryzhanovskaya, A. E. Zhukov, M. V. Maximov, E. I. Moiseev, I. I. Shostak, A. M. Nadtochiy, Y. V. Kudashova, A. A. Lipovskii, M. M. Kulagina, and S. I. Troshkov, “Room temperature lasing in 1-µm microdisk quantum dot lasers,” J. Sel. Top. Quantum Electron. 21, 709–713 (2015).
[Crossref]

Lipson, M.

Liu, B.

Q. Song, W. Fang, B. Liu, S.-T. Ho, G. S. Solomon, and H. Cao, “Chaotic microcavity laser with high quality factor and unidirectional output,” Phys. Rev. A 80, 041807 (2009).
[Crossref]

Liu, H.

N. V. Kryzhanovskaya, E. I. Moiseev, Y. S. Polubavkina, M. V. Maximov, D. V. Mokhov, I. A. Morozov, M. M. Kulagina, Y. M. Zadiranov, A. A. Lipovskii, M. Tang, M. Liao, J. Wu, S. Chen, H. Liu, and A. E. Zhukov, “Elevated temperature lasing from injection microdisk lasers on silicon,” Laser Phys. Lett. 15, 015802 (2017).
[Crossref]

Livshits, D.

N. Kryzhanovskaya, E. Moiseev, Y. V. Kudashova, F. Zubov, A. Lipovskii, M. Kulagina, S. Troshkov, Y. M. Zadiranov, D. Livshits, M. Maximov, and A. Zhukov, “Continuous-wave lasing at 100°C in 1.3 µm quantum dot microdisk diode laser,” Electron. Lett. 51, 1354–1355 (2015).
[Crossref]

Marsault, F.

Maximov, M.

I. Reduto, M. Fetisova, N. Kryzhanovskaya, K. Kotlyar, A. Raskhodchikov, S. Scherbak, Y. Guseva, A. Lipovskii, M. Maximov, and A. Zhukov, “Lasing in III–V microdisk core–TiO2 shell lasers,” J. Opt. Soc. Am. B 36, 2285–2291 (2019).
[Crossref]

N. Kryzhanovskaya, E. Moiseev, Y. V. Kudashova, F. Zubov, A. Lipovskii, M. Kulagina, S. Troshkov, Y. M. Zadiranov, D. Livshits, M. Maximov, and A. Zhukov, “Continuous-wave lasing at 100°C in 1.3 µm quantum dot microdisk diode laser,” Electron. Lett. 51, 1354–1355 (2015).
[Crossref]

Maximov, M. V.

N. V. Kryzhanovskaya, E. I. Moiseev, Y. S. Polubavkina, M. V. Maximov, D. V. Mokhov, I. A. Morozov, M. M. Kulagina, Y. M. Zadiranov, A. A. Lipovskii, M. Tang, M. Liao, J. Wu, S. Chen, H. Liu, and A. E. Zhukov, “Elevated temperature lasing from injection microdisk lasers on silicon,” Laser Phys. Lett. 15, 015802 (2017).
[Crossref]

N. V. Kryzhanovskaya, A. E. Zhukov, M. V. Maximov, E. I. Moiseev, I. I. Shostak, A. M. Nadtochiy, Y. V. Kudashova, A. A. Lipovskii, M. M. Kulagina, and S. I. Troshkov, “Room temperature lasing in 1-µm microdisk quantum dot lasers,” J. Sel. Top. Quantum Electron. 21, 709–713 (2015).
[Crossref]

Mechet, P.

Michael, C. P.

C. P. Michael, K. Srinivasan, T. J. Johnson, O. Painter, K. H. Lee, K. Hennessy, H. Kim, and E. Hu, “Wavelength- and material-dependent absorption in GaAs and AlGaAs microcavities,” Appl. Phys. Lett. 90, 051108 (2007).
[Crossref]

Moiseev, E.

N. Kryzhanovskaya, E. Moiseev, Y. V. Kudashova, F. Zubov, A. Lipovskii, M. Kulagina, S. Troshkov, Y. M. Zadiranov, D. Livshits, M. Maximov, and A. Zhukov, “Continuous-wave lasing at 100°C in 1.3 µm quantum dot microdisk diode laser,” Electron. Lett. 51, 1354–1355 (2015).
[Crossref]

Moiseev, E. I.

