A study of vertical coupling conditions between microdisk based resonators and waveguides is presented using an analytical model. The coupling efficiency optimization is investigated and compared to 3D FDTD computations. We also demonstrate that coupling losses can be exploited to favor high quality factor modes in circular resonators. In addition, we propose to modify the shape of the coupled waveguide to enhance mode selectivity and obtain a very compact structure with mode hoping capacities. Lower thresholds and modulation are also expected.
©2010 Optical Society of America
High speed optical connections in integrated chips require at least light emitting, routing [1–3], waveguiding and detecting devices . Since Si is not compatible with light emission at Telecom wavelengths, molecular bonding of III-V sub-wavelength thick membranes [5–7] onto Si or SOI wafers can be used for the gain material. The presence of Quantum Wells  (or Quantum Dots ) and of highly doped layers  prevent this III-V layer from being used to collect light into a waveguide. As the efficiency of a lateral coupling to a different material waveguide (like a-Si) heavily depends on lithography misalignment, vertically coupling the laser to the waveguiding elements  is one of the best solutions [6, 8].
Figure 1 shows an electrically driven microdisk coupled to a waveguide with a thin and doped layer (called “slab”) of high index material for the bottom contact. A cladding material (silica) is deposited after the microdisk fabrication, and this low index material is locally etched to contact the device with metal layers. For a detailed fabrication process, see .
In such devices, most confined modes (with the highest quality factors) propagate at the edge of the resonator. They are called Whispering Gallery Mode (WGM) and characterized for each polarization (Transverse Electric [TE] or Transverse Magnetic) by a set of 3 integers (l,m,n) denoting the number of nodes on the radial direction (l), of periods at the edge (azimuthal order m) and of nodes (n) on the height. A 3D semi-analytical model of the WGM  can be profitably exploited to calculate the resonant frequency f (l,m,n) with an accuracy better than 1 nm in a few micrometers radius microdisk at telecom wavelengths. Modes with the same radial l and vertical n orders are very regularly positioned (Fig. 2) in the frequency domain, leading to different combs (denoted P (l,n)) with a characteristic frequency distance called Free Spectral Range defined as FSR(l,n)= f (l,m,n)−f (l,m+1,n).
Fabrication limitations (position of the electrical contacts, roughness , slanted edges …) lead to disk radii larger than ~2 µm with Q factors limited to few thousands. Since the emission range in quantum wells is higher than the FSR, mode competition occurs between each comb P (l,n) and between modes from a same comb with different azimuthal orders (Fig. 2). To select a particular WGM, a solution consists in increasing the optical losses of the unwanted ones. This is partially done by the microdisk itself as losses of modes with small l and n are lower . To go further, the geometry of the whole device can be efficiently exploited. Thin III–V membranes (typ. ≤550 nm) lead to n≤1 and replacing the inner part of the microdisk by a low index material prevents modes from having high radial orders . Absorption through the top contact can also be profitably used to favor the most confined modes .
In order to restrict lasing regime to (0,m,0) modes, we propose to exploit the vertical coupling between a microdisk and a straight waveguide. We will explain how to efficiently couple modes from the P (0,0) comb at the expense of those with a higher vertical order. In the second part of this paper, we will increase the azimuthal order sensitivity and demonstrate that controlling the optical index of the waveguide can be advantageously used for mode hoping applications or to adjust the resonant frequency.
2. Vertical coupling of a microdisk to a waveguide
As fields from WGMs  and waveguides  are exponentially decreasing in the interstitial low index area, their interaction is necessary limited to areas where their energy densities are important. For this reason, WGMs (resp. waveguided modes) will be represented by a circle of effective radius Reff (resp. by its axial line) that corresponds to the maximum energy density on the radial direction for l=0 (resp. in the cross-section plane). If the center of the waveguide is close enough to the center of the resonator, coupling mainly occurs at two different points (A and B), along a local distance L (Fig. 3).
Consequently, the energy transfer depends on the position xg of the center of the waveguide since it changes the two optical paths from A to B (i.e. in the resonator and in the waveguide).
In the following sections, we propose to develop an analytical model in order to find the position with the best coupling efficiency when the thickness of the interstitial low index medium between the waveguide and the resonator is given.
2.1. Model used for the local coupling efficiency
Due to the Quantum Wells (QW) emission polarization , our study is restricted to TE modes. The “intrinsic losses” (denoted τ −1 0) term will stand for all types of cavity losses (like contact induced ones [11,16] or fabrication defaults like roughness , slant of edges  …), except for coupling to the waveguide ones.
