A perturbative analysis is proposed to estimate optical losses for electrically pumped micro-disk lasers. The optical field interaction with the electrical contacts and the optimization of their implementation is investigated. Our model shows a good agreement with 3D Finite Difference Time Domain (FDTD) computation and can be used for designing contacts for thin micro-disks, with a considerably reduced calculation time.
We also demonstrate that losses induced by the contacts can be exploited to select the optical mode of a micro-laser.
©2009 Optical Society of America
Low cost, robust and efficient light sources are needed for optical high speed communications in integrated circuits. Micro-disk resonator lasers correspond to one of the most adapted solution in regard to their performances and their processing easiness [1, 2, 3, 4]. They combine very low surface (radius below 10 μm) and low power consumption (threshold below 50 μW) . Using dies of InP membranes bonded onto 200 mm SOI wafers [6, 7, 8] allows the fabrication of complete optical links, with a source, high index Si waveguides and photodetectors for signal collection . A vertical PIN diode is commonly used to bring carriers in the active region of the micro-laser. A thin conductive bottom layer, called slab, corresponds to the bottom contact : its thickness mainly controls the serial resistance. All the structure is surrounded by a low index media, like silica, and a conductive material is placed on top for the top contact (Fig. 1).
Electrically contacting such sources implies the use of metals, or more generally optical absorbing elements, which may results in significant optical losses and, consequently, in an increased threshold power. These deleterious effects have to be minimized in order to limit the consumed power. Design of the contacts must meet the seemingly incompatible requirements of setting a minimum distance between the Whispering Gallery Modes (WGMs) and the absorbing conductive material, and of confining electrical current lines within the high electromagnetic field area . As these contacts do not always respect the cylindrical geometry of the resonator, 3D simulations are required, that can lead to long computation time. In this paper, we propose and validate a simple modeling method to evaluate contact induced optical losses based on a perturbative approach. Its validation is performed using FDTD simulations. In the last part, we estimate how contacts can be used to strengthen optical mode selectivity.
2. Description of the model
In this part, we develop a model to evaluate absorption losses induced by the top contact (the effect of the bottom one will be discussed in part 3.2). First, we will try to find an approximative 3D solution of the electromagnetic field for TE like modes of a micro-disk. As high quality factors are expected, we can assume that the field distribution is not significantly changed by the presence of absorbing materials that will have to be far enough from the WGM. So we can use this solution to evaluate the top contact induced losses. This approximation is more valid when ITO is used for the conductive material since the real part of its optical index is close to the silica one with a value of 1.51 at 1.55μm when not annealed. Moreover, this material can be used with standard Si equipments, and may act as an intermediate medium between standard metalization for Si processes and InP, without using gold based contacts.
The studied structure is a micro-disk (radius R, thickness H) made of a high optical index material (ndisk) in a lower index cladding medium (nclad), like silica. The top contact can be divided into 2 elementary volumes (see Fig. 2):
- the “via” which contacts the central area of the disk, and allows carriers injection from the top surface. The smaller is the radius Rc of this via, the smaller are the absorption losses in the conductive material and the greater is the global Q factor...but, at the same time, the electrical injection becomes less efficient.
- the “tab” (thickness Hm) which is separated from the WGM by silica (distance Hc). ΔL can be used for misalignment compensation in fabrication.
Both tab and via are made of the same absorbing medium and optical losses in the top contact will be mainly governed by parameters Hc and Rc.
WGM 3D distribution for a transverse electric field (TE) mode is fully described by three coefficients (l,m,n), that respectively correspond to the number of nodes along the r axis, the number of periods on the circumference, and the number of nodes on the z axis. For symmetry reasons, we will focus on a separated variables problem to describe the field in the r ≤ R area using cylindrical coordinates. Such approximation is commonly used to study thin micro-disks , but can not take into account the diffraction of the Whispering Gallery Modes at the edge of the disk. In this paper, we are dealing with optical losses like absorption of the evanescent field by a top contact. Therefore, as the electromagnetic field is mostly situated in the microdisk, we can ignore the optical losses for r > R. In each homogeneous region of the structure, the electric field obey the d’Alembert vectorial equation given by:
where ni is the index of the considered medium and k the wavenumber. In our model, we consider a perfect TE mode neglecting the vertical component of the electric field. Then, the equation 1 in cylindrical coordinates can be solved:
where Um is a linear combination of Bessel functions of the first kind (Jm and Ym) and E(z) the solution of the electric field for a TE mode of effective index neff guided in the corresponding infinite membrane. Continuity conditions are automatically satisfied at z = ±H/2 for both the electric field E⃗ and the magnetic induction H⃗, obtained from the Faraday’s law. Unfortunately, the 5 equations corresponding to continuity conditions at r = R have no trivial solution, except if setting aside the θ component of H⃗ (then we get 2 redundant equations) and the z equation from the d’Alembert for r > R. This limitation proves that a separated variables problem only returns an approximation of a WGM. Then, the resonant wavelength λres can be easily computed  since the equations 3 lead to solve the 2D solution of an infinite cylinder of the same index neff in a medium of index nclad. For better results, effective index is not considered as a constant parameter, but is adjusted according to the wavelength.
with N 2 Disk = ndisk 2 - neff 2 and Nclad 2 = neff 2 - nclad 2 and coefficients (j,y) to satisfy continuity conditions at r = R. The given expression of E(z) corresponds to even values of n : replace cos functions by sin ones for odd values and preserve continuity conditions at z = +H/2 changing if necessary the sign of the expression when |z| > H/2.