N. V. Kryzhanovskaya, E. I. Moiseev, Y. S. Polubavkina, M. V. Maximov, D. V. Mokhov, I. A. Morozov, M. M. Kulagina, Y. M. Zadiranov, A. A. Lipovskii, M. Tang, M. Liao, J. Wu, S. Chen, H. Liu, and A. E. Zhukov, “Elevated temperature lasing from injection microdisk lasers on silicon,” Laser Phys. Lett. 15, 015802 (2017).
[Crossref]

N. V. Kryzhanovskaya, A. E. Zhukov, M. V. Maximov, E. I. Moiseev, I. I. Shostak, A. M. Nadtochiy, Y. V. Kudashova, A. A. Lipovskii, M. M. Kulagina, and S. I. Troshkov, “Room temperature lasing in 1-µm microdisk quantum dot lasers,” J. Sel. Top. Quantum Electron. 21, 709–713 (2015).
[Crossref]

Mokhov, D. V.

N. V. Kryzhanovskaya, E. I. Moiseev, Y. S. Polubavkina, M. V. Maximov, D. V. Mokhov, I. A. Morozov, M. M. Kulagina, Y. M. Zadiranov, A. A. Lipovskii, M. Tang, M. Liao, J. Wu, S. Chen, H. Liu, and A. E. Zhukov, “Elevated temperature lasing from injection microdisk lasers on silicon,” Laser Phys. Lett. 15, 015802 (2017).
[Crossref]

Monnier, P.

Morgenroth, L.

Morozov, I. A.

N. V. Kryzhanovskaya, E. I. Moiseev, Y. S. Polubavkina, M. V. Maximov, D. V. Mokhov, I. A. Morozov, M. M. Kulagina, Y. M. Zadiranov, A. A. Lipovskii, M. Tang, M. Liao, J. Wu, S. Chen, H. Liu, and A. E. Zhukov, “Elevated temperature lasing from injection microdisk lasers on silicon,” Laser Phys. Lett. 15, 015802 (2017).
[Crossref]

Morthier, G.

Nadtochiy, A. M.

N. V. Kryzhanovskaya, A. E. Zhukov, M. V. Maximov, E. I. Moiseev, I. I. Shostak, A. M. Nadtochiy, Y. V. Kudashova, A. A. Lipovskii, M. M. Kulagina, and S. I. Troshkov, “Room temperature lasing in 1-µm microdisk quantum dot lasers,” J. Sel. Top. Quantum Electron. 21, 709–713 (2015).
[Crossref]

Narimanov, E. E.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[Crossref]

Nöckel, J. U.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[Crossref]

Ozdemir, S. K.

L. He, S. K. Ozdemir, and L. Yang, “Whispering gallery microcavity lasers,” Laser Photon. Rev. 7, 60–82 (2012).
[Crossref]

Painter, O.

C. P. Michael, K. Srinivasan, T. J. Johnson, O. Painter, K. H. Lee, K. Hennessy, H. Kim, and E. Hu, “Wavelength- and material-dependent absorption in GaAs and AlGaAs microcavities,” Appl. Phys. Lett. 90, 051108 (2007).
[Crossref]

K. Srinivasan, M. Borselli, O. Painter, A. Stintz, and S. Krishna, “Cavity Q, mode volume, and lasing threshold in small diameter AlGaAs microdisks with embedded quantum dots,” Opt. Express 14, 1094–1105 (2006).
[Crossref]

Pélisset, S.

S. Pélisset and M. Roussey, “Strip-loaded nano-photonics on horizontal slot waveguide,” J. Eur. Opt. Soc. Rapid Publ. 15, 6 (2019).
[Crossref]

S. Pélisset, J. Laukkanen, M. Kuittinen, S. Honkanen, and M. Roussey, “Modal properties of a strip-loaded horizontal slot waveguide,” J. Eur. Opt. Soc. Rapid Publ. 13, 37 (2017).
[Crossref]

M. Roussey, L. Ahmadi, S. Pélisset, M. Häyrinen, A. Bera, V. Kontturi, J. Laukkanen, I. Vartiainen, S. Honkanen, and M. Kuittinen, “Strip-loaded horizontal slot waveguide,” Opt. Lett. 42, 211–214 (2017).
[Crossref]

Polubavkina, Y. S.