To describe the interaction between the resonator and the waveguide, let us define the local coupling efficiency ηc as the portion of the energy that is transferred from the resonator to the waveguide at each turn of a propagative mode. Thus, if I represents the energy flux just before the coupling area, an energy ΔI = −ηc · I is transferred in the coupling area. For a (l,m,n) propagative WGM, this variation occurs each Δt = FSR(l,n)−1. This is the time duration of a lap in the cavity. Then, using the definition of the coupling loss rate τ −1 c, we get:
At each coupling point, we propose to use the same model as for two straight waveguides, but with a crossing angle . Then, we can estimate ηc with the same equations as in :
with L=Reff·(sin(θ 2)−sin(θ 1)) and using a projection of the WGM on the waveguide direction with . In this equation set, βd (resp. βg) is the propagative constant in the microdisk (resp. waveguide). C i→j represents the energy from element i that interferes with element j. It can be expressed using :
where ui,j corresponds to the distribution of the electromagnetic field along the vertical direction. These functions can be easily approximated using the effective index calculation of an infinite membrane of index ndisk for the microdisk (resp. ng for the waveguide) and height Hi, in a cladding material of optical index nclad. From these equations, when the waveguide is close to the center of the resonator (xg~0), the local coupling loss rate τ −1 c tends to 0, due to orthogonal propagative vectors.
Theoretically, the maximum coupling transfer between two straight waveguides corresponds to two different conditions:
- an effective index adaptation between the two elements;
- an interaction distance which is an odd multiple of the transfer distance Lt =π/(2γ′).
Unfortunately, these two conditions are not compatible with our structure. For representative separating distance (ie. more than 50 nm due to fabrication limitations) and few micrometers radius resonators, Lt is generally longer than L, so the contribution of the interfering distance L is maximum when the waveguide is close to the edge of the resonator. On the other hand, index adaptation is obtained at a particular position xgAdapt equal to:
For the most confined modes, βg<βd leads to xgAdapt <Reff: given that L is smaller than Lt, the maximum coupling efficiency is achieved for xg in the range [xgAdapt,Reff].
2.2. Effective coupling losses of the system
The phase difference ΔϕAB between the two optical paths from A to B can be easily estimated:
This later modulates the coupling losses of the system (τ′c)−1, its global quality factor Qr and its resonant frequency ωr. Using the Coupled Mode Theory (CMT) , we get:
Frequency ωr can theoretically vary in a range that is only limited by the local coupling loss rate τ −1 c and centered around the resonant frequency of the device with no waveguide (ω 0). Depending on the relative position between the waveguide and the microdisk xg, the phase conditions at the two coupling areas strongly impact the global quality factor Qr which vary:
- from Qc/2: interferences at the coupling points are constructive, and the quality factor is the same as if the resonator were coupled to two independent waveguides.
- to Q 0: with destructive interferences, no light is transferred to the waveguide.
2.3. Unwanted losses due to the presence of the waveguide
Due to a local distortion of the WGM close to the edges of the resonator, the presence of the waveguide induces unwanted losses that are not taken into account by our model. They are strengthened by an important overlap between the microdisk and the its coupled waveguide and if they are not negligible compared to the coupling ones, the coupling efficiency is deteriorated.
An illustrative sequence for different relative positions of the waveguide is depicted in Fig. 4 with the location (in red) where these diffraction losses mainly occur. If Reff is not far enough from the tangency position of the waveguide (xg=R−Wg/2), we have to take them into account to find the position with the best coupling efficiency since they can significantly increase the total losses of the system. In other words:
- either R−Reff > Wg/2 [Fig. 4(a)]: our model should give a pretty good description of the coupling properties if the unwanted diffraction losses can be neglected.
- or R−Reff ≲ Wg/2 [Fig. 4(b)]: both coupling and diffraction losses should increase for xg ∈[R−Wg/2,Reff]. FDTD is required to find the position with the best coupling efficiency.
2.4. Analytical results and comparison to FDTD when − > /2
In order to validate the model, we made 3D simulations (using home-made Tessa FDTD ) with a single mode Si waveguide and a large enough resonator (4 µm radius) to ensure that unwanted losses effect are reduced. To reduce spectral densities and ease the post treatment with Harminv , the resonator is replaced by a 1 µm wide ring with the same external radius as the wanted microdisk and an adapted excitation source is used.