From now we are able to evaluate a 3D field for a TE mode with a comparable to FDTD magnetic induction while 2D models do not return its θ and z components. This model takes the most from two 2D models thanks to the use of both separated variable functions and continuity conditions. Last, the cladding material influence on wavelength is taken into account since the E-field is not trivial at the r = R boundary [13, 12].
Knowing the 3D field distribution and considering that this solution is not significantly modified by the absorbing contacts, we can calculate the mean (over θ) energy density e <θ> as a function of z and r:
with ε 2D equal to neff 2 since r < R, and ε 3D equal to ndisk 2, nclad 2 or ncont 2, depending on the value of z. Then, by integration of e <θ>, we can easily obtain the energy in our system without contacts, Etot. Two kinds of losses are considered: first, the optical intrinsic losses τo -1 due to the diffraction at the edge of the micro-disk, and absorption losses in the electrical contact which will be calculated using its contained electromagnetic energy Econt, associated to a loss rate τa -1. This loss rate is proportional to the absorption constant of the contact medium α 0 since τa -1 = α 0 ∙ vg with vg the photons group velocity in this material. Then we can link the energy and global losses τ -1 with:
and deduce that contact losses τc -1 are given by:
As we are only considering the fields in the area r < R, absorption losses should be under estimated, but the strong confinement of WGM should ensure a quite good estimation if the real part of the index for the top contact is close to that of the cladding material.
In order to evaluate the influence of the size of the via (Rc) and the distance between the tab and the membrane (Hc), we compared our model to 3D FDTD calculations (TESSA  and Harminv  for post treatment) taking contacts in account. Contact losses τc -1 are estimated from the loss rate difference between with (τ -1) and without the contact (τo -1).
3. Application of the model to a contacted micro-disk laser
3.1. Influence of the top contact
The main goal consists in ensuring high quality factors, so that when a waveguide is added, coupling losses are the dominating source of losses [8, 10]. The geometrical parameters R, H, Hm and ΔL are set to respectively 2.5 μm, 0.55 μm, 0.9 μm and 0.5 μm. The following results only concern the (0,26,0) mode at 1.51 μm wavelength for nclad = 1.44 (silica), ndisk = 3.17 (InP) and an ITO based top contact, with an absorption constant α 0 = 1.83 × 106 m -1.
Field distributions (without any contact) as a function of z, as derived from FDTD simulations and from our model are shown in Fig. 3. Outside of the resonator, the effective index approximation of our model results in a slower decay of the field and we can consider that losses through the top contact should be over estimated with our model. For |z| < 500nm, we can observe a good agreement.
Since WGMs are located at the edge of the disk, optimal condition for electrical pumping corresponds to a central top contact that is large enough to allow current lines in the high electromagnetic field area, but not too large to limit the absorption. Figure 4 shows a good agreement of contact losses versus Rc as simulated by FDTD (dots) and as derived from our analytical model (bold blue line). It is possible to separate tab losses from the via ones and for a given distance Hc, optical losses occur mainly in the tab (resp. the via) if Rc is smaller (resp. larger) than a characteristic value (~1.58 μm in Fig. 4). It is observed that there is no need to reduce further beyond 1.7 μm the via radius, since intrinsic losses become dominant, and the top contact is uselessly too small.
As the top contact partially absorbs the evanescent field from the WGM, a too small distance Hc can lead to very high loss level, more particularly due to interactions in the portion 2 (Fig. 2). The impact of Hc on optical losses is shown in Fig. 5, using again FDTD simulation and the present model. When contact losses dominate the intrinsic losses of the contact free micro-disk, the discrepancy between FDTD and our model increases when the tab to micro-disk distance reduces. For quality factors higher than 10 000, our model lead to an over-estimation of the contact to membrane distance from 60 nm to 25 nm.
A quick design can be achieved determining a couple (Hc,Rc) for which top contact losses do not affect too much the global losses. With such a condition, Rc values are limited by Rcmax which gives the maximum acceptable radius of the via and corresponds to Hc=∞. A simple way to choose these 2 parameters consists in getting comparable losses in the tab and the via. Last, we will compensate misalignment between the micro-disk and the via during fabrication choosing a smaller value of Rc if necessary. On the contrary, we can keep the computed Hc value since this parameter is over-estimated.