N. V. Kryzhanovskaya, E. I. Moiseev, Y. S. Polubavkina, M. V. Maximov, D. V. Mokhov, I. A. Morozov, M. M. Kulagina, Y. M. Zadiranov, A. A. Lipovskii, M. Tang, M. Liao, J. Wu, S. Chen, H. Liu, and A. E. Zhukov, “Elevated temperature lasing from injection microdisk lasers on silicon,” Laser Phys. Lett. 15, 015802 (2017).
[Crossref]

Pomerene, A.

R. Ding, T. Baehr-Jones, W.-J. Kim, B. Boyko, R. Bojko, A. Spott, A. Pomerene, C. Hill, W. Reinhardt, and M. Hochberg, “Low-loss asymmetric strip-loaded slot waveguides in silicon-on-insulator,” Appl. Phys. Lett. 98, 233303 (2011).
[Crossref]

Raineri, F.

Raj, R.

Raskhodchikov, A.

Reduto, I.

Regreny, P.

Reinhardt, W.

R. Ding, T. Baehr-Jones, W.-J. Kim, B. Boyko, R. Bojko, A. Spott, A. Pomerene, C. Hill, W. Reinhardt, and M. Hochberg, “Low-loss asymmetric strip-loaded slot waveguides in silicon-on-insulator,” Appl. Phys. Lett. 98, 233303 (2011).
[Crossref]

Roelkens, G.

Rojo-Romeo, P.

Roussey, M.

S. Pélisset and M. Roussey, “Strip-loaded nano-photonics on horizontal slot waveguide,” J. Eur. Opt. Soc. Rapid Publ. 15, 6 (2019).
[Crossref]

M. Roussey, L. Ahmadi, S. Pélisset, M. Häyrinen, A. Bera, V. Kontturi, J. Laukkanen, I. Vartiainen, S. Honkanen, and M. Kuittinen, “Strip-loaded horizontal slot waveguide,” Opt. Lett. 42, 211–214 (2017).
[Crossref]

S. Pélisset, J. Laukkanen, M. Kuittinen, S. Honkanen, and M. Roussey, “Modal properties of a strip-loaded horizontal slot waveguide,” J. Eur. Opt. Soc. Rapid Publ. 13, 37 (2017).
[Crossref]

M. Häyrinen, M. Roussey, A. Bera, M. Kuittinen, and S. Honkanen, “Atomic Layer Re-Deposition for Nanoscale Devices,” in Encyclopedia of Plasma Technology, J. L. Shohet, ed. (Taylor & Francis, 2016), Vol. 1, pp. 143–151.

Sagnes, I.

Scherbak, S.

Seassal, C.

Shinohara, S.

T. Harayama and S. Shinohara, “Two-dimensional microcavity lasers,” Laser Photon. Rev. 5, 247–271 (2011).
[Crossref]

Shostak, I. I.

N. V. Kryzhanovskaya, A. E. Zhukov, M. V. Maximov, E. I. Moiseev, I. I. Shostak, A. M. Nadtochiy, Y. V. Kudashova, A. A. Lipovskii, M. M. Kulagina, and S. I. Troshkov, “Room temperature lasing in 1-µm microdisk quantum dot lasers,” J. Sel. Top. Quantum Electron. 21, 709–713 (2015).
[Crossref]

Sivco, D. L.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[Crossref]

Solomon, G. S.

Q. Song, W. Fang, B. Liu, S.-T. Ho, G. S. Solomon, and H. Cao, “Chaotic microcavity laser with high quality factor and unidirectional output,” Phys. Rev. A 80, 041807 (2009).
[Crossref]

Song, Q.

Q. Song, “Emerging opportunities for ultra-high Q whispering gallery mode microcavities,” Sci. China Phys. Mech. Astron. 62, 74231 (2019).
[Crossref]

Q. Song, W. Fang, B. Liu, S.-T. Ho, G. S. Solomon, and H. Cao, “Chaotic microcavity laser with high quality factor and unidirectional output,” Phys. Rev. A 80, 041807 (2009).
[Crossref]

Spott, A.

R. Ding, T. Baehr-Jones, W.-J. Kim, B. Boyko, R. Bojko, A. Spott, A. Pomerene, C. Hill, W. Reinhardt, and M. Hochberg, “Low-loss asymmetric strip-loaded slot waveguides in silicon-on-insulator,” Appl. Phys. Lett. 98, 233303 (2011).
[Crossref]

Spuesens, T.

Srinivasan, K.