Both loss rates and the resonant frequency for the TE(0,45,0) mode of the resonator are represented on Fig. 5. Positions where we get destructive interferences according to the analytical model (ΔϕAB ≡ π, at xg~3.46 µm and xg~3.68 µm) are in very good agreement with FDTD [Fig. 5(a)]. When inhibiting the energy transfer, losses should be extremely low but FDTD shows that loss rates variations is lower than one decade. For this reason, we can reasonably assume that the difference mainly comes from diffraction losses due to the presence of the waveguide itself. Their existence is confirmed in Fig. 5(b) since the difference between the model and FDTD results increases with xg.
2.5. FDTD analysis and model limitations when R−R ≲ W/2
As only the effective coupling loss rate is considered in our model, we provide further numerical experiences with FDTD to find positions with the best coupling efficiency.
Using the local Poynting equation and Green-Ostrogradski theorem, beamed power at the boundary of the studied volume in FDTD can be linked to the energy in the system E, since:
with em the electromagnetic energy density and S⃗ the Poynting vector.
As the transferred power Pg(ωr) is proportional to the square of the amplitude ag(ωr) of the guided mode, we only need a local sensor in the waveguide to measure the coupling efficiency. Then, the coupling loss rate (τ′c)−1 is given by:
Figure 6 represents both total losses and coupling ones (τ′c)−1 for different positions xg, maintaining the sensor in the center of the waveguide, far from the interaction area.
From our model, coupling losses should be minimal around xg~1.95 µm and FDTD confirms that this position corresponds to a minimum light transfer. A maximum coupling efficiency is obtained at position (a) where xg < Reff, and we fulfill 2 contradictory conditions corresponding to a maximum transferred energy and a minimum global loss rate. For higher values of xg, coupling losses are decreasing (red curve) contrary to the total ones (black curve): losses induced by the presence of the waveguide itself are dominating. Consequently, the coupling efficiency is maintained at low values for xg between Reff and R, and is divided by 10 at position xg≈2.35 µm.
Due to the presence of a thin high index layer (a 100 nm thick III-V membrane) used for the bottom contact (Fig. 1 and inserted picture in Fig. 6), an effective index adaptation between the waveguide and the extended part of WGM in this membrane can also correspond to an interesting coupling condition. This later (referred as (b) in Fig. 6) can compete with the best coupling position with xg<Reff (a), depending on the distance between the bottom surface of the resonator and the waveguide since it strongly affects the transfer distance Lt.
2.6. Influence of the vertical mode order
As the absorption in the top contact mainly affects the less confined modes in the radial direction , we will only focus on the vertical order influence on the coupling loss rate. Values obtained with our model for modes with n≤1 in a R=4 µm radius microdisk are displayed in Fig. 7.
As βd < βg for the TE(0,34,1) mode, the position with destructive interferences (around 2.15 µm in Fig. 7) is much farther from xg=Reff than for the TE(0,42−44,0) modes. Their coupling loss rate is maintained at a very high value on a large range of positions xg, while for the modes of interest, coupling is modulated by the position of the waveguide (range [2.5 µm, Reff] in Fig. 7). Consequently, having βd < βg corresponds to a more robust design since a misalignment error at fabrication does not significantly change the coupling efficiency. Unfortunately, such a condition cannot be achieved for the P (0,0) comb since the need of doped layers and QWs  leads to a minimal thickness of the III–V membrane corresponding to too high propagative constants βd.
Black curves in Fig. 7 show that coupling loss rates do not significantly vary with the azimuthal order, due to close optical paths between the two coupling areas. The coupling efficiency of the TE(0,m,0) modes is expected as being optimized at position xg=Reff, but this position competes with xg = 3.45 µm. In both cases, we get high enough quality factors to reach the lasing regime, while for the higher vertical order ones (green curve in Fig. 7), losses are significantly increased to limit this later to few hundreds.
From our analysis, vertical coupling can be used to enhance mode selectivity since in a large range of relative positions losses for modes with a vertical order are much more higher than for the TE(0,m,0) modes. For a given WGM, the effective coupling loss rate (τ′c)−1 depends on the propagative constants: in order to prevent WGMs from coupling to a low effective index waveguided mode with unwanted phase conditions, we suggest to use single mode waveguides.