3.2. The fully contacted structure
We are now considering that a bottom contact is present [6, 7]. Due to its high optical index, this thin membrane can be compared to a weakly confined waveguide that brings photons out the micro-disk. For this reason, it leads to slab induced losses and a maximum quality factor much lower than the intrinsic one. For a 100 nm thick slab and a 550 nm III-V membrane, FDTD results show that quality factors are limited to around 30 000. Bigger structures (thicker membranes or larger radius) are less affected by the bottom contact , and quality factors of ~ 2 ∙ 105 are theoretically reachable. The FDTD field distribution in Fig. 6 shows that the WGM extends laterally into the slab over a few wavelengths distance, which is obviously very different from the perfect micro-disk. Then, at position r > R, it is expected that the top contact absorbs additional photons.
In figure 7, the E(z) functions in the micro-disk and in the slab are normalized with respect to the field inside the micro-disk. The effective index approximation still shows a good agreement with FDTD simulations in both regions. Only the greatest positive values of z show a slight disparity. For negative values of z (i.e. z < -600 nm in our case), losses induced by the slab can dominate the disk ones due to a lower confinement of the field. For this reason, it is necessary to increase the distance between the bottom face of the micro-disk and underlying high index materials like the substrate. It is also observed that for high values of Hc (z > 0), the intensity of the field that meets the contact is close over the micro-disk (diamonds in Fig. 7) and in the r > R region (triangles in Fig. 7). We can deduce that the shape and size of the top contact when (r > R) can significantly affect the contact induced losses.
FDTD simulations shows that the influence of the top contact is greater with a bottom slab, that leads our model to under-estimate losses (Fig. 8 and Fig. 9). In comparison to the simulations without a slab (Fig. 5 and Fig. 7), losses are increases by a factor of about 5. This effect cannot be explained by the weak difference of the optical indexes between sections 3.1 and 3.2. Varying the size of the via Rc, the difference between our model and FDTD almost reaches one decade, and the crossing point between tab and via losses is lower for FDTD (~ 1.35 μm) than for our model (~ 1.53 μm), that means that the WGM is also less confined in the plane (e⃗r,e⃗θ).
In order to reduce losses, the bottom slab should not be left below the top contact since its lower optical confinement increases losses by absorption in the top contact. Then, our model should give more adequate results. For an optimized design, losses induced by the top contact should not exceed the bottom contact ones.
3.3. Mode selection induced by contact losses
WGMs with different radial, azimuthal and vertical orders (l,m,n) can compete to reach lasing regime, especially if quality factors are close and spectral overlaps between the optical mode and the gain are similar. Low threshold is expected when the chosen WGM quality factor for lasing is significantly larger than that of other modes. In the following, we show that top contacts can be used to get significantly different quality factors between the mode chosen for lasing and other ones. Table 1 gives orders and effective index of different modes computed as explained in section 2, with λres around 1.55 μm.
As the (0,m,0) modes correspond to lower intrinsic losses, the (0,25,0) mode will be considered as the desired mode because of its wavelength close to 1.55 μm. For all the modes mentioned in table 1, we estimate losses using our model with α 0 = 1.83 × 106 m -1 (ITO) and ncont = 1.52. Only the (0,19,1) mode has a neff value significantly different from the other solutions. As we can see in Fig. 10, this leads to an important shift in losses for this particular mode when studying Hc, due to a lower vertical confinement. Increasing the radial mode number (l) slightly increases losses.
Losses in the via are investigated in Fig. 11. Comparing modes (1,21,0) and (2,17,0) to the (0,25,0) one, we can observe that via losses are increased by at least two decades per variation of the radial order. Reducing the vertical confinement (i.e. reducing neff from ~ 2.99 to 2.37) also leads to a significant increase of losses with a difference of more than one decade.
We can deduce that the top via contact can be adjusted to favor only (0,m, 0) family modes and the distance from the WGM, Hc, to select (l,m,0) modes. Both via and tab contributes to a deep quality factor contrast between the (0,m,0) family modes and the other ones. A numeric application with our model for Rc=1.8 μm,Hc=0.40 μm and the parameters of part 3.1 shows that quality factors are limited to 5 300 except for the desired mode which should reach 46 000. In comparison, FDTD simulations leads to similar quality factors: respectively 5 400 and 56 000.
A model based on both a 3D solution of the d’Alembert equation in cylindrical coordinates, and continuity conditions is used for a very fast estimation of losses in a top contact for thin micro-disks based lasers. For the same computational power, only few seconds are required against a day for only one FDTD computation. The results are compared to 3D FDTD simulations, and we demonstrate that it leads to a good agreement for a perfect micro-disk.
Even if the field intensity in the slab is much lower than inside the micro-disk, its lower confinement increases losses by absorption in the overpassing part of the top contact. Then a well designed structure should limit the presence of a slab under a top contact.
Optimizing the top contact geometry can be used to enhance mode selectivity and (0,m,0) modes can be favored by considerably lowering associated losses (1 decade or more), still granting very high quality factors (typically more than 50 000).
This work has been supported by the European project FP6-2002-IST-1-002131-PICMOS and the European project FP7-ICT STREP WADIMOS.
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