C. P. Michael, K. Srinivasan, T. J. Johnson, O. Painter, K. H. Lee, K. Hennessy, H. Kim, and E. Hu, “Wavelength- and material-dependent absorption in GaAs and AlGaAs microcavities,” Appl. Phys. Lett. 90, 051108 (2007).
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[Crossref]

Stintz, A.

Stone, A. D.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[Crossref]

Tang, M.

N. V. Kryzhanovskaya, E. I. Moiseev, Y. S. Polubavkina, M. V. Maximov, D. V. Mokhov, I. A. Morozov, M. M. Kulagina, Y. M. Zadiranov, A. A. Lipovskii, M. Tang, M. Liao, J. Wu, S. Chen, H. Liu, and A. E. Zhukov, “Elevated temperature lasing from injection microdisk lasers on silicon,” Laser Phys. Lett. 15, 015802 (2017).
[Crossref]

Thourhout, D. V.

Troshkov, S.

N. Kryzhanovskaya, E. Moiseev, Y. V. Kudashova, F. Zubov, A. Lipovskii, M. Kulagina, S. Troshkov, Y. M. Zadiranov, D. Livshits, M. Maximov, and A. Zhukov, “Continuous-wave lasing at 100°C in 1.3 µm quantum dot microdisk diode laser,” Electron. Lett. 51, 1354–1355 (2015).
[Crossref]

Troshkov, S. I.

N. V. Kryzhanovskaya, A. E. Zhukov, M. V. Maximov, E. I. Moiseev, I. I. Shostak, A. M. Nadtochiy, Y. V. Kudashova, A. A. Lipovskii, M. M. Kulagina, and S. I. Troshkov, “Room temperature lasing in 1-µm microdisk quantum dot lasers,” J. Sel. Top. Quantum Electron. 21, 709–713 (2015).
[Crossref]

Ulin, V.

Vartiainen, I.

Verstuyft, S.

Wu, J.

N. V. Kryzhanovskaya, E. I. Moiseev, Y. S. Polubavkina, M. V. Maximov, D. V. Mokhov, I. A. Morozov, M. M. Kulagina, Y. M. Zadiranov, A. A. Lipovskii, M. Tang, M. Liao, J. Wu, S. Chen, H. Liu, and A. E. Zhukov, “Elevated temperature lasing from injection microdisk lasers on silicon,” Laser Phys. Lett. 15, 015802 (2017).
[Crossref]

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S. Koseki, B. Zhang, K. D. Greve, and Y. Yamamoto, “Monolithic integration of quantum dot containing microdisk microcavities coupled to air-suspended waveguides,” Appl. Phys. Lett. 94, 051110 (2009).
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L. He, S. K. Ozdemir, and L. Yang, “Whispering gallery microcavity lasers,” Laser Photon. Rev. 7, 60–82 (2012).
[Crossref]

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N. V. Kryzhanovskaya, E. I. Moiseev, Y. S. Polubavkina, M. V. Maximov, D. V. Mokhov, I. A. Morozov, M. M. Kulagina, Y. M. Zadiranov, A. A. Lipovskii, M. Tang, M. Liao, J. Wu, S. Chen, H. Liu, and A. E. Zhukov, “Elevated temperature lasing from injection microdisk lasers on silicon,” Laser Phys. Lett. 15, 015802 (2017).
[Crossref]

N. Kryzhanovskaya, E. Moiseev, Y. V. Kudashova, F. Zubov, A. Lipovskii, M. Kulagina, S. Troshkov, Y. M. Zadiranov, D. Livshits, M. Maximov, and A. Zhukov, “Continuous-wave lasing at 100°C in 1.3 µm quantum dot microdisk diode laser,” Electron. Lett. 51, 1354–1355 (2015).
[Crossref]

Zhang, B.

S. Koseki, B. Zhang, K. D. Greve, and Y. Yamamoto, “Monolithic integration of quantum dot containing microdisk microcavities coupled to air-suspended waveguides,” Appl. Phys. Lett. 94, 051110 (2009).
[Crossref]

Zhukov, A.

I. Reduto, M. Fetisova, N. Kryzhanovskaya, K. Kotlyar, A. Raskhodchikov, S. Scherbak, Y. Guseva, A. Lipovskii, M. Maximov, and A. Zhukov, “Lasing in III–V microdisk core–TiO2 shell lasers,” J. Opt. Soc. Am. B 36, 2285–2291 (2019).
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N. Kryzhanovskaya, E. Moiseev, Y. V. Kudashova, F. Zubov, A. Lipovskii, M. Kulagina, S. Troshkov, Y. M. Zadiranov, D. Livshits, M. Maximov, and A. Zhukov, “Continuous-wave lasing at 100°C in 1.3 µm quantum dot microdisk diode laser,” Electron. Lett. 51, 1354–1355 (2015).
[Crossref]

Zhukov, A. E.