Using a simple model of the vertical coupling, we demonstrate that the position of the waveguide has a deep influence on the coupling efficiency. For small microdisks (typ. R<3 µm), an optimization is required to find the most interesting position.
Choosing xg ≲ Reff ensures much more important coupling losses for modes with a high vertical order, that can be used to increase the selectivity between the TE(0,m,0) modes and the TE(0,m,n) ones. This behavior can be strengthened using the top contact of the microdisk .
Unfortunately, for the comb of the most confined WGMs (P (0,0)), we have almost no dependence on the azimuthal order m and on the wavelength. This behavior comes from too short optical paths between the two effective coupling areas.
3. Modulated coupling efficiency and azimuthal order control
As the FSR is generally smaller than the gain range of InP based QWs (Fig. 2), we need to select one particular mode in a given comb of resonant frequencies. In this part, we will investigate how to modulate the coupling loss rate depending on the azimuthal order, still using a vertical coupling between the waveguide and the resonator.
To increase the frequency sensitivity of ΔϕAB, we propose to strongly couple the resonator to the waveguide and to lengthen the optical paths between the two coupling areas (Fig. 8):
- in the resonator using two diametrically opposed points;
- using a longer Si waveguide to strengthen the impact of the propagative constant.
The symmetry of the structure (Fig. 8) ensures that the same equations rule both the Clock-Wise (CW) and Counter-ClockWise (CCW) modes in the resonator, and consequently the stationary ones. For this reason, we can restrict our analysis to CW propagative solutions.
In order to take into account losses in the waveguide (due to its local curvature for instance), we will introduce ε to measure the attenuation of the electromagnetic field between A and B. Results from CMT applied to a propagative mode are very close to the previously presented ones since only the following expressions are changed:
with βi·di the phase difference induced by the photons travel in each elementary portion of the waveguide (bent parts, straight ones…).
Table 1 describes the evolution of the coupling loss rate, the resonant frequency ωr and associated quality factor Qr. For low loss waveguided modes (ε close to 0), variations of these parameters are similar to the case of the vertical coupling studied in the previous part.
From this variation table, slight losses (ε ≠ 0) in the Si waveguide ensures optical light in the output waveguide whatever the value of ΔϕAB, since the coupling losses (τ′c)−1 cannot be canceled [Eq. (9)], preventing Qr from reaching Q 0. Both a high coupling efficiency and a very high quality factor when ΔϕAB≡π are ensured if τ−1 0≪ε·τ −1 c.
Last, the waveguide portion between the two coupling areas acts as a feedback control of the coupling loss rate, since its optical index directly affect the ΔϕAB parameter.
3.1. 2D simulations and optimization with the analytical model
To demonstrate the coupling inhibition and frequency variation in very compact devices, we only need locally strongly coupled waveguides (high τ −1 c values). This can be achieved using xg close to Reff as explained in the first part. Then, only 2D simulations are required to study the feedback loop influence on the device. In this model, the optical path in the feedback loop is controlled by the length of 2 straight waveguide portions (like in Fig. 8).
Once the optical propagative constants in each part of the optical circuit are determined by simulation [Fig. 9(a)], we calculate the quality factor and the frequency for 3 different modes with azimuthal order m between 25 and 27 for a 2.5 µm radius ring resonator.
Due to different optical paths, modes in a same comb do not share the same phase conditions. Using the CMT based model, we can optimize the feedback loop length to favor the TE(0,26) mode, as shown in Fig. 9(d). At the particular value of Z=9.69 µm, the quality factor contrast between the chosen mode and its neighbors is mainly limited by the high value of ε, due to the narrowness of the waveguides. For a 200×500 nm Si waveguide with a curvature radius > 2 µm, lower attenuation is expected with ε = 0.1 dB/90° .
Optical properties of the device are efficiently depicted with the CMT: this later can be used for a fast design of the structure, and then the size of the feedback loop can be refined with a more precise simulation (like FDTD).
3.2. Spectral properties depending on the feedback loop length
In the previous part, we demonstrate that one particular mode can be selected with a very compact device. In this part, we propose to study the influence of the feedback loop length for modes in a same comb P (l,n) (with a frequency distance FSR(l,n)) and demonstrate that different regimes can be achieved.