N. V. Kryzhanovskaya, E. I. Moiseev, Y. S. Polubavkina, M. V. Maximov, D. V. Mokhov, I. A. Morozov, M. M. Kulagina, Y. M. Zadiranov, A. A. Lipovskii, M. Tang, M. Liao, J. Wu, S. Chen, H. Liu, and A. E. Zhukov, “Elevated temperature lasing from injection microdisk lasers on silicon,” Laser Phys. Lett. 15, 015802 (2017).
[Crossref]

N. V. Kryzhanovskaya, A. E. Zhukov, M. V. Maximov, E. I. Moiseev, I. I. Shostak, A. M. Nadtochiy, Y. V. Kudashova, A. A. Lipovskii, M. M. Kulagina, and S. I. Troshkov, “Room temperature lasing in 1-µm microdisk quantum dot lasers,” J. Sel. Top. Quantum Electron. 21, 709–713 (2015).
[Crossref]

Zubov, F.

N. Kryzhanovskaya, E. Moiseev, Y. V. Kudashova, F. Zubov, A. Lipovskii, M. Kulagina, S. Troshkov, Y. M. Zadiranov, D. Livshits, M. Maximov, and A. Zhukov, “Continuous-wave lasing at 100°C in 1.3 µm quantum dot microdisk diode laser,” Electron. Lett. 51, 1354–1355 (2015).
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Appl. Phys. Lett. (4)

C. P. Michael, K. Srinivasan, T. J. Johnson, O. Painter, K. H. Lee, K. Hennessy, H. Kim, and E. Hu, “Wavelength- and material-dependent absorption in GaAs and AlGaAs microcavities,” Appl. Phys. Lett. 90, 051108 (2007).
[Crossref]

M. Lebental, J. S. Lauret, R. Hierle, and J. Zyss, “Highly directional stadium-shaped polymer microlasers,” Appl. Phys. Lett. 88, 031108 (2006).
[Crossref]

R. Ding, T. Baehr-Jones, W.-J. Kim, B. Boyko, R. Bojko, A. Spott, A. Pomerene, C. Hill, W. Reinhardt, and M. Hochberg, “Low-loss asymmetric strip-loaded slot waveguides in silicon-on-insulator,” Appl. Phys. Lett. 98, 233303 (2011).
[Crossref]

S. Koseki, B. Zhang, K. D. Greve, and Y. Yamamoto, “Monolithic integration of quantum dot containing microdisk microcavities coupled to air-suspended waveguides,” Appl. Phys. Lett. 94, 051110 (2009).
[Crossref]

Electron. Lett. (1)

N. Kryzhanovskaya, E. Moiseev, Y. V. Kudashova, F. Zubov, A. Lipovskii, M. Kulagina, S. Troshkov, Y. M. Zadiranov, D. Livshits, M. Maximov, and A. Zhukov, “Continuous-wave lasing at 100°C in 1.3 µm quantum dot microdisk diode laser,” Electron. Lett. 51, 1354–1355 (2015).
[Crossref]

J. Eur. Opt. Soc. Rapid Publ. (2)

S. Pélisset and M. Roussey, “Strip-loaded nano-photonics on horizontal slot waveguide,” J. Eur. Opt. Soc. Rapid Publ. 15, 6 (2019).
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S. Pélisset, J. Laukkanen, M. Kuittinen, S. Honkanen, and M. Roussey, “Modal properties of a strip-loaded horizontal slot waveguide,” J. Eur. Opt. Soc. Rapid Publ. 13, 37 (2017).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Sel. Top. Quantum Electron. (1)

N. V. Kryzhanovskaya, A. E. Zhukov, M. V. Maximov, E. I. Moiseev, I. I. Shostak, A. M. Nadtochiy, Y. V. Kudashova, A. A. Lipovskii, M. M. Kulagina, and S. I. Troshkov, “Room temperature lasing in 1-µm microdisk quantum dot lasers,” J. Sel. Top. Quantum Electron. 21, 709–713 (2015).
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T. Harayama and S. Shinohara, “Two-dimensional microcavity lasers,” Laser Photon. Rev. 5, 247–271 (2011).
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L. He, S. K. Ozdemir, and L. Yang, “Whispering gallery microcavity lasers,” Laser Photon. Rev. 7, 60–82 (2012).
[Crossref]