The quality factor Qr of a structure based on a 3 µm radius microdisk is given in Fig. 10(d) as a function of the feedback loop length Z. As we need ΔϕAB≡0 to reduce the effective coupling losses and to achieve the lasing regime with a low threshold, we will introduce the comb (in the frequency domain) for which destructive interferences are available (blue combs in Fig. 10). Then, depending on its spacing (that decreases with Z) and on the FSR of the resonator, different regimes are available:
- if all TE(0,m,0) modes almost share the same phase condition [Fig. 10(c)]. Then the mode with the higher quality factor is changed for a slight optical index modification in the waveguide (by thermal or electrical actuation for instance). Such a feedback loop length is well adapted to mode hoping: for instance, we can switch from λr=1.643 µm to λr=1.490 µm changing the optical path (i.e. the optical index) by a few ‰.
- if M modes are regularly distributed [Fig. 10(b) with M=5]: each mode has a dominating quality factor on a 2π/M range of ΔϕAB. The resonant wavelength can be adjusted still granting a high quality factor contrast between the chosen mode and the other ones. Such values of Z are quite interesting for wavelength tuning, more particularly for small M values, since the margin on ΔϕAB is higher.
Consequently, an active control of the optical path in the feedback loop leads to two kinds of applications:
- we can adjust the resonant wavelength with a maximum range ~±τ −1 0.
- we can switch from one resonant wavelength to another one, with a frequency step corresponding to the FSR.
In both cases, strongly coupled structures are required to control with precision the lasing wavelength or to increase the quality factor contrast (Table 1).
Last, due to the smaller effective index in the silicon waveguide than in the InP resonator, the frequency spacing between two destructive conditions can be higher than the FSR. Such a design prevents a higher number of unwanted modes from having a potentially high quality factor (if ΔϕAB≡π).
3.3. Tunable laser with a single output waveguide
Adding a 2 to 1 coupler (like a MMI) between the two optical outputs of the device, the CW mode can interfere with the CCW one and light collection is done in a single output (Fig. 11).
If the resonator only has propagative modes, then the device functionalities are not changed and the Y coupler is useless. If we are confronted to stationary modes, then we have to take care about the phase at A and B and the parity of the azimuthal order that induces a m·π phase difference in the resonator for two diametrically opposed points. For symmetrical paths, only even (or odd) modes are available at the output of the MMI, depending on the symmetry at the coupling points A and B. Table 2 summarizes the types of interferences at the single output.
We can observe that the FSR is doubled at the output of the coupler if the symmetry is preserved. Previous results demonstrated that the symmetric and antisymmetric coupling have slightly different coupling constants, but the quality factor of the structure must be at least 30 000 to discriminate these two solutions . This is not compatible with strongly coupled structures with optically absorbing contacts. So the main issue consists in predicting the symmetry of the coupling areas. Different solutions are under investigation to get stationary WGMs with a phase control:
- add a Bragg grating at the peripheral of the microdisk . In this case, the feedback loop is used to modulate the light in the waveguide, since there will only be one mode per P (l,n) comb in respect to the periodicity of the Bragg grating. This corresponds to a single wavelength operating laser with an extremely compact modulation system (with the feedback loop).
- use a local default in the microdisk or in the waveguide to set the phase. Then, depending on the length of the feedback loop, we get a laser with mode hoping, wavelength adjustment or modulation capacities.
Using an active control of the optical index in the paths between the coupling areas (A and B) and the Y coupler can also be used to modulate the output signal. Meanwhile, switching from even to odd azimuthal order modes with the feedback loop leads to much more compact structures, with a probably higher bandwidth.
In the first part, we studied the light transfer from a ring resonator to a straight waveguide. We demonstrated that the optical coupling rate is modulated by the relative position of the waveguide. The effective coupling loss rate can be inhibited (no light transfer) or strengthened (strongly coupled structure). In order to make this behavior wavelength dependent, we proposed to elongate the Si waveguide, so this later forms an outer loop for the resonator.
Strongly coupling a WGM based laser (like a ring or a microdisk) to this silicon feedback loop can be used to assign a high quality factor to a chosen mode with a high coupling efficiency. The lasing mode can be selected using an active control of the waveguide optical index. Weak losses in the feedback loop ensure that optical power is always available in the output waveguide.
Last, such micro-resonators with a feedback loop are extremely compact structures, compatible with mode hoping operation. The resonant wavelength of the source can be adjusted to compensate fabrication default for instance and modulation can also be achieved switching from one resonant mode of the resonator to another one.
The authors would like to thank Thijs Spuesens for his interesting discussions regarding the vertical coupling.
This work has been supported by the European project FP7-ICT STREP WADIMOS.
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