Laser Phys. Lett. (1)

N. V. Kryzhanovskaya, E. I. Moiseev, Y. S. Polubavkina, M. V. Maximov, D. V. Mokhov, I. A. Morozov, M. M. Kulagina, Y. M. Zadiranov, A. A. Lipovskii, M. Tang, M. Liao, J. Wu, S. Chen, H. Liu, and A. E. Zhukov, “Elevated temperature lasing from injection microdisk lasers on silicon,” Laser Phys. Lett. 15, 015802 (2017).
[Crossref]

Opt. Express (3)

Opt. Lett. (3)

Optica (1)

Phys. Rev. A (1)

Q. Song, W. Fang, B. Liu, S.-T. Ho, G. S. Solomon, and H. Cao, “Chaotic microcavity laser with high quality factor and unidirectional output,” Phys. Rev. A 80, 041807 (2009).
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Q. Song, “Emerging opportunities for ultra-high Q whispering gallery mode microcavities,” Sci. China Phys. Mech. Astron. 62, 74231 (2019).
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Science (1)

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
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Other (2)

http://microchem.com/pdf/SU-82000DataSheet2025thru2075Ver4.pdf .

M. Häyrinen, M. Roussey, A. Bera, M. Kuittinen, and S. Honkanen, “Atomic Layer Re-Deposition for Nanoscale Devices,” in Encyclopedia of Plasma Technology, J. L. Shohet, ed. (Taylor & Francis, 2016), Vol. 1, pp. 143–151.

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Figures (5)

Fig. 1.
Fig. 1. Concept and initial considerations. (a) Sketch of a microdisk laser under a strip-loaded horizontal slot waveguide structure. The MD is pumped with an external laser source, and the response is observed using scanning near-field optical microscopy. (b) Cross section of the waveguide device in the center of the MD. The waveguide is a five-layer horizontal slot waveguide including the rail layers and the slot. The MD laser is placed in another low-index dielectric material, SU-8. The loading strip is used to provide the lateral light confinement and its propagation in the $z$ direction. (c) Calculated distribution of the normal component of the electric field ( ${E_x}$ ) of the fundamental TM mode propagating in the $z$ direction. Calculations were made using the following parameters: ${\text{TiO}_2}$ layer widths are ${h_1} = 135\; \text{nm}$ and ${h_2} = 160\; \text{nm}$ , ${\text{SiO}_2}$ slot width is ${h_s} = 70\; \text{nm}$ , wavelength is 1290 nm, refractive indices are ${n_h} = 2.30$ , ${n_s} = 1.44$ , ${n_c} = 1.60$ , and ${n_{ph}} = 1.55$ .
Fig. 2.
Fig. 2. Top view of the waveguide-integrated microdisk. Optical microscope image of a 10 µm MD laser under the strip-loaded horizontal slot waveguide. The ring surrounding the MD laser is a result of the difference in etching rate between the materials of the QDs and the bottom ${(\text{AlGa})_x}{\text{O}_y}$ layer insulating the cavity from the substrate.
Fig. 3.
Fig. 3. (a) MD laser spectrum at low optical pump and (b) integrated intensity of the dominant mode line (squares) and mode linewidth (circles) versus the optical pump power for the 10-µm-diameter microlaser of the virgin MD without the SLSW (half-filled) and under the SLSW. The arrow denotes the “kink” showing the lasing threshold.
Fig. 4.
Fig. 4. Spectral and spatial near-field investigation of the microlaser and its coupling to the strip-loaded slot waveguide. Maps of (a) AFM and (b), (c) SNOM (top images), together with (a) height and (b), (c) intensity profiles of the 10 µm microdisk laser under the SLSW structure. SNOM images are taken (b) for the entire spontaneous emission spectral range ( $1280\; \text{nm} \lt \lambda \lt 1285\; \text{nm}$ ) and (c) for the lasing mode only ( $\lambda = 1290\;\text{nm}$ ). The location of the MD is shown by blue arrow in (a), and the profile measurement lines are marked with solid white lines on the corresponding top maps.
Fig. 5.
Fig. 5. Near-field investigation for a 7°-tilted sample. (a) An optical microscope image of two 10 µm microdisk lasers under the strip-loaded slot waveguide structure (top) and a near-IR camera image taken from the cleaved optical waveguide. (b) Near-field signal map of spontaneous emission from InAs/InGaAs/GaAs quantum dots over the emission spectral range of the QDs ( $1280\; \text{nm} \lt \lambda \lt 1285\; \text{nm}$ ) (top) and photoluminescence signal observed in a cross section along the loading strip (white line). (c) Near-field mapping of the lasing WGM ( $\lambda = 1290\; \text{nm}$ ) line (top) and its corresponding profile. The contour of the loading strip is highlighted by dashed white lines and the cross sections are marked with solid lines on the maps.

Equations (13)

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E x = A 0 { 1 n p h 2 e γ p h x , x < 0 1 n h 2 [ cos ( k h x ) + B 2 sin ( k h x ) ] , 0 < x < a 1 n s 2 [ A 1 cosh ( γ s ( x a ) ) + A 2 sinh ( γ s ( x a ) ) ] , a < x < b 1 n h 2 [ C 1 cos ( k h ( x b ) ) + C 2 sin ( k h ( x b ) ) ] , b < x < c 1 n c 2 E e γ c ( x c ) , x > c ,
H y = e y H y exp [ i ( β z ω t ) ] ,
2 H y z 2 ( β 2 k 0 2 ε i ) H y = 0 ,
k 0 2 n s 2 , k 0 2 n c 2 , k 0 2 n p h 2 < β 2 < k 0 2 n h 2 ,
E x = β ω ε 0 ε i H y ,
E x = A 0 { 1 n p h 2 e γ p h x , x < 0 1 n h 2 [ cos ( k h x ) + B 2 sin ( k h x ) ] , 0 < x < a 1 n s 2 [ A 1 cosh ( γ s ( x a ) ) + A 2 sinh ( γ s ( x a ) ) ] , a < x < b 1 n h 2 [ C 1 cos ( k h ( x b ) ) + C 2 sin ( k h ( x b ) ) ] , b < x < c 1 n c 2 E e γ c ( x c ) , x > c ,
B 2 = γ p h n h 2 n p h 2 k h ,
A 1 = cos ( k h a ) + γ p h n h 2 n p h 2 k h sin ( k h a ) ,
A 2 = k h n s 2 n h 2 γ s ( sin ( k h a ) + γ p h n h 2 n p h 2 k h cos ( k h a ) ) ,
C 1 = ( cos ( k h a ) + γ p h n h 2 n p h 2 k h sin ( k h a ) ) cosh ( γ s ( b a ) ) + ( k h n s 2 n h 2 γ s ( sin ( k h a ) + γ p h n h 2 n p h 2 k h cos ( k h a ) ) ) sinh ( γ s ( b a ) ) ,
C 2 = γ s n h 2 n s 2 k h [ ( cos ( k h a ) + γ p h n h 2 n p h 2 k h sin ( k h a ) ) sinh ( γ s ( b a ) ) + ( k h n s 2 n h 2 γ s ( sin ( k h a ) + γ p h n h 2 n p h 2 k h cos ( k h a ) ) cosh ( γ s ( b a ) ) ] ,
E = [ ( cos ( k h a ) + γ p h n h 2 n p h 2 k h sin ( k h a ) ) cosh ( γ s ( b a ) ) + ( k h n s 2 n h 2 γ s ( sin ( k h a ) + γ p h n h 2 n p h 2 k h cos ( k h a ) ) sinh ( γ s ( b a ) ) ] cos ( k h ( c b ) ) + [ γ s n h 2 n s 2 k h ( cos ( k h a ) + γ p h n h 2 n p h 2 k h sin ( k h a ) ) sinh ( γ s ( b a ) ) + ( k h n s 2 n h 2 γ s ( sin ( k h a ) + γ p h n h 2 n p h 2 k h cos ( k h a ) ) cosh ( γ s ( b a ) ) ] × sin ( k h ( c b ) ) .
k h n c 2 n h 2 γ c = tan [ k h ( c a ) + arctan ( n s 2 k h γ s n h 2 n h 2 γ s k h n s 2 tan ( k h a + arctan ( n p h 2 k h γ p h n h 2 ) ) + tanh ( γ s ( b a ) ) 1 + n p h 2 k h γ p h n h 2 tan ( k h a + arctan ( n p h 2 k h γ p h n h 2 ) ) tanh ( γ s ( b a ) ) ) ] .

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